Self Consistent Polarization Electronic Structure Gregory K. Schenter Pacific Northwest National Laboratory

“Chemical Dynamics: Challenges and Approaches” Institute for Mathematics and Its Applications, University of Minnesota January 12-16, 2009

Funding: DOE BES Chemical Sciences

Self Consistent Polarization (SCP) Electronic Structure = (NDDO, DFT, HF, …)

SCP-NDDO: Danny Chang (US EPA) Bruce Garrett (PNNL) JCP 128, 164111 (2008).

SCP-DFT (Water): Chris Mundy (PNNL) Garold Murdachaew (PNNL)

SCP-DFT (Argon): J. Ilja Siepmann (U. Minn.) Katie Maerzke (U. Minn.) JPCA (in press)

Future development in CP2K: Juerg Hutter (Zurich) Teo Liano (Zurich) Joost Vandevondele (Zurich)

 Characterize molecular interaction

 Represent molecular interaction  Construct a Hamiltonian  Statistical sampling and

dynamics of an ensemble  Predict and understand

properties

Approach: Molecular Simulation

€

VBO RN( )( )

€

H =Pi2

2mi

+V R N( )( )i=1

N

∑

€

Tr e−βH[ ], R N( ) t( ),P N( ) t( )( )

€

A , A t( )B 0( )

  Hard Sphere   Lennard-Jones (LJ)   Sphere + Mulitipole   Rigid Molecule + LJ + q   Rigid + LJ + q + POL   Flexible + LJ + q + POL   … + Dissociating   Semiempirical electronic

structure   Direct Density

Functional Theory   Direct Ab Initio

electronic structure   Interpolation

  Classical Molecular Dynamics

  Classical Monte Carlo   Quantum Mechanical,

Feynman Path Integral   Approximate Quantum

Mechanical Dynamics   Wave packets   Semiclassical

  Reaction path analysis   Ensembles of reaction

paths

“Potential” Description of Molecular

Interaction

“Simulation” Configuration sampling

(phase space) Statistical Mechanics

  Structure   Thermodynamic   Kinetic   Dynamic   Transport coefficients   Rare events   Reaction mechanisms   Reaction Rates

“Properties” + =

Mixing and Matching Representations of Molecular Interaction is Desired. It is practical, more accurate, more efficient.  QM/MM  Semiempirical NDDO = HF electronic structure

with empirical terms replacing “real” terms.  MM/POL  MM/1e- QM, e- QM/H+ QM (NEO)  Multilevel, Layered Electronic Structure  DFTB, Harris functional, Empirical Valence Bond  Duel level dyamics, SRP  SCP-NDDO, SCP-DFT  … What is the proper way to do this?

Hartree- Fock Electronic Structure

Semi-Empirical Electronic Structure (NDDO)

MM/POL Empirical Molecular Potentials With Polarization

Density Functional Theory

Common Elements: Charge Density Responding to Nuclear Configuration

The distinction between these approaches can become blurred.

A key component is the dependence of the energy of the system on the details of the charge density, described by an energy functional:

Coulomb is King

€

E ρ[ ] = dr∫ dr 'ρ r( )ρ r'( )r − r '

€

E ρ[ ]

Quantum mechanics can be implemented in terms of variational principles.

€

δE ρ[ ]δρ

= 0 Subject to constraints.

Mathematical Challenge: Partitioning Charge Density and Corresponding Energy Functional Space

€

E ρ1 + ρ2[ ] = E1 ρ1[ ] + Eapprox ρ1,ρ2[ ] + E2 ρ2[ ]

€

ρ1

€

ρ2

Self-consistent Polarization (SCP)-NDDO   Use a polarization density matrix pµν {s,x,y,z}   Charge polarization pss   Dipole polarization psx, psy, psz   Quadrupole polarization {pxx, pyy, pzz, pxy, pxz, pyz}   Dispersion via 2nd-order perturbation theory

Semiempirical SCF methods + Ability to treat the formation and breaking of chemical bonds

– Poor description of H-bonds and weak electrostatic complexes

Semiempirical MOs

Empirical methods – Poor description of chemical reactions (e.g., bond-breaking and formation) + Ability to add missing elements of H-bonding using classical electrostatic models

Classical Electrostatics

Representation of The Charge Density

€

ρA r( ) = ZAδµν ,ss − Pµν ;AAT( )φµ r( )φν r( )

µν

∑ + MAl( ) ⋅ ∇A

l( )

l∑ f r − rA( )

From Semiempirical NDDO: Localized Basis Set, Core + Density Matrix

From Empirical Potentials: Multipole Polarizability

Screened Coulomb Interactions

€

drd ′ r ρA r( )ρB ′ r ( )

r − ′ r ∫ =

MA0( ) + MA

1( ) ⋅ ∇A + MA2( ) :∇A∇A +( ) MB

0( ) + MB1( ) ⋅ ∇B + MB

2( ) :∇B∇B +( ) s rAB( )rAB€

A ≠ B

€

s rAB( )rAB

=1

rAB2 + a2

€

s rAB( )rAB

=1rAB

1− 1+12a

e−arAB

Klopman-Ohno: Slater:

Self-Interaction

€

A = B

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Eself = Pµµ;AAT Uµ

A

µ

∑

A

∑ +12

Pµν ;AAT Gµν ;µ ν

A Pµ ν ;AAT − Pµν ;AA

γ Hµν ;µ ν A Pµ ν ;AA

γ

γ

∑

µ ν

∑µν

∑

NDDO Atomic Hamiltonian

€

+12

MAl( ) ⋅

l, ′ l ∑

A∑ 1

a l , ′ l ( ) ⋅MA′ l ( )

Generalized Multipole Polarization (linear response)

Bonding

€

Eres =12

Pµλ ;ABT (βµ

A + βλB )Sµλ ;AB

µλ

∑B≠A∑

A∑

€

Eex = −12

Pµλ ;ABγ Pνσ ;AB

γ µAνA( | λBσB)γ

∑λσ

∑B≠A∑

µν

∑A∑

NDDO Resonance and Exchange

SCP-NDDO provided an accurate description of water clusters, (H2O)n

We have extended the ideas of SCP-NDDO to DFT and the condensed phase

"Self-consistent polarization neglect of diatomic differential overlap: Application to water clusters," Daniel T. Chang, Gregory K. Schenter, and Bruce C. Garrett, J. Chem. Phys. 128, 164111 (2008).

TTM2-R: Burnham and Xantheas, JCP 115, 15000 (2002)

Energy

(H2O)2 (H2O)3 (H2O)4 (H2O)5 (H2O)6

1 2 1 2

SCP-DFT

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EDFT ρDFT[ ] = T ρDFT[ ] + EH ρDFT[ ] + EXC ρDFT[ ]

€

ESCP−DFT ρDFT ,ρSCP[ ] = T ρDFT[ ] + EH ρDFT + ρSCP[ ] + EXC ρDFT[ ] + ESE ρSCP[ ]

€

EH ρ[ ] = dr∫ dr 'ρ r( )ρ r'( )r − r '

SCP approach to improving DFT has its origins in linear response theory

CJ Mundy et al JCP 123, 074108 (2005) •  “Parameter free” based on pure DFT

CJ Mundy et al JCP 117, 1416 (2002) •  “Parameter free” based on empirical potential

DFT/

DFT-DFT

SCP-SCP

DFT-SCP

Coulomb Integrals:

Screening, exclude same site interaction

Currently:

or

Self-consistent polarization (SCP)-DFT: •  Polarization functions provide “correction” •  Only a single parameter, aα •  Polarization will yield a straight forward path to a self-consistent determination of dispersion

Subject to wavefunction orthonormality constraints Gives conventional Density Functional Theory

SCF SCP Procedure

We obtain a self-consistent formula for multipole dispersion by considering a quantum mechanical

interaction between auxiliary densities

We obtain the “Generalized London” formula with the introduction of two additional parameters

Our perturbation:

2nd order expression

€

Edisp = α β( ) fαβ α β ( ) fα β

0 ˆ c α k 0 ˆ c β l k ˆ c α 0 l ˆ c β

0E0

α + E0β − Ek

α − Elβ

l≠0∑

k≠0∑

Casimir-Polder:

SCP-DFT parametrization procedure for water

1.  Use CP2K with BLYP/GTH pseudopotential/TZV2P-MOLOPT basis sets on O and H atoms (low BSSE, BSTE)

2.  Use single auxiliary basis function of p symmetry on O and H atoms

3.  O, H atoms: set ap (same for O and H currently) to “reasonable” value

4.  Molecular cohesion energies of (H2O)n clusters with optimized geometries: choose Ip (same for O and H currently) to recover the MP2 and TTM-2F cohesion energies of clusters, n=2-16

5.  (later we compared “rigidified” clusters to those from CC-pol-8s potential)

G Murdachaew, CJ Mundy & GK Schenter (in preparation).

CP2K: Our state-of-the-art computational tool   Features

  KS-DFT (OR classical force fields)   Ab initio molecular dynamics (Born-Oppenheimer)   Dual basis (Gaussian + PW) allows for good scaling on many processors   GTH pseudopotentials (OR all e- using Gaussian augmented plane waves, GAPW)   QM/MM   Semi-empirical methods recently implemented (MNDO, NDDO, PM3, . . . )   XAS (x-ray absorption spectra for e.g. studies of hydrogen bond network in water, core excitations)   SIC (self-interaction correction to correct exchange in DFT; sometimes needed for oxides)   Metadynamics for rare events, NEB for transition state searches, . . .   Classical and quantum dynamics of nuclei   Our recent contributions: SCP-NDDO and SCP-DFT

  CP2K (http://cp2k.berlios.de, code freely available); main developers are   Juerg Hutter (U Zurich)   Teodoro Laino (IBM Zurich)   Joost VandeVondele (U Zurich)   Matthias Krack (ETH Zurich)   Axel Kohlmeyer (U Penn)   Chris Mundy (PNNL) (GKS and GM work with him to make improvements in SCP-DFT)   William Kuo (LLNL)

  Code allows for readability, modification, and rapid development   Written in Fortran 95 with modular programming philosophy

(modules, user-defined types, dynamic memory and pointers, excellent portability, . . . )   Highly efficient and fully parallelized

  Discussion forum exists (http://groups.google.com/group/cp2k)

J VandeVondele M Krack F Mohamed M Parrinello T Chassaing & J Hutter ComputPhysCommun 167 103 (2005).

R Bukowski et al Science 315 1249 (2007). CC-POL = Empirical Potential fit to CCSD(T) electronic structure SS Xantheas, CJ Burnham, & RJ Harrison, JChemPhys 116 1493 (2002). = MP2/CBS

R Bukowski et al Science 315 1249 (2007).

R Bukowski et al Science 315 1249 (2007).

R Bukowski et al Science 315 1249 (2007).

ΔHvap classical (kcal/mol)

DFT -8.80

SCP-DFT -10.20

CC-pol+NB -10.89 (rigid)

QM correction (est.) 0.86

Exp -9.92 ±0.3 R Bukowski et al Science 315 1249 (2007).

(H2O)2

(H2O)3

(H2O)4

(H2O)5

(H2O)2

(H2O)3

(H2O)4

(H2O)5

Argon

Try to be more systematic about parameterization:

 Solid  Liquid  Gas

KA Maerzke, G Murdachaew, CJ Mundy, GK Schenter, JI Siepmann, J Phys Chem A (in press).

SCP-DFT parametrization procedure for argon

1.  Use CP2K with BLYP/GTH pseudopotential/DZVP basis set on Ar atom

2.  Use single auxiliary basis function of p symmetry on Ar atom 3.  Ar atom: use finite-field approach with charges to fit ap so as

to reproduce known polarizability 4.  Ar dimer: fit Ip to recover the Aziz interaction energy (at 3.6

Angstrom)

KA Maerzke, G Murdachaew, CJ Mundy, GK Schenter, JI Siepmann, JPhysChemA (in press).

Ar2 intermolecular potential

RA Aziz JChem.Phys 99 4518 (1993). K Patkowski GM CM Fou & K Szalewicz MolPhys 103 2031 (2005).

Ar2 intermolecular potential: the tail

Second virial coefficients of argon

JH Dymond & EB Smith The Virial Coefficients of Pure Gases and Mixtures: A Critical Compilation; OUP: Oxford 1980.

We are working on fixing the tail and that will improve virials

Cohesion energies per atom of argon clusters

FY Naumkin & DJ Wales MolPhys 96 1295 (1999). DJ Wales et al The Cambridge Cluster Database http://www-wales.ch.cam.ac.uk/CCD.html.

Aziz is 2b only (Wales calculation) LJ is empirical: effectively includes man-body effects SCP-DFT includes many-body effects

Structure of liquid argon at 85 K

JL Yarnell et al PhysRevA 7 2130 (1973).

Cohesion energy per atom of solid fcc argon

C Tessier et al Physica (Amsterdam) 113A 286 (1982). VF Lotrich & K Szalewicz PhysRevLett 79 1301 (1997).

Collaborators: SCP-NDDO: Danny Chang (US EPA) Bruce Garrett (PNNL) JCP 128, 164111 (2008).

SCP-DFT (Water): Chris Mundy (PNNL) Garold Murdachaew (PNNL)

SCP-DFT (Argon): J. Ilja Siepmann (U. Minn.) Katie Maerzke (U. Minn.) JPCA (in press)

Future development in CP2K: Juerg Hutter (Zurich) Teo Liano (Zurich) Joost Vandevondele (Zurich)

Funding: DOE BES Chemical Sciences

Summary:  We have enhanced Semiempirical (NDDO) and Density Functional Theory (DFT) electronic structure using Molecular Mechanics-like polarization. (SCP)

 We are still exploring how to make this approach more systematic.