1 computerexperiment*12:electroncorrelation*in*the*helium...
TRANSCRIPT
1 Computer Experiment 12: Electron Correlation in the Helium
Atom
1.1 Background Electrons move in a molecule in a very complicated way because they experience
attraction from the nuclei but repulsion from the other electrons. This repulsion
exists due to the classical Coulomb interaction of two equally charged particles plus
a pure quantum effect – an exchange ‘interaction’. The exchange contribution arises
due to the fact that two electrons can not exist in one point of space with equally
oriented spin (Pauli exclusion principle). Therefore, the movement of electrons is
not independent and the movement of one electron influence the movements of all
others.
1.1.1 Hartree-‐Fock and Electron Correlation The Hartree-‐Fock method describes the movement of an electron in the field of the
nuclei and the average field of all other electrons. This is a surprisingly good
approximation and typically yields more than 99.5% of the correct total energy of a
molecule. However, in this method the correlation of the electron (their
“instantaneous” interaction) is not properly accounted for and consequently, the
remaining errors of the HF approximation are still very large on a chemical energy
scale (several hundred kcal/mol). Thus, the HF method can not provide the exact
total energy and is usually referred to an “uncorrelated approach”.1 As a result, the
correlation energy is defined as the difference between the exact total energy and
the energy of the Hartree-‐Fock calculation:
Ecorr= E
exact!E
HF
(1) A big challenge for theoretical chemistry is to calculate this correlation energy to
high precision. Unfortunately, there is always a second source of error in the
calculations which stems from the need to introduce a finite basis set in order to
1 This is not entirely fair since the HF method properly includes the effects of the “Fermi correlation” (thus, the exchange correction arising from the Pauli principle).
represent the one-‐electron MOs on which the calculations are based. It is obvious
that the more basis functions we bring in, the better the variational principle will be
able to model the shape of the “real” HF MOs. At the same time as the orbitals
approach the true HF orbitals in the limit of an infinite basis set, the HF energy
approaches the basis set HF limit.2 However, in practice the basis set limit can not be
reached since the number of basis functions that can be included in the calculations
is limited by the computational time.3 At the same time the error of introducing a
non-‐complete basis set – the basis set truncation error – should be made as small as
possible. In the limit of vanishing error one obtains results in the basis set limit.
Consider for example a series of HF calculations on the Ne atom with basis sets of
increasing size (energies are given in Eh):4
One can see that the HF limit can be approached closely with the sufficiently large
basis sets and for the chemical accuracy of ~1 mEh (with respect to the HF-‐limit!)
can be reached for the TZV and QZV basis sets which are already small enough such
that they can be used in molecular calculations. Note however, that even in the HF-‐
limit the total energy is still far from the exact energy due to the missing correlation.
2 Mathematically speaking, the basis set should approach “completeness”. 3 In fact, the computing time increases (formally) with the fourth power of the basis set size (for HF and worse for more accurate methods). However, there are also issues concerning the finite accuracy with which numbers can be represented by a digital computer which also puts constraints concerning numerical stability on the size of the basis set that can be used. 4 The details of how these basis sets come to their names is not important in the context of this course. They are simply basis sets of increasing size and accuracy.
STO-3G : -126.60453 6s 3p (33/3) 3-21G : -127.80382 6s 3p (321/21) SV : -128.37641 7s 4p (511/31) TZV : -128.54149 11s 6p (62111/411) QZV : -128.54685 15s 9p (8211111/6111) Partridge-1 : -128.54695 14s 9p (uncontracted) Partridge-2 : -128.54708 16s11p (uncontracted) Partridge-3 : -128.54709 18s13p (uncontracted) HF-limit : -128.54710 Exact Energy (exp) : -129.056
1.1.2 Calculation of the Correlation Energy In order to recover a significant fraction of the missing correlation calculation, we
can employ a more complicated expansion of the wavefunction by adding terms
where 1,2,…, N electrons in the HF Slater determinant (i,j,k) are replaced by virtual
orbitals (a,b,c) (excited configurations). Thus, the expansion of the exact N-‐electron
wavefunction can be written in the form:
! =C0"
HF+ C
i
a"i
a
a
#i
#Singles
! "#### $####+
12!
$
%&&&&
'
())))
2
Cij
ab"ij
ab
a,b#
i,j#
Doubles
! "####### $#######+
13!
$
%&&&&
'
())))
2
Cijk
abc"ijk
abc
a,b,c#
i,j ,k#
Triples
! "######## $########+% (2)
This way of writing the exact N-‐electron wavefunction is known as the “full-‐CI”
method. The number of terms in this configuration interaction method expansion is
extremely large and – in fact – grows in factorial way with the number of electrons
and basis functions. Hence, the method is only suitable for benchmark calculations
on very small systems. Using todays best programs and computers, CI calculations
can be (and have been) done with up to ~1010 terms in the determinantal expansion
within the restrictions of the employed one-‐particle basis set. The N-‐particle
wavefunction is an object of extreme complexity and contains all information that
can be determined on the quantum system.
1.1.3 Reduced Density Matrices In practice, we need to know much less about the N-‐particle wavefunction as long as
we are only interested in the calculation of energies and observables. In fact, the
Born-‐Oppenheimer Hamilton operator only contains one and two-‐body terms. Thus,
the exact energy can already be calculated from the so-‐called reduced one-‐particle
and two-‐particle density matrices. They are defined as:
! 1( ) x
1, !x
1( ) = N "* !x1,...,x
N( )" x1,...,x
N( )dx2...dx
N#
(3)
! 2( ) x
1,x
2; !x
1, !x
2( ) = N N "1( ) #* !x1, !x
2,...,x
N( )# x1,x
2...,x
N( )dx3...dx
N$ (4)
These quantities are very important in quantum chemistry since they contain all the
necessary information for calculating the total energy and all kinds of properties of
the system. They called here “matrices” assuming that two arguments x1 and !x1
serve as continuous „indices“ and take the place of the discrete integer indices i and
j. These quantities represent generalizations of the electron density function
! 1( ) x
1( ) = N !" x1,...,x
N( )! x1,...,x
N( )dx2...dx
N#
(5)
If we integrate over the spin of the first electron, we obtain the spin-‐free first-‐order
density function:
P 1( ) r
1( ) = ! 1( ) r1s
1( )! ds1
(6)
which represent the probability to find an electron at the space part r1 in the
infinitely small volume dr1. The electron density is observable and can be measured
by the X-‐ray crystallography.
Similar, the two-‐electron density matrix is a generalization of the of two-‐body
density function:
! 2( ) x
1,x
2( ) = N N !1( ) "* x1,x
2,...,x
N( )" x1,x
2...,x
N( )dx3...dx
N#
(7)
Integrating over spins of the first and the second electrons, we obtain:
P 2( ) r
1,r
2( ) = ! 2( ) r1s
1,r
2s
2( )!! ds1ds
2
(8)
The last expression defines the probability to find one electron of an arbitrary spin
at the point r1 and volume dr1 with any spin and the second at the point r2
and in
the volume dr2 while the remaining N-‐2 electrons can have arbitrary positions and
spins. Density matrices allow a very useful statistical approach to the analysis of the
electron correlation which is not related to the total energy.
In statistics two variables x and y with probability distributions f(x) and g(y) are
uncorrelated if the joint probability h(x,y) is given by:
h x,y( ) = f x( )g y( )
(9)
For our case this implies that the pair density should be a simple product of the one
particle densities.
P 2( ) r
1,r
2( ) = N N !1( ) 1N
P 1( ) r1( )
"#$$
%$$
&'$$
($$
1N
P 1( ) r2( )
"#$$
%$$
&'$$
($$=
N !1N
P 1( ) r1( )P 1( ) r
2( ) (10)
1.1.4 Hole Functions In order to obtain a parameter of the correlation “strength” we can introduce a
quantity which defines the deviation of our density from the uncorrelated case:
P 2( )!! r
1,r
2( ) = P 1( )! r1( )P 1( )! r
2( ) 1+ f !! r1,r
2( ){ }
(11)
P 2( )!" r
1,r
2( ) = P 1( )! r1( )P 1( )" r
2( ) 1+ f !" r1,r
2( ){ }
(12)
The f -‐functions called the parallel and antiparallel spin correlation functions.
The probability of finding the first electron at r1 with the condition that the second
electron is located at r2 with the same spin is defined by
P 2( )!! r1,r
2( )P 1( )! r
2( ). The difference
between the last quantity and the uncorrelated probability of finding the first
electron at r1
hX
r1,r
2( ) =P 2( )!! r
1,r
2( )P 1( )! r
2( )!P 1( )! r
1( ) = P 1( )! r1( ) f !! r
1,r
2( )
(13)
is called the Fermi hole. In the same way one can define similar quantity for
antiparallel spins:
hC
r1,r
2( ) =P 2( )!" r
1,r
2( )P 1( )! r
2( )!P 1( )! r
1( ) = P 1( )! r1( ) f !" r
1,r
2( )
(14)
which is called the Coloumb hole. The idea of the Fermi and Coulomb holes allow a
very pictorial and intuitive approach for understanding of how exchange and
Coulomb interaction affect the electron distribution in an atom or molecule. One can
image an electron digging a hole around so that the probability to find another
electron nearby is diminished. The form of the Fermi hole is generally known only
for a few cases such as a uniform electronic gas, but at the short electronic distances
the hole behaves as r122 . The shape of the Coulomb hole function is complicated and
not well known. However, it is known that at the small electronic distances the hole
function has a cusp.
1.1.5 Natural Orbitals The one-‐electron density matrix can be written for a complete set of spin-‐orbitals
as:
! 1( ) x
1, !x
1( ) = !pq
1( )"p
x1( )"q
* !x1( )
p,q"
(15)
Since a unitary transformation applied for a set of spin-‐orbitals leaves the wave
function unchanged, we can transform !
pq
1( ) to the diagonal form:
U+! 1( )U = n =
n1
0 ! 0
0 n2! 0
" " # "0 0 ! n
!" 0
#
$
%%%%%%%%%%%%%
&
'
(((((((((((((((
np
p) = N
(16)
The numbers ni are the occupation numbers of the natural spin orbitals. They
can be shown to be 0 ≤ np ≤ 1. They are usually arranged in order of decreasing
occupation number. The natural spin orbitals are:
!1
NSO( )
!2
NSO( )
!
!!
NSO( )
"
#
$$$$$$$$$$$$$$
%
&
'''''''''''''''''
= U
!1
!2
!!!
"
#
$$$$$$$$$$$$$
%
&
'''''''''''''''
(17)
The NOs usually come as N orbitals with an occupation number close to 1 (strongly
occupied NSOs) and the remaining NSOs with small occupation numbers (weakly
occupied NSOs). The first order density becomes:
! 1( ) x
1, !x
1( ) = np"
p
NSO( ) x1( )"
p
NSO( )* !x1( )
p
"
(18)
Similarly one can obtain a set of orbitals which diagonalize the spin-‐free first-‐order
reduced density matrix. They are called Natural Orbitals (NOs):
P
1( ) r1, !r
1( ) = np!
p
NO( ) r1( )!
p
NO( )* !r1( )
p
" (19)
The occupation numbers here are confined to 0 ≤ np ≤ 2. An important theorem
states that CI expansions built on natural orbitals show the fastest possible
convergence. That is, the natural orbitals with negligible occupation numbers can be
omitted from the one-‐electron space with negligible consequence on the accuracy of
the CI expansion.
For example, the full-‐CI calculations on the He-‐atom with the large aug-‐cc-‐pV6Z(-‐h)
basis set (Figure 1) show that the convergence initially is very rapid. According to
this picture, only a few orbitals (up to 4p which is about 18 correlating orbitals for
this electron pair) need to be included in the correlation treatment in order to reach
chemical accuracy (~1 mEh) in the correlation energy.
HF 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5d 5f 5p 5g 5s all
-0,040
-0,030
-0,020
-0,010
0,000
Chemical AccuracyReached (1 kcal/mol)
Tota
l Ene
rgy
(Eh)
-E(H
F=--
2.86
1673
13)
NOs included in the CI
Exact Correlation Energy
Figure 1: Full-‐CI calculations on the He atom in a basis of natural orbitals derived from the aug-‐cc-‐pV6Z(-‐h) basis set. Plotted is the error in the calculation made by only including the indicated NOs on the x-‐axis in the CI expansion.
1.2 Description of the Experiment In this experiment you will carry out a few full-‐CI calculations on the He-‐atom – the
simplest two-‐electron system. In this case a CI calculation with single-‐ and double
excitations already represents the full-‐CI problem and we concentrate on studying
the basis set convergence of the HF energy and the correlation energy.
JOB: • Study the He-‐atom with the cc-‐pVXZ (X=2,3,4,5,6) basis sets, the ORCA
program and the MDCI module. The input for the ORCA program can be used
as follows:
Run calculations using the basis sets cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, cc-pV6Z.
# Full CI He calculation ! RHF cc-pVDZ TightSCF %mdci CIType CISD Ewin -4,1e10 End * xyz 0 1 He 0.0 0.0 0.0 *
• Plot the correlation energy as a function of X and observe its convergence.
Estimate the basis set limit. In order to accomplish this proceed as follows:
Make an input file energy.dat in the form of 2 Correlation energy cc-pVDZ 3 … 4 … 5 … 6 Correlation energy cc-pV6Z
For visualisation with xmgrace type: xmgrace –legend load energy.dat
The relation between the correlation consistent energies and the energy of
the basis set limit may be written as
Eexact! E
X+ AX"3
(20)
where X denotes the correlation consistent basis set cc-‐pVXZ and EXis the
correlation energy calculated with the basis set cc-‐pVXZ. Now calculating two
energies EX and EY
with the basis sets X and Y we have
Eexact
= EX
+ AX!3
Eexact
= EY
+ AY !3
(21)
Solving this for Eexactwe obtain the following expression for the extrapolated
correlation energy and the parameter A:
Eexact
=X 3E
X!Y 3E
Y
X 3!Y 3
A =!E
X!E
Y
X!3!Y !3
(22)
Calculate the basis set limit for the correlation energy for X=2, Y=3.
• Plot the SCF energy as a function of X and observe its convergence. Estimate
the basis set limit. Using the same procedure, it is possible to extrapolate the
total energy to the basis set limit. Having done two calculations with the
basis sets cc-‐pVXZ and cc-‐pVYZ with X<Y we obtain:
E
exact
tot = EX
HF +X
3E
X
corr !Y3E
Y
corr
X3!Y
3
(23)
Calculate the exact total energy for X=2 and Y=3. Compare your result to the
accurate CCSD(T)-‐R12 calculation of Klopper of E
r12!CCSD(T )tot =!2.9037061 E
h
and E
r12!CCSD(T )corr =!42.037 mE
h.
• Determine an experimental total energy of the He atom. This can be
calculated as a sum of ionization potentials
E
totHe( ) = I He! He
+( )+ I He+ ! He
2+( ) . From the NIST tables the first
ionization potential is I He! He
+( ) = 24.5874 eV . The second ionization
potential can be obtained as the energy of the hydrogen-‐like atom with the
charge 2 using the fact that the energy of the hydrogen-‐like atom scales
quadratically with respect to the nuclear charge.
• Determine the natural orbitals of the system for the largest basis set. Redo
the calculation with limited set of natural orbitals and observe which fraction
of the correlation energy is recovered after terminating the CI at the 2s, 3s,
3p, 3d, 4s, 4d, 4f, 5s, 5p, 5d, 5f, 5g … natural shell.
Natural orbitals are stored in the file *.mdci.nat. They can be loaded from
file with the options moread and moinp
# load natural orbitals ! RHF cc-pVDZ TightSCF ! moread noiter %moinp “orbitals.mdci.nat” * xyz 0 1 He 0.0 0.0 0.0 *
Look into the orbital energies and determine 1s, 2s, 2p etc. natural shells. The
orbitals included into CI calculation can be selected by adjusting the
parameter ewin (=orbital window) in the mdci block.
• Do radial plots of the 1s, 2s, 3s, 4s, 5s natural orbitals. Is there are regularity? What do you think you describe with these natural orbitals? What are the
higher angular momentum NOs good for? Can you distinguish several types
of correlation that are described with different NOs? Compare the 1s and 2s
natural orbitals to the 1s (occupied) and 2s (unoccupied) HF orbitals. What
do you observe? Are the virtual HF orbitals a suitable basis for introducing
correlation?
Produce the data file containing the 1s, 2s, 3s, 4s and 5s natural orbitals of He
using the following input file for ORCA:
The resulting data file “He-‐nat1s.dat” is a standard two-‐column ASCII file and
can be plotted with any plotting program like xmgrace or gnuplot. Plot the
radial distribution functions for 1s, 2s, 3s, 4s and 5s orbitals in the form of
! r( )
2r 2 .
# Plot 1s natural orbital of He ! RHF cc-pV6Z TightSCF ! moread noiter %moinp "orbitals.mdci.nat" %plots dim1 512 dim2 1 min1 0 max1 10 min2 0 max2 1 Format origin; MO("He-nat1s.dat",0,0); v1 0,0,0; v2 0,0,1; v3 0,1,0; end * xyz 0 1 He 0.0 0.0 0.0 *
2 Computer Experiment 13: Potential Energy Surfaces Using Correlated ab initio Methods
2.1 Background In this computer experiment we are going to calculate the potential energy surfaces
of some very small diatomic molecules using multireference ab initio methods. It is
seen how the geometric and electronic structure of the molecules changes from one
state to the other which will be related to the bonding in the molecule. Secondly, it
will be studied how the electronic structure in the equilibrium region correlates
with the free-‐atom dissociation products. Thirdly, it will be studied how rigorous
multireference ab initio methods achieve comparatively high accuracy in the
prediction of the electronic structure of small molecules.5
2.1.1 Potential Energy Curves of Diatomic Molecules In diatomic molecules there is only a single degree of freedom – the internuclear
separation. Therefore, the potential energy curves can be readily vizualized and a lot
of insight into the electronic structure of the electronic states being studied can be
gained. We will not enter a detailed discussion of the theoretical background here
and refer to the classical text of Herzberg for this purpose.6 In these books the
theory is discussed in detail on an accessible level. In addition, Herzerg’s books
provide a comprehensive overview of the experimental data available at the time and
provides an invaluable resource for anybody who performs calculations on diatomic
molecules.
In the Born-‐Oppenheimer approximation, the energy of a given rovibrational state
of a diatomic molecule nicely separates into contributions from the electronic
energy, the vibrational energy and the rotational energy.7 For a given electronic
state I it reads:
5 Again, we will use basis sets here that are smaller than would be „state-‐of-‐the-‐art“. Consequently, the full potential of the ab initio methods will not be realized here. 6 Herzberg, G. Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules. Van Nostrand, New York, 1950; Huber, K.P.; Herzberg, G.; Molecular Spectra and Molecular Structure. IV Constants of Diatomic Molecules. Van Nostrand, New York, 1979. 7 There are even finer contributions coming from relativistic effects, the electron spin as well as the hyperfine and quadrupole interactions that were partly studied in chapter Error! Reference source not found. and will be
EI
R,!,J( ) = Eel
I( ) + Evib
I( ) + Erot
I( )
!Te
I( ) R( )+G I( ) !( )+ F I( ) !,J( ) (24)
The electronic contribution Te is the “raw” electronic energy that we are trying so
desperately to calculate to good accuracy with our electronic structure methods.
The vibrational energy function G is approximately represented by:
G I( ) !( ) = "
e
I( ) ! + 12( )!"e
I( )xe
I( ) ! + 12( )2
+ "e
I( )ye
I( ) ! + 12( )3
+ ...
(25)
Where ν=0, 1, 2, … is the vibrational quantum number. Thus, in general it contains
beyond the harmonic vibrational energy !e
I( ) (which changes from state to state),
corrections for „anharmonicities“ that are described by !e
I( )xe
I( ),!e
I( )ye
I( ) . The
rotational energy is given by:
F I( ) !,J( ) = B
!
I( )J J +1( )!D!
I( )J 2 J +1( )2+ ...
(26)
(J = 0,1,2,…). Here B!I( ),D
!
I( ) are the rotational constants that depend – in general –
on the electronic and vibrational states of the system. For our purposes, the
contributions of the rotations are so small that they will be neglected. In high
resolution gas phase spectroscopy transition between the different vibronic states
of the system are observed in either absorption or fluorescence. From the data one
can derive – in principle – a number of spectroscopic constants:
• The symmetry of a given state.
• The equilibrium distance RI( ) for each electronic state I.
• The vibrational frequency !e
I( ) for each electronic state I.
• The anharmonicities !e
I( )xe
I( ),!e
I( )ye
I( ) for each electronic state I.
disregarded here. There are also various couplings between the different interactions that we will also ignore for the present purposes.
Thus, relative to the ground state one can determine:
• The adiabatic excitation energy Te
I( ) for each excited electronic state. This is
neither the energy of the 0-‐0 transition8 nor the energy of the vertical
transition 9 but simply represent the difference in energy between the
minima of the two curves. It is given by: T
e
I( ) R I( )( )!Te
0( ) R 0( )( ) .
• The shift of the equilibrium distance RI( )!R 0( ) . This is the most important
quantity in determining the intensity distribution in the vibronic spectra.10
A typical set of potential energy surfaces (PESs) are shown in Figure 2 for the C-‐H
molecule. While the calculations leading to these curves have been slightly
oversimplified they already show some important features:
• The shapes of the potential energy surfaces can be quite complicated. In addition
to the minima in the “quadratic region” near R=1.1 Angström, there are
crossings of curves, local maxima and related phenomena.
• The curves “re-‐unite” at “infinite” distances to a combination of the states of the
free atoms.
• The symmetry of each state is determined by inspection of the orbital
configurations that dominate the electronic structure of the state (vide infra)
together with group theoretical tables.
• The equilibrium distance is different for each state and it is already obvious from
inspection that the curvature around the minimum changes from one state to the
other.
8 The 0-‐0 transition is the energy for a transition between the two lowest vibrational levels (ν=0) of the two electronic states involved in the transition. 9 The „vertical transition“ is the electronic energy difference between the two vibronic states at the equilibrium distance of the electronic ground state (or, more generally, the initial state involved in the transition). 10 It is of course also possible to study the electric, magnetic and quadrupolar transition probabilities between vibronic levels. We will not pursue this here.
• The potential energy surfaces were calculated pointwise. Thus, the electronic
structure problem was solved on successive internuclear distances and a smooth
curve is obtained from interpolation. This is characteristic of the Born-‐
Oppenheimer approach – we approximately solve the electronic Schrödinger
equation for each fixed nuclear arrangement separately.
1.0 1.5 2.0 2.5 3.0 3.5
-38.34
-38.32
-38.30
-38.28
-38.26
-38.24
-38.22
-38.20
-38.18
-38.16
-38.14
0
5000
10000
15000
20000
25000
30000
35000
40000
Rel
ativ
e E
nerg
y (c
m-1)
C(1D)+H(2S)
C(3P)+H(2S)
C-2Σ+
B-2Σ−
A-2Δ
a-4Σ
Tota
l Ene
rgy
(Eh)
C-H Distance (Angström)
X-2Π
Figure 2: The potential energy surfaces of the low-‐lying excited states of the C-‐H molecule as calculated with the MRCI+Q method and the SVP basis set.
Suppose that we have calculated a number of such potential energy surfaces E
el
I( ) R( ) and we whish to determine the spectroscopic constants. How can you do that?
Below is a simple recipe:
• First, a smooth curve is fitted through the available points. In practice this may
be a cubic spline or Chebyshev polynomials or similar techniques.
• Second, a search for the minima on each potential energy surface is carried out.
These are found by examing triples of points on the PES surface. If one of the
three points is lower in energy then the two neighbouring points to the left and
the right, there must be a minimum close by. This may then be located precisely
using the interpolated curves and standard techniques of numerical analysis.
• Once the equilibrium distance is found, the second derivative of the interpolated
curve is calculated. From the value of !2E
el
I( ) /!R2
R=RI( ), (in Eh/(au2)) and the
reduced mass of the molecule ( µ = M
AM
B/ M
A+ M
B( ) , in Dalton) the harmonic vibrational frequency (in wavenumbers) is calculated by:11
! I( ) =
12"c
!2E!R2
R=RI( )
1µ
= 5140.487!2E!R2
R=RI( )
1µ
(27)
• From the value of the electronic energy at the equilibrium distance, the “term
energy” Te is determined. Subtraction of the ground state Te and conversion to
wavenumber units12 results in a number that can be compared with Herzberg’s
tables.
• From the difference in equilibrium distance relative to the ground state
RI( )!R 0( ) (in atomic units), the dimensionless displacement parameter !
I( ) is
calculated:
! I( ) =!R
2!cµ" I( )
h= 0.091139!R µ" I( )
(28)
( !R in atomic units, µ in Dalton and !I( ) in cm-‐1) These dimensionless
displacements are quite useful. First of all they define the so-‐called Huangh-‐Rhys
factor S I( ) = 1
2! I( )( )
2
. In terms of these parameters the Franck-‐Condon factors for
vibrational progressions can be given in closed form provided that both PESs are
assumed to be harmonic – this is a rough approximation but a useful first
11 The conversion is a little painful. You want to give the energy second derivative in Eh/(au2) and the reduced mass in Dalton. The best way is to convert everything to SI units and finally back to wavenumbers. The units reduce as follows: 1 J = 1 Nm = 1 m2/s2 kg so J/m2*1/kg = m2/s2 kg/m2*1/kg=1/s2. Thus, the prefactor has the unit s/m and the integral reduces to 1/s and the final prefactor is 1/m = 1/100*cm-‐1 12 Using 1 au = 219474 cm-‐1
orientation. For transitions out of the lowest vibrational level of the ground state
into the m’th vibrational level of the excited states PES the FC factors are given by:
0 m m 0 =
S I( )( )m
m !e!S
I( )
(29)
The maximum intensity in the progression will occur roughly for m ≈ S.
An alternative to the polynomial fitting, is a fit to a Morse potential curve. Such a
more restricted, parameterized form has an advantage if the calculated data contain
some numerical noise (as is the case with the MRCI variant that we will use below).
The Morse potential is:13
E
MorseR( ) = E
0+ D
e1!e!! R!R( )( )
2
(30)
Figure 3: The Morse function and its parameters.
The Morse function has the following derivatives at R = R :
!EMorse
!RR=R
= 0
(31)
13 An excellent short discussion is given by A.C. Hurley: Introduction to the Electron Theory of Small Molecules, Academic Press, London, New York, 1976, chapter 1.
!2EMorse
!R2
R=R
= 2!2De
(32)
!3EMorse
!R3
R=R
="6!3De
(33)
!4EMorse
!R4
R=R
= 14!4De
(34)
The Morse function has four fit parameters: E0,D
e,!,R . After these have been
determined by least-‐square fit to the calculated surface, the harmonic frequency is
calculated from (27) and (32) while the anharmonic constant is given by:
!
ex
e=
h"2
8#2µc= 60.16623
"2
µ[cm!1 ]
(35)
Here, De is Eh, ! is in au
-‐1 and µ is in Dalton.
Finally, the most accurate method is to directly solve for the vibrational energy
levels of a given PES. In this case, one can write the unknown vibrational
wavefunction as a series expansion in harmonic oscillator wavefunctions and apply
the variational principle in order to find the vibrational eigenstates in an arbitrary
potential. The required integrals can either be calculated by numerical integration
techniques or may also be obtained in closed form by assuming certain shapes for
the PES. If the harmonic oscillator basis is large enough, one obtains accurate results
for vibrational energy levels as well as vibrational eigenfunctions that can then be
used to calculate accurate Franck-‐Condon factors.
2.1.2 Single reference ab initio Methods As explained in chapter 1 (page 1), the Hartree-‐Fock method provides in many cases
a reasonable 0th order description of molecules. As “the” HF method we interprete
methods in which a single determinant is taken as Ansatz for the N-‐electron
wavefunction and the energy of this determinant is minimized through variation of
the orbitals in order to achieve the lowest possible energy. As described below,
there are molecules and states that can not be described to 0th order by a single
determinant. For these molecules and states one must use the “multireference
methods” described below. For a large class of molecules, in particular for many
closed-‐shell molecules close to their equilibrium geometry, the single determinantal
HF wavefunction is a reasonable approximation. What remains to be done in order
to improve on the HF approximation up to the point where it gives quantitative
agreement with experiment, is to take care of the dynamic correlation.14 One way
to approach this problem is to pass from a single determinant Ansatz to a many
determinant Ansatz for the N-‐particle wavefunction. However, one should choose
the many determinants in the expansion in a systematic fashion. The most popular
of the systematic approaches is to arrange the possible determinants15 by excitation
level. As an “excitation” one does not understand an actual excited state of the
system but simply the construction of determinants in which occupied HF orbitals
(labels i,j,k) are replaced with virtual orbitals (a,b,c). The exact N-‐particle
wavefunction can then be written as:
! = !
HF+ C
a
i!i
a
ia
" + 14
Cab
ij!ij
ab
ijab
" + 136
Cabc
ijk!ijk
abc
ijkabc
" + ...
(36)
The different terms represent the HF “reference” determinant, single-‐excitations,
double excitations, triple-‐excitations and “…” indicates that this scheme is to be
extended up to N-‐tuple excitations.16 It is, unfortunately, highly impractical to base
actual calculations on such a “brute” force scheme since it is evident that the
number of terms in the expansion grows so quickly that even the largest computers 14 Essentially the notion that the movement of the electrons depend in a very complicated way on the movements of the other electrons. In the Hartree-‐Fock methods, each electrons just “swims” in an average potential “sea” created by the remaining N-‐1 electrons. 15 There are on the order of N! possible determinants in a N-‐electron system which immediately leads to the conclusion that it is impossible to introduce them all in the calculation (except for the smallest benchmark systems where this has been extensively pursued). 16 One useful way to think about the configuration expansion coefficients C is that they (actually their square) describe the probability of electrons to temporarily “jump” out of their HF orbitals and partly occupy virtual orbitals. These jumps reduce the electron-‐electron repulsion since since the average electron-‐electron distance is larger if the electrons occupy different orbitals.
and most efficient programs can not handle them. Thus, the number of singles is
proportional to O(N2), the number of doubles to O(N4), the number of triples to
O(N6) etc. Fortunately, the importance of the excitation classes diminish with
increasing excitation level. The by far most important effects are brought in by the
double excitations,17 the triple-‐excitations provide important refinements and the
quadruple excitations are only important in special contexts.18 Since the number of
doubles is “only” proportional to O(N4) chemically significant calculations can be
done for up to double excitations.19 The treatment of triple-‐excitations is restricted
to small molecules and in an approximate way for at most medium sized
molecules.20
The different classes of single-‐reference correlation methods mainly differ in the
way that the expansion coefficients Cai , Cab
ij , Cabcijk ,… in eq (36) are determined. One
possibility is to approach the problem by perturbation theory. In this case the
complete Born-‐Oppenheimer Hamiltonian is divided into a 0th order part which has
the HF solution as its eigenfunction and the remainder V = H BO ! H 0( ) (sometimes
called the “fluctuation potential”).21 Such methods are quite successful and can be
implemented and applied roughly up to the 4th order in perturbation theory.
However, electron correlation is not a small perturbation and failures of these
methods due to a lack of convergence of the perturbation expansion are not
uncommon. Very popular is the 2nd order method (MP2) which represents a major
17 The singles to contribute very little to the correlation energy due to “Brillouin’s theorem”. 18 The so-‐called “disconnected” quadruples restore the size-‐consistency of the CISD expansion which is of major importance. These excitations may be thought of as independent pair excitations. “Genuine” quadruple excitations contribute very little to the correlation energy. 19 However, in practice we need to compute the interactions between double excitations. While there are formally O(N8) doubles/doubles interactions, “only” O(N6) of them are nonzero. Thus, methods that require the calculation of doubles/doubles interactions scale as O(N6) unless further approximations are introduced. This puts severe limits on the size of the molecules that can be treated with such methods. 20 This may change with the advent of the now emerging “linear scaling local correlation” methods. Here the calculations are arranged such that terms of negligible size can be identified and neglected. The remaining “significant” contributions to the correlation energy only scale linearly with the size of the molecule since electron correlation is a rather short-‐range phenomenon. 21 There are many ways in which “reasonable” definitions of H(0) and V can be accomplished. The most popular choice is the Möller-‐Plesset (MP) choice in which H(0) is the sum of the Fock operators for the N-‐electrons.
improvement over HF theory and has a comparatively low cost (proportional to
O(N5)).22
Methods that are of infinite order in the fluctuation potential are frequently the
methods of choice since they do not assume smallness of electron correlation
effects. However, straightforward application of the variational principle to an
expansion that is truncated at the level of single and double excitations (called
CISD) is not a successful method because it has an important defect: it lacks size
consistency. That is, for noninteracting fragments A and B a calculation of the
supermolecule AB yields a different result than the sum of calculations on the
isolated fragments A and B. Thus, the method behaves inconsistently as the size of
the molecule changes and this is deadly for its application to chemistry.
The size consistency is restored by including higher excitations but their full
inclusion is impractical (as pointed out above). However, the situation is not so bad
since in order to “repair” the size-‐inconsistency of the CISD method one “only”
needs to approximately incorporate the doubles/quadruples interaction. Of key
importance are the quadruple excitations that can be described as products of
double excitations (“simultaneous pair excitations”). Their coefficients are
essentially given by products of double excitation coefficients. The earliest attempts
to include this important excitation class are known as “coupled electron pair”
models (CEPA).23 These methods are size consistent, but no longer variational (as
the CISD method). The size-‐consistency is the more important property!
The CEPA methods make some approximations that prevent them from including
the disconnected quadruples rigorously.24 The simplest method that does so is the
22 Very efficient and accurate approximations to the exact MP2 method exist. They lead to calculations where for most molecules of practical size (~1000 basis functions), the initial HF calculation takes more time than the computation of the MP2 correlation energy. 23 W. Meyer Int. J. Quant. Chem. (1971), 5, 341; W. Meyer J. Chem. Phys. (1972), 58, 1017; W. Meyer Theoret. Chim. Acta (1974), 35, 277; W. Meyer, P. Rosmus J. Chem. Phys. (1975), 63, 2356; R. Ahlrichs, F. Driessler, H. Lischka, V. Staemmler, W. Kutzelnigg J. Chem. Phys. (1975), 62, 1235; R. Ahlrichs Comp. Phys. Comm. (1979), 17, 31; For a lucid introduction into the field of electron correlation see Hurley, A.C. Electron Correlation in Small Molecules, Academic press, London, 1976 24 For example CEPA methods are not “unitarily invariant”, that is, they yield different results if the occupied or virtual orbitals are unitarily mixed among themselves which should not change the outcome of a calculation.
so-‐called “quadratic CI” (QCISD) method.25 It is very successful but also not
variational. While it is more complex than the CISD method, in practice one can
apply it in every situation where a CISD or CEPA calculation is feasible. However,
QCISD can also be thought of as an approximation to an even more general and
powerful method: the so-‐called “coupled-‐cluster” method26 with single-‐ and
double-‐excitations (CCSD). The CC hierarchy is based on a completely different27
Ansatz for the correlated wavefunction that is size-‐consistent from the start.28 It
leads, however, to a very complex set of equations. With proper programming, CCSD
takes a limited additional time over QCISD and since it is more rigorous some
workers prefer CCSD over QCISD. However, the general accuracy of both methods is
very similar.
If a perturbative correction for the triple-‐excitations is added to these methods (the
CCSD(T) and QCISD(T) methods) one comes to the approaches that are known to be
the most accurate practical single reference correlation methods. They typically
yield results of chemical accuracy under two conditions: (a) The HF determinant
provides a valid starting point and (b) large basis sets are used.29
2.1.3 Multireference ab initio Methods In order to arrive at consistent potential energy curves it is necessary to calculate a
number of electronic states over large areas of the potential energy surface and to
describe these states in a balanced manner. There are quite a number of
complications that theory has to face in accomplishing this goal. First of all the
character of a state may change very quickly in certain areas of the PES. Secondly,
the states may be orbitally degenerate or nearly degenerate and this needs to be
adequately taken into account in the electronic structure calculations. Third, the
states are also, in general, of variable multiplicity and consequently, a balanced
25 Pople, J.A.; Head-‐Gordon, M.; Raghavachari, K. J. Chem. Phys. (1987), 87, 5968. This paper also introduced the perturbative triples correction that is widely used in conjunction with the CCSD method 26 The literature on CC methods is large and complicated. The closest to an elementary introduction can be found in Hurley’s book and in a more recent review: Crawford, T.D.; Schaefer, H.F. in: Reviews in Computational Chemistry. Vol 14, K.B. Lipkowitz, D.B. Boyd (Eds), Wiley-‐VCH, New-‐York, 2000, pp 33-‐136. 27 The Ansatz is exponential rather than linear as in the CI method. Details will not be provided in this course. 28 But CC methods can not be made variational, at least not in a straightforward way 29 The (slow) convergence of the correlation energy with basis set size has been studied in chapter 1
treatment of states with different numbers of unpaired electrons is necessary.
Fourth, states that cross each other or have an avoided crossing pose particular
challenges. Finally, the electronic structure methods used must be able to correctly
describe the dissociative region, that is, they must be able to correctly track the
PES’s all the way from the bonding region to the multiplets of the isolated atoms.
Presently, there is only one line of approach known that allows one to achieve all of
these objectives simultaneously. This is provided by so-‐called multireference
methods. The basic idea is the following: The calculation of accurate electronic
energies is divided into two steps – in the first step, 0th order approximations to the
states of interest are constructed that are qualitatively correct. Thus, there is a
limited set of electronic configuration state functions (CSFs)30 that is necessary to
describe the physics of the system correctly. That is, this set of CSFs contains all the
configurations that are necessary to construct all target states with their correct
multiplicity, spatial symmetry as well as all those electronic configurations that are
necessary to properly dissociate the molecule into its fragments (atoms). Given a set
of suitable CSFs, the energy of the many electron wavefunction for each state is
minimized:
E I( );0 =! I( );0 | H
BO|! I( );0
! I( );0 |! I( );0
(37)
Here ! I( );0 is the 0th order wavefunction for the I’th electronic state and HBO
is the
many-‐electron Born-‐Oppenheimer Hamilton operator and the superscript ‘0’
reminds us that we are only trying to generate an energy and wavefunction that is
qualitatively correct (e.g. to ‘0th order’). As discussed above, we write ! I( );0 as a
linear combination of the ‚qualitative set’ of CSFs !{ } :
30 A configuration state function consists of essentially two parts: First, the electronic configuration, i.e. the distribution of electrons among the available orbitals, and secondly a spin-‐coupling of the unpaired electrons to produce the desired total spin. The spin coupling is perhaps best pictured by a spin-‐branching diagram as for described elaborately for example by Pauncz (Pauncz, R. Spin Eigenfunctions. Construction and use. Plenum Press. New York and London, 1979)
! I( );0 = C
JI"
JJ#
(38)
It is important to recall that the CSFs !{ } are themselves built from a set of one-‐
electron orbitals !{ } that in turn are taken as a linear combination of one-‐electron
basis functions !{ } . Thus, let us assume for simplicity that each
!
J is a single
Slater determinant; then one could write them as:
!
J= !
1
J( )....!N
J( )
(39)
Where !i
J( ) is the i’th orbital occurring in the N-‐electron Slater determinant for CSF
J. The orbitals are:
!
i= c
µi"
µµ!
(40)
The energy in eq (37) is now a function of two sets of wavefunction parameters: the
CSF coefficients C and the MO coefficients c. Thus we can write E I( );0 = E I( );0 C,c( ) .
The minimization of this energy with respect to both C and c leads to the so-‐called
“Multiconfiguration Self-‐Consistent Field” (MCSCF) method. This method is fully
variational.31
We have thus far been deliberately vague on the subject which set of !{ } is
„suitable“ to provide the desired 0th order description. There is no general best
answer to this question that is universally agreed upon. This then constitutes a
strength and a weakness of the MCSCF method: the choice of expansion functions
!{ } depends on the insight that the user of the method has into the nature of the
problem and is therefore subjective. Since one is limited in the number of 0th order
31 The minimization techniques are somewhat involved and will not be discussed here.
CSFs for computational reasons, it is not uncommon that several sets are tried
before the “best” one is found. If calculations on small molecules are performed a
“safe” choice is to include all valence orbitals and electrons in the active space as
long as this is possible. Technical limitations restrict the maximum size of the active
space to about 14 orbitals. If the system of interest has more valence orbitals, a
choice of the “important” orbitals for the problem at hand must be made.32
A general and systematic way to choose a set of !{ } ’s is the following: (1) divide the
set of MOs into three subsets: (a) the “inactive” orbitals that are doubly occupied in
all !{ } , (b) the „active“ orbitals that are partly occupied in the
!{ } ’s. These are the
important orbitals the occupation and shape of which changes during the chemical
reaction, spectroscopic transition,… (c) the “virtual” orbitals that are empty in all
!{ } ’s. This description implies that the number of electrons and orbitals in each subspace is fixed. Since we know the total number of electrons (N), we only need to
specify how many of them are active (n) and in how many orbitals (m). (2) In order
to avoid any ambiguity in the choice of CSFs, perform a full-‐CI with the n-‐electrons in
m-‐orbitals. Thus, one constructs all !{ } ’s with n-‐electrons in m-‐orbitals that are
compatible with the requested multiplicity and spatial symmetry and uses this set
as trial set fort he MCSCF procedure. Since the set of !{ } is complete in this many-‐
electron subspace one speaks of the „complete active space self-‐consistent field”
(CASSCF(n,m)) method.33
One subtlety is the following: the optimal MO coefficients c are different for different
states. It may therefore become very laborious to optimize a new set of c’s for each
32 It may not always be possible to include all of the valence orbitals for another reason: lack of convergence! If an active orbital becomes almost doubly occupied or almost empty, convergence problems will result in the CASSCF. In this case it will be advantageous to not include these orbitals into the active space. 33 For CASSCF and MRPT methods (CASPT2) see K. Andersson, B. O. Roos, in: D. Yarkony (Ed.), Modern Electronic Structure Theory. World Scientific, Singapore, 1995, p. 55
state of interest.34 Consequently, one tries to determine a single set of c’s that
represent the best compromise for all states of interest. Thus, one writes:
ESA!CASSCF = w
IE I( );0
I"
(41)
With constant, user-‐supplied, weights wI that sum to unity. Minimization of this
energy expression results in a set of “democratic” orbitals for the states of interest
as well as 0th order approximations for all states of interest.
In quantum chemical language, the correct 0th order description that was thus
generated takes care of all static (near-‐degeneracy) correlation effects. What is left to
be done is to try to recover as much as possible of the remaining (dynamic)
correlation energy. This can be approached either by perturbation theory (leading
to “multireference perturbation theory” MRPT methods) or by configuration
interaction techniques (leading to “multireference configuration interaction”,
MRCI).35 What is done in these techniques is to regard either the entire ! I( );0 or its
constituent parts !
J as „reference“ wavefunctions and to perform single-‐ and
double-‐excitations relative to these functions. This produces a set of “excited” CSFs
!
J which are used as expansion sets of the many-‐electron problem. Thus one
writes:
! I( ) = B
JI"
JJ#
(42)
With parameters BJI that are determined by variational techniques (MRCI) or
perturbational techniques (MRPT). Since the set of !
J may include the set of
!
J,
34 In addition one would have to calculate transition properties from sets of non-‐orthogonal orbitals used for the expansions of the states involved in the transition. This is possible but much more laborious than calculations based on a single orthonormal set of orbitals. 35 The “ultimate goal“ of a consistent and efficient „multireference coupled cluster“ (MRCC) theory has so far been plagued by massive technical problems and it is presently not clear whether such methods can be developed into large-‐scale research tools for non-‐experts.
we do not write: ! I( ) = ! I( );0 + B
JI"
JJ# -‐ in general, the coefficients C of those
CSFs !
J that are contained in the
!
J set are “revised” by the dynamic
correlation treatment.36 The set of !
J’s may be very large and may have hundreds
of millions of members. While such large scale calculations are barely feasible using
todays most advanced programs and computers it is highly desirable to come to as
compact sets of CSFs as possible and this is still, even after decades of research, a
topic for method development. We will not go into further detail here, but simply
note that the ORCA program is of the “individually selecting” type in its MRCI and
MRPT modules – a technique that was pioneered and extensively practised in Bonn
for a long time and with major success.37 Thus, the program contains two sets of cut-‐
offs:
• A parameter Tsel (typically 10-‐5-‐10-‐7 Eh) that determines which !
J are
important an thus are “allowed” to enter the variational space in a MRCI
calculation. These are the ones that interact most strongly with the reference
wavefunction ! I( );0 and, in practice, are selected on the basis of second-‐
order perturbation theory. This selection procedure selects typically ≈104-‐
106 CSFs for the variational treatment. The remaining (much larger) number
of CSFs are simply treated by second-‐order perturbation theory.
• A second parameter Tpre (typically 10-‐2-‐10-‐5) determines which !
J are
allowed to act as reference configurations in a MRCI or MRPT calculation.
36 This “revision” may be quite important since for a number of problems the CASSCF method does not produce accurate initial values. An important case is met in the calculation of exchange coupling constants as discussed for example by Malrieu and co-‐workers (Miralles, J.; Castell, O.; Caballol, R.; Malrieu, J. P. Chem. Phys. 1993, 102, 103; Miralles, J.; Daudey, J. P.; Caballol, R. Chem. Phys. Lett. 1992, 198, 555; Castoll, O.; Miralles, J.; Caballol, R. Chem. Phys. 1994, 179, 377) and Staemmler and co-‐workers.(Fink, K.; Fink, R.; Staemmler, V. Inorg. Chem. 1994, 33, 6219; Fink, K.; Wang, C.; Staemmler, V. Int. J. Quant. Chem 1997, 65, 633; Fink, K.; Wang, C.; Staemmler, V. Inorg. Chem. 1999, 38, 3847) 37 R. J. Buenker, S. D. Peyerimhoff Theoret. Chim. Acta 35 (1974) 33; R. J. Buenker, S. D. Peyerimhoff Theoret. Chim. Acta 39 (1975) 217
Each !
J that contributes with a weight of at least Tpre to any of the
! I( );0
will be part of this “privileged” set of CSFs.
The final subject to be mentioned is a fundamental weakness of MRCI – its lack of
size consistency. That is, calculated energies of noninteracting subsystems do not
exactly sum to the same energy as the same system calculated as a “supersystem”.38
As discussed in detail in the lectures on quantum chemistry, this weakness is shared
by all CI based approaches and is related to the neglect of higher than doubles
excitations. A straightforward, approximate and essentially empirical remedy for
this problem is the so-‐called “multireference Davidson correction”39 which is
applied to the MRCI energies to yield the so-‐called “MRCI+Q” method which will be
used in this experiment.
2.2 Description of the Experiment In the actual experiment, we will study the CH, NH and OH molecules (five valence
orbitals and 5-‐7 valence electrons) that will be correlated in this experiment with
multireference methods. We will also treat F2 and N2 using single reference methods
(14 and 10 valence electrons and 10 valence orbitals respectively). The
experimental data for these molecules is reproduced in Table 1. Table 1: Experimental data for some low-‐lying states of some diatomic molecules.
Molecule State Te (cm-‐1) ωe (cm-‐1) ωexe (cm-‐1) Re ( Å) CHa 2Σ+ 31801 2840 126 1.1143 2Σ-‐ (26044) (1795) -‐ (1.1975) 2Δ 23189 2931 97 1.1019 4Σ-‐ (5844) (3145) (72) (1.085) 2Π 0 2859 63 1.1199 NHb 1Π (43744) (2122) -‐ 1.1106 3Π 29807 3231 99 1.0369 3Σ+ 21202 3352 74 1.0360 1Δ (12566) 3188 (68) 1.0341 3Σ-‐ 0 3282 78 1.0362 OHc 2Σ+ 32684 3179 93 1.0121
38 In multireference CI calculations the problems are not overwhelming if the reference space is chosen wisely. Single reference CI calculations suffer severely from the problem. 39 G. Hirsch, P. J. Bruna, S. D. Peyerimhoff, R. J. Buenker Chem. Phys. Lett. 52 (1977) 442
2Π 0 3738 85 0.9697 HFd 1Σ+ 0 4138 90 0.9171 F2e 1Σg+ 0 917 11 1.4119 N2f 1Σg+ 0 2358 14 1.0977 a – Ionization potential = 10.64 eV; D0= 3.46 eV b – Ionization potential = (13.36) eV D0 ≤ 3.47 eV c – Ionization potential = 12.9 eV, D0= 4.39 eV d – Ionization potential = 16.06 eV, D0= ≤6.4 eV e – Ionization potential = 15.686 eV, D0= 1.602 eV f – Ionization potential = 15.580 eV, D0= 9.759 eV
2.2.1 Single Reference Calculations In this section, we will compare some single reference calculations amongst each
other and with experiment. In order to provide a first orientation, use a small SVP
basis set and try to calculate the entire potential energy surface up to the
dissociation limit. We have deliberately chosen a molecule with a single bond (F2) as
well as a molecule with a triple bond (N2) in order to demonstrate the possibilities
and limitations of the methods used. We will compare the RHF, UHF, MP2, QCISD(T)
and CCSD(T) methods.
• Calculate a PES for the dissociation process with the RHF method using the
input file below; repeat in a similar way for F2 and N2. What do you observe?
Is the result physically acceptable? Calculate the isolated F and N atoms using
the UHF method and compare with your dissociation limit in as far as you
can define one!
The result of this calculation will be a table of numbers that list the value of the
parameter that you have scanned and the energy that you have calculated. This
# # Potential energy surface of HF with RHF # ! RHF SVP TightSCF # This line define a parameter R that is to be varied from 1.2 to 3.5 # Angström in a total of 35 equidistant steps. Other definitions of PES # are of course possible (please refer to the ORCA manual) %paras R= 0.85,3.5,40 end * xyz 0 1 F 0 0 0 H 0 0 {R} *
appears in two places in the ORCA output: (a) in the output file itself, and (b) to
make you life easy, there will also be a file written that is called “*.trjscf.dat”.
“TRJ” stands for “trajectory” and “SCF” for SCF energy. If you do other calculations as
well there will be additional files “*.trjxxx.dat” where xxx may be “mdci” (for
single reference correlation methods) “mp2” (for the MP2 method), “mrci” (for
calculations with the MRCI module) “casscf” (for individual states in a state-‐
averaged CASSCF calculation) or “cis” (for CIS or TD-‐DFT calculations). These files
can be directly imported into plotting or data analysis programs.
As you will find in your calculations above that the RHF method does not lead to
potential curves which dissociate correctly, it is possible to enforce this on the
method using a special type of UHF wavefunction, namely one of “broken
symmetry”. In this case, the spin-‐up and spin-‐down orbitals of the UHF solution are
forced to localize on one of the atoms. The solution has the correct physics and
energy but an erroneous spin density. It is, nevertheless, a feasible way to simulate
some static correlation effects with single determinant methods.40 The input for this
method is shown below.
JOB
• Plot the two PESs and compare them. Determine the dissociation energy for
the UHF solution.
40 In inorganic chemistry this is a quite important subject in the framework of exchange coupling constant prediction based on DFT methods. See chapter Error! Reference source not found. on page 226.
# # Potential energy surface of HF with UHF # ! UHF SVP TightSCF # Now we scan the opposite direction since we need to find the correct # solution for the dissociation limit before “moving in” %paras R= 3.5,0.8,40 end # This instructs the program to flip the spin on the second fluorine # following an initial spin triplet calculation. We have a single bond, # therefore we look for a broken symmetry solution with one localized # electron per site; in the N2 case it is three electrons. Brokensym 1,1 * xyz 0 1 F 0 0 0 H 0 0 {R} *
Now that we have a HF solution, let us try to do the correlated calculations. At the
moment this is only feasible with the ORCA program based on a RHF solution.
JOB:
• Re-‐run the RHF calculation but include the keywords (in turn): QCISD(T),
CCSD(T) and CISD.
• Repeat the calculations for N2 and F2.
Compare your findings – do the correlated methods dissociate correctly? If not,
what is your interpretation?
Now that the applicability of different methods is understood let us turn to a more
serious calculation and examine the vicinity of the minimum in order to predict
spectroscopic constants. From your initial calculations you have a fairly good idea
where the minimum occurs on the PES of the two molecules with which method
(see also Table 1). Now, do a PES scan around the minimum with a large basis set in
order to obtain accurate results. On the order of 10 steps should be sufficient to give
a reliable fit.41
JOB:
41 In fact, if you are serious about science you always CHECK on everything you do as carefully as possible -‐ This is independent of whether one does experiments or performs calculations! In the present case this would mean that you check (with a small basis set and perhaps with the HF method) how many points you need and how far a region around the minimum you have to study in order to get stable results. If you choose a region roughly ±0.1-‐0.2 Å around the minimum and have spacings between your points of roughly 0.01-‐0.02 Å the results should be reasonably converged. If you choose the region too small you can not calculate reliable anharmonic constants since you are well in the quadratic region; if you choose the spacing too large, the region around the minimum is not well described by a Morse potential.
# # Potential energy surface of HF with correlation methods # ! RHF TZVPP QCISD(T) TightSCF Conv # %paras R= 0.81,1.01,12 end * xyz 0 1 F 0 0 0 H 0 0 {R} *
• Do the calculations with RHF, MP2, QCISD, CCSD, QCISD(T), CCSD(T) and the
B3LYP-‐DFT method for HF, F2 and N2. To include MP2 is instructive for
comparing higher and lower level correlation methods.
• Compare first the MP2 with the CCSD results. How important are higher
order correlation effects brought in by the iterative nature of the CCSD
method?
• Compare the CCSD with the CCSD(T) results. How important are the
connected triple excitations?
• For the CCSD(T) method, study the basis set dependence by choosing a
smaller basis set (SVP), a basis set lacking the f-‐polarization set (TZV(2d,2p)),
a basis set with only one set of polarization functions (TZVP) and finally a
very large basis set (QZVP). Since you get the SCF results automatically too –
what is your conclusion about the basis set dependence of correlated
methods? What types of correlation effects are you describing by enlarging
the basis?
• Determine the spectroscopic constants and compare them with experiment.
Produce tables with your results? Are the ab initio calculations reliable?
Which method shows the best agreement with experiment?42
The latter fit you can either do yourself with xmgrace (chapter Error! Reference
source not found. on page Error! Bookmark not defined.). Alternatively, there is
a small fitting program in the ORCA package that can do the job (orca_fitpes)
and that is called automatically at the end of a surface scan for a diatomic
molecule.43
42 In fact, in the case of HF you can walk in the footsteps of the great pioneers of accurate calculations: Meyer, Ahlrichs, Kutzelnigg and Staemmler who developed the CEPA method in the early 1970’s and applied them with great success and impressive results to small molecules. To do CEPA calculations best choose CEPA/1 instead of CCSD(T) in the input line. See CEPA references given on page 174. 43 You can also call this module in a stand-‐alone mode. To this end you have to edit the file *.trjxxx.dat” and add a some information on the first line: 1 11 1 14 14 <<<<---- additional required information
2.2.2 Multireference Calculations We will now turn to multireference methods which can be applied to the ground-‐
and excited states over the entire PES. In general we scan the PES from “out-‐to-‐in” in
order to help the program to initially select the correct set of orbitals that must be
correlated. We start by a SA-‐CASSCF calculation and include the states listed in
Table 1. It is a good idea to start with a small basis set and only do the final
calculations with the largest (target) basis set. A suitable ORCA input for the CASSCF
calculations is:
JOB:
• Plot the result contained in the “*.trjcasscf.dat” file.
Now we scan the region around the minima using the more accurate MRCI+Q
method. In order to do this, you have to add a block like:
1.20000000 -108.80922141 <<<<---- ORCA output
1.18000000 -108.82097001
1.16000000 -108.83134251
Etc
The new title contains the following items: NPES NPTS ISANG MA MB. Where: NPES – Number of potential energy surfaces (columns of data) NPTS – Number of points (rows with data) ISANG – Set to unity if the distance is in Angström, otherwise units of Bohr are assumed MA,MB – Masses of the two nuclei (in units of the proton mass, e.g. C=12). This new file is run through orca_fitpes in order to determine the various spectroscopic constants which can then be compared to the experimental data. (orca_fitpes myjob.trjmdci.dat >myjob.trjmdci.out)
! RHF SVP TightSCF Conv %casscf nel 5 # number of active electrons Norb 5 # number of active orbitals mult 2,4 # multiplicities doublet and quartet nroots 6,1 # 6 doublet roots, 1 quartet root switchstep nr # second order converger ON end %paras R= 3.5,1.0,40 end # scan all the way inwards * xyz 0 2 C 0 0 0 H 0 0 {R} *
JOB:
• Use the orca_fitpes program to analyze the results of the CASSCF
calculations 44 (in the “*.trjcasscf.dat” files) and of the MRCI
calculations (in the “*.trjmrci.dat” files).
• Compare your results with the experimental data in Table 1. Document your
mean error, maximum error and mean absolute error relative to experiment.
• Increase the basis set to TZVPP. Do your results get better?
• Vary the values of the parameters Tsel and Tpre. How do your results react to
these changes?
ADDITIONAL CALCULATIONS I (VOLUNTARY)
• Use eq (29) to calculate the Franck-‐Condon factors for transition between the
lowest ground state vibrational levels and vibrational levels of the
electronically excited states in the harmonic approximation.
• Plot the intensity distribution in the vibrational progression. If you “broaden
it out” by convoluting the calculated transitions with Gaussian curves -‐ where
is the band maximum? How much does it deviate from the 0-‐0 transition and
44 If you have crossings of levels in your PES you have to be careful. In the case of discontinuities the Morse-‐fits fail. Thus, you have to make sure that your columns contain the correct data that correctly tracks the state of interest. You have to look at the results of the Morse fits. Sometimes they converge to nonsensical values.
%mrci ewin -3, 1e10 citype mrci tsel 1e-6 # Selection parameter tpre 1e-4 # pre-selection parameter intmode fulltrafo newblock 2 * # calculate doublet states nroots 6 # six roots (as in CASSCF) refs cas(5,5) end # the reference space end newblock 4 * # calculate quartet states nroots 1 # one root (as in CASSCF) refs cas(5,5) end # SAME reference space! end end
the vertical transition energy? What does this implicate for comparison of
calculated vertical transition energies and band maxima for larger
molecules?
ADDITIONAL CALCULATIONS II (VOLUNTARY)
• Compare single-‐ and multireference methods for N2. To this end perform a
CASSCF(6,6)45 calculation on N2 and perform a MRCI+Q PES scan as in the
first part of the experiment.
• Compare CASSCF with RHF
• Compare MRCI+Q with QCISD and CEPA/2.
ADDITIONAL CALCULATIONS III (VOLUNTARY):
• How would you calculate the ionization potential of the molecules using
multireference methods?
• How would you consistently calculate the dissociation energies De´and D0?
45 This is not perfectly straightforward since close to the equilibrium distance the program may have difficulties to “find” the high-‐lying σ* orbital which is required to complete the 2p-‐valence space. To this end, start the calculation from the natural orbitals of a MP2 calculation (a commonly used trick to start CASSCF calculations): ! RHF MP2 SVP TightSCF
%mp2 density relaxed natorbs true end
%paras R= 1.20 end
* xyz 0 1
N 0 0 0
N 0 0 {R}
*
$new_job
! SVP TightSCF moread
%moinp "N2-CASSCF-01.mp2nat"
%casscf nel 6 norb 6 switchstep nr end
%paras R= 1.20,1.0,12 end
* xyz 0 1
N 0 0 0
N 0 0 {R}
*
• Do these calculations and compare your results with the experimental data in
Table 1. Also, compare the D-‐values that you get with the ones obtained from
the Morse fits. Are the Morse-‐fit values reliable?
ADDITIONAL CALCULATIONS IV (VOLUNTARY):
• The only of these molecules where DFT is expected to provide a reasonable
description is NH.
• Do a DFT/TD-‐DFT calculations on the ground-‐ and excited states of NH and
compare your results with the ab initio calculations. How accurate is the DFT
methodology?