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Miroslav Kohout

DGrid 5.0

– User’s Guide –

Springer

Preface

DGrid-5.0 computes and evaluates gridded fields utilizing of the information aboutthe wavefunction as given in special wavefunction file. All the values, gradients,and Hessian matrices are computed analytically (as far as possible). This is notalways the case for solid state calculations. Usually, only the precomputed grid wereevaluated numerically. The preceeding version, DGrid-4.6, was able to access thedata from the full potential solid state code Elk. Dr. Alexey I. Baranov succeeded tocreate of an interface to this (open source) program. This not only enabled DGrid-4.6to determine the critical points and interconnection lines for solid state compound,but also to compute the overlap integrals and thus to evaluate the fluctuation and thedelocalization indices. DGrid-5.0 has no access to the Elk modules, however, it isnow possible to create a wavefunction file for the solid state code FHI-aims, thanksto an interface written by Dr. Alim Ormeci.

Starting with DGrid-5.0 the control files can perform several jobs in a sequence.This enables the user to perform a series of evaluation jobs with a single control file.Moreover, as the file names a created during the run, there will be clear connectionbetween the results. With this it is easier to evaluate which files are derived fromanother files in a given task.

Rudimentary help for certain parts of DGrid is given by calling:

dgrid -h

which prints the following information:

---------------------------------------------------dgrid -p print property keywordsdgrid -s print spin keywordsdgrid -t print type keywordsdgrid -u utilitiesdgrid -v print DGrid version---------------------------------------------------

i

ii Preface

The manual was written with Latex using the SVMono style. The permission ofSpringer is acknowledged.

Dresden, January 2017 M. K.

Contents

1 DGrid 5.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.1 Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Linux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Wavefunction file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Control file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1 Control file for a job sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Data assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Evaluation sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 CBO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.3 compute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.4 DI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.5 domain natural orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.6 Fermi orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.7 ICL graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.8 integrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.9 intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.10 isect analyze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3.11 localized natural orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.12 overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3.13 refine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3.14 search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.15 superbasin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

iii

iv Contents

4 Grid files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.1 Field file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Naming of the field files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3 Basin file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Parallel calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Grid slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 dgrid para-5.0 script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1 Conversion of QM package output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.1.1 ADF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1.2 GAMESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.1.3 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.1.4 Molpro, Molcas, Turbomole . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2 Wavefunction file conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3 Structure file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.4 AO contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.5 Completion of sliced fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.5.1 Evaluation of the Source Function . . . . . . . . . . . . . . . . . . . . . . 1046.6 Operations on single grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.7 Operations on two grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.8 Cropped basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Comments to grid field calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.1 Evaluation of FHI-aims calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.2 Grid mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.3 Spin density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.4 ELI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.5 ELF, LOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.6 Overlap integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.6.1 Fluctuation and delocalization indices . . . . . . . . . . . . . . . . . . . 117A.6.2 DNOs / Fermi orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.6.3 Decomposition of the delocalization index . . . . . . . . . . . . . . . 125

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

List of Tables

2.1 Wavefunction file name extender subject to QM package used . . . . . . 6

3.1 Data descriptors utilized in the control file . . . . . . . . . . . . . . . . . . . . . . . 123.2 Evaluation sections utilized in the control file . . . . . . . . . . . . . . . . . . . . 163.3 Keywords and parameters available for the basin determination . . . . . 173.4 Keywords and parameters available for the CBO calculation . . . . . . . . 253.5 Keywords and parameters available for the compute section . . . . . . . 273.6 Available fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.7 Conditional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.8 Descriptors for critical point search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.9 Format descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.10 Space representation descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.11 Keywords and parameters available for the DI calculation . . . . . . . . . . 483.12 Keywords and parameters available for the DNO calculation . . . . . . . 503.13 Keywords and parameters available for the FO calculation . . . . . . . . . 523.14 Keywords and parameters available for the ICL graph . . . . . . . . . . . . . 553.15 Keywords and parameters available for the integration . . . . . . . . . . . . . 603.16 Keywords and parameters available for the intersections . . . . . . . . . . . 623.17 Keywords and parameters available for the intersection analysis . . . . 643.18 Keywords and parameters available for the LNO calculation . . . . . . . . 693.19 Keywords and parameters available for the orbital overlap integrals . . 713.20 Keywords and parameters available for the data refinement . . . . . . . . . 743.21 Keywords and parameters available for the DI calculation . . . . . . . . . . 773.22 Core regions used by the ELID core keyword . . . . . . . . . . . . . . . . . . . . 803.23 Keywords and parameters available for the superbasins . . . . . . . . . . . . 82

4.1 Grid file name extender subject to field type . . . . . . . . . . . . . . . . . . . . . 894.2 Grid file name extender subject to field spin . . . . . . . . . . . . . . . . . . . . . 904.3 Grid file name extender subject to field pair spin . . . . . . . . . . . . . . . . . . 90

6.1 Operations on two grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

v

vi List of Tables

A.1 Localization and delocalization indices . . . . . . . . . . . . . . . . . . . . . . . . . 118

List of Figures

3.1 Electron density basins for C3H3NO. . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 ELI-D for C3H3NO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Usage of the top keyword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Oxazole orbitals in real and momentum space . . . . . . . . . . . . . . . . . . . . 333.5 Definition of grid region with the mesh assignment . . . . . . . . . . . . . . . 423.6 Basins and molecular graph for C3H3NO . . . . . . . . . . . . . . . . . . . . . . . . 563.7 ELI-D ICL graphs for C3H3NO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.8 ELI-D basins and ICL graphs for C3H3NO . . . . . . . . . . . . . . . . . . . . . . 583.9 Intersection of ELI-D basins for C3H3NO . . . . . . . . . . . . . . . . . . . . . . . 663.10 Intersection of electron density basin for C3H3NO . . . . . . . . . . . . . . . . 673.11 Density superbasins for C3H3NO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.12 Electron density isosurfaces for C3H3NO . . . . . . . . . . . . . . . . . . . . . . . 84

A.1 Fermi orbitals for C3H3NO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

vii

Acronyms

AO atomic orbitalCP critical pointDAFH domain-averaged Fermi-holeDFT density functional theoryDI delocalization indexDNO domain natural orbitalELI electron localizability indicatorELI-D electron localizability indicator based on D-restricted partitioningFO Fermi orbitalHF Hartree-FockICL interconnection lineLNO localized natural orbitalMO molecular orbital

ix

Chapter 1DGrid 5.0

1.1 Introduction

DGrid is a program for the generation of field values on an equidistant grid (respec-tively at a single point only) and the analysis of those gridded fields, inclusive thebasin search. For the calculation of field values two input files are essential:

1. wavefunction file with the information about the basis set and the orbital repre-sentation (for a CI calculation additionally also with the density matrices) for theevaluated atom, molecule or solid. The corresponding data files supplied by oneof the quantum chemical packages GAUSSIAN [24], MOLPRO [49], MOLCAS[7], TURBOMOLE [6], GAMESS [4], ADF [1], and FHI-aims[3] respectively,need to be converted into the DGrid wavefunction file format. This conversion isdescribed in the next section.

2. control file (written by the user) with keywords controlling the field value calcu-lation.

Both wavefunction and control file must be written in a special format. The in-formation is read in by the parsing routine rdcard that obeys few simple rules:

• Empty lines and lines starting with a colon are skipped.• Strings and numbers terminated by a colon are skipped.• Strings and numbers must be separated by one or more blanks.• Two colons at the beginning mark the line as a text input – the line is read in

without parsing (e.g., as a title).• All lines following line star ‘=’ is a variable. The string or number following the

equal sign is assigned to this variable.• Line starting with ‘>’ marks the start of a block of lines being skipped inclusive

the line starting with ‘<’ which terminate the block.

The output of particular evaluation is either a grid file with the field values (eachtype of a grid file has designated file extension) and/or a log file (usually marked by

1

2 1 DGrid 5.0

an file extension given by capital letters) with the information about the calculationrespectively the results.

1.2 Disclaimer

The program DGrid (inclusive the source code) is distributed free of charge. Thecode remains in the ownership of the author. It is allowed to change the DGrid codefor own purposes, whereby the copyright must not be changed when adapting thecode. DGrid with code changed by the user must not be redistributed. The sameapply to the original DGrid code. Refer the prospective user to download the DGridpackage form the official webpage (given below).

The program DGrid is distributed ‘as is’ without warranties of any kind. No re-sponsibility of any kind is assumed from the use of DGrid. Use the following citationfor the program DGrid:

M. Kohout, DGrid, version 5.0, Dresden, 2017

1.3 Distribution

The current version as well as all updates with changes and corrections implementedlater can be downloaded from the web page:

http://www.cpfs.mpg.de/∼kohout/dgrid.html

The User’s guide as well as the source code, precompiled executables, modules forthe visualization programs, and examples can be downloaded separately.

1.4 Installation

Executable versions of DGrid 5.0 are available from the DGrid web page for Linux(openSUSE 13.2) and Windows. Otherwise, the source code must be compiled. Theprogram DGrid is written in C++, thus, an appropriate compiler must be available.The installation procedures for a Unix system respectively for Windows PC (usingMS Visual Studio) are described.

1.4 Installation 3

1.4.1 Windows

In this section the compilation of DGrid with Microsoft Visual Studio 2013 is de-scribed. For Windows 10 only the Visual Studio 2015 is available. In this case, usethe distribution file dgrid-5.0-VS2015.zip and proceed like for VS 2013.

Change to the directory where you wish to save the Visual Studio Project and un-pack there the distribution file dgrid-5.0-VS2013.zip with an appropriate Windowstool (like Wintar or 7-zip). Change into the directory DGrid-5.0. You should findthere the project file DGrid-5.0.vcproj together with the directory src containing allthe source code files.

Start the Visual Studio and open the project DGrid-5.0 using the project fileDGrid-5.0.vcproj. Build the project DGrid-5.0. This compiles all the files andcreates the executable DGrid-5.0.exe which can be finally found in the directoryC:\Mydir\DGrid-5.0\Release. Copy the executable DGrid-5.0.exe to a suitableplace on your computer, for instance, C:\Mydir and change the name of the exe-cutable to dgrid. Run the DGrid calculation with the command:

C:\Mydir\dgrid.exe controlfilename

The output will be written to specifically named output files, unless the commandconsole is given in the control file, cf. page 10.

1.4.2 Linux

Unpack the distribution tar file dgrid-5.0.tar.gz using the commands:

gunzip dgrid-5.0.tar.gztar xf dgbas-5.0.tar

This creates new directory DGrid-5.0. Change to the directory DGrid-5.0. Youshould find there the script files dgrid para-5.0, dgrid dglog, dgrid log,the directory src with all the source code files and a makefile. View the makefile –you may have to change the name of the compiler (for instance the Intel compilericpc) and the compiler options. Create DGrid with the command:

make

The executable dgrid-5.0 will be written to the directory DGrid-5.0 (this can bechanged in the makefile). For convenience, enter the directory where the executableresides and link it with the name dgrid:

4 1 DGrid 5.0

ln -s dgrid-5.0 dgrid

Run the DGrid calculation with the command:

dgrid controlfilename

The control file name is given as a parameter (i.e., without ’<’). The output will bewritten to specifically named output files, unless the command console is given inthe control file, cf. the Sec. 3, ‘Control file’, page 9).

Chapter 2Wavefunction file

Each of the quantum chemical packages GAUSSIAN, MOLPRO, MOLCAS, TUR-BOMOLE, GAMESS, ADF, and FHI-aims, respectively, provide an option to writethe information about (among others) the structure geometry, basis set, i.e., theatomic orbitals (AO), and the resulting orbitals (MO) to a separate output file. Thoseoutput files are written in different formats. Additionally, ADF is based on Slater-type orbitals, whereas the other molecular programs use Gauss-type orbitals andFHI-aims utilizes numerical atomic orbitals together with real spherical harmonics.

For the GAMESS all the information is recovered form the output file (the preci-sion of some output variables must be increased in few GAMESS routines). GAUS-SIAN calculations supply the formatted name.wfn file as well as the binary check-point file name.chk, where the latter first needs to be converted into the format-ted check-point file name.fchk using the GAUSSIAN utility formchk. Similarly,the density functional program ADF writes the required information to a binaryfile TAPE21, that needs to be converted into a formatted file using the ADF utilitydmpkf. For MOLPRO, MOLCAS, and TURBOMOLE the necessary informationneed to be written to a file in the molden format. In case of TURBOMOLE includein the molden file as the second line the string:

[TURBOMOLE]

otherwise the normalization of d and f functions will be wrong (it is anyway a goodidea to check the proper normalization, for instance, by charge integration of the Xeatom).

For the FHI-aims program the procedure is slightly more complex. First, theadapted FHI version (implementation written by Alim Ormeci) must be used tocreate the necessary output data. The ‘.dat’ output is forced by the presence of theempty file input.eli. This creates several files, ready for the conversion into the wave-function file.

5

6 2 Wavefunction file

DGrid reads the necessary information from the wavefunction file written in aspecial format. Thus, the quantum chemical packages output files have to be con-verted into this format by the conversion routines included in the DGrid package:

dgrid formatted-qm-file

The resulting DGrid wavefunction file includes all orbitals from the quantum chem-ical calculation (occupied as well as the virtual ones). Of course, in the DGrid wave-function file the unoccupied orbitals have occupations equal zero. To include theseorbitals into the property calculation change the orbital occupations using the key-word occupation in the control file.

To test the procedure, find in the examples directory (downloaded from theDGrid web page) the formatted GAUSSIAN03 checkpoint file C3H3NO HF 6-31G.fchk for the oxazole molecule (geometry from Ref. [45] re-optimized withGAUSSIAN03). Using the command:

dgrid C3H3NO HF 6-31G.fchk

yields the DGrid wavefunction file file C3H3NO HF 6-31G.g03 with 51 orbitals (6-31G basis set). Of course, only 18 orbitals are (doubly) occupied. Converting datafrom other QM packages yields wavefunction files with corresponding extender, cf.Table 2.1.

Table 2.1 Wavefunction file name extender subject to QM package used

Program descr. Source Extender

ADF TAPE21 7→ adfADF TAPE21 (loc. orbitals) 7→ adf locGAMESS output file 7→ gmsGAUSSIAN94 Checkpoint file 7→ g94GAUSSIAN98 Checkpoint file 7→ g98GAUSSIAN03 Checkpoint file 7→ g03GAUSSIAN09 Checkpoint file 7→ g09GAUSSIAN WFN file 7→ gwfMOLPRO molden file 7→ mpMOLCAS molden file 7→ mcMOLDEN molden file 7→ mdFHI-aims special files 7→ fhiCLEMENTI-ROETTI Tables [18] 7→ CR

The first line of the wavefunction file starts with the keyword ‘WAVEFUNCTION-DATA’. Additionally, the version of DGrid creating the data is included. The wave-function file contains the information about the title, date and time of the calcula-

2 Wavefunction file 7

tion, as well as the package used for the quantum chemical calculation, time and,followed by the coordinates of the involved atoms and the assignment of the basissets to the atoms. After the header ‘AO expansion labels’ a table of atomic orbital(AO) descriptors involved in the LCAO expansion is written. In the AO expansiontable the number and the descriptor point to the atom and the corresponding AO,respectively, in the basis set list. For example, ‘8 s’ is the s-AO on the atom number8 (hydrogen), with the exponents and coefficients of the first s-AO on this atom (thesecond s-AO on this atom is written as ‘8 s a’).

The AO expansion descriptors are followed by a table with MO descriptors givenas orbital number in the symmetry, symmetry label (symmetry NONE means or-bital output without symmetry descriptors) and total orbital number. After the MOdescriptor table the LCAO expansion coefficients are written for each MO. Thewavefunction file C3H3NO HF 6-31G.038 from the above example reads:

WAVEFUNCTION-DATA created by DGrid version 5.0 01-11-2016 ...

/=========\| title |+---------+

+-----------------------------------------------------------------------------+::C3H3NO HF 6-31G+-----------------------------------------------------------------------------+

+----------------------------------+| date= Wed Nov 2 11:07:39 2016 |+----------------------------------+

/============================================\| calculation= RHF program= GAUSSIAN03 |+--------------------------------------------+

+----------------------------------------------------------------+| energy= -244.502729 hartree || || electrons= 18 alpha + 18 beta |+----------------------------------------------------------------+

/=====================\| coordinates= bohr |+---------------------+

atom # X Y Z charge+----------------------------------------------------------------------+| O 1 -2.06183559 0.57482826 0.00000000 8.00 || C 2 0.00000000 2.12106164 -0.00000000 6.00 || N 3 2.08694733 0.89554761 -0.00000000 7.00 || C 4 1.41345421 -1.67482606 -0.00000000 6.00 |...| H 8 -2.43100862 -3.36800124 0.00000000 1.00 |+----------------------------------------------------------------------+

/=====================================\| basis= 6-31G cartesian GTO |+-------------------------------------+

atom # AO exponents and coefficients+------------------------------------------------------------------------------------+| O 1 s exp: 5.48467166e+03 8.25234946e+02 1.88046958e+02 || coe: 1.83107443e-03 1.39501722e-02 6.84450781e-02 || exp: 5.29645000e+01 1.68975704e+01 5.79963534e+00 || coe: 2.32714336e-01 4.70192898e-01 3.58520853e-01 |

...| H 6 - 8 s exp: 1.87311370e+01 2.82539436e+00 6.40121692e-01 || coe: 3.34946043e-02 2.34726953e-01 8.13757326e-01 || H 6 - 8 s exp: 1.61277759e-01 |+------------------------------------------------------------------------------------+

8 2 Wavefunction file

/=======================\| AO expansion labels |+-----------------------+

+-------------------------------------------------------------------------+| 1 s 1 s_a 1 px ... 1 py_a || ... ... ... ... ... |...| 5 pz 5 s_b 5 px_a ... 7 s || 7 s_a 8 s 8 s_a | |+-------------------------------------------------------------------------+

/==================================\| molecular orbitals= restricted |+----------------------------------+

orbital ## energy+---------------------------------------------------+| 1 NONE 1 -20.6520649000 |

...| 51 NONE 51 2.1610599100 |+---------------------------------------------------+

/=============================\| AO expansion coefficients |+-----------------------------+

MO= 1+------------------------------------------------------------------+0.9958683260 0.0234678988 0.0021574144 ... 0.00129300260.0000000000 0.0000464216 0.0001404648 ... -0.0009640913-0.0031038085 0.0000000000 -0.0000113092 ... 0.00012488760.0003120335 -0.0003269614 0.0000000000 ... 0.0000000000-0.0045956519 0.0024333778 -0.0024785540 ... 0.00042345750.0000000000 0.0064537170 0.0021246829 ... -0.0001443667-0.0009808851 0.0001005512 0.0014130199+------------------------------------------------------------------+

/====================\| occupation= SDet |+--------------------+

ALPHA BETAorbital ## occupation orbital ## occupation

+-----------------------------------------------------------------------------------+| 1 NONE 1 1.00000000000 | 1 NONE 1 1.00000000000 |...| 18 NONE 18 1.00000000000 | 18 NONE 18 1.00000000000 |+-----------------------------------------------------------------------------------+

Notice that the molecular orbital number 51 has positive orbital energy (like all othervirtual orbitals in the wavefunction file). The last information in the wavefunctionfile is the section with the occupations for each orbital (‘SDet’ in the header standsfor single determinant). Observe that only the 18 occupied orbitals are written to thesection.

The name of the particular wavefunction file is known by the DGrid programthrough the assignment ‘wfn 1=basisfilename’ in the control file described in nextsection.

Chapter 3Control file

For the control file any name can be chosen. The control file supplies all the infor-mation concerning the calculation. The data must comply with the rules describedin the introduction, cf. page 1. All keywords are read case insensitive. Often thekeywords can be written in any order. It is recommended to put every group of key-words in a separate line, because sometimes additional data are assumed per default.Some data even need to be written in rigorous sequence. All available keywords aredescribed in detail in following sections.

The control file is organized in data assignments and evaluation sections. Thedata assignments are used to connect files with descriptors that are subsequentlyused in the evaluation sections. Following is the example rho elid.inp of the controlfile for the grid field calculation organized in three sections:

1. Title2. Data assignment3. evaluation section

Here is an example of a control file:

::C3H3NO HF 6-31G:------------------------------------------------|wfn_1=C3H3NO_HF_6-31G.g03

compute:-----------------------------------------

using wfn_1

ELI-D alpha-alpharhorho Laplacian

mesh=0.10 rho=0.001 parallel:-----------------------------------------compute_end

console

9

10 3 Control file

• The first readable information in the control file must be the title (remember thatall empty lines as well as lines beginning with a single colon will be skipped;thus, they can be omitted). The title line starts with two colons, i.e., the line willnot be parsed. The two colons are obligatory, whereas the title text can be omitted(i.e., empty title). If some title text is found in this line, it will be written into thetitle-section of each resulting file.

• The data assignment is accomplished by invoking the keyword wfn 1 followedby the name of the (first) wavefunction file (described in the previous section).This can also be a UNIX path to this file. The descriptor wfn 1 can be utilizedin any subsequent evaluation section. All possible data descriptors for differentevaluation sections are compiled in Tab. 3.1 in Sec. 3.2.

• The evaluation section, in the above example defining the grid calculations, be-gins with the keyword compute and ends with the keyword compute end (thesection delimiting lines ‘:- - - - -’ are present just to highlight the section andcan be omitted). With the assignment using wfn 1 the chosen wavefunction fileC3H3NO HF 6-31G.g03 is disclosed to the grid evaluation routine.

• The log information and the results of the DGrid calculation are written to logfiles (usually specified by a capital letters file-extension), respectively saved indesignated grid files, refinement files, or overlap files. To write the log informa-tion directly to the console, instead to specific files, use the command:

console

given on a separate line, for instance at the end of the control file, cf. the aboveexample.

3.1 Control file for a job sequence

Usually, the grid calculation is followed by subsequent evaluation of the resultinggrid field, for instance by the determination of the basins or the search for criti-cal points,calculation of overlap integrals over basins, etc. Each of the jobs can bestarted from separate control files started after the preceding job finished and thenecessary data are available.

However, this can be performed also by means of a single control file. DGridexamines successively each evaluation section, possibly using data files saved frompreceding evaluation sections. For instance, we wish to compute for the moleculeC3H3NO the electron density, then determine the QTAIM basins, compute the over-lap integrals over the basins and finally determine the delocalization indices betweenthe basins. All this will be accomplished by the following single control file:

3.1 Control file for a job sequence 11

::C3H3NO HF 6-31G:--------------------------------------------------|wfn_1=C3H3NO_HF_6-31G.g03

compute:-----------------------------------------

using wfn_1rhosave field_1

x: y: z:----------------------------:origin= -6.0 -7.0 -4.0 intervals:----------------------------------------:i-vector: 12.0 0.0 0.0 120j-vector: 0.0 14.0 0.0 140k-vector: 0.0 0.0 8.0 80----------------------------------------:

:-----------------------------------------compute_end

basin:---------------------

using field_1save basin_1crop using field_1 0.0001

:---------------------basin_end

overlap:--------------

using wfn_1over basin_1save sij_1

:--------------overlap_end

DI:--------------

using sij_1:--------------DI_end

The wavefunction file C3H3NO HF 6-31G.g03 is used to compute the electron den-sity grid saved in the file C3H3NO HF 6-31G.g03.rho r (the name is generated in-ternally by DGrid). In the subsequent evaluation sections of the control file thisdensity grid file is addressed by the descriptor field 1, cf. the command save field 1following the rho assignment.

The compute section is followed by the basin section determining the QTAIMbasins based in the electron density field (cf. the command using field 1 withinthe section) cropped by the 0.0001 density isosurface. The resulting basin fileC3H3NO HF 6-31G.g03.rho r.bsn will be created together with the correspond-ing log file C3H3NO HF 6-31G.g03.rho r.bsn.LOG. This basin file is known in the

12 3 Control file

subsequent evaluation sections of the control file by the descriptor basin 1, cf. thecommand save basin 1.

Then, in the section overlap the overlap integrals between the molecular orbitalstaken from the wavefunction file C3H3NO HF 6-31G.g03 (cf. the using wfn 1command) is computed over the QTAIM basins (cf. the over basin 1 command).The resulting overlap integrals is saved in the file C3H3NO HF 6-31G.g03.rho r.bsn.sij.The corresponding log file C3H3NO HF 6-31G.g03.rho r.bsn.sij.LOG shows someinfo about this section. The overlap file is known in the subsequent evaluation sec-tions of the control file by the descriptor sij 1, cf. the command save sij 1.

Finally, the overlap file C3H3NO HF 6-31G.g03.rho r.bsn.sij is used in the DIsection creating the log file C3H3NO HF 6-31G.g03.rho r.bsn.sij.DI with the tableof delocalization indices.

3.2 Data assignments

The data assignments are used to create in the control file a reference between spe-cific descriptor and an existing file (containing data written in special format). Thedata descriptors recognized by DGrid are compiled in Tab. 3.1. Each assignmentmust be written in separate input line, for instance:

wfn 1= /home/test/molecule.adf

The descriptor number discriminates between different data files of the same format(a wavefunction file in the above example). Usually, new files are created in theevaluation sections. The the actual filenames are unknown before the job starts.With the save command within an evaluation section, for instance save basin 2enables to use the saved basin file in another subsequent evaluation section utilizingthe descriptor basin 2.

Table 3.1 Data descriptors utilized in the control file

Descriptora File format

basin 1 7→ basin gridcps 1 7→ critical pointsfield 1 7→ field gridnorm 1 7→ norm partsrfn 1 7→ refinement datasij 1 7→ overlap integralsstr 1 7→ structure datawfn 1 7→ wavefunction

a The number discriminates between different files of the same data format

3.2 Data assignments 13

3.2.1 Descriptors

Below, all available descriptors are described in alphabetical order. The descriptorsare read case insensitive as a variable, i.e., followed by ‘=’. Exemplary, the number‘ 1’ is used, corresponding to the first descriptor of given format.

3.2.1.1 basin 1

The descriptor basin 1 is followed by the name of the (first) basin file. This can alsobe a UNIX path to this file:

basin 1= /home/test/molecule.adf.rho r.bsn

After stripping of the preceding path name the name of the basin file is accessiblein the control file by the descriptor basin 1. Consequently, all other files derivedfrom this basin file will be written into the same directory where the control file islocated. Computing for the above basin file the overlap integrals yields the overlapfile named molecule.adf.rho r.bsn.sij located in the active directory (which need notbe the directory ‘/home/test’).

3.2.1.2 cps 1

The descriptor cps 1 is followed by the name of the (first) file with the critical. Thiscan also be a UNIX path to this file:

cps 1= /home/test/molecule.adf.rho r.cps

After stripping of the preceding path name the name of the cps file is accessible inthe control file by the descriptor cps 1. Consequently, all other files derived fromthis cps file will be written into the same directory where the control file is located.Computing for the above cps file the interconnection lines yields the file namedmolecule.adf.rho r.bsn.cps.graph.str with molecular graph data located in the activedirectory (which need not be the directory ‘/home/test’).

3.2.1.3 field 1

The descriptor field 1 is followed by the name of the (first) field file. This can alsobe a UNIX path to this file:

14 3 Control file

field 1= /home/test/molecule.adf.rho r

After stripping of the preceding path name the name of the field file is accessiblein the control file by the descriptor field 1. Consequently, all other files derivedfrom this field file will be written into the same directory where the control file islocated. Computing for the above field file the basins yields the basin file namedmolecule.adf.rho r.bsn located in the active directory (which need not be the direc-tory ‘/home/test’).

3.2.1.4 norm 1

The descriptor norm 1 is followed by the name of the (first) norm parts file (con-taining the integrals of the squared orbitals over basins). This can also be a UNIXpath to this file:

norm 1= /home/test/molecule.adf.rho r.bsn.sij.lno.norm

After stripping of the preceding path name the name of the norm parts file is accessi-ble in the control file by the descriptor norm 1. Consequently, all other files derivedfrom this field file will be written into the same directory where the control file is lo-cated. From the filename in the above example it can be inferred that the norm partswere determined for the LNOs, which were created by the localization procedurebased on the overlap integrals read from the overlap sij-file for the QTAIM basins.Computing for the above norm parts file the delocalization indices yields the log filenamed molecule.adf.rho r.bsn.sij.lno.norm.DI located in the active directory (whichneed not be the directory ‘/home/test’).

3.2.1.5 rfn 1

The descriptor field 1 is followed by the name of the (first) refinement file. This canalso be a UNIX path to this file:

rfn 1= /home/test/molecule.adf.rho r.rfn

After stripping of the preceding path name the name of the refinement file is acces-sible in the control file by the descriptor rfn 1. The data in the rfn file are used formore accurate integration.

3.2 Data assignments 15

3.2.1.6 sij 1

The descriptor sij 1 is followed by the name of the (first) overlap file (containingthe overlap integrals of the orbital products over basins). This can also be a UNIXpath to this file:

sij 1= /home/test/molecule.adf.rho r.bsn.sij

After stripping of the preceding path name the name of the overlap file is accessiblein the control file by the descriptor sij 1. Consequently, all other files derived fromthis overlap file will be written into the same directory where the control file is lo-cated. From the filename in the above example it can be inferred that the overlap in-tegrals were computed for the canonical orbitals by the integration over the QTAIMbasins. Computing for the above overlap file the delocalization indices yields thelog file named molecule.adf.rho r.bsn.sij.DI located in the active directory (whichneed not be the directory ‘/home/test’).

3.2.1.7 str 1

The descriptor str 1 is followed by the name of the (first) structure file. This canalso be a UNIX path to this file:

str 1= /home/test/molecule.adf.rho r.str

After stripping of the preceding path name the name of the structure file is accessiblein the control file by the descriptor str 1. The data in the str file are used to accessthe atomic coordinates, critical points, connection sticks and polygons.

3.2.1.8 wfn 1

The descriptor wfn 1 is followed by the name of the (first) wavefunction file. Thiscan also be a UNIX path to this file:

wfn 1= /home/test/molecule.adf

In case of a grid calculation the name of the resulting grid file name will be gen-erated from the name of the wavefunction file after stripping of the preceding pathname. Consequently, the grid files will be written into the same directory wherethe control file is located. Computing for the above wavefunction file the electrondensity yields the grid file named molecule.adf.rho r.

16 3 Control file

3.3 Evaluation sections

In the evaluation sections of the control different keywords are utilized to performthe desired job. Some of the keywords are specific only the particular section. Thekeywords are given either as variables, like e.g. ‘origin=’, or as stand-alone ex-pressions, like ‘using’. Some of them are followed by 1 or more numbers, e.g.,‘mesh=0.05 rho=0.001’. The parsing routine discriminates between float and inte-ger numbers. Thus, the command ‘point= 0.5 1.0 3’ would not be valid, because ‘3’is an integer number, whereas three floats are expected for the coordinates. Oftenmore keywords can be written in single line. But it is better to put every keywordsequence in separate line, because sometimes additional data are assumed per de-fault. Some data even need to be written in rigorous sequence, e.g., the definition ofa grid using the origin= keyword or the choice of occupied orbitals.

Each evaluation section starts with a specific keyword written in a separate line.The section ends with the same keyword (on separate line), but with the string endappended, for instance:

DI:---------------------

using sij_1:---------------------DI_end

Bear in mind that the delimiting strings ‘:- - - - -’ are used only to highlight the startand end of the section. They are not parsed (first character is the colon ‘:’) and can beomitted. The keywords defining the begin of the evaluation sections are compiledin Tab. 3.2. In some cases another sections can be included within the evaluationsection.

Table 3.2 Evaluation sections utilized in the control file

First line Description

basin basin evaluationCBO Cioslowski bond ordercompute field valuesDI delocalization indexdomain natural orbitals DNO determinationFermi orbitals Fermi orbital determinationICL graph interconnection graphintegrate integration of a field over basinsintersections intersections between basinsisect analyze analysis of basin intersectionLNO localized natural orbitalsoverlap overlap between orbitals over basinsrefine calculation of additional field valuessearch search for critical pointssuperbasin collect basins into superbasin

3.3 Evaluation sections 17

Below, all evaluation sections and the keywords available for the section are de-scribed in alphabetical order. The keywords are read case insensitive whereas thedata (for instance the filenames) are utilized case sensitive.

3.3.1 basin

The section for the basin determination is delimited by separate lines with the key-words basin and basin end. The evaluation is controlled by keywords summarizedin Tab. 3.3 and described in detail below.

Table 3.3 Keywords and parameters available for the basin determination

Keyword Description Value

compactify= merging of basins <distance> <valdiff>crop using crop basins by isovalue field <n> isovalueELID core unify basins in ELI-D core eventually core sectionlocalization domains= domains defined by isovalue <isovalue>save basin grid to be saved basin <n>symmetrize in cell symmetrize by translationtop= unify basins by isovalue <isovalue>using field grid to be used field <n>

Example:

:--------------------------------------------------|::C3H3NO HF 6-31G:--------------------------------------------------|wfn_1=C3H3NO_HF_6-31G.g03

basin:---------------------

using field_1crop using field_1 0.001top=0.4save basin_1

:---------------------basin_end

The basin data are written to a basin grid file, cf. Sec. 4.3. The result for the abovecontrol file is shown in Fig. 3.1. The information about the search for the basins,the number of basins found as well as the basin volumes are written to the log filewith the same name as the basin grid file but with the string ‘.LOG’ appended. Itincludes the info about the files read in, the calculation progress and some statistics.The progress during the basin search is shown by a scale, giving the number ofpoints which are tested. This check is repeated till the basins reach the final shapes.

18 3 Control file

Fig. 3.1 Electron density basins for C3H3NO, cropped by 0.001 bohr−3 electron density isosur-face. Blue: oxygen basin; green: nitrogen basin; black: carbon basins; gray: hydrogen basins

For each basin its volume, the field value at the attractor position are given. Basinstouching the grid-box border are marked by the ‘x’ sign:

/=================\| V O L U M E S |+-----------------+

BASIN VOLUME INTEGRAL MAXIMUM X Y Z ATOMS DIST ECCNT+-------------------------------------------------------------------------------------+| x 1 40.490 - 0.4279 -0.308 4.102 -0.000 H_1 0.00 - || x 2 41.158 - 0.4249 -2.431 -3.368 0.000 H_3 0.00 - || 3 42.720 - 0.4277 2.805 -3.122 -0.000 H_2 0.00 - || 4 62.982 - 118.3135 0.000 2.121 -0.000 C_1 0.00 - || x 5 79.034 - 118.3130 1.413 -1.675 -0.000 C_2 0.00 - || x 6 79.391 - 118.3147 -1.110 -1.860 0.000 C_3 0.00 - || x 7 99.748 - 291.3237 -2.062 0.575 0.000 O_1 0.00 - || x 8 116.278 - 192.1030 2.087 0.896 -0.000 N_1 0.00 - |+-------------------------------------------------------------------------------------+| TOTAL | 561.801 |+-------------------------------------------------------------------------------------+| DIFF | -471.539 |+-------------------------------------------------------------------------------------+

If the wavefunction file is not present then the search for the attractors is not possible.In this case the maximal field values and positions correspond to the closest grid


point (this situation is highlighted by the ‘+’ sign next to the ‘MAXIMUM’ value).For each attractor the closest atom (or atoms, see below) as well as the distance tothis atom are given (in the above example the distance is always zero because theattractors of electron density coincide with the atomic positions).

Remark 3.1. When the basins are cropped by an isosurface (keyword crop, cf.Sec. 3.3.1.2) then the sum of the basin volumes is less than the total grid regionvolume. In that case the line ‘DIFF’ is appended.

Additionally, let us have a look at the ‘log’ file for the basins created from theELI-D field for the C3H3NO molecule (again cropped by the 0.001 bohr−3 densityisosurface):

/=================\| V O L U M E S |+-----------------+

BASIN VOLUME INTEGRAL MAXIMUM X Y Z ATOMS DIST ECCNT+--------------------------------------------------------------------------------------------+| 1 0.272 - 13.6041 -2.062 0.575 0.000 O_1 0.00 - || 2 0.464 - 15.5280 2.087 0.896 -0.000 N_1 0.00 - || 3 0.814 - 17.7186 1.413 -1.675 -0.000 C_2 0.00 - || 4 0.820 - 17.8594 -1.110 -1.860 0.000 C_3 0.00 - || 5 0.826 - 17.8381 0.000 2.121 -0.000 C_1 0.00 - || 6 5.420 - 1.5848 -1.633 -0.561 -0.000 O_1-C_3 1.21-1.40 0.014 || 7 5.719 - 1.5798 -1.101 1.296 -0.000 O_1-C_1 1.20-1.38 0.001 || 8 10.802 - 1.7353 1.797 -0.386 0.000 N_1-C_2 1.31-1.34 0.044 || 9 40.700 - 1.7374 1.051 1.501 -0.000 C_1-N_1 1.20-1.22 0.003 || x 10 71.915 - 7.9200 -2.501 -3.442 0.000 H_3 0.10 - || 11 73.849 - 7.8468 2.883 -3.195 -0.000 H_2 0.11 - || x 12 78.069 - 8.2609 -0.329 4.216 0.000 H_1 0.12 - || x 13 85.780 - 1.7354 -3.091 0.870 0.000 LP_O_1 1.07 - || x 14 91.939 - 1.9968 3.314 1.377 0.000 LP_N_1 1.32 - || x 15 94.412 - 1.8219 0.145 -1.876 0.000 C_3-C_2 1.26-1.28 0.108 |+--------------------------------------------------------------------------------------------+| TOTAL | 561.801 |+--------------------------------------------------------------------------------------------+| DIFF | -471.539 |+--------------------------------------------------------------------------------------------+

Now there are much more basins than in the previous example using the electrondensity field (where each ρ-basin can be attributed to one of the 8 atoms). Thefirst 5 basins show the atomic cores of oxygen, nitrogen, and carbon, with ELI-Dattractors at the corresponding atoms, i.e., DIST=0.00, cf. also the colored basins inFig. 3.2. Observe that those basins do not touch the grid-box border.

The basins 6–9 correspond to the O-C and N-C bonds, respectively, with ELI-Dattractors each located between the corresponding atomic nuclei. The column DISTshows the distance of the attractor position to the closest nuclei. The attractor num-ber 6 is located 1.21 bohr from the oxygen and 1.40 bohr from the carbon. It isnot located exactly at the straight line between the atoms, as shown by the eccen-tricity (perpendicular distance to the O1-C5 line) ECCNT=0.014 bohr (whereas theattractor number 7 is located practically at the interatomic line).

The basins number 10-12 are attributed to the C-H bonds, because of the ab-sence of hydrogen cores in the ELI-D representation. Thus, from a simple ELI-Danalysis one cannot distinguish between the part that could be attributed separately

20 3 Control file

to the hydrogen atom. Such analysis can be performed using the intersections, cf.Secs. 3.3.9 and 3.3.10.

Similarly to the basins number 10-12 the ELI-D attractors number 13 and 14are located in proximity to the oxygen and nitrogen nucleus, respectively. A closerinspection of the data reveals that the corresponding basins describe lone-pairs, cf.the ELI-D diagram in Fig. 3.2. Interestingly, there is just single oxygen lone-pairbasin (with the attractor located in the molecular plane) populated by 4.75 electrons(i.e., not split into two separate parts, like in water molecule).

The distance of the field maximum (attractor) to closest atoms together with theeccentricity is used to set up the basin descriptors (respectively the ‘ATOMS’ col-umn in the output). It works in the following way. If the basin property maximum(i.e., the basin attractor) lies within the tabulated ELI-D core radius, then the cor-

Fig. 3.2 ELI-D for C3H3NO. Gold colored (irreducible) 1.45-localization domains (‘H’ and ‘Lp’mark the hydrogen and lone-pair domains, respectively. Blue: oxygen core basin; green: nitrogencore basin; black: carbon core basins


responding atomic symbol is assigned to the respective basin. If the basin attractoris positioned between two atoms and the angle between the atom-atom straight lineand the attractor-atom line is less than 40 then the basin descriptors is formed bythe corresponding atomic symbols with a period in between. Otherwise, in the basinfile, the symbol ‘LP’ (for lone-pair) is used.

Remark 3.2. The basin descriptors should serve as a hint or help. It is not intendedto be used as a rigid classification.

In some cases the number of basins can be much larger than requested (for instancelarge number of core ELI-D basins for heavier elements). Then, the total number ofbasins can be reduced in three ways:

• using the keyword ‘compactify dist vdiff’:grid maxima closer than the distance dist and differing in the function values byless than vdiff get the same basin number. If omitted vdiff is set to the propertyvalue range.

• using the keyword ‘top=iso’:grid maxima inside iso-localization domains get the same basin number.

• using the keyword ELID core:grid maxima inside the respective atomic core defined by ELI-D (internal table)get the same basin number.

3.3.1.1 compactify

If the wavefunction file is not available the field attractors cannot be determined an-alytically. Then only the discrete grid maxima will be used to determine the basins.This procedure can create many spurious small basins for close grid maxima. Suchbasins can be collected into single basin by the command:

compactify= <dist> <valdiff>

Here, all basins corresponding to grid maxima closer than <dist> and with fieldvalues differing by less than <valdiff> from each other (default is all value differ-ences) will be unified.

3.3.1.2 crop using

In case of a molecule the outer basins extend to infinity. Thus, the basins are croppedby the grid box. Instead, the basins can be cropped by an isosurface of chosen field(usually the electron density) by the command:

22 3 Control file

crop using field <n> <isoval>

where the field grid file need to be assigned to the field <n> descriptor in a preced-ing statement. The isovalue <isoval> is given by a float number.

The basins created by the above command will be cropped by the isosurface withthe isovalue <iso> based on the field <n> data. As an example cf. the ρ-basins inFig. 3.1 on page 18 cropped by the 0.001 bohr−3 isosurface of the oxazole electrondensity.

Remark 3.3. If the cropping procedure is not used then the outer basins for a molec-ular system extend to the borders of the computed region. Already computed basinscan be conveniently cropped after the basin search using the DGrid utility ‘crop’,see Sec. 6.8.

Of course, with the cropping procedure part of the volume (outside the croppingisosurface) will not be considered for the integration. Consequently, the sum of thebasin volumes will not recover the total volume of the computed region.

Remark 3.4. If a large grid-box around the molecule is used and electron densitybasins are searched, then due to the very low density (and density gradient) thesearch takes much more time because of corrections. In this case it is better to cutoff the regions of low density with the crop command.

3.3.1.3 ELID core

For fields strongly oscillating in core regions, e.g., the electron density Laplacianor ELI-D, the basin determination can fail, respectively large number of inner shellbasins will be created. Such behavior can be avoided by the command:

ELID core

whereby all grid maxima within tabulated ELI-D core radii around the atomic posi-tions will be unified to create a single basin. The ELID core command can be usedalso as a block section:

ELID corescale r= <val><elem 1> <elem 2> ...ELID core end


Here, <val> is a multiplicative factor to scale the tabulated ELI-D core radii. Thiscan be followed by a list of element symbol for which the scale factor applies (other-wise it is valid for all the elements defined in the evaluation (e.g., in the wavefunctionfile).

3.3.1.4 localization domain

Instead of the basins (i.e., regions given by all trajectories terminating at the givenattractor) the f -localization domains can be determined by the command:

localization domain= <f 1> <f 2>

The f -localization domains are regions of space bounded by the isosurfaces forthe isovalue <f 1> of the field chosen by the using field <n> command. Thus anf -localization domain encloses regions with field values less or equal the <f 1>isovalue. The second isovalue <f 2> is optional, in which case the domains areregions between the two isosurfaces, i.e., enclosing regions of space with field valuesless than <f 1> and larger than <f 2>.

3.3.1.5 save

The basins determined in the basin section must be assigned to specific basin de-scriptor by the command:

save basin <n>

The data descriptor basin <n> can be used in subsequent evaluation sections toutilize the data in the saved basin file.

3.3.1.6 symmetrize in cell

In case of a solid state calculation the grid field can be computed for the unit cellregion. The basins for such grid data can be joined together by the translationalsymmetry operation using the command:

symmetrize in cell

Without this command all basin cut by the grid box will be given as a separateunique basin.

24 3 Control file

Fig. 3.3 Usage of the top keyword. The vertical lines mark the basin borders (saddle points)

3.3.1.7 top

Sometimes it is difficult to perform the basin search. This is the case when too manygrid maxima are found or, especially, when the wavefunction file is not available.Then, the basin search can be assisted by choosing a field isovalue for the startingf -localization domain. This isovalue is given by the command:

top= <val>

All grid maxima within each f -localization domain will be assigned to the samebasin.

The keyword top can also be used to merge basins into larger sets. Then, applyingthe top value <val> DGrid ‘cuts’ out all grid points with field values higher thanthe top value and assign a basin to each separate closed region. Thus, choosing thevalue V 1 in the Fig. 3.3 will mark all values above the V 1 line and assign the pointsto the first basin. After that the basin search proceed the usual way to determinethe remaining basins. Choosing as the top value V 2 would first mark two separateregions and assign basins to it, thus avoiding possible spurious basins around thesecond attractor (if there were no spurious basins then the result would be the sameas for the top value V 1).

Sometimes it is desirable to merge few basins into a single one (creating abasin set, when the basins have common scalar field isosurface, for superbasins see


Sec. 3.3.15). With V 3 as the top value the third basin would form a separate region.However, the first and second basin would merge together (the values above the V 3line yield a single region). In this way it is possible conveniently merge together lotof basins and analyze such sets of basin as autonomous parts.

3.3.1.8 using

The obligatory data for the basin determination is a grid file containing the field inwhich the basins will be searched. This field grid file is chosen by the command:

using field <n>

in case of the <n>-th field grid file. The field used, for instance the second one,must be either declared by the data assignment field 2=<filename> or be createdby preceding compute section (with save field 2).

3.3.2 CBO

The section for the calculation of bond order according to Cioslowski [17] is de-limited by separate lines with the keywords CBO and CBO end. The calculation iscontrolled by keywords summarized in Tab. 3.4 and described in detail below.

Table 3.4 Keywords and parameters available for the CBO calculation


basin set= basin set choice from intersection <1|2>decomp basin= decomposition for selected basin <bsn>select basins basin selection selection blockusing norm parts to be used norm <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|norm_1=C3H3NO_HF_6-31G.g03.rho_r.bsn.sij.lno.norm

CBO:-----------------------------------------

using norm_1:-----------------------------------------CBO_end

26 3 Control file

3.3.2.1 basin set

For an examined system basins based on different fields can be determined, forinstance, the QTAIM basins based on the electron density and the basins based onthe ELI-D, respectively. A special technique to analyze the bonding situation is tointersect the basins of one set (e.g., QTAIM basins) with the other set of basins (e.g.,the ELI-D basins), cf. the description on page 62. With the command:

basin set= <1|2>

the analyzed basin set is chosen. For the above mentioned fields it means that withbasin set=<1> the QTAIM basins, cut in pieces by the ELI-D basins, are analyzed.With basin set=<2> the focus is on the analysis of the ELI-D basin cut by theatomic basins.

3.3.2.2 decomp basin

If the command:

decomp basin= <bsn>

is invoked then the Cioslowski bond orders between the basin number <bsn> andall other basins are decomposed into LNO contributions.

3.3.2.3 select basins

For large molecules many basin (for instance QTAIM basins) will be created. Then,the bond orders table can be very difficult to survey, because it will contain dataeven for very distant basins. The analysis can be reduced to basins of interest by theblock section:

select basins<bsn 1> <bsn 2> ...select basins end

where <bsn 1><bsn 2> ... is a blank separated list of basin numbers for which thetable of bond orders will be created. The numbers can be also specified by hyphenseparated range descriptor, e.g., <bsn from>–<bsn to>.


3.3.2.4 using

The obligatory data for the CBO calculation is a file containing the norm parts. Thenorm file number <n> is chosen by the command:

using norm <n>

The required norm <n> file must be either declared by the data assignmentnorm <n>=<filename> outside the CBO section or be created by preceding lo-calized natural orbital section (using save norm <n> within that section).

3.3.3 compute

The section for the calculation of the field values is delimited by separate lines withthe keywords compute and compute end. The calculation is controlled by key-words summarized in Tab. 3.5 and described in detail the following subsections.

Table 3.5 Keywords and parameters available for the compute section

Keyword Description Valuea

contributions= orbital contributions at position <x> <y> <z>cp= critical point search <min|max|saddle|ring>

→ <x> <y> <z>curv decomp decompose curvatures at position <x> <y> <z>ev projection= project Hessian EV at position <x> <y> <z>format= grid output format <cube|dgrid|grace|lmto>mesh= grid definition <dist> <field>=<isoval>

→ paralleloccupation orbital occupations definition blockorigin= grid definition block <x> <y> <z>overlap matrix= overlap file to be used sij <n>par flt= up to 5 float parameters <flt1> <flt2> ...par int= up to 5 integer parameters <int1> <int2> ...point= values at position <x> <y> <z> virialref point= reference point position (spin) <x> <y> <z>

→ <alpha|beta|both>save field grid to be saved field <n>select orbitals select orbitals selection blockslices= grid slice calculation <slices> <slice>space= space representation <position|momentum>using wavefunction file to be used wfn <n>

aDefault values are given in Typewriter font

28 3 Control file

Example:


compute:-----------------------------------------

using wfn_1

ELI-D triplet-pairrhorho Laplacian


In the compute section must be an assignment specifying an existing wavefunctionfile (with the command using wfn <num>) and at least one field must be chosen.The descriptors for the available fields are compiled in Tab. 3.6 and Tab. 3.7.

Some setup information and the timing as well as the calculation progress ofthe DGrid job will be written to the log file named rho elid.inp.DG LOG (all logfiles have capital letters extender). This behavior can be changed with the keywordconsole in which case the output will be written to the console.

3.3.3.1 Field descriptors

At least one assignment choosing the appropriate descriptors for the field to be com-puted is obligatory within the compute section. For each field the correspondingdescriptors must be on a separate input line. In the above example the ELI-D as wellas the electron density and its Laplacian will be computed. Let as have a closer lookon the field assignments. The general form is:

<fld> <type> <spin> <pair> <cmp>

Per default all fields are computed in the real (coordinate) space. With the assign-ment space=momentum it is possible to perform the calculation of the desired fieldsin the momentum space. In DGrid-5.0 the transformation into the momentum spaceis available only for Gauss-type orbitals.

Remark 3.5. Some descriptors can be omitted in the field descriptor for the grid cal-culation. Then the default values for <type>, <spin>, and <pair>, respectively,will apply. As default for <type> the field values will be computed. For <spin>and <pair> the defaults depend on the chosen field. For a 1-particle field (like theelectron density, kinetic energy density, local source, etc.) the <spin> descriptor isset to both, except for ‘phi’ in which case the majority spin alpha is chosen. For


Table 3.6 Available fields

Keyword Property Description Ref. File ext.

ALIE 1ρ ∑εi|φi|2 average local ionization

energyalie

ELF[1+χ2

]−1 ELF: ’spin-polarized’ [15, 42, 34] elfELF-cs

[1+χ2

cs]−1 ELF: ’closed-shell’ [15, 42, 34] elf cs

ELF-Kirzhnits[1+χ2

kirzh

]−1 ELF: Tsirelson [46] elf kirzhELI-D ρ VD ELI-D [29, 32] elidELI-q ρ

(s)2 V 2

q ELI-q [36] eliqELIA ρ

αβ

2 /(ρα ρβ ) ELIA [33] eliahole-curvature ∇2ραα

2 Fermi-hole curvature hole curvLOL τLSDA

τ+τLSDA localized orbital locator [44] lol

LS − 14π

∇2ρ(r)|r−r′| local source [10, 26] ls

OEP ∇2√ρ

2√

ρone-electron potential [27] oep

PUC −∇2 lnρ position uncertainty cur-vature

puc

pair-volume-function VD pair-volume function [30, 47] pair volpELI-D ρorb VD orbital resolved ELI-D [30, 47] p elidphi φi orbitals phicharge-volume-function Vq charge-volume function [36] q volrho ρ electron density rhoon-top-density ρ2(r,r) on-top density rho ontoprho-spin ρs spin density rho spntau τ kin. energy density tautP τ− tw Pauli kin. energy dens. tp

tW 18(∇ρ)2

ρWeizsacker term tw

vP tp +ALIE−HOMO Pauli potential vp

ρs = ρα −ρβ

gσ = ∑σi< j ∑

σk<l Pi j,kl [φi∇φ j−φ j∇φi].[φ

∗k ∇φ ∗l −φ ∗l ∇φ ∗k ] Fermi hole curvature

VD = [12/gσ ]3/8

Vq = 1/ρ(s); ρ(s) density of singlet coupled electrons

cF = 310 (3π2)2/3 Fermi constant

τ = 12 ∑i |∇φi|2 kinetic energy density of non-interacting system

χ =

[τ− 1

8(∇ρα )

2

ρα− 1

8(∇ρβ )

2

ρβ

]/[22/3cF (ρ

5/3α +ρ

5/3β

)]

χcs =[τ− 1

8(∇ρ)2

ρ

]/[cF ρ5/3

]χkirzh =

[cF ρ5/3− 1

9(∇ρ)2

ρ+ 1

6 ∇2ρ

]/[cF ρ5/3

]τLSDA = 22/3cF ρ5/3 originally defined for single spin channel, i.e., ρσ [44]

30 3 Control file

2-particle fields the <pair> descriptor is set in accordance with the <spin> descrip-tor, i.e., alpha-alpha pair spin is used if alpha spin is given, etc. (if <spin>given, otherwise spinless is used). If only <pair> descriptor is given, then the<spin> descriptor is set accordingly (for the <pair> descriptor spinless the<spin> descriptor both is used).

Table 3.6 summarizes the available fields (the conditional fields are given in Table3.7 on page 31). For almost all fields the derivatives can be computed using thecorresponding <type> descriptors, cf. page 89. Observe that ELI-D is written inTable 3.6 as ϒD = ρVD, i.e., as the electron density ρ multiplied by the pair-volumefunction VD. As described in Ref. [30, 47] ELI-D can be exactly decomposed intoorbital contributions, the same way as the electron density is given by the sum oforbital contributions.

The calculation of the ELI-D orbital contributions is performed with the keyword‘pELI-D’, for instance with ‘pELI-D alpha’. In this case the ϒ α

D , i.e., ELI-D for theα-spin pairs is considered. The orbitals for the contribution are chosen within ablock starting with the keyword select orbitals, cf. Sec. 3.3.3.18. The pair-volumefunction VD for pELI-D is of course computed from the total pair density. The infor-mation about the chosen orbitals is written in the log file as well as in the grid file.Summing up the pELI-D contributions of all orbitals recovers the total ELI-D.

Remark 3.6. pELI-D contributions are possible only for occupied orbitals. DGriddoes not compute contributions of virtual orbitals.

Remark 3.7. Using the assignment ‘eli-d alpha’ together with the keyword occupa-tion, instead of ‘peli-d alpha’, yields another quantity, because in this case also thepair-volume function VD is calculated from the chosen orbitals. This quantity willNOT add up to total ELI-D.

3.3.3.2 Conditional fields

Some fields, like the electron pair density depend on two coordinates. In this case,to compute a 3-dimensional grid field, it is necessary to fix one coordinate at areference position. Table 3.7 summarizes the available conditional fields.

Let us consider the following example for the calculation of the conditional pairdensity:


compute:-----------------------------------------

using wfn_1

cond-pair-density alpha-alpha


ref_point= 0.0 0.0 0.5


With the assignment ‘ref point= x y z’, cf. Sec.3.3.3.16, the position of the refer-ence electron is fixed at Cartesian coordinates [x y z] (the electron pair density isa 6 dimensional field). Similar background follows the calculation of the domain-averaged Fermi-hole (DAFH) [37] in which case one of the pair density coordinatesis confined to given region Ω (for instance a basin). To compute DAFH the overlapintegrals over the confinement region are needed, cf. Sec. 3.3.12. In case that some1-particle field would be computed within the same compute section, the ref pointcommand have no influence on this 1-particle field.

Table 3.7 Conditional properties

Keyword Property Description File ext.

cond-pair-density Pcond(1,2) conditional pair density pd cond

DAHD f σσav (1) domain-averaged hole density da hole density

hole-density ρσex(1) hole density hole density

pair-density ρ2(1,2) pair density pd

ρσσ2 (1,2) = 1

2 ∑σi< j ∑

σk<l Pi j,kl

∣∣φi(1)φ j(2)∣∣ ∣∣φ ∗k (1)φ ∗l (2)∣∣

f σσav (1) = ρ(1)

∫Ω

ρ(2)d2 − 2∫

Ωρσσ

2 (1,2)d2

Pσσcond(1,2) = ρσσ

2 (1,2)/ρσ (2)

ραβ

2 (1,2) = 12 ∑

αi, j ∑

β

k,l Pi j,kl∣∣φi(1)φ j(2)

∣∣ ∣∣φ ∗k (1)φ ∗l (2)∣∣Pσσ ′

cond(1,2) = ρσσ ′2 (1,2)/ρσ ′ (2)

ρ2(1,2) = ραα2 (1,2)+ρ

ββ

2 (1,2)+ραβ

2 (1,2)+ρβα

2 (1,2) = ρ(s)2 (1,2)+ρ

(t)2 (1,2)

Pcond(1,2) = ρ2(1,2)/ρ(2)

ρσex(1) = 2Pσσ

cond(1,2)−ρσ (1)

3.3.3.3 Orbitals

The input for the calculation of an orbital deviates somewhat from the assignmenton page 28:

32 3 Control file

phi <num> sym=<lbl> <spin> <part>

The descriptors can be written in any order. Here, <num> is the number of thechosen orbital in the symmetry labeled <lbl>. <spin> can be one of ‘alpha’ or‘beta’. With the descriptor <part> the real (descriptor ‘real’, which is the de-fault) or imaginary (‘imag’) part of the orbital can be computed (the imaginary partis usually present for orbitals in momentum space or in certain cases for crystal or-bitals in real space). Other possibility is given for complex orbital values expressedin polar form, where the descriptors ‘amplitude’ and ‘phase’ computes the cor-responding parts. The orbital symmetry and number as well as the chosen spin andpart form the extender of the resulting grid file.

Figure 3.4 shows the orbitals number 17 and 18 (which is the HOMO) for theoxazole molecule in different representations. As can be seen from the usual co-ordinate space representations in diagrams 3.4a the MO 17 is roughly the nitrogenlone pair and MO 18 is a π orbital (the red colored isosurfaces have negative iso-value). Using the wavefunction file C3H3NO HF 6-31G.g03 the grid for the orbital17NONE was computed with the assignment ‘phi 17 sym=NONE alpha’ in the con-trol file (the part ‘real’ was set by default). The diagrams 3.4b-d show isosurfacesfor the orbitals computed in the momentum space representation. Each orbital isgenerally given by a complex function:

φ(p) = ϕ(p) + i χ(p) (3.1)

with the real and imaginary parts ϕ and χ , respectively. In the diagrams 3.4b theisosurfaces for the real parts of the corresponding orbitals are depicted (the graycolored isosurfaces have negative isovalue; observe the inversion symmetry). Thediagrams 3.4c show the imaginary of the orbitals. Here the orbital 17NONE wascomputed with the assignment ‘phi 17 sym=NONE alpha imag’ with the command‘space=momentum’ included in the compute section.

Both parts of an complex orbital can be unified into single diagram using thepolar form of the orbital:

φ(p) = ϑ(p) ei ω(p)ϑ =

√ϕ2 +χ2 ω = sgn(χ) arccos(ϕ/ϑ) (3.2)

with the amplitude ϑ and phase ω . The diagrams 3.4d show the isosurface for theamplitudes of orbitals number 17 and 18. Each isosurface is colored by the phase(the range for the color scale is ±π). Interestingly, the isosurface for amplitudeof the orbital 17 resembles the isosurface for the imaginary part of the MO, cf.Fig. 3.4c, whereas the amplitude for the orbital number 18 is close the isosurfacefor the corresponding real part of the MO 18. This is also confirmed by the coloringof the amplitude isosurfaces (deep blue and white correspond to the phase±π whichstands for the real part (similarly, green color means phase ω ≈ 0, i.e., again the realpart). In case of MO 17 the overall coloring of the amplitude isosurface is around±π/2, i.e., phase value for the imaginary part.


Fig. 3.4 Coordinate (a) and momentum space (b-d) representations of the orbitals number 17 and18 (HOMO) for oxazole C3H3NO (C black, H gray, N green, O blue). a: ±0.16 and ±0.10 isosur-faces for MO 17 and 18, respectively; b: real parts of the MOs (±0.20 isosurfaces); c: imaginaryparts of the MOs (±0.26 and ±0.05 isosurfaces); d: amplitudes for the MOs (0.26 and 0.20 iso-surfaces colored by the phase)

34 3 Control file

3.3.3.4 contributions

For the electron density and ELI-D, respectively, the keyword contributions allowsto decompose the field value into its molecular orbital contributions. The keywordcontributions= is followed by the coordinates of the position at which the decom-position is performed:

contributions= <x> <y> <z>

The fields must be declared within the compute block with the ‘rho’, respectively‘ELI-D’ descriptors, cf. the description in Sec 3.3.3.1. For example, using the con-trol file C3H3NO elid contrib.inp:


compute:-----------------------------------------

using wfn_1

ELI-D alpha-alpha

contributions= 1.05053 1.50085 0.0:-----------------------------------------compute_end

yields the log file C3H3NO elid contrib.inp.DG LOG including the result of theanalysis:

----------------------------------------------------------------------------contributions at point [ 1.050530, 1.500850, 0.000000]----------------------------------------------------------------------------

spin : alphapair : alpha-alpha

----------ELI-D[r] : 1.73736

--------------------MO %

--------------------7 7 37.614 14 18.69 9 11.08 8 9.317 17 6.76 6 5.910 10 4.415 15 4.311 11 1.3

--------------------99.0


The orbital contributions are given at the position of the ELI-D bond attractor be-tween the nitrogen and carbon (found using the keyword cp=, see page 36 below).The analysis shows that the ELI-D value 1.73736, which is proportional to thepopulation of α-spin electrons needed to form a fixed fraction of an αα-spin pair[36, 47], is mainly the sum of 4 contributions. The two highest contributions orig-inate from the canonical orbitals φ7 (37.6%) and φ14 (18.6%). The orbital indicescorrespond to the numbering in the wavefunction file set in the control file by thecommand wfn 1= C3H3NO HF 6-31G.g03. In the above case the sum of the or-bital contributions does not yield 100% because contributions smaller than 1% arenot listed.

If the keyword contributions is used, any grid definitions will be skipped, i.e.,only the given position will be evaluated.

3.3.3.5 cp

The keyword cp= is followed by the descriptor <crit> for the critical point and thecoordinates of the start point where the search for the critical point begins:

cp= <crit> <x> <y> <z>

The descriptor <crit> can be one of min, max, saddle, and ring, respectively,cf. Table 3.8. The Cartesian coordinates [x y z] are given by 3 float numbers.

Table 3.8 Descriptors for critical point search

<crit> description signature

min 7→ minimum (3, +3)max 7→ attractor (3, -3)saddle 7→ saddle point (3, -1)ring 7→ ring critical point (3, +1)

The search for the critical point starts from the position given by the Cartesian co-ordinates [x y z] (3 float numbers). The field for which the critical point will besearched must be given within the compute section. The position of the criticalpoint (if found) is printed to the log file, together with the property value at thispoint, the gradient (which will be close to zero) and the Laplacian. Additionally,the eigenvectors and curvatures are determined from the Hessian. If you see the in-formation ‘using Taylor’ then the Hessian needed for the search was not computedanalytically. For instance, with the control file:

36 3 Control file


compute:-----------------------------------------

using wfn_1

ELI-D alpha-alpha

cp= max 1.5 2.0 0.0:-----------------------------------------compute_end

the search of an ELI-D attractor will be started from the position [1.5, 2.0, 0.0 ]. Theresulting attractor (C-C bond descriptor) position is [1.05053, 1.50085, 0.00000] ascan be seen by the inspection of the log file C3H3NO elid cp.inp.DG LOG:

--------------------------------------------------------------------------search for (3,-3) critical point at [ 1.50000 2.00000 0.00000]--------------------------------------------------------------------------

(3,-3) critical point at the position [ 1.05053 1.50085 0.00000]


----------ELI-D[r] : 1.73736

gradient : 5.628e-11components : [ 0.00000, 0.00000, 0.00000 ]

Laplacian : -7.69090

Hessian : | -4.47886, 1.27616, 0.00000 || 1.27616, -2.79455, 0.00000 || 0.00000, 0.00000, -0.41749 |

Eigenvectors curvature-------------------------------- -----------

1. ev : [ 0.88057, -0.47392, 0.00000 ] -5.165692. ev : [ 0.47392, 0.88057, 0.00000 ] -2.107723. ev : [ 0.00000, 0.00000, 1.00000 ] -0.41749

x y z

--------------------------------------------------------------------------

The corresponding search path (the trajectory) is saved in the file C3H3NO HF 6-31G.g03.ELI-D.search.str. This file is written in special STR format.

3.3.3.6 curv decomp

The keyword curv decomp is followed by the coordinates of the position at whichELI-D will be decomposed:


curv decomp= <x> <y> <z>

The decomposition procedure applies only to ELI-D. However, the ELI-D field mustbe chosen in the compute section (need to specify the ELI-D ”flavor”). At the po-sition given by the Cartesian coordinates [x y z] (3 float numbers) the ELI-D value,gradient, Laplacian and Hessian, together with the curvature eigenvalues and eigen-vectors are computed and written to the log file. Along the principal ELI-D curva-tures the ELI-D Laplacian is decomposed into the electron density and pair-volumefunction dependent terms [48]. Using the control file C3H3NO elid decomp.inp:


compute:-----------------------------------------

using wfn_1

ELI-D alpha-alpha

curv_decomp= 1.05053 1.50085 0.0:-----------------------------------------compute_end

the decomposition will be performed at the ELI-D attractor position (C-C bond de-scriptor). The decomposed values are written to the file C3H3NO elid decomp.inp.DG LOG:

----------------------------------------------------------------------------ELI-D decomposition at point [ 1.050530, 1.500850, 0.000000]----------------------------------------------------------------------------


----------ELI-D[r] : 1.73736

gradient : 0.00002components : [ 0.00002, -8.400e-06, 0.00000 ]


Hessian : | -4.47888, 1.27618, 0.00000 || 1.27618, -2.79456, 0.00000 || 0.00000, 0.00000, -0.41749 |


1. ev : [ 0.88056, -0.47393, 0.00000 ] -5.165732. ev : [ 0.47393, 0.88056, 0.00000 ] -2.107713. ev : [ 0.00000, 0.00000, 1.00000 ] -0.41749

x y z

----------------------------------------------------------------------------

38 3 Control file

Data projected onto the ELI-D eigenvectors----------------------------------------------------------------------------

Y’’/Y rho’’/rho V’’/V 2rho’/rho*V’/V----------------------------------------------------------------------------11 | -2.97333 1.79670 -4.52212 -0.2479022 | -1.21317 -2.26058 1.05509 -0.0076833 | -0.24030 -1.95648 1.71617 0.00000----------------------------------------------------------------------------sum | -4.42680 -2.42035 -1.75086 -0.25558----------------------------------------------------------------------------

Here the 11, 22, and 33 rows contain data for curvatures along the correspondingprincipal curvatures, e.g., 11 for the curvature along the first eingenvector (‘1. ev’ inthe list above). ELI-D is given by the symbol Y (meaning ϒD) and the pair-volumefunction by the symbol V.

3.3.3.7 ev projection

The keyword ev projection= is followed by the coordinates (3 float numbers) of theposition where the projection will be performed:

ev projection= <x> <y> <z>

The eigenvector projection works as follows. At least two fields must be chosen byprevious field assignment, cf. Sec. 3.3.3.1. As an example let us consider the controlfile C3H3NO evproj.inp:


compute:-----------------------------------------

using wfn_1

rho alphaELI-D alpha-alpha

ev_projection= 0.773035 1.624593 0.000000:-----------------------------------------compute_end

For the first field (here, α-spin electron density) the value, gradient, Laplacian andHessian, together with the curvature eigenvalues and eigenvectors will be computedat the position given by the Cartesian coordinates [0.773035, 1.624593, 0.000000],which is the bond critical point between the nitrogen and carbon, and written to thelog file C3H3NO evproj.inp.DG LOG:


----------------------------------------------------------------------------eigenvectors at point [ 0.773035, 1.624593, 0.000000]----------------------------------------------------------------------------

spin : alphapair : --

----------rho[r] : 0.18699

gradient : 1.279e-06components : [ -1.142e-06, 5.759e-07, 0.00000 ]


Hessian : | 0.03443, -0.23782, 0.00000 || -0.23782, -0.30102, 0.00000 || 0.00000, 0.00000, -0.35080 |


1. ev : [ 0.88779, -0.46024, 0.00000 ] 0.157722. ev : [ 0.46024, 0.88779, 0.00000 ] -0.424303. ev : [ 0.00000, 0.00000, 1.00000 ] -0.35080

x y z

Observe that the density gradient (1e-6) is close to zero (as expected for a criti-cal point). The gradients of fields given by the successive field assignments (in theexample ELI-D) are projected onto the principal curvatures of the first field (i.e.,rho):

gradient projections onto the eigenvectors:----------------------------------------------------------------------------


----------ELI-D[r] : 1.43893

gradient : 2.07444components : [ 1.76091, -1.09658, 0.00000 ]

component projections : [ 2.06801, -0.16309, 0.00000 ]


Hessian : | -3.96020, 3.13776, 0.00000 || 3.13776, -2.17892, 0.00000 || 0.00000, 0.00000, 1.20839 |

rotated Hessian : | -6.14707, 1.08063, 0.00000 || 1.08063, 0.00794, 0.00000 || 0.00000, 0.00000, 1.20839 |

----------------------------------------------------------------------------

First the ELI-D value and the gradient components along the Cartesian axes aregiven, followed by the gradient components projected onto the principal densitycurvatures (of course, the gradient vector itself remains unchanged). It can be seenthat the ELI-D gradient vector is oriented mainly along one of the principal densitycurvatures. Similar projection is performed for the ELI-D Hessian components.

40 3 Control file

3.3.3.8 format

The keyword format is followed by the descriptor <fmt> (case insensitive) for theformat of the resulting grid file:

format=<fmt>

Table 3.9 Format descriptors

<fmt> Format

cube 7→ CUBE [2]dgrid 7→ DGrid grid filegrace 7→ Grace [5]lmto 7→ TB-LMTO-ASA [28]

Remark 3.8. In both the Cube and Grace formats the field values are given by floatnumbers. Routines which expect grids of integer values (like the basin evaluation)could fail.

3.3.3.9 mesh

The command mesh= is used to define the grid region computed by DGrid. In caseof periodic structure calculations it is recommended to specify the grid region bythe command origin= instead. The keyword mesh= is followed by the mesh-size<size> and the specification of the isovalue <iso> for the field <fld>:

mesh=<size> <fld>=<iso> parallel

The mesh-size (float number) is the distance between neighboring grid points (theunits from the wavefunction file are used which is usually set to bohr). The string<fld> can be any of the available field descriptors (cf. Tab. 3.6), followed by theisovalue of the field isosurface that will be enclosed by the grid box. For instance:

mesh=<0.10> rho=0.001

states that the grid region will accommodate the density isosurface 0.001. As al-ready mentioned, any other field can be chosen to determine the grid region, but theelectron density is a reasonable choice as the extent of a molecule is connected withthe extent of the ‘electronic cloud’ [9].


The program works as follows (cf. Fig. 3.5). From the atomic positions the aver-age coordinate is computed. The ‘initial box’ (the smallest box around the C3H3NOmolecule enclosing all atoms) is created. Observe, that the ‘initial box’ is not paral-lel along the Cartesian axes. Instead, the box is oriented along the principal axes ofthe molecular rotational ellipsoid, i.e., slightly rotated in the xy (molecular) plane.The ‘initial box’ is enlarged independently in all three directions by the border untilthe density isosurface with the isovalue 0.001 is inside the grid box, see the lowerdiagram in Fig. 3.5 (whereby the grid region dimensions remain multiple of themesh-size of 0.10 bohr).The keyword parallel forces the box sides to be aligned parallel with the Cartesianaxes, for instance:

mesh=<0.10> rho=0.001 parallel

Remark 3.9. For the calculation of overlap integrals (see Sec. 3.3.12) from theGauss-type orbitals, used for the fluctuation and delocalization indices, it is nec-essary that the grid is oriented parallel to the Cartesian axes.

3.3.3.10 occupation

The keywords occupation and occupation end delimit a block where orbitals andtheir occupations can be chosen for further evaluation within the compute sectionin the control file.

Normally, the fields are computed from all occupied orbitals. For each orbital ofparticular symmetry the occupations are read from the wavefunction file. To com-pute the fields for selected orbitals only (for instance the electron density based oncertain orbitals) the occupations must be specified in the occupation block:

occupation--------------------------------:#: sym: alpha: beta:-------------------------------:17 NONE 1.0 1.018 NONE 1.0 1.0

--------------------------------:occupation_end

For each chosen orbital the data must be written in single line. Each orbital is char-acterized by its number in the symmetry and the symmetry descriptor (these dataare accessible from the wavefunction file) followed by the α-spin and β -spin occu-pations (new choice is possible). In the above example the orbitals number 17 and18 of the symmetry ‘NONE’ (GAUSSIAN writes the orbitals without symmetry),both double occupied, are selected for the calculation.

42 3 Control file

grid regiony

x

initial box

y

x

ρ=0.001bohr−3

Fig. 3.5 Grid region for oxazole C3H3NO (C black, H gray, N green, O blue) created by thecommand ‘mesh=0.05 rho=0.001’


Remark 3.10. Bear in mind that all fields will be computed from the chosen set oforbitals. This may lead to not properly defined fields, for instance in case of ELI.

Remark 3.11. If only a subset of orbitals is chosen by the occupation block then itis better to specify the grid region by grid vectors (i.e., with the origin block) andnot by the mesh= command. The reason is that the orbital density isosurface wouldbe used which does not always converge to a reasonable grid box.

Remark 3.12. To compute the amplitude for single orbital, (i.e., not the orbital den-sity) use the keyword ‘phi’ in the compute section, for instance, ‘phi 2 sym=NONEalpha’ for the amplitude of the second alpha MO of the symmetry NONE, cf.Sec 3.3.3.1. As for any other field, there can be more than one ‘phi’ assignmentin the control file.

3.3.3.11 origin

The keyword origin starts a block of 4 lines with readable information defining thegrid region for the field calculation (the lines can be interrupted by empty lines orcomments). The command origin= is followed by 3 float numbers for the Cartesiancoordinates of the grid-box origin:

x: y: z:----------------------------:origin= 0.0 -4.0 0.0 intervals:----------------------------------------:i-vector: 3.0 4.0 0.0 100j-vector: 4.0 3.0 0.0 100k-vector: 0.0 0.0 5.0 100----------------------------------------:

The strings ‘x:’, ‘y:’, ‘z:’, ‘intervals’, ‘i-vector:’, etc. in the control file are not parseddue to the colon at the end of the strings – they used just for clarity.

The next 3 lines after the origin define 3 vectors spanning the grid mesh. Each linecontains 3 float numbers for the vector components along the x, y, z axes, followedan integer number for the number of intervals the vector will be divided in. Thus,the above block will generate a grid mesh starting at the position [0, −4, 0]. The 3vectors (i, j,k) are 5 bohr long. The direction of vector k coincides with z axis. Thevectors i, j are perpendicular to each other. The endpoint of vector i is at [3, 0, 0](this information is not explicitly given in the input, but can easily be deduced fromthe data – from the origin position [0, −4 ,0] one moves 3 bohr in x direction and4 bohr in y direction). The grid mesh has points spaced by 0.05 bohr along eachvector (i.e., 101×101×101 points in total).

Remark 3.13. To avoid numerical instabilities the mesh point distance should notexceed 0.1 bohr.

44 3 Control file

3.3.3.12 overlap matrix

The keyword overlap matrix= is followed by the overlap descriptor sij <n>:

overlap matrix= s ij<n>

in case of the <n>-th overlap file. The overlap file used, for instance the second one,must be either declared by the data assignment sij 2=<filename> or be created bypreceding overlap section (with save sij 2).

The overlap matrix for all molecular orbitals over the chosen basin will be pre-pared in DGrid. The data are needed for instance to compute the domain-averagedFermi-hole.

3.3.3.13 par flt

The keyword par flt= is followed by up to 5 float parameter:

par flt= <f1> <f2> <f3> ...

The parameter can be addressed in subroutines of the DGrid code.

3.3.3.14 par int

The keyword par int= is followed by up to 5 integer parameter:

par int= <i1> <i2> <i3> ...

The parameter can be addressed in subroutines of the DGrid code.

3.3.3.15 point

The keyword point= is followed by the Cartesian coordinates [x y z] (3 float num-bers) at which the chosen fields will be evaluated:

point= <x> <y> <z>

The fields must be specified by separate field assignments, cf. Sec 3.3.3.1. For eachfield the value at the position [x y z] is written to the log file (respectively to the con-sole when invoking the console command outside the compute section), together


with the field gradient and the field Laplacian. Additionally, the eigenvectors andcurvatures are determined from the Hessian. If you see the information ‘using Tay-lor’ then the data were not computed analytically.

If the keyword point is used any grid definitions will be ignored, i.e., only thegiven position will be evaluated, and no grid file will be written.

An additional feature is the information about indices connected with local virial[9]. The feature is invoked with the keyword virial following the [x y z] coordi-nates:

point= <x> <y> <z> virial

At the end of the above described output the following data will be included (fol-lowing are the data for the C3H3NO at the position [0.77304, 1.62459, 0.00000],which is the N-C bond critical point in electron density):

---------------------------------------------------------------Virial indices: L V G |V|/G H---------------------------------------------------------------

-1.2348 -0.8647 0.2780 3.1104 -0.5867---------------------------------------------------------------

Here, L is the total electron density Laplacian, V = 14 ∇2ρ−2τ is the local potential

energy density, G is the positive definite kinetic energy density and H = G+V is theenergy density. |V |/G is the corresponding local virial ratio at the chosen position[x y z].

3.3.3.16 ref point

The keyword ref point is followed by the position given by the Cartesian coordi-nates [x y z] (float numbers) for a reference position (reference electron):

ref point= <x> <y> <z> <spn>

The assignment is needed only for 2-particle fields, see Table 3.7 on page 31. In thiscase the coordinate of one electron (with the spin <spn>) is fixed at the position [xy z], whereas the position of the other electron runs through the grid (as defined bythe origin or mesh assignments). Additionally, it is used for the reference positionwhen calculating the local source.

3.3.3.17 save

The fields computed in the compute section can be assigned to specific field descrip-tor by the command:

46 3 Control file

save field <n>

The command must follow immediately in the next line after the specification of thecomputed field, for instance:


compute:-----------------------------------------

using wfn_1

ELI-D triplet-pairsave field_1rhosave field_2


Here, the ELI-D grid is assigned to the descriptor field 1 and the electron densitygrid to the descriptor field 2. The data descriptor field <n> can be used in subse-quent evaluation sections to utilize the data in the saved grid files.

3.3.3.18 select orbitals

The keywords select orbitals and select orbitals end delimit a block where or-bitals can be chosen for further evaluation within the compute section in the controlfile.

Normally, the fields are computed from all occupied orbitals, respectively fromthe orbital set chosen by user (cf. Sec 3.3.3.10). However, for certain fields (e.g.,partial ELI-D) it is necessary to chose a selected subset of orbitals. This orbitalsmust be specified in the select orbitals block:

select_orbitals--------------------------------:5-10 16 18--------------------------------:select_orbitals_end

The orbitals are selected by the number given in the wavefunction file. There aretwo possibilities: either by blank separated orbital numbers (16 18) or by a range ofnumbers (5-10; read as single string without spacing). The numbers can be written


in more lines as the block is read until the keyword select orbitals end. The occu-pations of the orbitals are read from the wavefunction file and cannot be changed.

3.3.3.19 slices

The field grid defined by ni×n j×nk points in each direction can be divided intoslices along the k direction (in DGrid the loop over the grid points runs first over thei direction, followed by j and k), cf. Sec. 5. The grid division into slices includes thegrid border as well. The calculation of a grid slice is accomplished with the keywordslices= followed by the number <div> of slices and the slice number <slc>:

slices=<div> <slc>

For each slice a separate control file is needed (thus, <div> jobs must be sub-mitted to get all the slices for the whole grid). To the grid field file for each slicethe string ‘.slice-<slc>’ is appended to discriminate between the slices. Having allslices computed the whole grid can be build by merging the slices together callingthe ‘complete’ utility:

dgrid gridname.slice-1 complete

The resulting completed field grid gridname includes all the points. How to run allthe slices in parallel using a script is described in Sec. 5.

3.3.3.20 space

The keyword space= is followed by the descriptor <spc> determining in whichspace representation the field data will be evaluated:

space=<spc>

Table 3.10 Space representation descriptors

<spc> Representation

position 7→ coordinate space [x y z]momentum 7→ momentum space [px py pz]

The default is the evaluation in the position space representation. In case of the mo-mentum space representation the grid definition is using the momentum coordinates.

48 3 Control file

Remark 3.14. DGrid computes the properties in momentum space using the sameformulas as in the coordinate space. The only difference is that it utilizes Fourier-transformed orbitals (momentals). Thus, the charge given by the modulus of squaredorbitals has in momentum space similar meaning as in the real space. However, thisis not the case for the kinetic energy density, because this property is computed byDGrid from squared orbital gradients (of course, in momentum space the kineticenergy is proportional to the expectation value of p2).

3.3.3.21 using

The obligatory data for the field calculation is a wavefunction file. The wavefunctionfile is chosen by the command:

using wfn <n>

in case of the <n>-th wavefunction file. The wavefunction file used, for instance thesecond one, must be either declared by the data assignment wfn <n>=<filename>or be created, e.g., by preceding domain natural orbitals, Fermi orbitals, or lo-calized natural orbitals section (with save wfn <n>).

3.3.4 DI

The section for the calculation of the delocalization indices (DI) [21] is delimitedby separate lines with the keywords DI and DI end. The calculation is controlledby keywords summarized in Tab. 3.11 and described in detail below.

Table 3.11 Keywords and parameters available for the DI calculation


basin set= basin set choice from intersection <1|2>select basins basin selection selection blockusing overlap file to be used sij <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|sij_1=C3H3NO_HF_6-31G.g03.rho_r.bsn.sij


DI:-----------------------------------------

using sij_1:-----------------------------------------DI_end

The resulting table with the DIs between the basins is written to the log file withthe extender ‘.DI’ appended. The whole procedure is described in more detail inSec. A.6.1.

3.3.4.1 basin set

For an examined system basins based on different fields can be determined, forinstance, the QTAIM basins based on the electron density and the basins based onthe ELI-D, respectively. A special technique to analyze the bonding situation is tointersect the basins of one set (e.g., QTAIM basins) with the other set of basins (e.g.,the ELI-D basins), cf. the description on page 62. With the command:

basin set= <1|2>

the analyzed basin set is chosen. For the above mentioned fields it means that withbasin set=<1> the QTAIM basins, cut in pieces by the ELI-D basins, are analyzed.With basin set=<2> the focus is on the analysis of the ELI-D basin cut by theatomic basins.


For large molecules many basin (for instance QTAIM basins) will be created. Then,the table of delocalization indices can be very difficult to survey, because it willcontain data even for very distant basins. The analysis can be reduced to basins ofinterest by the block section:



50 3 Control file

3.3.4.3 using

The obligatory data for the DI calculation is a file containing the overlap integrals.The n-th overlap file number sij <n> is chosen by the command:

using sij <n>

The required sij <n> file must be either declared by the assignment sij <n>=<filename>outside the DI section or be created by preceding overlap section (using savesij <n> within that section).

3.3.5 domain natural orbitals

The section for the calculation of domain natural orbitals (DNOs) [37, 38] is de-limited by separate lines with the keywords domain natural orbitals and do-main natural orbitals end. The calculation is controlled by keywords summarizedin Tab. 3.12 and described in detail below.

Table 3.12 Keywords and parameters available for the DNO calculation


pair= pair descriptor <A-B>save wavefunction/norm file to be saved wfn <n> / norm <n>select basins domain basin selection selection blockusing overlap file to be used sij <n>

Example:


domain_natural_orbitals:-----------------------------------------

using sij_1

select_basins_domain7

select_basins_end

save wfn_1save norm_1

:-----------------------------------------domain_natural_orbitals_end


The resulting AO expansion for the DNOs of the above example is written to thewavefunction file named C3H3NO HF 6-31G.g03.rho r.bsn.sij.dno-b7-O 1, i.e. thefile name is derived from the name of the overlap file sij 1 with the appendix ‘dno’,together with the basin number and name of the domain, which was used for thediagonalization. Also, the norm parts will be written to the file with an additional‘.norm’ appended. The log file with the DNO occupation numbers will be writtenti the file with the extender ‘.LOG’ appended. The whole procedure is described inmore detail in Sec. A.6.2.

3.3.5.1 pair

In case of a closed-shell wavefunction the DNOs are determined by the diagonal-ization of the α-spin overlap matrices and the resulting DNOs are occupied by bothspin components. For open-shell wavefunction DNOs for separate same-spin pairswill be determined with the command:

pair= <A-B>

The resulting α and β spin DNOs will have different occupations.

3.3.5.2 save

The DNOs determined in the domain natural orbitals section can be assigned tospecific wfn descriptor by the command:

save wfn <n>

The data descriptor wfn <n> can be used in subsequent evaluation sections toutilize the data in the saved wfn file. Additionally, for the decomposition of thedelocaliazation index into the DNO contributions [14] the corresponding norm partscan be saved by the command:

save norm <n>

and used in subsequent evaluation sections.

52 3 Control file

3.3.5.3 select basins domain

The DNOs are created by the diagonalization of the overlap matrices over chosendomain. The corresponding domain is defined by a list of one or more basins givenin the block section:

select basins domain<bsn 1> <bsn 2> ...select basins end

where <bsn 1> <bsn 2> ... is a blank separated list of basin numbers. The num-bers can be also specified by hyphen separated range descriptor, e.g., <bsn from>–<bsn to>.

3.3.5.4 using

The obligatory data for the DNO determination is a file containing the overlap inte-grals. The n-th overlap file number sij <n> is chosen by the command:

using sij <n>

The required sij <n> file must be either declared by the assignment sij <n>=<filename>outside the domain natural orbitals section or be created by preceding overlapsection (using save sij <n> within that section).

3.3.6 Fermi orbitals

The section for the calculation of the Fermi orbitals (FOs) [37, 38] is delimitedby separate lines with the keywords Fermi orbitals and Fermi orbitals end. Thecalculation is controlled by keywords summarized in Tab. 3.13 and described indetail below.

Table 3.13 Keywords and parameters available for the FO calculation


pair= pair descriptor <A-B>save wavefunction/norm file to be saved wfn <n> / norm <n>select basins domain basin selection selection blockselect basins isopyc basin selection (isopycnic transformation) selection blockusing overlap file to be used sij <n>


Example:


Fermi_orbitals:-----------------------------------------

using sij_1


select_basins_end

select_basins_isopycall

select_basins_end


:-----------------------------------------Fermi_orbitals_end

The resulting AO expansion for the FOs of the above example is written to thewavefunction file named C3H3NO HF 6-31G.g03.rho r.bsn.sij.fo-b7-O 1, i.e. thefile name is derived from the name of the overlap file sij 1 with the appendix ‘.fo’,together with the basin number and name of the domain, which was used for thediagonalization. Also, the norm parts will be written to the file with an additional‘.norm’ appended. The log file with the FO occupation numbers has the extender‘.LOG’ appended.

Observe that besides the domain for which the diagonalization is performed alsothe basins for the isopycnic transformation must be specified (all basins in the aboveexample). The whole procedure is described in more detail in Sec. A.6.2.

3.3.6.1 pair

In case of a closed-shell wavefunction the FOs are determined by the diagonaliza-tion of the α-spin overlap matrices followed by the isopycnic transformation. Theresulting FOs are occupied by both spin components. For open-shell wavefunctionthe FOs for separate same-spin pairs will be determined with the command:

pair= <A-B>

The resulting α and β spin FOs will have different occupations.

54 3 Control file

3.3.6.2 save

The FOs determined in the Fermi orbitals section can be assigned to specific wfndescriptor by the command:

save wfn <n>

The data descriptor wfn <n> can be used in subsequent evaluation sections toutilize the data in the saved wfn file. Additionally, for the decomposition of thedelocaliazation index into the FO contributions [14] the corresponding norm partscan be saved by the command:

save norm <n>


3.3.6.3 select basins domain

The FOs are created by the diagonalization of the overlap matrices over chosendomain. The corresponding domain is defined by a list of one or more basins givenin the block section:

select basins domain<bsn 1> <bsn 2> ...select basins end


3.3.6.4 select basins isopyc

After the diagonalization of the overlap matrices over the chosen domain (yieldingthe domain natural orbitals) the FOs are created by isopycnic transformation of theDNOs [16]. The domains over which the isopycnic localization is performed aredefined by a list of basins given in the block section:

select basins isopyc<bsn 1> <bsn 2> ...select basins end



All domains of the basin file can be selected with the command:

select basins isopycallselect basins end

3.3.6.5 using

The obligatory data for the FO determination is a file containing the overlap inte-grals. The n-th overlap file number sij <n> is chosen by the command:

using sij <n>

The required sij <n> file must be either declared by the assignment sij <n>=<filename>outside the Fermi orbitals section or be created by preceding overlap section, cf.Sec. 3.3.12 (using save sij <n> within that section).

3.3.7 ICL graph

The section for the determination of the graph of the interconnection lines (ICLs)is delimited by separate lines with the keywords ICL graph and ICL graph end.The calculation is controlled by keywords summarized in Tab. 3.14 and describedin detail below.

Table 3.14 Keywords and parameters available for the ICL graph


rings paths from minima to ringssaddles paths from saddles to attractorsusing CPs file to be used cps <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|cps_1=C3H3NO_HF_6-31G.g03.rho_r.cps

56 3 Control file

ICL_graph:-----------------------------------------

using cps_1

saddlesrings

:-----------------------------------------ICL_graph_end

The ICL graph is determined on the basis of the critical points read from the cpsfile. The gradient and Hessian of the field is computed from the wavefunction filegiven in the grid file to which the cps file points. The data for the interconnectionlines (in the STR format) are written to a file with the same name as the cps file, butthe string ‘.graph.str’ appended. This file can be visualized by appropriate tool. Thelog file for the job is written to a file with additional string ‘.LOG’ appended.

The above evaluation section direct DGrid to create the interconnection linesbetween the saddle points and the attractors as well as between the ring criticalpoints and the minima. The resulting data are written to the file C3H3NO HF 6-31G.g03.rho r.cps.graph.str in the STR format. In case of electron density the di-agram formed by the lines from saddles to the attractors is called molecular graph[9]. Fig. 3.6 shows (program Avizo) the molecular graph for oxazole, together withthe nitrogen and hydrogen basins (cropped with 0.001 bohr−3 isodensity surface). It

Fig. 3.6 ρ-basins (cropped by the 0.001 bohr−3 isosurface of electron density) and the moleculargraph for C3H3NO. Hydrogen basins are in gray, the nitrogen basin is green colored (the oxy-gen and carbon basins are not shown). Red spheres: attractors; green spheres: saddle points; bluesphere: ring critical point


Fig. 3.7 ELI-D ICL graphs for C3H3NO. The small colored spheres have following encoding:red - attractors; green - saddle points; blue - ring critical points; black - minima. The hydrogenpositions are marked by gray spheres. The gold colored ICLs run toward the attractors, whereasthe blue ones run toward the ELI-D minima. The large rose-colored spheres are the respective coreregions. The rose-colored stick are used to highlight the molecular structure

can be seen that the saddle points (small green spheres) are located in the zero-fluxsurfaces (basin borders). The ring critical point (blue sphere) is located within themolecular ring. The blue line follows the gradient path to the minimum in the direc-tion of the negative principal curvature (only the part till the grid border is drawn asit is a long way into the infinity). It can nicely be seen that the trajectory runs alongthe edge of the nitrogen basin.

Remark 3.15. DGrid is working on an equidistant grid of data. Thus, it cannot re-solve details hidden between the grid points. Strong oscillations in atomic regionscan be taken into account only by appropriately fine mesh. At least 0.05 bohr meshshould be used to properly resolve the core-valence separation for heavier elements.Even then, due to the very high gradients and curvatures, the determination of criti-cal points can fail, cf. Sec. 3.3.14.

In case of ELI-D there are much more basins than in case of electron density. Es-pecially if heavy elements are involved one encounters the problem of resolving thelarge number of inner shell basins connected with huge number of critical points.The core regions can be removed from the critical point search using the keywordELID core in the search section, cf. Sec. 3.3.14. As an example of the ELID core

58 3 Control file

Fig. 3.8 ELI-D basins and ICL graphs for C3H3NO. The small colored spheres have followingencoding: red - attractors; green - saddle points; blue - ring critical points; black - minima. Thehydrogen positions are marked by gray spheres. The gold colored ICLs run toward the attractors,whereas the blue ones run toward the ELI-D minima. The large rose-colored spheres are the re-spective core regions. The rose-colored stick are used to highlight the molecular structure. The3 ELI-D basins (corresponding to the C-O and C-N bonds) are colored according the the ELI-Dvalue at the zero-flux surface

keyword utilization the ICL graph for the ELI-D of the oxazole molecule is shown inFig. 3.7. Of course, in this case each core region just replace a single attractor (thus,the same result would be given for the evaluation without ELID core. The resultingELI-D ICL-graphs are not very complicated thus, both ICL graphs (from saddles toattractors and from rings to minima) can be shown at once, cf. Fig. 3.7. The largerose-colored core spheres in the figure have the radius given in the Table 3.22 onpage 80.

Fig. 3.8 shows the same ICL graphs like Fig. 3.7. Additionally, 3 ELI-D basinsare shown (corresponding to the C-O and C-N bonds). The borders of the basins(the ELI-D zero-flux surfaces) are colored according to the ELI-D value at the givenposition. It can nicely be seen, that the ELI-D maxima (2-dimensional; rank 2, signa-ture -2) in the zero-flux surfaces correspond to ELI-D saddle points (3,-1), whereasthe ELI-D saddle points in the zero-flux surface (again 2-dimensional; rank 2 signa-ture 0) correspond to the ring critical points (3,+1).

Remark 3.16. The search for critical points is using certain thresholds for gradient,curvatures, etc. in DGrid. If the critical points are in regions below the thresholds,


then they cannot be detected. For instance in almost spherical regions around atomsin a molecule, which of course cannot be perfectly spherical, however the changesin gradient and curvatures are close to numerical instabilities. Even if the Euler char-acteristics is the proper one, there still could be some critical points missing (notethat if both an attractor and a saddle point were not found the Euler characteristicsremains unchanged).

3.3.7.1 saddles rings

Each of the possible keywords saddles and rings, respectively must be written tosingle line within the search section.

saddlesrings

With the command saddles DGrid will create the interconnection lines starting fromthe saddle points and ending at the corresponding attractors. In case of the electrondensity such a graph is termed the molecular graph. With the command rings eachICL starts at a field minimum and terminate at the ring critical point. When bothkeywords are used the complete ICL graph will be created (the two graphs havedifferent colors in the STR format).

3.3.7.2 using

The obligatory data for the determination of the interconnection graph is a cps filewith the critical points. The n-th cps file number cps <n> is chosen by the com-mand:

using cps <n>

The required cps <n> file must be either declared by the assignment cps <n>=<filename>outside the icl graph section or be created in the preceding search block, cf. Sec.3.3.14 (using save cps <n> within that section).

3.3.8 integrate

The section for the integration of the field grid is delimited by separate lines with thekeywords integrate and integrate end. The integration is controlled by keywordssummarized in Tab. 3.15 and described in detail below.

60 3 Control file

Table 3.15 Keywords and parameters available for the integration


core charge replace core regionsover basin file basin <n>using grid field file to be integrated field <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|field_1=C3H3NO_HF_6-31G.g03.rho_rbasin_2=C3H3NO_HF_6-31G.g03.elid_r_a_aa.bsn

integrate:-----------------------------------------

using field_1over basin_2

:-----------------------------------------integrate_end

The field in the grid file specified by the field <n> descriptor will be integrated overdomains read from the basin file specified by the basin <m> descriptor. Instead ofthe grid file the file with the refinement data, cf. Sec. 3.3.13, can be utilized (usingthe descriptor rfn <n>). With the command using empty only the basin volumeswill be determined. If the basin <m> descriptor is omitted then the total integralof the whole grid region is computed.

The resulting integrals of the field over the basins will be written into a logfile. The name of the log is derived from the name of the basin file with the fieldname and the string ‘ ITG’ appended. For the above example it will be the fileC3H3NO HF 6-31G.g03.elid r a aa.bsn.rho r ITG.

In DGrid the integration of a property is done numerically on an equidistant grid.Thus, the precision depends on the mesh size and can be subject to large errors.In case of electron density – the standard charge determination – especially in theatomic core regions. There are 3 possibilities to compensate for the errors:

• using the keyword core charge:in a small sphere around each atom the electronic charge is replaced by a valuefrom an internal table (data computed from the basis sets of Clementi and Roettifor the atoms N - Xe, as well as from relativistic ADF [1] calculations for theatoms Cs-Lu).

• using the refinement procedure (see Sec. 3.3.13). The file name from the refine-ment procedure is then used in the integrate section.

• For Gauss-type orbitals see remark 3.17 below.


3.3.8.1 core charge

For heavier elements, respectively grids with mesh that is not fine enough, the in-tegrals can be imprecise. For such grids the integral precision can be enhanced byso-called refinement procedure, cf. Sec. 3.3.13, for which the wavefunction must beavailable. Another possibility, in case of the integration of the electron density, toincrease the integral precision is to utilize the following command:

core charge

Here, the grid voxels are replaced by a predefined (spherical) core regions and theintegrals within the regions are replaced by (precalculated) atomic charges. Thisapproach substantially depends on the precision of the voxel replacement, i.e., howgood the spheres match the (rectangular) voxels.

Remark 3.17. For Gauss-type orbitals and grids parallel with the Cartesian axes theanalytical integration can be performed using the section overlap, cf.Sec 3.3.12.

3.3.8.2 over

The integration of the grid field values is usually performed over basins. The corre-sponding basin file is chosen by the command:

over basin <n>

The required basin <n> file must be either declared by the assignment basin <n>=<filename>outside the integrate section or be created in the preceding basin block, cf. Sec.3.3.1 (using save basin <n> within that section). If the command over is not giventhen the integration yields the total integral of the field over the grid box.

3.3.8.3 using

The grid file with the field values to be integrated is declared by the command:

using field <n>

The required field <n> file must be either declared by the assignment field <n>=<filename>outside the integrate section or be created in the preceding compute block, cf. Sec.3.3.3 (using save field <n> within that section).

If the grid field file is not used then the integration yields the basin volumes only.However, also in that case the using command must be used in the form:

62 3 Control file

using empty

3.3.9 intersections

Basins computed for two different scalar fields yield two different sets of space par-titioning. It can be of interest to examine, how a particular basin from the first setis intersected by the basins of the second set [41]. Of course, not all basins of thesecond set will intersect the chosen basins. The block computing the intersection be-tween two basin grids is delimited by separate lines with the keywords intersectionsand intersections end. The intersection is controlled by keywords summarized inTab. 3.16 and described in detail below.

Table 3.16 Keywords and parameters available for the intersections


save intersection basin file basin <n>using basin file to be intersected basin <n>with intersecting basin file basin <n>

Example:::C3H3NO HF 6-31G:------------------------------------------------|basin_1=C3H3NO_HF_6-31G.g03.rho_r.bsnbasin_2=C3H3NO_HF_6-31G.g03.elid_r_a_aa.bsn

intersections:-----------------------------------------

using basin_1with basin_2

:-----------------------------------------intersections_end

According to the QTAIM approach the electron density basins, cf. Fig. 3.1 on page18, describe the atoms in molecule [9]. Because non-nuclear maxima are not presentin case of the C3H3NO molecule, the number of the density basins correspondsto the number of atoms. In case of ELI-D the number of basins is larger than forthe electron density, because there are separate basins corresponding to the ELI-D attractors between the atoms (which can be assumed as bond descriptors) andaround the oxygen and nitrogen core basins (lone-pair descriptors), cf. Fig. 3.2 onpage 20.

There are two possibilities for the intersection analysis depending on the choiceof the intersecting set (A intersect B, or B intersect A). In the intersections block, all


domains in the basin file specified by the basin <n> descriptor given in the usingcommand will be intersected by all domains read from the basin file specified bythe basin <m> descriptor given in the with command. The resulting intersecteddomains will be written into separate basin file. The name of this file is derivedfrom the name of the ‘using’ basin file with the string ‘isect-’ and the end-string ofthe ‘with’ basin file appended. In the above example the resulting file will be namedC3H3NO HF 6-31G.g03.rho r.bsn.isect-elid r a aa.bsn. The basin numbers in thisfile are coded according with the format given in the ‘field=’ block of the basin fileheader. There the line ‘intersections with’ ends with hash characters (#), the numberof which gives the number of digits for the first basins (the remaining digits describethe second basin). The basin file with the intersections can be either visualized oranalyzed with the isect analyze block, cf. Sec. 3.3.10.

The name of the log is derived from the name of the resulting basin file with thestring ‘.LOG’ appended.

3.3.9.1 save

The resulting it basin file with the intersected basins determined in the intersectionsblock can be connected to a basin descriptor by the command:

save basin <n>

The descriptor basin <n> can be used in subsequent evaluation sections to utilizethe data in the saved basin file.

3.3.9.2 using

The grid file with the basins to be intersected is declared by the command:

using basin <n>

The required basin <n> file must be either declared by the assignment basin <n>=<filename>outside the intersections block or be created in the preceding basin section, cf. Sec.3.3.1 (using save basin <n> within that section).

3.3.9.3 with

The grid file with the basins which intersect the basins specified in the using com-mand is declared by the command:

64 3 Control file

with basin <n>

The required basin <n> file must be either declared by the assignment basin <n>=<filename>outside the intersections block or be created in the preceding basin section, cf. Sec.3.3.1 (using save basin <n> within that section).

3.3.10 isect analyze

The block for the analysis of the basin intersection file (i.e., the result of the inter-sections section) is delimited by separate lines with the keywords isect analyze andisect analyze end. The analysis is controlled by keywords summarized in Tab. 3.17and described in detail below.

Table 3.17 Keywords and parameters available for the intersection analysis


basin set= basin set for grouping <1|2>core charge replace core regionintegrate integrate field file field <n>select basins basin selection selection blockusing basin file to be analyzed basin <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|basin_1=C3H3NO_HF_6-31G.g03.rho_r.bsn.isect-elid_r_a_aa.bsnrfn_1 =C3H3NO_HF_6-31G.g03.rho_r.rfn

isect_analyze:-----------------------------------------

using basin_1integrate rfn_1

select_basins6 7

select_basins_end

basin_set=2:-----------------------------------------isect_analyze_end

The above example analyzes the basin file with the domain intersections from theexample on page 62. The file is declared by the basin <n> descriptor preceding


the analyze block. The intersections are grouped together with respect to the ELI-D basin membership (basin set number 2). The intersections of two ELI-D basins(basins number 6 and 7, attributed to the O-C bonds) is analyzed. Without the se-lect basins block the intersections of all basins would be analyzed.

The resulting analysis will be written to log file with the string ‘.ISECT’ ap-pended. There, the volume percentage of the intersection as well as the integralpercentage (using the refined field file ‘.rfn’) is listed:

/======================================================\| Intersections of rho[r] basins by ELI-D[r] basins |+------------------------------------------------------+

basin descriptor volume % integ %+----------------------------------------------------------------------------+| 6-06 C_3xO_1-C_3 0.940 17.34 0.1811 15.20 || 7-06 O_1xO_1-C_3 4.480 82.66 1.0106 84.80 |+----------------------------------------------------------------------------+| total | 5.420 100.00 1.1917 100.00 |+----------------------------------------------------------------------------+

basin descriptor volume % integ %+----------------------------------------------------------------------------+| 4-07 C_1xO_1-C_1 1.128 19.72 0.2201 17.43 || 7-07 O_1xO_1-C_1 4.591 80.28 1.0429 82.57 |+----------------------------------------------------------------------------+| total | 5.719 100.00 1.2630 100.00 |+----------------------------------------------------------------------------+

The intersections for the ELI-D basins number 6 and 7 are grouped together. TheELI-D basin number 6 (the O 1-C 3 bond basin) is intersected by the QTAIM (elec-tron density) basins number 6 (C 3 atom) and 7 ((O 1 atom), cf. the basin numbering‘6-06’ and ‘7-06’, cf. the left diagram in Fig. 3.9. The oxygen QTAIM basin takes82.66% of the bond basin volume. The intersection charges can be connected withthe bond polarity [41]. Because the oxygen basin encloses 84.80% of the ELI-Dbasin population of 1.19e the bond can be regarded as polar, with the oxygen as themore negative bonding partner.

/======================================================\| Intersections of rho[r] basins by ELI-D[r] basins |+------------------------------------------------------+

basin descriptor volume % integ %+----------------------------------------------------------------------------+| 7-01 O_1xO_1 0.272 0.27 2.1275 23.63 || 7-10 O_1xH_3 0.773 0.77 0.0122 0.14 || 7-12 O_1xH_1 0.951 0.95 0.0146 0.16 || 7-09 O_1xC_1-N_1 1.182 1.18 0.0183 0.20 || 7-15 O_1xC_3-C_2 1.719 1.72 0.0262 0.29 || 7-06 O_1xO_1-C_3 4.480 4.49 1.0106 11.22 || 7-07 O_1xO_1-C_1 4.591 4.60 1.0429 11.58 || 7-13 O_1xLP_O_1 85.780 86.00 4.7511 52.77 |+----------------------------------------------------------------------------+| total | 99.748 100.00 9.0034 100.00 |+----------------------------------------------------------------------------+

The above analysis of the oxygen QTAIM basin (nr. 7) shows that it is intersectedmainly by 3 ELI-D basins, with the largest contribution of the ELI-D oxygen lone-pair (basin nr. 13; 86%). The two other contributions of roughly 5% each are theintersections of the oxygen basin with the surrounding O-C ELI-D bond basins. Ob-serve that the ELI-D core basin of the oxygen take up just a small fraction (ca. 0.3%)

66 3 Control file

Fig. 3.9 ELI-D basins (cropped by the 0.001 bohr−3 isosurface of electron density) of C3H3NOintersected by electron density basins. Left: intersection of ELI-D bond basin O1-C3. Right: in-tersection of ELI-D bond basin N1-C1. The color coding is blue (oxygen), green (nitrogen), black(carbon), gray (hydrogen)

of the QTAIM basin volume (however, containing 2.1 electrons). The 3 intersectionsare shown in Fig. 3.10.

3.3.10.1 basin set

The it basin file with the intersected basins was determined in the intersectionsblock, cf. Sec. 3.3.9 for two sets of basin. With the command:

basin set= <1|2>

the data for the desired basin set are grouped together in the resulting log file.

3.3.10.2 core charge

The analysis of the basin intersections can include also the integration of speci-fied grid field (usually the electron density) utilizing the command integrate, seenext subsection. For heavier elements, respectively grids with mesh that is not fineenough, the integrals can be imprecise. For such grids the integral precision can beenhanced by so-called refinement procedure, cf. Sec. 3.3.13, for which the wave-function must be available. Another possibility, in case of the integration of theelectron density, to increase the integral precision is to utilize the following com-mand:


Fig. 3.10 Electron density basin (cropped by the 0.001 bohr−3 isosurface of electron density) forthe oxygen in the C3H3NO molecule intersected by ELI-D basins. The blue intersection correspondto oxygen lone-pair, the red and green intersections are due the respective C-O bond basins

core charge

Here, the grid voxels are replaced by a predefined (spherical) core regions and theintegrals within the regions are replaced by (precalculated) atomic charges. Thisapproach substantially depends on the precision of the voxel replacement, i.e., howgood the spheres match the (rectangular) voxels.

3.3.10.3 integrate

The analysis of the basin intersections can be extended by the integration of speci-fied grid field (usually the electron density) utilizing the following command:

integrate field <n>

The required field <n> file must be either declared by the assignment field <n>=<filename>outside the isect analyze section or be created in the preceding compute block, cf.Sec. 3.3.3 (using save field <n> within that section).

68 3 Control file

If the precision of the integration is not high enough, the refinement procedurecan be used, cf. Sec. 3.3.13. The resulting ‘.rfn’ file can be utilized instead of thefield file:

integrate rfn <n>

The required rfn <n> file must be either declared by the assignment rfn <n>=<filename>outside the isect analyze section or be created in the preceding refine block, cf. Sec.3.3.13 (using save rfn <n> within that section).


The intersections block (cf. Sec. 3.3.9) performs the basin intersections between allthe basins in the two utilized basin files. The analysis of the often very large numberof basins can be reduced to basins of interest by the block section:


where <bsn 1><bsn 2> ... is a blank separated list of basin numbers (valid for thebasin set chosen by the command basin set, Sec. 3.3.10.1) for which the analysiswill be performed. The numbers can be also specified by hyphen separated rangedescriptor, e.g., <bsn from>–<bsn to>.

3.3.10.5 using

The grid file with the basin intersection to be analyzed is declared by the command:

using basin <n>

The required basin <n> file must be either declared by the assignment basin <n>=<filename>outside the isect analyze section or be created in the preceding intersections block,cf. Sec. 3.3.1 (using save basin <n> within that section).

3.3.11 localized natural orbitals

The section for the calculation of the localized natural orbitals (LNOs) [16, 17] isdelimited by separate lines with the keywords localized natural orbitals and local-


ized natural orbitals end. The calculation is controlled by keywords summarizedin Tab. 3.18 and described in detail below.

Table 3.18 Keywords and parameters available for the LNO calculation


pair= pair descriptor <A-B>save wavefunction/norm file to be saved wfn <n> / norm <n>select basins isopyc basin selection (isopycnic transformation) selection blockusing overlap file to be used sij <n>

Example:


localized_natural_orbitals:-----------------------------------------

using sij_1


select_basins_end


:-----------------------------------------localized_natural_orbitals_end

The resulting AO expansion for the LNOs of the above example is written tothe wavefunction file named C3H3NO HF 6-31G.g03.rho r.bsn.sij.lno, i.e. the filename is derived from the name of the overlap file sij 1 with the appendix ‘.lno’.Also, the norm parts will be written to the file with an additional ‘.norm’ appended.The log file with the FO occupation numbers will be written ti the file with theextender ‘.LOG’ appended.

3.3.11.1 pair

In case of a closed-shell wavefunction the resulting LNOs are occupied by both spincomponents. For open-shell wavefunction LNOs for separate same-spin pairs willbe determined with the command:

pair= <A-B>

The resulting α and β spin LNOs will have different occupations.

70 3 Control file

3.3.11.2 save

The LNOs determined in the localized natural orbitals section can be assigned tospecific wfn descriptor by the command:

save wfn <n>

The data descriptor wfn <n> can be used in subsequent evaluation sections toutilize the data in the saved wfn file. Additionally, for the decomposition of thedelocalization index into the LNO contributions the corresponding norm parts canbe saved by the command:

save norm <n>


3.3.11.3 select basins isopyc

The LNOs are created by isopycnic transformation of the orbitals over domains [16].The domains over which the isopycnic localization is performed are defined by a listof basins given in the block section:

select basins isopyc<bsn 1> <bsn 2> ...select basins end


All domains of the basin file can be selected with the command:

select basins isopycallselect basins end

3.3.11.4 using

The obligatory data for the LNO determination is a file containing the overlap inte-grals. The n-th overlap file number sij <n> is chosen by the command:


using sij <n>

The required sij <n> file must be either declared by the assignment sij <n>=<filename>outside the localized natural orbitals section or be created by preceding overlapsection, cf. Sec. 3.3.12 (using save sij <n> within that section).

3.3.12 overlap

The section for the calculation of the orbital overlap integrals is delimited by sepa-rate lines with the keywords overlap and overlap end. The calculation is controlledby keywords summarized in Tab. 3.19 and described in detail below.

Table 3.19 Keywords and parameters available for the orbital overlap integrals


over basins specifying the domains basin <n>precision= precision for numerical integration float numbersave overlap file sij <n>select basins basin selection selection blockselect orbitals orbital selection selection blockusing wavefunction with the orbitals wfn <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|wfn_1 =C3H3NO_HF_6-31G.g03basin_1=C3H3NO_HF_6-31G.g03.rho_r.bsn

overlap:-----------------------------------------

using wfn_1over basin_1

save sij_1:-----------------------------------------overlap_end

The example computes the overlap integrals between the orbitals from the wave-function file specified by the descriptor wfn 1. The integration runs over the basinstaken from the basin file given by the descriptor basin 1. The resulting overlap in-tegrals are saved in the overlap file named C3H3NO HF 6-31G.g03.rho r.bsn.sij,i.e., the string ‘.sij’ is appended to the name of the basin file. The log file has theextender ‘.LOG’ appended. Check also the information in Sec. A.6.

72 3 Control file

3.3.12.1 over

The calculation of the overlap integrals runs over basins. The corresponding basinfile is chosen by the command:

over basin <n>

The required basin <n> file must be either declared by the assignment basin <n>=<filename>outside the overlap section or be created in the preceding basin block, cf. Sec. 3.3.1(using save basin <n> within that section).

3.3.12.2 precision

In DGrid only for Gauss-type orbitals the overlap integrals are computed analyti-cally. For all other orbital types a numerical procedure is used to evaluate the in-tegrals. Then, the precision of the integration must be specified by the keywordprecision= which is followed by the float number <prec>:

precision= <prec>

specifying the precision of the numerical integration.

3.3.12.3 save

The overlap integrals computed in the overlap section can be assigned to specificsij descriptor by the command:

save sij <n>

The data descriptor sij <n> can be used in subsequent evaluation sections to utilizethe data in the saved sij file, for instance, to compute the delocalization indices.


For large molecules many basin (for instance QTAIM basins) will be created. Then,the calculation of the overlap integral can be very time consuming. To avoid this,the calculation can be performed for basins of interest by the block section:




3.3.12.5 select orbitals

The keywords select orbitals and select orbitals end delimit a block where or-bitals can be chosen for further evaluation within the overlap section in the controlfile.

Normally, the overlap integrals are computed from all occupied orbitals. How-ever, sometimes it is convenient to chose a selected subset of orbitals. This orbitalsmust be specified in the select orbitals block, for instance:

select_orbitals--------------------------------:5-10 16 18--------------------------------:select_orbitals_end

The orbitals are selected by the number given in the wavefunction file. There aretwo possibilities: either by blank separated orbital numbers (16 18) or by a range ofnumbers (5-10; read as single string without spacing). The numbers can be writtenin more lines as the block is read until the keyword select orbitals end.

3.3.12.6 using

The obligatory data for the calculation of the overlap integrals is a wavefunction filechosen by the command:

using wfn <n>

The required wfn <n> file must be either declared by the data assignment wfn <n>=<filename>outside the overlap section or be created by preceding domain natural orbitals,Fermi orbitals, or localized natural orbitals section (using save wfn <n> withinthat section).

74 3 Control file

3.3.13 refine

DGrid utilizes equidistant grid meshes. The grid point spacing is often not fineenough for the integration. In this case, additional points can be added to enhancethe integration precision. In DGrid this procedure is termed ‘refinement’.

The data refinement section is delimited by separate lines with the keywordsrefine and refine end. The calculation is controlled by keywords summarized inTab. 3.20 and described in detail below.

Table 3.20 Keywords and parameters available for the data refinement


over basins specifying the domains basin <n>precision= refinement precision float numbersave refined integral data file rfn <n>select basins basin selection selection blockslices= slice calculation <slices> <slice>using grid field file field <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|field_1=C3H3NO_HF_6-31G.g03.rho_r

refine:-----------------------------------------

using field_1precision=0.01

:-----------------------------------------refine_end

The example computes additional grid points for the electron density field usingthe wavefunction file specified in the field 1 file. The procedure select voxels in thegrid field 1 that needs additional points. The additional points are computed untilthe desired precision is reached. The resulting integrals for the voxels are saved inthe refinement file named C3H3NO HF 6-31G.g03.rho r.rfn, i.e., the string ‘.rfn’ isappended to the name of the field file. The log file has the extender ‘.LOG’ appended.

3.3.13.1 over

The calculation of the additional points for the integral refinement runs over thewhole grid box. To reduce the computation time the procedure can be performed forselected basins. The corresponding basin file is chosen by the command:


over basin <n>

The required basin <n> file must be either declared by the assignment basin <n>=<filename>outside the refine section or be created in the preceding basin block, cf. Sec. 3.3.1(using save basin <n> within that section).

3.3.13.2 precision

The precision of the integration result must be specified by the keyword precision=which is followed by the float number <prec>:

precision= <prec>

specifying the precision of the refinement procedure (the field integral over the gridbox, respectively the chosen basins).

3.3.13.3 save

The voxel integrals computed the refine section can be assigned to specific rfn de-scriptor by the command:

save rfn <n>

The data descriptor rfn <n> can be used in subsequent evaluation sections to uti-lize the data in the saved rfn file, usually, to compute the field integrals over basins.


For large molecules many basin (for instance QTAIM basins) will be created. Then,the calculation of the overlap integral can be very time consuming. To avoid this,the calculation can be performed for basins of interest by the block section:



76 3 Control file

3.3.13.5 slices

The refinement can be very time consuming procedure. The computation time canbe reduced by the slicing procedure submitting selected number of voxels to berefined to dedicated processors. The calculation of such a slice is accomplished withthe keyword slices= followed by the number <div> of slices and the slice number<slc>:

slices=<div> <slc>

For each slice a separate control file is needed (thus, <div> jobs must be submittedto get all the slices for all tested points). To the cps file for each slice the string‘.slice-<slc>’ is appended to discriminate between the slices. Having all slicescomputed the whole grid can be build by merging the slices together calling the‘complete’ utility:

dgrid rfnfile.slice-1 complete

The resulting completed rfn file includes all refined voxel integrals. How to run allthe slices in parallel using a script is described in Sec. 5.

3.3.13.6 using

The obligatory data for the data refinement is a grid field file chosen by the com-mand:

using field <n>

The required field <n> file must be either declared by the data assignment field <n>=<filename>outside the refine section or be created by preceding compute section (using savefield <n> within that section).

3.3.14 search

The section for the critical points search is delimited by separate lines with thekeywords search and search end. The calculation is controlled by keywords sum-marized in Tab. 3.21 and described in detail below. The critical points are searchedon a grid field file. The gradient and Hessian of the field is computed from the wave-function data based on file which name is given in the grid file.


The data for the critical points are written to a file with the same name as thefield file, but the string ‘.cps’ appended. This file can be further processed by DGrid(after the assignment cps <n>=filename), for instance, to create an interconnectiongraph. The critical points are also written to a log file with additional string ‘.LOG’appended.

Table 3.21 Keywords and parameters available for the DI calculation


all cps search for all critical pointsattractors search for attractorscrop using crop field by isovalue field <n> isovalueELID core remove ELI-D core from search eventually core sectionminima search for minimarings search for ringssaddles search for saddlessave CPs file to be saved cps <n>slices= slice calculation <div> <slice>using grid file to be used field <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|field_1=C3H3NO_HF_6-31G.g03.rho_r

search:-----------------------------------------

using field_1crop using field_1 0.001

attractorssaddles

:-----------------------------------------search_end

The resulting data for the above critical point search will be written to the file namedC3H3NO HF 6-31G.g03.rho r.cps, i.e., with name derived from the grid field filename and the string ‘.cps’ appended. Some information about the critical points iswritten to the log file with additional ‘.LOG” appended, for instance, the info aboutthe attractors:

/===========================\| attractors in region= 8 |+---------------------------+

# Bsn x y z value Laplacian curvature+--------------------------------------------------------------------------------------------------+| 1 1 -2.4310 -3.3680 0.0000 0.4249 - H3 || 2 2 2.8049 -3.1219 -0.0000 0.4277 - H2 || 3 3 -1.1101 -1.8596 0.0000 118.3147 - C3 || 4 4 1.4135 -1.6748 -0.0000 118.3130 - C2 || 5 5 -2.0618 0.5748 0.0000 291.3237 - O1 || 6 6 2.0869 0.8955 -0.0000 192.1030 - N1 || 7 7 0.0000 2.1211 -0.0000 118.3135 - C1 || 8 8 -0.3077 4.1025 -0.0000 0.4279 - H1 |+--------------------------------------------------------------------------------------------------+

78 3 Control file

For each of the 8 attractors the Cartesian coordinates are printed, followed by thefield value at the attractor position. In the above example the value of the field Lapla-cian is not given, because the attractors correspond to positions closer than 0.01 bohrto a nucleus. Instead of the three principal curvatures only the atomic symbols areprinted. The remaining output shows that no minima were found in the grid-boxregion:

/=======================\| minima in region= 0 |+-----------------------+

# x y z value Laplacian curvature |V|/G H+---------------------------------------------------------------------------------------------------------------++---------------------------------------------------------------------------------------------------------------+

/========================\| saddles in region= 8 |+------------------------+

# x y z value Laplacian curvature ellip |V|/G H+------------------------------------------------------------------------------------------------------------------+| 1 0.7730 1.6246 -0.0000 0.3740 -1.2348 0.3154 -0.8486 -0.7016 0.21 3.110 -0.587 || 2 0.2026 -1.7737 0.0000 0.3307 -0.8850 0.3332 -0.6833 -0.5349 0.28 3.773 -0.346 || 3 -0.2059 3.4012 0.0000 0.2858 -1.0034 -0.7896 0.5492 -0.7631 0.03 8.853 -0.287 || 4 -1.9621 -2.8230 0.0000 0.2824 -0.9697 -0.7670 0.5333 -0.7360 0.04 8.256 -0.281 || 5 1.6778 -0.6138 0.0000 0.2802 -0.7097 -0.5396 0.3407 -0.5107 0.06 3.340 -0.310 || 6 2.2982 -2.5930 0.0000 0.2798 -0.9334 -0.7398 0.5286 -0.7221 0.02 7.582 -0.275 || 7 -0.6803 1.5630 0.0000 0.2619 -0.1784 0.6952 -0.4217 -0.4519 0.07 2.142 -0.359 || 8 -1.4000 -1.0226 0.0000 0.2446 -0.0688 -0.3527 0.6642 -0.3803 0.08 2.056 -0.323 |+------------------------------------------------------------------------------------------------------------------+

/======================\| rings in region= 1 |+----------------------+

# x y z value Laplacian curvature ellip |V|/G H+------------------------------------------------------------------------------------------------------------------+| 1 0.0141 0.0039 -0.0000 0.0514 0.4363 0.2378 0.2551 -0.0566 0.07 0.910 0.009 |+------------------------------------------------------------------------------------------------------------------+

/===============================================\| Euler characteristics (inside cropped grid) |+-----------------------------------------------+

+--------------------------------------------------------------------+| 8 (Attr) - 8 (Saddle) + 1 (Ring) - 0 (Min) = 1 |+--------------------------------------------------------------------+

This is followed by the data for the saddle and ring points, including the ellipticity,virial ratio, and the value of the energy density at the respective critical points. Theellipticity at the saddle point is given as c1/c2−1, with the two (negative) curvaturesc1 and c2 perpendicular to the interaction path, whereby |c1| ≥ |c2|. The virial ratio|V |/G is computed from the positive definite kinetic energy density G = ∇∇′Γ (r,r′)and the local potential energy density [20] is set to V = 1

4 ∇2ρ − 2G. The energydensity at the saddle point is given by H = G+V , which is strictly valid only forequilibrium geometry. Last information is the Euler characteristic, showing that thePoincare-Hopf sum is fulfilled. This need not be the case if just part of the sys-tem was calculated. As mentioned above, the sum applies only to the computed(cropped) region.

Remark 3.18. The data in the cps file can be converted into the STR format forvisualization with the utility (i.e., command on console):

dgrid gridfilename.cps str

yielding the file gridfilename.cps.str with data in STR format.


3.3.14.1 all cps

The keyword is written to single line within the search section.

all cps

DGrid will search for all critical points, i.e., for the attractors and saddle points aswell as the minima and ring points.

3.3.14.2 attractors, saddles, minima, rings

Each of the selected keywords attractors, saddles, minima, and rings, respectivelymust be written to single line within the search section.

attractors saddles

DGrid will search only for the selected critical point types. For instance, in the aboveexample, for the attractors and saddles used in case of electron density field to createthe bond paths.

3.3.14.3 crop using

In case of a large molecule there is lot of ‘empty’ space in the grid box surroundingthe molecule. To speedup the search for critical points the search can restricted toregion cropped by the isosurface of chosen field (usually the electron density) by thecommand:

crop using field <n> <isoval>

where the field grid file need to be assigned to the field <n> descriptor in a preced-ing statement. The isovalue <isoval> is given by a float number.

3.3.14.4 ELID core

DGrid is working on an equidistant grid of data. Thus, it cannot resolve detailshidden between the grid points. Strong oscillations in atomic regions can be takeninto account only by appropriately fine mesh. At least 0.05 bohr mesh should be usedto properly resolve the core-valence separation for heavier elements. Even then, dueto the very high gradients and curvatures, the determination of critical points can

80 3 Control file

fail, e.g., in case of the electron density Laplacian or ELI-D. This can be avoided bythe command:

ELID core

whereby all grid maxima within the tabulated ELI-D core radii around the atomicpositions will be removed from the search. The ELID core command can be usedalso as a block section:

ELID corescale r= <val><elem 1> <elem 2> ...ELID core end

Here, <val> is a multiplicative factor to scale the tabulated ELI-D core radii. Thiscan be followed by a list of element symbol for which the scale factor applies (other-wise it is valid for all the elements defined in the evaluation (e.g., in the wavefunctionfile). Then, the procedure will not search for critical points within the core region,cf. Table 3.22, of the chosen atoms.

Table 3.22 Core regions used by the ELID core keyword

Atoms Core shells

Li–Ne 7→ 1st

Na–Ar 7→ 1st +2nd

K 7→ 1st +2nd +3rd

Ca–Cu 7→ 1st +2nd

Zn–Kr 7→ 1st +2nd +3rd

Rb 7→ 1st +2nd +3rd +4th

Sr–Ag 7→ 1st +2nd +3rd

Cd–Xe 7→ 1st +2nd +3rd +4th

Cs–La 7→ 1st +2nd +3rd +4th

Ce–Lu 7→ 1st +2nd +3rd

Hf–Au 7→ 1st +2nd +3rd +4th

Hg–Rn 7→ 1st +2nd +3rd +4th +5th

The radii can be found in the constructor of the class Per Sys, respectively in Ref. [34]

The reason for the choice of the 1st and 2nd shell as the core region for the series Ca–Cu is the observation that the structurization of the penultimate atomic shell seemsto be important [35] (and similarly for other series). The critical points within thecore-valence separation will be searched, whereas each core region will be replacedby a single entity, called ‘core’ and treated as an attractor in the determination ofEuler characteristic.


3.3.14.5 save

The critical points data determined in the search section a written to specific filewhose name is assigned during the run. The file can be connected to a cps descriptorby the command:

save cps <n>

The descriptor cps <n> can be used in subsequent evaluation sections to utilize thedata in the saved cps file.

3.3.14.6 slices

The search for the critical points is initiated either by the determination of grid max-ima (grid points with value higher than the 26 neighbors) or from set of grid pointswhere the routine assumes a saddle/ring. Especially for the saddles/rings the numberof tested points can be large. This set can be divided into slices and each slice eval-uated separately. The calculation of such a slice is accomplished with the keywordslices= followed by the number <div> of slices and the slice number <slc>:

slices=<div> <slc>

For each slice a separate control file is needed (thus, <div> jobs must be submittedto get all the slices for all tested points). To the cps file for each slice the string‘.slice-<slc>’ is appended to discriminate between the slices. Having all slicescomputed the whole grid can be build by merging the slices together calling the‘complete’ utility:

dgrid cpsfile.slice-1 complete

The resulting completed cps file includes all critical points. How to run all the slicesin parallel using the dgrid para-5.0 script is described in Sec. 5.2.

3.3.14.7 using

The obligatory data for the critical point search is a field grid file. The n-th field filenumber field <n> is chosen by the command:

using field <n>

82 3 Control file

The required field <n> file must be either declared by the assignment field <n>=<filename>outside the search section or be created in the preceding compute section (usingsave field <n> within that section).

3.3.15 superbasin

In some cases it is desirable to unify basins into single (connected or disconnected)object, so called superbasin. Be it all the core basins of an atom, some basins in thevalence region or even basins of different chemical groups. As a prerequisite to thecreation of superbasins DGrid must identify all the basins in the computed regionand write the basin data into the basin file, cf. the basin section 3.3.1.

The data refinement section is delimited by separate lines with the keywordssuperbasin and superbasin end. The calculation is controlled by keywords sum-marized in Tab. 3.23 and described in detail below.

Table 3.23 Keywords and parameters available for the superbasins


save superbasin file basin <n>using basin file basin <n>

Example:

::C3H3NO HF 6-31G:------------------------------------------------|basin_1=C3H3NO_HF_6-31G.g03.rho_r.bsn

superbasin:-----------------------------------------

using basin_1save basin_2

C1-H1 = 4 1C2-H2 = 5 3C3-H3 = 6 2

:-----------------------------------------superbasin_end

The format for the choice of the basins creating superbasins is as follows:

<sbname1>= <bsn 1> <bsn 2> ...... <bsn 3> <bsn 4> ...<sbname2>= <bsn 5> <bsn 6>


where <sbname1> is a string for the name of the first superbasin. <bsn 1><bsn 2> ... is a blank separated list of basin numbers collected together to form thesuperbasin. The numbers can be also specified by hyphen separated range descrip-tor, e.g., <bsn from>–<bsn to>. The list is read until all superbasin is specifiedand the string superbasin end is found. In principle, any sequence of basins fromthe basin file can be chosen to form superbasins.

In the above example the basin file specified by the basin 1 descriptor is utilized.The atomic basins number 1 and 4 are constitute the superbasin with the descriptorC1-H1. The name was chosen because the superbasin represents this fragment. Ad-ditionally, the basins 3,5, and 2,6 are unified into the superbasins C2-H2 and C3-H3,respectively. The superbasins are visualized in Fig 3.11.

Remark 3.19. In contrast to a superbasin, the so-called basin set is given by spe-cial choice of basins [35]. Basin set is join of sets of interconnected basins wherethe basin interconnection points are above or equal to a specified value (the basininterconnection point is the defined by the position and property value for the sad-

Fig. 3.11 Density superbasins for C3H3NO, cropped by 0.001 bohr−3 electron density isosurface.Blue: oxygen basin; green: nitrogen basin; dark gray: three C-H superbasins;

84 3 Control file

Fig. 3.12 Electron density isosurfaces for C3H3NO. Left: 0.3-localization domains; the N-C andC-C groups are within separate reducible localization domains. Right: 0.281-localization domains;now the N-C-H and C-C-H groups are within separate reducible localization domains

dle point connecting two basins). Thus, the basin set is always an object formedby connected basins. Not every superbasins is at the same time a basin set. In theleft diagram of Fig. 3.12 the 0.3-localization domains are shown (isosurfaces for thedensity value 0.3 bohr−3). It can be seen that none of the C and H density basins,as chosen in the above control file, can form separate basin set (by lowering thedensity to connect the C-H one cannot avoid that C-N is already connected as well).However, the N and C as well as the two carbon basins form (2 membered) basinsets to the interconnection value ρic = 0.3, whereas the hydrogen and oxygen basinsremain separated. Lowering the interconnection value to ρic = 0.281 gives two 3-membered basin sets (NCH and CCH) and two separate basins (H and O). For thecorresponding basins cf. Fig. 3.1.

The basins are saved in the basin file named C3H3NO HF 6-31G.g03.rho r.bsn.sb,i.e., the string ‘.sb’ is appended to the name of the basin file. The log file has theextender ‘.LOG’ appended. In the header of the sb file (as well as in the ‘.log’ file)the correspondence between the original basins and the superbasins is given:

/=============\| basins= 8 |+-------------+

basin volume maximum i j k descriptor superbasin+---------------------------------------------------------------------------+| 1 103.472 0.4136 51 98 39 H_1 1 C1-H1 || 2 120.549 0.4095 30 24 39 H_3 3 C3-H3 || 3 121.754 0.4186 81 27 39 H_2 2 C2-H2 || 4 0.000 58.7254 54 79 39 C_1 1 || 5 0.000 64.0388 68 41 39 C_2 2 || 6 0.000 52.4968 42 39 39 C_3 3 || 7 99.748 06.3250 33 64 39 O_1 |


| 8 116.278 74.4581 74 67 39 N_1 |+---------------------------------------------------------------------------+

In the column ‘superbasin’ the descriptors C1-H1, C2-H2, and C3-H3 are givenand the basins forming the superbasin are marked by the superbasin number. Thevolumes of the basins belonging to the superbasin are set to zero, except the firstbasin in the list, which gets the total volume.

3.3.15.1 save

The resulting superbasin file can be assigned to specific basin descriptor by thecommand:

save basin <n>

The data descriptor basin <n> can be used in subsequent evaluation sections toutilize the data in the saved basin file.

3.3.15.2 using

The obligatory data for the superbasin creation is a basin file chosen by the com-mand:

using basin <n>

The required basin <n> file must be either declared by the data assignmentbasin <n>=<filename> outside the superbasin section or be created by precedingbasin section (using save basin <n> within that section).

Chapter 4Grid files

Each field is written to separate grid file distinguished by a descriptor accordingto the assignment in the compute section, see Tables 3.6 and 3.7. Every grid fieldfile has a header followed by the field values. The information is written in blocksstarting with the string ‘/====\’. DGrid is searching the text for these blocks andevaluate the data in the corresponding sections.

4.1 Field file

Following is an example of the grid field file header for the orbital density computedfrom wavefunction file based on data supplied by the GAUSSIAN03 package:

FIELD-DATA created by DGrid version 5.0 03-11-2016 build ...

/=========\| title |+---------+

+-------------------------------------------------------------------------------+::C3H3NO HF 6-31G+-------------------------------------------------------------------------------+

+----------------------------------+| date= Thu Nov 3 14:45:29 2016 |+----------------------------------+

/=================\| field= rho[r] |+-----------------+

+------------------------------------------------------------------------------+| wavefunction= C3H3NO_HF_6-31G.g03 program= GAUSSIAN03 || spin= both pair= --- |+------------------------------------------------------------------------------+

/========================\| selected orbitals= 2 |+------------------------+

sym occupation+----------------------------------------------------------------+| 17 NONE 1.00000000 1.00000000 || 18 NONE 1.00000000 1.00000000 |+----------------------------------------------------------------+

87

88 4 Grid files


atom X Y Z+------------------------------------------------------------+| O 1 -2.06183559 0.57482826 0.00000000 || C 1 0.00000000 2.12106164 -0.00000000 || N 1 2.08694733 0.89554761 -0.00000000 || C 2 1.41345421 -1.67482606 -0.00000000 || C 3 -1.11013465 -1.85958662 0.00000000 || H 1 -0.30771125 4.10249907 -0.00000000 || H 2 2.80485596 -3.12185095 -0.00000000 || H 3 -2.43100862 -3.36800124 0.00000000 |+------------------------------------------------------------+

/==============\| grid= bohr |+--------------+

origin I J K+--------------------------------------------------------------------+| X | -6.00000000 | 12.00000000 0.00000000 0.00000000 || Y | -7.00000000 | 0.00000000 14.00000000 0.00000000 || Z | -4.00000000 | 0.00000000 0.00000000 8.00000000 |+--------------------------------------------------------------------+| points | 121 141 81 |+--------------------------------------------------------------------+| mesh | 0.10000000 0.10000000 0.10000000 |+--------------------------------------------------------------------+| || <(IJ) = 90.000 <(IK) = 90.000 <(JK) = 90.000 |+--------------------------------------------------------------------+

/==============\| data= real |+--------------+

+----------------------------------------------------------------------------------+0.00000000e+00 0.00000000e+00 0.00000000e+00 ... 0.00000000e+00 0.00000000e+00

The header starts with the keyword ‘FIELD-DATA’ followed by the informationabout the DGrid version. Next lines contain the title extracted from the control fileand the date and time of the calculation.

The name of the calculated field is followed by the name of the quantum chemicalpackage that produced the wavefunction file and the name of the wavefunction fileand additional data concerning the calculation, like the spin and pair spin of thefield, selected orbitals (when the occupation section was involved) and the atomiccoordinates. The remaining part of the header contains the information about thegrid region, which is the description of the computed region with the coordinates ofthe grid origin together with the three vectors spanning the region and the number ofpoints in each direction and in case of a solid state calculation a block with the latticevectors. This information is necessary for DGrid when performing some subsequentevaluation.

Remark 4.1. In certain cases DGrid needs the access to the wavefunction file. Thus,do not rename this file (otherwise change the name in the required grid files aswell). If DGrid is not able to find the necessary wavefunction file it will possiblyfollow another evaluation route (the corresponding information will be written intothe output file).

4.2 Naming of the field files 89

Remark 4.2. In grid files generated with old DGrid versions, this information ismissing. Then some new features of the program will not be used! For instance,in previous version ‘ELI’ was used instead of ‘ELI-D’. Starting from version 4.4DGrid does not know the property ‘ELI’. Thus, e.g., DGrid will not be able to findthe exact attractor positions, or to evaluate the curvatures at attractor positions.

The values of the computed field are written in the format 15.8e. The grid points runin the x direction first (in contrast to the LMTO format where z runs first).

If the gradient vector of the property prop is computed with the assignment‘fld gradient-vec’ then all 3 vector components will be written in separate line inthe result file. It is possible to write out only single component of the vector field(using, e.g., for the y component the command ‘fld gradient-vec d/dy’). Then thedescriptor ‘data=reals d/dY’, preceding the grid values, indicates that the data forthe d/dy component are following.

4.2 Naming of the field files

The grid field file-names are generated from the generic file name which is thewavefunction file name. For each field a unique file extension, given in Table 3.6,will be appended to the generic file name. Additionally, for properties computed incoordinate space the descriptor ‘ r’ will be appended (for calculations performed inmomentum space – using the keyword ‘space=momentum’ in the compute section– the descriptor ‘ p’ would be appended).

In the control file at page 28 the computed fields are given by the assignments‘ELI-D triplet-pair’, ‘rho’, and ‘rho Laplacian’, i.e., the ELI-D for triplet-coupledelectrons [36], the total electron density (summing up both spins) and the to-tal density Laplacian will be computed. In case of the the electron density fieldsthe files C3H3NO HF 6-31G.g03.rho r and C3H3NO HF 6-31G.g03.lap rho r,respectively, will be created (due to the assignment wfn 1=C3H3NO HF 6-31G.g03).

Table 4.1 Grid file name extender subject to field type

Property type File namea

7→ filename.fld rgradient-mag 7→ filename.grad-mag fld rgradient-vec 7→ filename.grad fld rln-derivative 7→ filename.ln deriv fld rhessian 7→ filename.hess fld rLaplacian 7→ filename.lap fld rrelative-Laplacian 7→ filename.rel-lap fld r

a Exemplary for the generic name filename and field fld in real space

90 4 Grid files

Table 4.2 Grid file name extender subject to field spin

Property spin File namea

alpha 7→ filename.fld r abeta 7→ filename.fld r bboth 7→ filename.fld rsinglet 7→ filename.fld r striplet 7→ filename.fld r ta Exemplary for the generic name filename and field fld in real space

In any case, files written by DGrid will not overwrite already existing files. If a fileexists, DGrid will create new file name by appending ‘ 1’ to it (and successivelyhigher numbers). The grid field files will be written in the DGrid format (specialheader and 7 values per line).

Depending on the combination of the descriptors the resulting grid file name willbe affected. For instance, in case of the density Laplacian, the string ‘lap ’ will pre-cede the field name in the generated grid file name. With the descriptors <spin>and <pair> additional information will be appended to the grid file name, for in-stance ‘ a aa’ for the same spin ELI-D. In Tables 4.1 to 4.3 the possible descriptorsare given together with the strings (extenders) that will be appended to the grid filename (exemplary for the field fld and the generic name filename in coordinate space,i.e., with ‘ r’ appended).

If <type> is omitted, cf. the first row in Tab. 4.1 then the field values will becomputed (and the grid file will be named filename.fld r). The <type> descriptorscan be listed with the command ‘dgrid -t’.

Remark 4.3. If a field gradient or Hessian is computed, all values of the vector arewritten in single line in the grid file. With the respective <cmp> assignments d/dx,d/dy, d/dz in case of a gradient and d/dxx, d/dyy, d/dzz, d/dxy, d/dxz, d/dyz for theHessian a single component can be written to the grid file.

In case of the ELI-D the grid file will be named C3H3NO HF 6-31G.g03.elid r t tr,cf. Tab. 4.2 and Tab. 4.3 showing the ELI-D spin and the pair-spin dependency.

Table 4.3 Grid file name extender subject to field pair spin

Property pair spin File namea

alpha-alpha 7→ filename.fld r aabeta-beta 7→ filename.fld r bbalpha-beta 7→ filename.fld r abbeta-alpha 7→ filename.fld r basinglet-pair 7→ filename.fld r sgtriplet-pair 7→ filename.fld r trspinless 7→ filename.fld r

a Exemplary for the generic name filename and field fld in real space

4.3 Basin file 91

Remark 4.4. It can be seen that in case of the ‘ELI-D triplet-pair’ assignment the<spin> descriptor was set per default to ‘triplet’ (similarly, with ‘ELI-D triplet’ the<pair> descriptor would be set to ‘triplet-pair’ as default).

If <spin> is omitted, then the total spin will be used for 1-particle fields (i.e., thedefault value is both, cf. Tab. 4.2). For exceptions from this rule, as well as thechoice in case of 2-particle property see remark 3.5 below. The <spin> descriptorscan be listed with the command ‘dgrid -s’.

If <pair> is omitted, then the default value is either unknown (in which casethe file name is created as for spinless, cf. Tab. 4.3) or the value is spin depen-dent, see remark 3.5 below. The <pair> descriptors can be listed with the command‘dgrid -s’.

4.3 Basin file

After the basin search each grid point is assigned to its basin by an integer number.The data are written on separate basin file. It has a header followed by the basinvalues. The format of the header is similar to the one for the field file, describedin previous section. Following is the file C3H3NO HF 6-31G.g03.rho r.bsn createdby the control file given in Sec. 3.3.1:

BASIN-DATA created by DGrid version 5.0 24-11-2016 build ...

/=========\| title |+---------+

+-------------------------------------------------------------------------------+::C3H3NO HF 6-31G+-------------------------------------------------------------------------------+

+----------------------------------+| date= Thu Nov 24 20:03:10 2016 |+----------------------------------+

/=================\| field= rho[r] |+-----------------+

+----------------------------------------------------------------------------+| from file= C3H3NO_HF_6-31G.g03.rho_r || || wavefunction= C3H3NO_HF_6-31G.g03 program= GAUSSIAN03 || spin= both pair= --- || || basins cropped by 0.00100000 isovalue || using file= C3H3NO_HF_6-31G.g03.rho_r |+----------------------------------------------------------------------------+


atom X Y Z+------------------------------------------------------------+| O 1 -2.06183559 0.57482826 0.00000000 || C 1 0.00000000 2.12106164 -0.00000000 || N 1 2.08694733 0.89554761 -0.00000000 || C 2 1.41345421 -1.67482606 -0.00000000 |

92 4 Grid files

| C 3 -1.11013465 -1.85958662 0.00000000 || H 1 -0.30771125 4.10249907 -0.00000000 || H 2 2.80485596 -3.12185095 -0.00000000 || H 3 -2.43100862 -3.36800124 0.00000000 |+------------------------------------------------------------+

/==============\| grid= bohr |+--------------+

origin I J K+--------------------------------------------------------------------+| X | -5.36000000 | 11.00000000 0.00000000 0.00000000 || Y | -5.78000000 | 0.00000000 12.20000000 0.00000000 || Z | -3.85000000 | 0.00000000 0.00000000 7.70000000 |+--------------------------------------------------------------------+| points | 111 123 78 |+--------------------------------------------------------------------+| mesh | 0.10000000 0.10000000 0.10000000 |+--------------------------------------------------------------------+| || <(IJ) = 90.000 <(IK) = 90.000 <(JK) = 90.000 |+--------------------------------------------------------------------+

The first readable information is a line beginning with the string ‘BASIN-DATA’ andthe info about the DGrid version followed by the title and the date and time of thecalculation. In next section is the name of the field and the corresponding file usedto determine the basins, together with some information about the wavefunction fileas well as the cropping of the grid region. This is followed by the block with atomiccoordinates and the section with information about the grid region. In case of a solidstate calculation the lattice vectors are given as well.

The last part of the header contains some information about the basins. It showsthe volumes and the maximal grid value in the respective basins as well as a descrip-tor indicating which atom the respective basin can be attributed to:

/=============\| basins= 8 |+-------------+

basin volume maximum i j k descriptor+--------------------------------------------------------------------+| 1 40.490 0.4136 51 98 39 H_1 || 2 41.158 0.4095 30 24 39 H_3 || 3 42.720 0.4186 81 27 39 H_2 || 4 62.982 58.7254 54 79 39 C_1 || 5 79.034 64.0388 68 41 39 C_2 || 6 79.391 52.4968 42 39 39 C_3 || 7 99.748 106.3250 33 64 39 O_1 || 8 116.278 74.4581 74 67 39 N_1 |+--------------------------------------------------------------------+

This section is followed by integer values according to the basin the given pointcorresponds to. The grid points run in the x direction first. The data line below showsthat there is a ‘basin’ number zero. This corresponds to the grid points outside themolecular envelope.

/=================\| data= integer |+-----------------+

+---------------------------------------------------------------------------+0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0

4.3 Basin file 93

It is always a good idea to visualize the basins, cf. Fig. 3.1, to see whether thereare some unusual artifacts. Special visualization techniques should be utilized (thiscreates surfaces around blocks of constant data values, cf. the manual for the AvizoSTR-module).

Remark 4.5. Using isosurfaces of basin values (instead of the special techniques)to create the basins one should bear in mind that, for instance, the isosurface with(basin) value 3 is not only found around the basin 3, but also between the basins 1and 4, 2 and 4, etc.

In DGrid 5.0 there is a new section following the (integer) basin data. This sectioncontains (coded) data from the attractor search routine for each grid maximum.

Chapter 5Parallel calculation

The calculation of a field grid can be an extremely time consuming task that can lastfor weeks or even months. As nowadays usually more than one processor is avail-able in a computer it would be convenient to compute the grids in parallel. AlthoughDGrid 5.0 itself is not ready yet for automatic parallelization (except for the calcula-tion of overlap integrals) the script dgrid para-5.0, included in the DGrid pack-age, supplies a work around for such parallel runs. The script dgrid para-5.0creates control files for job computing just a part of the grid (a slice). The operat-ing system distributes the jobs among the processors. As the final step all slices aremerged together into the whole grid. Similar procedure can be applied to the searchfor the critical points.

5.1 Grid slices

The field grid is defined by three vectors i, j, k and the number of points ni×n j×nkalong each direction, cf. the previous Section. Around the grid box there is a 2-points wide border. In DGrid the loops for the grid values generation run first alongthe i vector, followed by j and k. With the control command, cf. Sec. 3.3.3.19:

slices=<div> <slc>

set within the compute section, the whole grid is divided into <div> slices alongthe k direction and the slice number <slc> will be computed. Thus, the slice covers(ni + 4)× (n j + 4) points in the i, j directions (i.e., inclusive the grid border) and(nk + 4)/div points in the k direction (some slices can include 1 point more). Theresulting slice of the field grid (with the string ‘.slice-<slc>’ appended to the filename) contains the same header as the full grid. However, before the array of gridvalues starts, there is an additional line with the information about the grid slice. Forinstance, the calculation of the ELI-D grid for the oxazole with the control file:

95

96 5 Parallel calculation


compute:-----------------------------------------

using wfn_1

ELI-D triplet-pair

x: y: z:----------------------------:origin= -6.0 -7.0 -4.0 intervals:----------------------------------------:i-vector: 12.0 0.0 0.0 120j-vector: 0.0 14.0 0.0 140k-vector: 0.0 0.0 8.0 80----------------------------------------:

slices=4 1:-----------------------------------------compute_end

yields the ELI-D grid named C3H3NO HF 6-31G.g03.elid r t tr.slice-1, which isthe first slice of the whole grid (defined by 121×141×81 points) after the divisioninto 4 slices. In the grid field file the (unchanged) grid definition is followed by datawith the slice information:

/=============\| slices= 4 |+-------------+

+----------------------------------------------------------------+| slice= 1 range= -2 - 19 border=2 |+----------------------------------------------------------------+

/==============\| data= real |+--------------+

+---------------------------------------------------------------------------+9.29858100e-02 9.57969355e-02 ... ... 1.06484323e-01 1.08974705e-01...

The slice info shows that this is the slice number 1 (of 4 slices). Because of the 2points wide border the slice starts in the k direction at the point number (-2) andends at point number 19 (i.e., in total 22 points in k direction). The grid itself, i.e.,without the border points starts in the k direction at point number 0 and ends at pointnumber 80. The slices number 2, 3, and 4 contain 21 points each in the k direction(20-40, 41-61, 62-82).

After each of the 4 slices is finished the whole grid is build by merging the slicestogether using the DGrid utility:

dgrid C3H3NO HF 6-31G.g03.elid r t tr.slice-1 complete

5.2 dgrid para-5.0 script 97

The utility must be called with the file for the slice number 1. Also, all the remainingslice files must be present in the active directory. The resulting grid will be saved inthe file C3H3NO HF 6-31G.g03.elid r t tr.

5.2 dgrid para-5.0 script

The submission of all the grid-slice jobs can be accomplished automatically usingthe script dgrid para-5.0 available in the DGrid package. The script is calledthe same way like an ordinary DGrid calculation:

dgrid para-5.0 controlfilename

where the control file does not include the slices= command (i.e., the example con-trol file at page 95 can be used after removing the slices= command). The scriptrequires on the command line the input of the number of slices (below, for controlfile named rho slc.inp, 4 slices were requested) and writes the message about thesubmitted jobs to the console:

------------------------------------input number of slices :4rho_slc.inp-01 job submittedrho_slc.inp-02 job submittedrho_slc.inp-03 job submittedrho_slc.inp-04 job submitted------------------------------------

The script creates 4 new control files by duplicating the control file rho slc.inpand including the command slices=4 1, ... slices=4 4, successively into each file.Additionally, the LOG files rho slc.inp-01.DG LOG, ..., rho slc.inp-04.DG LOGwill be created. The progress of the calculations can be observed with the scriptdgrid dglog):

dgrid dglog controlfilename

After all jobs are finished the 4 grid-slice files, with names terminated by ‘.slice-1’till ‘.slice-4’, are located in the active directory. The slice files can be collected intothe full grid file using the DGrid complete utility as described on page 96.

Remark 5.1. The number of slices chosen with the dgrid para script need notto coincide with the number of available processors. Often it is even better to usehigher number of slices (i.e., processor tasks) than processors. Especially for largegrids around complex molecules there is lot of ‘empty space’ where the primitivefunctions are almost zero valued, thus very fast evaluated. In this case some of the

98 5 Parallel calculation

slices can be done much earlier than others, leaving idle processors without tasks.Then it is favorable to occupy the processors with more than one task.

Chapter 6Utilities

The DGrid utilities are called from the command line with the following generalsyntax:

dgrid filename <kwr> <par>

The possible choices of keywords <kwr> and one or more parameters <par> de-pend on the input file (given by the filename), which can be an output file fromquantum chemical program, a wavefunction file, grid file, basin file, or a slice file,respectively. The DGrid utilities work without the control file. Quick reference tothe syntax is given by the command:

dgrid -u

6.1 Conversion of QM package output

As described in previous sections, the DGrid program needs a wavefunction file toevaluate the chosen properties. The wavefunction file is written in a specific for-mat, cf. chapter 2, and must be created in advance from data supplied by quantummechanical packages used.

6.1.1 ADF

In case of the program ADF [1] the formatted TAPE21 filename.kf (using the‘dmpkf’ utility of ADF) must be convert into the wavefunction file using the DGridcommand:

99

100 6 Utilities

dgrid filename.kf

yielding the wavefunction file named filename.adf. For localized orbitals computedwith ADF the command must be extended as follows:

dgrid filename.kf locorb

In this case the wavefunction file will be named filename.adf loc.

6.1.2 GAMESS

The program GAMESS [4] writes the wavefunction and MO data to an output file.DGrid extracts the necessary data from that file using the command:

dgrid filename.out

This creates the wavefunction file named filename.gms.

Remark 6.1. To create a GAMESS output readable by DGrid use the command‘dgrid=.true.’ in the GAMESS input file.

To read from the GAMESS output file the natural orbitals use the command:

dgrid filename.out natorb

Remark 6.2. The wavefunction file can be also created from the WFN data includedin the GAMESS ‘.dat’ file when the variable ‘AIMPAC=.true.’ was used in theGAMESS input. Then, the DGrid command reads:

dgrid filename.dat wfn

6.1.3 Gaussian

In case of Gaussian [24] the formatted Checkpoint file filename.fchk (using the‘formchk’ utility of Gaussian) must be convert into the wavefunction file using theDGrid command:

6.1 Conversion of QM package output 101

dgrid filename.fchk

yielding the wavefunction file named filename.g03, respectively filename.g09. An-other possibility is to use the information in the Gaussian WFN file:

dgrid filename.wfn

yielding the wavefunction file named filename.gwf.

Remark 6.3. In contrast to the wavefunction file created from the Checkpoint file thefilename.gwf does not contain virtual orbitals (as those are not present in the WFNfile).

Remark 6.4. Even for a Gaussian calculation yielding at once both the Checkpointand the WFN files the resulting wavefunction files will be different! The reason isthat in the WFN file the basis is already decontracted (and so will be the data in thecorresponding wavefunction file). In contrast, the wavefunction file created from theCheckpoint file has contracted basis and the decontraction is performed internallyduring the DGrid job. However, the property values from a DGrid calculation willbe the same, regardless which route is chosen.

Remark 6.5. IMPORTANT!In case of ROHF (restricted open-shell) calculation Gaussian03 does not write prop-erly the wavefunction set data to the WFN file. All orbitals are doubly occupied in aWFN file from such calculation, i.e., the information about the spin polarization isnot given. After the conversion the occupation for the single occupied orbitals mustbe changed manually. The data in the WFN file from an UHF calculation are fine.

6.1.4 Molpro, Molcas, Turbomole

For the programs Molpro [49], Molcas [7], and Turbomole the wavefunction dataoutput must be written in the Molden [43] format. Such output file filename.moldenin Molden format must be convert into the wavefunction file using the DGrid com-mand:

dgrid filename.molden

producing the wavefunction file named filename.md, respectively filename.mp, file-name.mc and filename.tm for the recognized programs. In case of the program Tur-bomole include in the Molden output file manually as the second line the string:

102 6 Utilities

[TURBOMOLE]

otherwise the normalization will not be the proper one.

Remark 6.6. For natural orbitals from spin-polarized calculation the occupation isdeclared only as the α-spin channel. In the corresponding DGrid wavefunction filethe orbital occupations are declared as ‘natural’, i.e., there is only one number be-tween 0-2 for each orbital. It is no more possible to extract the spin polarizationfrom the natural occupation (for this the natural spinorbitals were needed, whichresult from diagonalization separate for each spin channel).

6.2 Wavefunction file conversion

An older wavefunction file can be converted into the new format using the DGridcommand:

dgrid wavefunctionfilename wrt

This yields a copy of the wavefunction file named wavefunctionfilename 1 writtenin the new format. If the system energy or the number of electron were not presentin the older file then the corresponding data will be set to zero.

6.3 Structure file

With the DGrid command:

dgrid wavefunctionfilename str

a structure file will be created from the wavefunction file. It contains the informationabout the atomic positions. The resulting file named wavefunctionfilename.str canbe used by an external program to visualize the molecular structure.

The structure file can be also created from the grid files using the same command.Only atoms already present in the grid file will be written to the structure file.

6.4 AO contributions 103

6.4 AO contributions

The atomic orbital expansions of the molecular orbitals are printed by the DGridcommand:

dgrid wavefunctionfilename c=<coef>,<occ>

where <coef> (float number) is the threshold for the AO coefficients and <occ>the threshold for the MO occupation (with the default value 0.0). No blanks areallowed between c= and the numbers (separated by comma, without blanks). Forinstance:

dgrid C3H3NO HF 6-31G.g03 c=0.05

prints out each molecular orbital from the wavefunction file C3H3NO HF 6-31G.g03as a linear combination of atomic orbitals with coefficients of absolute value largeror equal 0.05 and MO occupation larger than 0.0 (which is the default value for<occ>):

******************************************************************************AO expansion coefficients above the threshold of 0.0500 (occup > 0.0000)

******************************************************************************

input from GAUSSIAN03 basis file C3H3NO_HF_6-31G.g03

entitled: C3H3NO HF 6-31G

Total energy = -244.5027 hartree

--------------------------CLOSED SHELL DATA--------------------------

NONE # 1: occup = 2.00000000 Energy = -20.6521

0.9959 O 1 ( s )


0.9962 N 3 ( s )

...

...


-0.0768 O 1 ( pz ) -0.0722 O 1 ( pz_a ) ... -0.2363 C 2 ( pz_a )-0.1675 N 3 ( pz ) -0.1454 N 3 ( pz_a ) ... 0.2474 C 4 ( pz_a )0.3764 C 5 ( pz ) 0.3033 C 5 ( pz_a )

The virtual orbitals will not be printed for the default <occ> value. The printing ofvirtual orbitals can forced by any negative value for <occ> (for instance -1.0).

104 6 Utilities

Another possibility is to print all orbitals (satisfying the AO contribution and occu-pation thresholds) within an energy range. Such output is produced by the command:

dgrid wavefunctionfile c=<coef>,<occ> ew=<emin>,<emax>

where <emin>and <emax> (float numbers) specify the respective minimum andmaximum for the allowed orbital energies.

6.5 Completion of sliced fields

The calculation of grids as well as the determination of critical points can be per-formed in parallel by submitting slices of the fields to each processor (for grids cf.Sec. 5 and for the critical points cf. Sec. 3.3.14.6). The slice submission can bedone with the script dgrid para, cf. Sec. 5.2. The resulting data for each slice aresaved in files named ‘*.slice-1’, ‘*.slice-2’, etc. The complete field is created with aDGrid utility using the slice number 1 as the first argument followed by the keywordcomplete:

dgrid filename.slice-1 complete

Note, that all the remaining slice files must be present in the active directory. Theresulting field will be saved in the file filename, i.e. with the slice numbering strippedoff.

6.5.1 Evaluation of the Source Function

The source function contribution S(r,Ω) is defined as the integral of the local sourceLS(r,r′) [10, 26]

LS(r,r′) =− 14π

∇2ρ(r′)|r− r′|

(6.1)

over given region Ω

S(r,Ω) =− 14π

∫Ω

∇2ρ(r′)|r− r′|

dr′ (6.2)

It shows, how the region Ω ‘participate’ on the reconstruction of the electron densityat the chosen reference position r. Positive values of S(r,Ω) mark the region Ω as asource, negative values as a sink.

6.5 Completion of sliced fields 105

To compute the local source field, the reference position needs to be specified inthe control file. The specification is done with the keyword ref point followed bythe Cartesian position. Following is an example for the oxazole molecule:

::C3H3NO HF 6-31G:--------------------------------------------------|wfn_1=C3H3NO_HF_6-31G.g03

compute:-----------------------------------------

using wfn_1

LSref_point= -1.4000 -1.0226 0.0000


Usually, a saddle point in the electron density is chosen as the reference position.The above position is the electron density saddle point between the atoms O1-C5.The mesh size is set to 0.05 bohr. This is the recommended value to get better basinborders (zero-flux surfaces). In the resulting grid file for the local source containsthe following header:

/==========================\| field= Local-source[r] |+--------------------------+

+-------------------------------------------------------------+| wavefunction= C3H3NO_HF_6-31G.g03 program= GAUSSIAN03 || || spin= both pair= --- || || reference point= -1.400000 -1.022600 0.000000 || || density at reference point= 0.24458005 |+-------------------------------------------------------------+

Thus, the grid file contains not only the information about the reference position, butalso the density value at that position. The (exact) integral of the local source overthe whole space should recover this value. As the computed region is just a part oftotal space and the integration is done numerically, the integral will deviate from thedensity at the reference position. As proposed by Gatti [10, 26], the source functioncontribution S(r,Ω), i.e., the local source integrals over the basins, are output byDGrid as percentage of source contribution to the density at the reference point.

Integrating the above local source field from the file C3H3NO HF 6-31G.g03.ls ryields value much lower than the density ρ(rb) = 0.24458 bohr−3 at the saddlepoint. It amounts to -2131% (!) of the expected value. Refining the local sourcegrid by the precision parameter 0.1 (computing additional 1 484 396 points) yieldsby integration 103.40% of the reference density. Increasing the precision with the

106 6 Utilities

refinement parameter 0.01 (evaluating 7 613 796 additional points) recovers by inte-gration 101.20% of the reference density. Further increase of the precision using therefinement parameter 0.001 does not significantly change the recovering of the den-sity at the reference position (101.30%). Bear in mind that at large distances there isa charge depletion, i.e., positive electron density Laplacian. Those remaining con-tributions from the missing volume, due to the finite grid box size, will reduce therecovered density value to the proper value of 100%.

The above procedure just describes a test of the quality of the local source in-tegration. In the analysis of the source function contribution one is interested, howparticular basins contribute to the density at the reference (saddle) point. For this,the (refined) local source grid will be integrated over the basins. The QTAIM basinscan be created within the control file by a separate evaluation section using theC3H3NO HF 6-31G.g03.rho r grid (0.05 bohr mesh size, cropped by 0.001 bohr−3

density isosurface), yielding the basin file C3H3NO HF 6-31G.g03.rho r.bsn. Thecontrol file for the source function contributions with the refined local source gridand the mentioned basin file creation reads as follows:

::C3H3NO HF 6-31G:--------------------------------------------------|field_1=C3H3NO_HF_6-31G.g03.rho_r

basin:---------------------

using field_1crop using field_1 0.001save basin_1

:---------------------basin_end

field_2=C3H3NO_HF_6-31G.g03.ls_r

refine:--------------

using field_2precision=0.01save rfn_2

:--------------refine_end

integrate:--------------

using rfn_2over basin_1

:--------------integrate_end

DGrid first creates the QTAIM basins (cropped by the 0.001 isodensity surface andreferred in the subsequent sections by the descriptor basin 1)), then refines the local-source grid (the descriptor rfn 2 is used only to show the correspondence to thefield 2 – any other number could be used instead). Finally, the refined data are inte-

6.5 Completion of sliced fields 107

grated over the QTAIM basins. For each evaluation section a separate log file willbe written. Following is a part of the ‘ITG’ file from the ‘integration’ section:

/=====================================\| integrals for 21486 refined cells |+-------------------------------------+

+----------------------------------------------------------------+| read from file C3H3NO_HF_6-31G.g03.ls_r.rfn || ==> || 7613796 values were computed for inhomogeneity > 0.0100000 |+----------------------------------------------------------------+

0.244580BASIN VOLUME RECOVER% MAXIMUM X Y Z ATOMS DIST ECCNT

+-------------------------------------------------------------------------------------+| x 1 R 40.490 1.4873 0.4136+ -0.260 4.020 0.050 H_1 0.11 - || x 2 R 41.158 5.2802 0.4095+ -2.360 -3.380 0.050 H_3 0.09 - || 3 R 42.720 1.8162 0.4186+ 2.740 -3.080 0.050 H_2 0.09 - || 4 R 62.982 2.2419 58.7254+ 0.040 2.120 0.050 C_1 0.06 - || x 5 R 79.034 6.8118 64.0388+ 1.440 -1.680 0.050 C_2 0.06 - || x 6 R 79.391 42.3318 52.4968+ -1.160 -1.880 0.050 C_3 0.07 - || x 7 R 99.748 41.1871 106.3250+ -2.060 0.620 0.050 O_1 0.07 - || x 8 R 116.278 3.6997 74.4581+ 2.040 0.920 0.050 N_1 0.07 - |+-------------------------------------------------------------------------------------+| TOTAL | 561.801 104.8560 |+-------------------------------------------------------------------------------------+| DIFF | -471.539 |+-------------------------------------------------------------------------------------+

Each basin number is followed by an ‘R’ showing that the integrated data wererefined for that basin. The volume difference at the end is a hint that the basinswere cropped. Thus, part of the volume of the box-region was not considered. Theimpact of the cropping, i.e., smaller integration volume, is that the recovering ofthe electron density at the reference position amounts to almost 105% (taking intoaccount the grid-box volume that was cropped away, reduces the integration error to101.2%, cf. page 105).

It can be seen that all the atomic basins are contributing as a source of the densityat the saddle point between the oxygen and carbon nr. 3, with main contributionsfrom the two neighboring basins (basins nr. 6 and 7).

In the above example the evaluation of the source function was not too expensive.With this advantage it was possible to refine the whole local source grid. This is notthe case when heavy atoms and large basis sets are involved. Such refinement couldtake for hours! Then it reasonable to perform the refinement for local source overthe desired basins only, cf. Sec. 3.3.13. As an example let us take the same grid dataas before, but perform the source function refinement for the basins number 7 and8, i.e., the oxygen and nitrogen atomic basins:

::C3H3NO HF 6-31G:--------------------------------------------------|field_1=C3H3NO_HF_6-31G.g03.ls_rbasin_2=C3H3NO_HF_6-31G.g03.rho_r.bsn

108 6 Utilities

refine:--------------

using field_1precision=0.01

over basin_2select_basins

7 8select_basins_end

save rfn_1:--------------refine_end

integrate:--------------

using rfn_1over basin_2

:--------------integrate_end

For the refinement over the two basins 1 775 328 additional points were computed(cf. the 7 613 796 points for the refinement with the same precision parameter com-puted previously over all the QTAIM basins). Taking closer look on the participa-tion of the two basins on the recovering of the saddle point electron density shows achange from 41.19% to 41.02% (oxygen) and 3.70% to 3.66% (nitrogen):

/====================================\| integrals for 3904 refined cells |+------------------------------------+

+----------------------------------------------------------------+| read from file C3H3NO_HF_6-31G.g03.ls_r.rfn_3 || ==> || 1775328 values were computed for inhomogeneity > 0.0100000 |+----------------------------------------------------------------+

0.244580BASIN VOLUME RECOVER% MAXIMUM X Y Z ATOMS DIST ECCNT

+--------------------------------------------------------------------------------------+| x 1 40.490 1.4829 0.4136+ -0.260 4.020 0.050 H_1 0.11 - || x 2 41.158 5.2748 0.4095+ -2.360 -3.380 0.050 H_3 0.09 - || 3 42.720 1.8115 0.4186+ 2.740 -3.080 0.050 H_2 0.09 - || 4 62.982 -161.2739 58.7254+ 0.040 2.120 0.050 C_1 0.06 - || x 5 79.034 -107.6556 64.0388+ 1.440 -1.680 0.050 C_2 0.06 - || x 6 79.391 -734.2907 52.4968+ -1.160 -1.880 0.050 C_3 0.07 - || x 7 R 99.748 41.0243 106.3250+ -2.060 0.620 0.050 O_1 0.07 - || x 8 R 116.278 3.6571 74.4581+ 2.040 0.920 0.050 N_1 0.07 - |+--------------------------------------------------------------------------------------+| TOTAL | 561.801 -949.9697 |+--------------------------------------------------------------------------------------+| DIFF | -471.539 |+--------------------------------------------------------------------------------------+

The reason for this change is that now the precision parameter is valid for the inte-gral of just two basins. The convergence of the integration should be checked withincreased precision of the refinement. Especially, when heavier atoms are involved.

6.6 Operations on single grid 109

6.6 Operations on single grid

This utility operates on data of a single grid field. It is invoked by the DGrid com-mand:

dgrid gridfilename op1

DGrid reads in the grid file and asks for the desired operation which can be one of:

Available operations:-------------------------------------------------------------------------------+ - x -> number/ ˆ over -> number

mirror translation inversion -> i, j, k, in, jn, knconvert -> cube, dgrid, grace, lmtospin -> alpha, beta, both, singlet, tripletpair -> alpha-alpha, beta-beta, triplet-pair

sqr crop reduceelf2eli eli2elftrp2aa aa2trp

save quit-------------------------------------------------------------------------------

The chosen operation is performed at each grid position. Some of the operationsneed additional parameters. After the operation is done, DGrid awaits the next oper-ation. The transformed grid data are held in the memory. The procedure is finishedeither by exiting the program without saving the data (command ‘quit’) or by sav-ing the data (command ‘save’) in which case the resulting grid will be written to thefile with the same name as the original file with an appendix reflecting the chosenoperation. Following is the description of the operations.

The operations ‘+, −, ×, /, ˆ, over’ need a number as a parameter inthe input line. For instance the command ‘× 5’ will multiply the property value ateach grid point by 5. With ‘over 2’ the value 2 will be divided by the property valueat each grid point. The operation sqr takes the square root of each grid value andneed no additional parameters.

The operations ‘mirror, translation, inversion’ perform the given symmetryoperation of the grid. An additional parameter, which is one of ‘i j k’, respectively‘in jn kn’is needed, determining at which position the operation is acting. Thus,the command ‘mirror j’ will mirror the grid in the j-direction at the origin of thej-axis, whereas ‘mirror jn’ will perform the mirroring at the end of the j-axis. Theresulting grid will be twice as large as the original one. The command ‘inversionk’ will perform an inversion of the grid in the k-direction around the grid midpointof the ij-plane. This utility can be used for molecules or solid with the mentionedsymmetry to save computational time, for instance, to compute just one octant for ahomonuclear dimer and enlarge the grid by mirroring.

The operations spin and pair are used to set the appropriate value for the spinand pair-spin variable in a grid file.

110 6 Utilities

The last operation which needs an additional parameter is convert. It is followedby the string determining to which format the grid file should be converted. Thepossible formats are given on page 40. It is also possible to use this utility to convertfiles written with older DGrid versions into the current 4.5 format (in some casesnot all data will be available, like e.g. the energy of the system, which will be set toa default value).

The crop operation will ask for the name of the file containing the molecularstructure (in STR or PIC format). Then all grid values within the atomic radii (givenin the structure file) will be set to zero (cropped).The reduce operation can be used only for grid files with odd number of pointsin each direction (i.e., even number of intervals). The number of grid points willbe reduced in such way, that every second point in each direction will be removedwithout changing the size of the region-box (doubling the mesh size).

With the operations elf2eli and eli2elf the ELF grid files can under specific con-ditions formally be converted into ELI-D grid files (and vice versa). The conditionsare that the property must stem from a single-determinantal wavefunction, the occu-pations must not be fractional and in case of triplet-coupling only restricted wave-function is allowed.

The operations trp2aa and aa2trp convert ELI-D files for closed-shell calcu-lation, in which case there is a simple conversion factor between the single spin-channel and the triplet-coupled form of ELI-D [36].

6.7 Operations on two grids

Sometimes it is useful to add, subtract, multiply, or divide the values in two grids, forinstance, to create a grid with promolecular electron density or compute a densitydifference maps. The operation can be accomplished by the DGrid command:

dgrid gridfilename1 <oper> gridfilename2

DGrid reads in the two grid files (which must have identical mesh size and griddimensions) and performs the required operation at each grid position. The possibleoperations <oper> are summarized in Table 6.1.

Table 6.1 Operations on two grids

<fmt> Format

+ 7→ add the grid values f1(x,y,z)+ f2(x,y,z)− 7→ subtract the grid values f1(x,y,z)− f2(x,y,z)x 7→ multiply the grid values f1(x,y,z)∗ f2(x,y,z)/ 7→ divide the grid values f1(x,y,z)/ f2(x,y,z)

6.8 Cropped basins 111

The resulting grid values, with the same grid dimensions as the original ones, willbe written to the file named grid.add, grid.sub, grid.mult, or grid.div, respectively.The header of the resulting grid file is identical with the header of the first grid fileon the input line (only the title is changed). This means that, for instance the spinlabels will be the ones from the first grid file. Such data, if desired, must be changedmanually.

6.8 Cropped basins

This utility works on basin files created by the DGrid program, i.e., grids of integervalues yielding the basins.

For a molecule the outer basins extend to infinity, i.e., such basins are boundedby the region-box chosen for the grid calculation. However, it might be useful tocrop the basins by an isosurface of another property field, for instance the elec-tron density. Using the 0.001 bohr3 density isosurface would correspond to basinscropped by the ‘molecular envelope’. The procedure is accomplished by the DGridcommand:

dgrid basinfilename crop <val> gridfilename

DGrid reads in both the basin file and the property file (the files must have theidentical grid dimensions and mesh size). Each basin grid point outside the <val>-isosurfaces (i.e., outside the <val>-localization domains) will be set to zero (i.e.,marked as no basin). The resulting cropped basin grid, together with the informationabout the cropping isovalue and the property file name, will be written to the filewith the same name as the basin file with the string ‘.crop’ appended.

The procedure yields the same result as a basin search with the command cropincluded in the control file, cf. Sec. 3.3.1.2. The utility can be used if one do notwant to perform a new basin search for different cropping isovalues.

Appendix AComments to grid field calculations

A.1 Evaluation of FHI-aims calculations

To evaluate the FHI-aims [3] wavefunction (using the FHI version adapted by Dr.Alim Ormeci) the output file ‘dgrid aims.dat’ needs to be converted:

dgrid dgrid aims.dat

The generated file ‘dgrid aims.fhi’ contains the LCAO expansion of the crystal or-bitals. The file should be renamed to reflect the calculated compound (this name willbe later propagated through the computed grid fields, etc.).

A.2 Grid mesh

For the visualization (using an external program) the mesh distance of 0.1 bohrbetween the grid points (0.1 bohr mesh size) is sufficient. However, only for lightelements (i.e., Li to F) the precision of numerical charge integration will be satis-factory in case of such grids. For a steep property with lot of critical points (likeELI-D or the density Laplacian) a 0.05 bohr mesh (or better) is recommended. Forheavier elements (for instance the d-elements) even much more dense mesh wouldbe necessary, resulting in huge grid files. In this case, use a 0.05 bohr mesh withadditional refinement performed with the refine utility, cf. section 3.3.13. Thisapproach is possible only if the wavefunction file is present (i.e., not possible forsolid state calculations, except with FHI-aims[3]). For molecular Gauss-type basisset an analytical charge integration will be used, cf. Sec. 3.3.12.

113

114 A Comments to grid field calculations

A.3 Spin density

For the spin density, computed with the assignment rho-spin in the compute section,cf. Table 2 on page 29, both spin parts are needed. Thus, the spin density will not becomputed if only a particular spin part is chosen. Note that with natural occupationthe spin density cannot be determined as there is no spin information (the occupationis between 0-2). To compute the spin density the corresponding natural spinorbitalsare needed.

A.4 ELI

In case of ELI-D, computed with the assignment eli-d <pair> in the compute sec-tion, three results are possible, depending on the choice of the pair-spin component.Choosing the keyword ‘alpha-alpha’ respectively ‘beta-beta’ for the pair-spin yieldsthe ELI-D for the corresponding same-spin electron pairs [32]. In each respectivecases the electron density of corresponding spin will be sampled.

ELI-D for triplet coupled electron pairs is computed with the keyword ‘triplet-pair’ for the pair-spin. In this case the charge of triplet-coupled electrons is sampled.Note that the value of ELI-D for triplet pairs is influenced by the total number ofelectrons of the system, see Ref. [36].

In ELI-D for triplet-coupled electrons both spin parts are included. The formulaused in DGrid is derived for restricted basis only. Bear in mind, that for unrestrictedbasis (like for spin-polarized system computed with ADF), the formula is not thecorrect one (as in this case the expectation value of S2 yields an improper value,contaminated by higher multiplets). In case of unrestricted basis ELI-D for triplet-coupled electrons will be evaluated, however, the density and the pair-volume func-tion of triplet-coupled electrons is computed according to Eqs. 5 and 37 of Ref.[36].

The other variant of ELI, namely ELI-q (cf. Ref. [36]), is derived from the spacepartitioning based on fixed charge. If both spins should be included it must be spec-ified that the partitioning is based on the fixed charge of singlet-coupled electrons,i.e., with the command ‘ELI-q singlet’ in the compute section, in which case thesinglet-coupled electron pairs are sampled. It is still possible to compute the variantELIA, which is derived from space partitioning based on fixed product of α-spinand β -spin charge, cf. Ref. [33].

A.5 ELF, LOL

The assignment ‘ELF alpha’, respectively ‘ELF beta’ in the compute section, re-sults in the calculation of ELF according the formula of Becke and Edgecombe[15]. If both spin channels are chosen (i.e., ‘ELF’), then ELF for the total density

A.6 Overlap integrals 115

will be computed according the ‘spin-polarized’ formula, cf. Table 2 as well as Eq.9 in Ref. [34]. The assignment ‘ELF-cs’ yields for the total density the ELF com-puted according the ‘closed-shell’ formula, cf. Table 2 as well as Eq. 7 in Ref. [34].The ‘spin-polarized’ formulation follows the idea that ELF is based on kinetic en-ergy densities, whereas the ‘closed-shell’ formula is more related to the electronpair densities and can be, in certain sense, rationalized from ELI for triplet-coupledelectron pairs [36].

With the assignment ‘ELF-Kirzhnits’ (with or without the spin descriptor) theELF variant of Tsirelson [46] is computed. It is based on kinetic energy densityapproximated by the Kirzhnits formula.

In case of LOL, computed with the assignment ‘LOL’, single spin channel shouldbe chosen to comply with the definition of Schmider and Becke [44].

Additional comments as well as an overwiev about the history of ELF and theelectron localization can be found in Ref. [31].

A.6 Overlap integrals

The integration of electron density over basins yields the basin populations (theaverage number of electrons within given region), which provides information aboutthe atomic charges, respectively speculative reference to the bond order (for ELI-D basins). For the examination of chemical bonding it is of interest to analyze theelectron pair density as well. In Sec. 3.3.3 it was already mentioned that it is possibleto compute some pair-density properties, cf. Table 3.7, however there the focus wason the conditional properties. The reason was that the pair density itself is a 6-dimensional function and one need to reduce 3 dimensions to get a 3D-propertygrid.

This section deals with integrals of electron pair density. For molecules DGridis using Slater-type or Gauss-type orbitals approximating the wave function by oneor more determinants build up from the orbitals. From this Ansatz the 1-matrix and2-matrix is formed, providing the pivot for all the property evaluations.

Remark A.1. In DGrid the 2-matrix is created formally by the reduction of the de-terminantal Ansatz. It is not guaranteed that the 2-matrix properly describe the pairinteraction. For instance, starting from a DFT result yielding wavefunction repre-sented by Kohn-Sham orbitals DGrid creates the 2-matrix the same way as for aHF calculation, even if the DFT wavefunction is not meant to properly describe thelocal pair interactions, but ‘just’ the (correlated) electron density.

The electron pair density ρ2(r1,r2) is the diagonal part of the reduced 2-matrixρ2(r′1r′2,r1r2). The 2-matrix can be decomposed into same-spin and opposite-spincontributions, which in the (restricted) orbital representation can be written as:

ρσσ2 (r′1r′2,r1r2) =

12

n

∑i< j

n

∑k<l

Pσσi j,kl |φi(r′1)φ j(r′2)| |φ ∗k (r1)φ

∗l (r2)| (A.1)


and

ρσσ ′2 (r′1r′2,r1r2) =

12

n

∑i, j

n

∑k,l

Pσσ ′i j,kl φi(r′1)φ j(r′2)φ

∗k (r1)φ

∗l (r2). (A.2)

Thus, the determination of the integrals of electron pair density over basins are re-duced to the calculation of the overlap integrals Sik =

∫Ω

φi(r)φ ∗k (r)dV of the molec-ular orbitals over the basins. To accomplish this the factorization of the Gauss func-tions can be utilized.

Remark A.2. This factorization is necessary to achieve adequate speed up for theevaluation. For this reasons the calculation of the overlap integrals is performed inDGrid only for Gauss-type orbitals and only if the grid is oriented parallel with theCartesian axes.

Let us compute for the oxazole molecule the electron density grid parallel withthe Cartesian axes, determine the QTAIM basins, compute the overlap integralsover the basins and write the resulting overlap integrals to the file C3H3NO HF 6-31G.g03.rho r.bsn.sij.

::C3H3NO HF 6-31G parallel mesh:------------------------------------------------|wfn_1=C3H3NO_HF_6-31G.g03

compute:-------------------------------------

using wfn_1rhosave field_1

mesh=0.05 rho=0.001 parallel:-------------------------------------compute_end

basin:-------------------------------------

using field_1crop using field_1 0.001

save basin_1:-------------------------------------basin_end

overlap:-------------------------------------

using wfn_1over basin_1

:-------------------------------------overlap_end

Remark A.3. It is possible to compute the basins with chosen basis set and thendetermine the overlap integrals for a different basis set over those basins. One couldeven create a basin file with artificial basins and then compute the desired overlapintegrals.


A.6.1 Fluctuation and delocalization indices

The resulting overlap integrals from the previous section are used to evaluate thelocalization (LI) and delocalization indices (DI). The indices are given by integralsover the exchange-correlation part of the pair density (exchange only for HF) [22,23]: ∫

Aρ(r1)dr1

∫B

ρ(r2)dr2−∫

Adr1

∫B

ρ2(r1r2)dr2 (A.3)

which for the HF case yields:

∑i, j

∫A

φi(r1)φ∗j (r1)dr1

∫B

φ j(r2)φ∗i (r2)dr2 = ∑

i, jSA

i j SBji (A.4)

With the following control file:

::C3H3NO HF 6-31G parallel mesh:------------------------------------------------|sij_1=C3H3NO_HF_6-31G.g03rho_r.bsn.sij

DI:-------------------------------------

using sij_1:-------------------------------------DI_end

the LI is obtained (integrating both electrons over the same basin, i.e., A = B) oth-erwise the DI results. In the log file first the table with the localization indices isgiven:

+------------------------+| Localization indices |+------------------------+

basin descriptor Q sigma2 LI LIaa LIbb+---------------------------------------------------------------------------+| 1 H_1 0.866 0.515 0.352 0.176 0.176 || 2 H_3 0.880 0.520 0.360 0.180 0.180 || 3 H_2 0.907 0.527 0.380 0.190 0.190 || 4 C_1 4.964 1.767 3.197 1.599 1.599 || 5 C_2 5.670 1.969 3.701 1.850 1.850 || 6 C_3 5.607 1.882 3.725 1.863 1.863 || 7 O_1 9.003 1.169 7.834 3.917 3.917 || 8 N_1 7.946 1.569 6.376 3.188 3.188 |+---------------------------------------------------------------------------+| 35.843 9.917 25.925 12.963 12.963 |+---------------------------------------------------------------------------+

For each basin the descriptor and the electron population (Q) is given. The popula-tions are computed analytically.

Remark A.4. The density integrals are computed analytically in this section. How-ever, the regions (basins) over which the integrations are performed are still the sameones formed from small cubes given by the grid mesh. Especially for small basinsare coarse mesh one should not mistake the populations with ‘exact’ basin popula-tions for which also an analytical determination of the zero-flux surfaces would benecessary (but the errors are very small).


Remark A.5. With natural orbitals (from correlated calculation) two different for-mulae are used. Using the idea of sharing index [25] the above overlap integralsSi j are multiplied with the square root of the product occupation numbers, whereasfollowing Angyan [8] the product of the occupation numbers is used.

The last 4 columns ‘sigma2’, ‘LI’, ‘LIaa’, and ‘LIbb’ shows the fluctuation (vari-ance) in the average population of the respective region and the localization indices(total as well as spin resolved). Note that in literature sometimes differing symbolscan be found (Fradera [23] used λ for the localization index LI, respectively σ2

for the fluctuation, whereas Bader [11, 12] used F(Ω ,Ω) for LI, Λ for the fluctua-tion (and λ for the relative fluctuation). Table A.1 should clarify the meaning of thesymbols as used by DGrid.

Table A.1 Localization and delocalization indices

Index Description Equation

N(Ω) average population in Ω∫

Ωρ(r)dr

Dαα (Ω) αα-pairs in Ω∫∫

Ωραα

2 (r1,r2)dr1dr2

Dαβ (Ω) αβ -pairs in Ω∫∫

Ωρ

αβ

2 (r1,r2)dr1dr2

Dβα (Ω) βα-pairs in Ω∫∫

Ωρ

βα

2 (r1,r2)dr1dr2

D(Ω) average number of pairs in Ω Dαα (Ω)+Dββ (Ω)+Dαβ (Ω)+Dβα (Ω)

F(Ω) N2(Ω)−2D(Ω)a

σ2(Ω) fluctuation in population in Ω 〈N2〉Ω −〈N〉2Ω = N(Ω)−F(Ω)

D(Ω ,Ω ′) r1 in Ω , r2 in Ω ′∫

Ωdr1

∫Ω ′ ρ2(r1,r2)dr2

F(Ω ,Ω ′) r1 in Ω , r2 in Ω ′ N(Ω)N(Ω ′)−2[D(Ω ,Ω ′)

Q charge N(Ω)Daa αα-pairs Dαα (Ω)

Dbb ββ -pairs Dββ (Ω)

Dab opposite-spin pairs Dαβ (Ω)+Dβα (Ω)sigma2 fluctuation σ2(Ω)lambda relative fluctuation σ2(Ω)/N(Ω)LI localization index F(Ω)

δ (A,B) delocalization index F(Ω ,Ω ′)b

D(A,B) pairs between regions A and B D(Ω ,Ω ′)

a Number of pairs normalized to N(N−1)/2. In Ω : D(Ω) = 12

[N2(Ω)−F(Ω)

]b often defined as F(Ω ,Ω ′)+F(Ω ′,Ω)

The information about localization is followed by table of delocalization indicesδ (A,B), summed up over the spin components. The delocalization indices can beconnected with the number of electron pairs shared between the basins (when con-sidering the number QAQB of pairs as not being shared – with the total number ofelectron pairs between the regions A and B equal to QAQB−δ (A,B)):


+------------------------------------------------+| Delocalization indices for total pair-matrix |+------------------------------------------------+

Basin 1 2 3 4 5 6 7 8+--------------------------------------------------------------------------------+| 1 x 0.0005 0.0004 0.4438 0.0050 0.0053 0.0231 0.0268 || 2 0.0005 x 0.0006 0.0048 0.0190 0.4585 0.0215 0.0052 || 3 0.0004 0.0006 x 0.0032 0.4638 0.0222 0.0060 0.0203 || 4 0.4438 0.0048 0.0032 x 0.0262 0.0535 0.4552 0.7587 || 5 0.0050 0.0190 0.4638 0.0262 x 0.7947 0.0713 0.5622 || 6 0.0053 0.4585 0.0222 0.0535 0.7947 x 0.4634 0.0578 || 7 0.0231 0.0215 0.0060 0.4552 0.0713 0.4634 x 0.1078 || 8 0.0268 0.0052 0.0203 0.7587 0.5622 0.0578 0.1078 x |+--------------------------------------------------------------------------------+

The delocalization indices δ (A,B) between the basins are given only for A 6= B.The diagonal values in the above table (A = B) are the localization indices LI. Thefluctuation equals the sum of DIs: σ2(A) = ∑A6=B δ (A,B), respectively half of thesum when using A,B as cumulative symbol for A,B+B,A, which depends on thedefinition of the delocalization index, see Table A.1. However, summing up thedelocalization indices for the basin number 8 (nitrogen) yields 1.539, which is lessthan σ2(8) = 1.569 in the table of the localization indices. This discrepancy is dueto the cropping of the basins. The missing part is the delocalization between thenitrogen basin 8 and the region outside the 0.001-envelope (which contains 0.157electrons, cf. the total charge in the localization table). Additionally, following sumrule is valid: QA = σ2(A)+λ (A)

Remark A.6. In case of natural orbitals the LI and DI indices are given first forthe Angyan form (∑i, j nin j SA

ji SBji) and then for the Fulton formulation using

∑i, j√nin j SA

ji SBji.

Remark A.7. For correlated wavefunctions the LI/DI indices are computed from theintegral over the whole correlated pair density followed by the subtraction of theelectron populations in the examined regions, cf. Eq. A.3. Subtracting from the totalcorrelated LI/DI indices the LI/DI in the Angyan form (form which first the naturalorbitals need to be created) yields the cumulant contribution to the indices.

The last two tables in the log file show the number of same-spin and opposite-spinpairs formed between the basins.

+-------------------+| Same-spin pairs |+-------------------+

Basin 1 2 3 4 5 6 7 8+--------------------------------------------------------------------------------+| 1 0.012 0.190 0.196 0.853 1.225 1.212 1.938 1.707 || 2 0.190 0.014 0.199 1.090 1.238 1.005 1.970 1.746 || 3 0.196 0.199 0.015 1.123 1.053 1.260 2.038 1.791 || 4 0.853 1.090 1.123 4.561 7.022 6.932 10.945 9.480 || 5 1.225 1.238 1.053 7.022 6.185 7.551 12.726 10.981 || 6 1.212 1.005 1.260 6.932 7.551 5.998 12.390 11.110 || 7 1.938 1.970 2.038 10.945 12.726 12.390 16.348 17.831 || 8 1.707 1.746 1.791 9.480 10.981 11.110 17.831 12.595 |+--------------------------------------------------------------------------------+| sum = 303.328 pairs |+--------------------------------------------------------------------------------+


The total number of same-spin pairs for oxazole is 2(18× 17)/2 = 306. The dif-ference between this value and the one found is again due to the cropping of thebasins. The same applies to the number of opposite-spin pair and the total numberof electron pairs.

A.6.2 DNOs / Fermi orbitals

The overlap integrals over basins used by the calculation of the domain-averagedFermi-hole (DAFH) [37] can be utilized to compute the domain natural orbitals andthe Fermi orbitals, for details cf. Secs 3.3.5 and 3.3.6 respectively. This analysis ofwhich was put forward by R. Ponec [38, 39]. In case of HF wavefunction the pairdensity can be written as:

ρ2(r1r2) =12

[ρ(r1)ρ(r2) − ∑

i, jφi(r1)φ

∗j (r1) φ j(r2)φ

∗i (r2)

](A.5)

Integrating the coordinate r2 over the region A (e.g., basin) yields:

∫A

ρ2(r1r2)dr2 =12

[ρ(r1)qA − ∑

i, jφi(r1)φ

∗j (r1) SA

ji

](A.6)

where qA is the average population (charge) in the region A and SAji the overlap

integrals of the MOs φ j and φi over the domain A. The domain-averaged Fermi-holeis the term:

DAFH = ∑i, j

φi(r1)φ∗j (r1) SA

ji (A.7)

The overlap matrix SAji is symmetric (possibly complex) and usually not diagonal.

The diagonal elements SAii are the orbital populations

∫A |φi|2dr within A, i.e., the

trace of S is the average population qA of the basin A. DAFH integrated over thewhole space yields the population qA in the basin A. Thus, it can be regarded askind of density distribution (different from the electron density) normalized to qA.Actually, (qAρ −DAFH) is proportional to twice the probability density to find anelectron at r when the other electron is confined to the region A. One can seek fora ‘natural’ representation of the DAFH, i.e., representation expressing DAFH bydiagonal terms only. This is done by diagonalization of the overlap matrix S.

The diagonalization of S leaves the trace unchanged creating the eigenvectors ϕi.This orbitals, ‘natural’ with respect to DAFH, have the property of being mutuallyorthogonal within the basin A (this is what the S diagonalization is about). The di-agonal elements Sii – the eigenvalues – describe the ‘contributions’ of the rotatedorbitals ϕi to the total population in basin A. The eigenvalues are the weighting fac-tors of the squared orbitals to yield the DAFH density (formally like the ‘naturaloccupations’). Each of the (spin) orbitals ϕi is of course occupied by single electron


– it is still the same HF wavefunction, but with rotated orbitals – recovering the totalelectron density asρ = ∑ϕ2

i . Additionally, the integration of the DAFH in the ‘nat-ural’ representation (using ϕi and the ‘natural occupations’ yields the localizationindex LI.

Remark A.8. Usually the above mentioned orbitals are termed the domain naturalorbitals (DNOs) and the eigenvalue the occupations.

/noindent For a ‘chemical’ interpretation one would like to have the orbitals lo-calized. The localization is accomplished with the isopycnic localization procedureof Cioslowski [16], which leaves the DAFH density unchanged and localizes theorbitals maximally into the basins (chosen for the localization procedure, cf. Sec.3.3.6.4. The resulting orbitals are called Fermi orbitals. It should be emphasizedthat after the isopycnic transformation the Fermi orbitals are no more orthogonalto each other and thus do not recover the electron density by the sum of squaredorbitals. The ‘FO occupations’ now describe the contributions of the Fermi orbitalsto the (domain averaged) hole density.

To compute a grid with the values for chosen Fermi orbital new wavefunctionfile with the Fermi orbitals must be created. Using the overlap file C3H3NO HF 6-31G.rho r.bsn.sij created with the control file on page 116 the Fermi orbitals can bedetermined for the atomic basin of the oxygen, which is the basin number 7:

::C3H3NO HF 6-31G parallel mesh:------------------------------------------------|sij_1=C3H3NO_HF_6-31G.g03rho_r.bsn.sij

Fermi_orbitals:-------------------------------------

using sij_1


select_basins_end


select_basins_end:-------------------------------------Fermi_orbitals_end

The overlap matrix for the oxygen basin is diagonalized and the resulting domainnatural orbitals are localized by the isopycnic procedure in all the available atomicbasins (at the same time rotating the orbital occupations). The resulting Fermi or-bitals are written to a wavefunction file. Additionally, the eigenvalues (i.e., the ro-tated occupations) are written also to the log file:


/==================\| Fermi orbitals |+------------------+

+------------------------------------------------------------------------------------+| reference electron located in basin No. || 7 O_1 || populated by 9.003411 electrons |+------------------------------------------------------------------------------------+| isopycnic localization procedure running over additional basins No. || 1 H_1 2 H_3 3 H_2 4 C_1 5 C_2 6 C_3 || 8 N_1 |+------------------------------------------------------------------------------------+

/===============\| eigenvalues |+---------------+

Orb Alpha occ Beta occ Total occ+------------------------------------------------------------------------------------+| 1 - - - - 1.99999701475 2.00 || 2 - - - - 0.00000015620 - || 3 - - - - 0.00026224851 - || 4 - - - - 0.00024868147 - || 5 - - - - 0.00000015611 - || 6 - - - - 1.97629828410 1.98 || 7 - - - - 0.00089674529 - || 8 - - - - 0.00002312824 - || 9 - - - - 1.52880477576 1.53 || 10 - - - - 0.00025378408 - || 11 - - - - 0.03086495873 - || 12 - - - - 1.75804964021 1.76 || 13 - - - - 0.03271104179 - || 14 - - - - 0.02716152938 - || 15 - - - - 1.51944598027 1.52 || 16 - - - - 0.05640965908 - || 17 - - - - 0.03138641913 - || 18 - - - - 0.04059688541 - |+------------------------------------------------------------------------------------+| - - - - 9.00341108850 8.79 |+------------------------------------------------------------------------------------+

The column with the total occupations shows the contributions of the Fermi orbitalsto the basin population rounded to 2 figures (contributions less than 0.05 are markedby period). It can be seen that the basin population of 9 electrons (i.e., representingnegatively charged atomic oxygen basin) is described mainly by 5 orbitals.

New wavefunction file C3H3NO HF 6-31G.g03.rho r.bsn.sij.fo-b7-O 1 is cre-ated with the 18 Fermi orbitals. Using this file the grids for the Fermi orbitals (cf.page 31 in Sec. 3.3.3) can be now computed for the 5 strongly occupied FOs:

::C3H3NO HF 6-31G:-----------------------------------------------------|wfn_1=C3H3NO_HF_6-31G.g03.rho_r.bsn.sij.fo-b7-O_1

compute:-----------------------------------------phi 1 sym=NONE alphaphi 6 sym=NONE alphaphi 9 sym=NONE alphaphi 12 sym=NONE alphaphi 15 sym=NONE alpha


X: Y: Z:----------------------------:origin= -6.0 -7.0 -4.0 INTERVALS:----------------------------------------:i-vector: 12.0 0.0 0.0 120j-vector: 0.0 14.0 0.0 140k-vector: 0.0 0.0 8.0 80----------------------------------------:

:-----------------------------------------compute_end

Observe that the origin command and 3 vectors were used for the grid-box defi-nition. Using the mesh command would not be helpful, because the box would bedetermined by the Fermi density.

The resulting strongly contributing Fermi orbitals are visualized in Fig. A.1 (rightcolumn). In the same figure the orbitals determined by the overlap matrix diagonal-ization alone (the DNOs, i.e., without the isopycnic localization) are shown in theleft column. In the bottom right diagram the atomic basin (blue colored) of the oxy-gen is depicted.

The 2 lowermost orbitals in Fig. A.1 are contributing with 2 electrons to the oxy-gen basin population. One of the orbitals can clearly be recognized as the oxygencore orbitals, the other as the oxygen ‘lone’ pair. The third orbital contributes by 1.78electrons and can be attributed also to the oxygen ‘lone’ pair. The isopycnic transfor-mation has almost no effect on these 3 orbitals (they are already properly localizedwithin the oxygen basin). In contrast, the two uppermost orbitals are changed bythe isopycnic localization. The resulting Fermi orbitals describe the two O-C bonds.They contribute by 1.52 to the hole density, showing the polarity of the bond (nonpolar bond would show Fermi orbital contribution close to 1).

The transformation to Fermi orbitals can be performed for open-shell systems aswell [40]. In case of restricted basis it can be chosen to analyze the Fermi orbitalsfor the total hole density (similar to ‘natural orbitals’), whereas for unrestricted basisonly Fermi orbitals for each spin channel separately are available. In DGrid this pos-sibilities are discriminated by the wavefunction descriptor ‘fo’ and ‘fso’ (meaningFermi orbitals and Fermi spin-orbitals, respectively).

Using overlap integrals evaluated for natural orbitals changes somewhat the pro-cedure. In this case the Fermi-hole can be approximated (pseudo-DAFH) by the sumincluding overlap integrals weighted by the square root of the product of natural oc-cupations√nin j, expression which conserves the basin charge [19]:

pDAFH = ∑i, j

φi(r1)φ∗j (r1)

√nin jSA

ji (A.8)


Fig. A.1 Strongly contributing Fermi orbitals for C3H3NO from the overlap diagonalization forthe oxygen atomic basin number 7 (blue colored). The numbers correspond to the contribution ofthe orbitals to the basin population (respectively to the hole density). Left: localization domainsfor the ‘natural’ orbitals (the isopycnic localization not active). Right: localization domains for theFermi orbitals. Only the two uppermost orbitals are affected by the isopycnic localization


A.6.3 Decomposition of the delocalization index

The the domain natural orbitals (DNOs), respectively the Fermi orbitals (FOs), fordetails cf. Secs 3.3.5 and 3.3.6 can be utilized for the decomposition of the de-localization index into orbital contributions [13]. In case of HF wavefunction thedelocalization index δ (A,B) is given by the integral of the exchange density overtwo domains A and B:

δ (A,B) =∫

Adr1

∫B

ρx(r1r2)dr2 = ∑i, j

∫A

φi(r1)φ∗j (r1)dr1

∫B

φ j(r2)φ∗i (r2)dr2

(A.9)Instead of the canonical orbitals φi the DNOs ϕi for the Fermi-hole averaged overthe domain A can be utilized to evaluate the delocalization index (DI):

δ (A,B) = ∑k

nAk

∫B|ϕk(r)|2dr (A.10)

where nAk is the occupation of the DNO ϕk (which is actually the contribution of the

DNO to the population of the domain A, i.e., nAk =

∫A |ϕk(r)|2dr). Thus, the DI can

b decomposed into DNO contributions:

δAk (B) = nA

k

∫B|ϕk(r)|2dr = nA

k pBk (A.11)

with the norm parts pBk of the DNO ϕk over the domain B. Such DI decomposition

can be performed with DGrid using the file with the corresponding norm parts fromthe DNO calculation, cf. Secs 3.3.5:


domain_natural_orbitals:-------------------------------------

using sij_1


select_basins_end

save norm_1:-------------------------------------domain_natural_orbitals_end

DI:-------------------------------------

using norm_1:-------------------------------------DI_end


Of course, it is also possible first compute the norm parts using a separate con-trol file with the domain natural orbitals evaluation section, yielding the fileC3H3NO HF 6-31G.g03.rho r.bsn.sij.dno-b7-O 1.norm, and utilize this file in nextcontrol file with the DI section. The result of the decomposition is written to the‘log’ file C3H3NO HF 6-31G.g03.rho r.bsn.sij.dno-b7-O 1.norm.DI:

+-------------------------------------------------------------+| DI contributions (%) of both-spin domain natural orbitals |+-------------------------------------------------------------+

DNO\Bsn 1 2 3 4 5 6 7 8 norm sum+------------------------------------------------------------------------------------------+| 1 - - - - - - : 25.5 : - | 2.0000 || 4 - - 0.2 - - - : - : 0.2 | 1.9946 || 6 4.5 5.3 4.0 1.5 1.2 1.4 : 24.9 : 1.1 | 1.9942 || 7 - - 0.2 - 0.3 - : - : 0.4 | 1.9958 || 8 - - 0.1 - - - : - : - | 1.9839 || 9 24.2 23.4 19.3 40.0 7.2 42.2 : 13.2 : 7.8 | 1.9977 || 10 - - 0.1 - - - : - : - | 1.9988 || 11 29.0 34.5 1.8 2.1 7.1 2.6 : - : 4.5 | 1.9836 || 12 5.4 4.0 12.6 18.8 26.5 15.1 : 19.7 : 24.7 | 1.9903 || 13 5.0 9.5 4.8 0.5 1.7 1.2 : - : 1.0 | 1.9864 || 14 15.7 0.8 11.8 1.2 9.3 1.1 : - : 6.9 | 1.9879 || 15 6.2 3.9 33.5 30.4 14.3 29.3 : 16.6 : 19.2 | 1.9961 || 16 0.2 0.2 2.7 0.7 10.8 0.6 : - : 12.1 | 1.9786 || 17 7.9 15.6 3.3 1.3 3.0 1.2 : - : 11.3 | 1.9858 || 18 1.9 2.8 5.7 3.4 18.5 5.3 : - : 10.7 | 1.9727 |+------------------------------------------------------------------------------------------+| DI | 0.046 0.043 0.012 0.910 0.143 0.927 7.834 0.216 | 29.8464 |+------------------------------------------------------------------------------------------+

The oxygen QTAIM basin nr. 7 is highlighted in the above table by colons, becausethe data correspond to the localization odex decomposition. The basin number 7 rep-resent the oxygen atom, whereas the basins number 4 and 6 represent the carbondatoms connected to the oxygen. From the above table it can be seen that the DI be-tween the basins 7 (oxygen) and 4 (carbon), respectively 7 (oxygen) and 6 (carbon),showing the bond order of 0.9 for the two C-O bonds, is given mainly by roughly 2contribution from the DNOs number 9, 15, and somewhat lower contribution of thepi-like DNO number 12, cf the DNOs in Fig. A.1.

Using instead of the DNO the Fermi orbitals (i.e., DNO localized by the isopy-cnic transformation) shows somewhat different picture for the DI decomposition.From Fig. A.1 it is clear that the two DNOs number 9 (occupation 1.44) and 15 (oc-cupation 1.61) participate equally to each of the two C-O bonds. The localizationprocedure creates two FOs, each serving mainly one C-O bond (FOs with occupa-tion 1.52 and 1.53). The control file:

::C3H3NO HF 6-31G:------------------------------------------------|norm_1=C3H3NO_HF_6-31G.g03.rho_r.bsn.sij.fo-b7-O_1.norm

DI:-------------------------------------

using norm_1:-------------------------------------DI_end

yields the decomposition of the DIs into the FO contributions:


+----------------------------------------------------+| DI contributions (%) of both-spin Fermi orbitals |+----------------------------------------------------+

FO\Bsn 1 2 3 4 5 6 7 8 norm sum+------------------------------------------------------------------------------------------+| 1 - - - - - - : 25.5 : - | 2.0000 || 3 - - 0.2 - - - : - : 0.2 | 1.9915 || 4 - - - 0.1 - - : - : - | 1.9997 || 6 4.5 5.2 4.2 1.5 1.3 1.4 : 24.9 : 1.2 | 1.9942 || 7 - - 0.4 - 0.4 - : - : 0.5 | 1.9949 || 8 - - 0.2 - - - : - : - | 1.9793 || 9 24.4 5.6 1.6 67.4 5.0 2.8 : 15.0 : 22.4 | 1.9969 || 10 - - 0.1 - - 0.1 : - : - | 1.9989 || 11 1.2 56.4 0.2 0.1 2.1 3.3 : - : 0.4 | 1.9782 || 12 5.4 4.0 12.6 18.8 26.5 15.1 : 19.7 : 24.7 | 1.9903 || 13 55.2 1.7 1.1 3.4 0.6 0.1 : - : 1.9 | 1.9778 || 14 - 2.2 20.1 - 16.6 2.6 : - : 0.8 | 1.9944 || 15 6.0 21.7 50.9 3.1 16.4 68.8 : 14.8 : 4.5 | 1.9969 || 16 0.8 2.7 8.3 1.2 28.7 5.4 : - : 1.6 | 1.9714 || 17 1.2 0.2 0.3 1.5 1.7 - : - : 20.6 | 1.9931 || 18 1.3 0.3 - 2.9 0.7 0.5 : - : 21.2 | 1.9784 |+------------------------------------------------------------------------------------------+| DI | 0.046 0.043 0.012 0.910 0.143 0.927 7.834 0.216 | 31.8359 |+------------------------------------------------------------------------------------------+

Now the DI between the oxygen and carbon (basin number 4, respectively 6) is duemainly to the contribution of single Fermi orbital. The electron sharing with basinnumber 4 is almost to 70% given by the contribution of FO number 9, whereas FOnumber 15 serves the sharing with the basin number 6.

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Index

ADFTAPE21, 5wavefunction file, 5, 99

AO contributionsprinting, 103

basinintersections

bond polarity, 65reduction, 21

basin file, 91, 93example, 91header, 92

basin section, 17–25compactify, 21crop, 21ELID core, 22localization domain, 23save, 23symmetrize in cell, 23top, 24using, 25

CBO section, 25–27basin set, 26decomp basin, 26select basins, 26using, 27

Checkpoint file, 100complete utility, 47, 76, 81, 104compute section, 27–48

conditional fields, 30contributions, 34cp, 35curv decomp, 36ev projection, 38field descriptors, 28

format, 40mesh, 40occupation, 41orbitals, 31origin, 43overlap matrix, 44par flt, 44par int, 44point, 44ref point, 45save, 45select orbitals, 46slices, 47space, 47using, 48

control file, 1data assignment, 12–15evaluation section, 16–85example, 9, 11, 17, 25, 28, 30, 34, 36–38,

46, 48, 50, 53, 55, 60, 62, 64, 69, 71, 74,77, 82, 95, 104, 106, 107, 116, 117, 121,122, 125, 126

grid calculation, 9grid field, 9grid field calculation, 28job sequence, 10–12title, 10

conversionQM files into wavefunction file, 6QM output, of, 99

delocalization indices, 117descriptor

default values for type, spin pair, 28file extension

basis, 6numbering, 90

133

134 Index

pair spin, 90space rep, 89spin, 89type, 90

DGridcitation, 2installation, 2–4

Linux, 3Windows, 3

web page, 2dgrid para, 97–98

completion, 104DI section, 48–50

basin set, 49select basins, 49using, 50

domain natural orbitals, 120DI decomposition, 125

domain natural orbitals section, 50–52pair, 51save, 51select basins domain, 52using, 52

ELI-Dorbital contributions, 30

Euler characteristic, 78example

basin file, 91control file, 9, 11, 17, 25, 28, 30, 34, 36–38,

46, 48, 50, 53, 55, 60, 62, 64, 69, 71, 74,77, 82, 95, 104, 106, 107, 116, 117, 121,122, 125, 126

field file, 87wavefunction file, 7

Fermi orbitals, 120DI decomposition, 125natural orbitals, 123

Fermi orbitals section, 52–55pair, 53save, 54select basins domain, 54select basins isopyc, 54using, 55

FHI-aimsevaluation, 113output files, 113wavefunction file, 5

fieldconditional

reference point, 30field file, 87file descriptor

basin 1, 13cps 1, 13field 1, 13norm 1, 14rfn 1, 14sij 1, 15str 1, 15wfn 1, 15

file systembasin file, 93

fluctuation, 117format

conversionQM packages output, 5

input files, 1rules, 1

molden, 5TURBOMOLE, 5

property fileDGrid, 90

wavefunction fileDGrid, 6

GAMESSwavefunction file, 5, 100

GaussianCheckpoint file, 5

conversion, 6WFN file, 5

gradientcomponents, 90

gridbox definition

mesh, 40vectors, 43

definitionmesh, 40

meshrecommended size, 43, 78

parallel calculation, 95–98dgrid para, 97

source function, 104grid field file, 91

header, 88grid file

naming, 89–91grid files, 87

basin file, 91field file, 87

Hessiancomponents, 90

icl graph section, 55–59

Index 135

rings, 59saddles, 59using, 59

integrate section, 59–62core charge, 61over, 61using, 61

intersections section, 62–64save, 63using, 63with, 63

isect analyze section, 64–68basin set, 66core charge, 66integrate, 67select basins, 68using, 68

keywordbasin, 17–25

compactify, 21crop, 21ELID core, 22localization domain, 23save, 23symmetrize in cell, 23top, 24using, 25

basin 1, 13CBO, 25–27

basin set, 26decomp basin, 26select basins, 26using, 27

compute, 10, 27–48conditional fields, 30contributions, 34cp, 35curv decomp, 36ev projection, 38field descriptors, 28format, 40mesh, 40occupation, 41orbitals, 31origin, 43overlap matrix, 44par flt, 44par int, 44point, 44ref point, 45save, 45select orbitals, 46slices, 47

space, 28, 47using, 48

console, 28core charge, 60cps 1, 13data descriptors, 13–15DI, 48–50

basin set, 49select basins, 49using, 50

domain natural orbitals, 50–52pair, 51save, 51select basins domain, 52using, 52

Fermi orbitals, 52–55pair, 53save, 54select basins domain, 54select basins isopyc, 54using, 55

field 1, 13icl graph, 55–59

rings, 59saddles, 59using, 59

integrate, 59–62core charge, 61over, 61using, 61

intersections, 62–64save, 63using, 63with, 63

isect analyze, 64–68basin set, 66core charge, 66integrate, 67select basins, 68using, 68

localized natural orbitals, 68–71pair, 69save, 70select basins isopyc, 70using, 70

norm 1, 14overlap, 71–73

over, 72precision, 72save, 72select basins, 72select orbitals, 73using, 73

refine, 74–76

136 Index

over, 74precision, 75save, 75select basins, 75slices, 76using, 76

rfn 1, 14search, 76–82

all cps, 79attractors, 79crop, 79ELID core, 79minima, 79rings, 79saddles, 79save, 81slices, 81using, 81

sij 1, 15space, 89str 1, 15superbasin, 82

save, 85using, 85

wfn 1, 10, 15

LMTOheader order, 89

localized natural orbitals section, 68–71pair, 69save, 70select basins isopyc, 70using, 70

MO expansion, 103virtual orbitals, 103

MOLCASwavefunction file, 5

molden file, 101molecular graph, 56MOLPRO

wavefunction file, 5momentum space

orbitals, 32

natural orbitalsFermi orbitals, 123

orbitals, 43grid calculation, 31momentum space, 32polar form, 32virtual, 6

ELI-D, 30

output filebasin evaluation

example, 18QM packages, 5

overlap integrals, 115overlap section, 71–73

over, 72precision, 72save, 72select basins, 72select orbitals, 73using, 73

parallel calculation, 95dgrid para, 97

pELI-D, 30property file

vector field, 89

refine section, 74–76over, 74precision, 75save, 75select basins, 75slices, 76using, 76

refine superbasin, 85refinement

in basin, 106remarks

2-matrix, 115basins

cropping, 19, 22descriptors, 21overlap integrals, 116, 118visualization, 93

critical points, 58conversion to STR format, 78

Cube format, 40DAFH

DNO/FO, 121descriptors, 28dgrid para, 97field

components, 90fluctuation

correlated wavefunction, 119cumulant part, 119natural orbitals, 119

GAMESSwavefunction file, 100WFN file, 100

integrationbasin mesh, 117

Index 137

Gauss functions, 116mesh size, 57

momentum space, 48occupation block

ELI, 43mesh, 43

orbitals, 43, 89overlap integrals, 41, 61pELI-D, 30

virtual orbitals, 30, 43STR format

conversion from critical points, 78superbasins, 83wavefunction file access, 88

search section, 76–82all cps, 79attractors, 79crop, 79ELID core, 79Euler characteristic, 78minima, 79rings, 79saddles, 79save, 81slices, 81using, 81

source functionevaluation of, 104

structure file, 102superbasin

refine, 85superbasin section, 82

save, 85using, 85

tablebasin section, 17CBO section, 25compute section, 27

fields, 28DI section, 48domain natural orbitals section, 50evaluation sections, 16Fermi orbitals section, 52icl graph section, 55

integrate section, 59intersections section, 62isect analyze section, 64localized natural orbitals section, 69overlap section, 71refine section, 74search section, 77superbasin section, 82

TAPE21, 99TURBOMOLE

molden format, 5wavefunction file, 5

using Taylor, 35utility

crop, 111operation on single grid, 108operation on two grids, 110syntax, 99

wavefunction file, 1, 5–8ADF

localized orbitals, 100TAPE21, 99

conversion into, 6conversion of old format, 102creation from QM packages, 99format

DGrid, 6example, 7

GAMESS, 100Gaussian

Checkpoint file, 100WFN file, 101

moldennatural orbitals, 102

molden file, 101normalization

TURBOMOLE, 5QM output, from, 99Turbomole

molden file, 101virtual orbitals, 8, 101

WFN file, 101ROHF calculation, 101

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