Practical QUantum

Practical Quantum Chemistry

Script of a crash course held by

Florian Muller-Plathe

March 16th–20th, 1998

Lecture notes prepared by

Patrick Ahlrichs, Roland Faller, Oliver Hahn, Mathias P¨utz, Dirk Reith, Yannick Rouault, HeikoSchmitz and Thomas Soddemann

Contents

1 Introduction 1

2 Quantum Mechanics 2

2.1 Some useful postulates . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 The Born-Oppenheimer approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Units, Symbols and Notations . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3.1 Atomic units . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3.3 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.3.1 Dirac bras and kets: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.3.2 Two electron integrals notation: . . . . . . . . . . . . . . . . . . . . . . . 4

3 Hartree-Fock 5

3.1 Hartree approximation . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Pauli’s antisymmetry principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Hartree-Fock . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.4 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.5 Hartree-Fock methods . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6 Hartree-Fock-Roothaan equations (for RHF) . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.6.1 Digression: How to solve the matrix equation? . . . . . . . . . . . . . . . . . . . . 9

3.7 Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.8 Iterative scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.9 Hartree-Fock-Roothaan implementation . . . . . . . . .. . . . . . . . . . . . . . . . . . . 10

3.9.1 Classical SCF . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.9.2 Direct SCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.9.3 Speed up Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.10 Koopman’s theorem . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Basis functions - basis sets 13

4.1 Introduction . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

i

Contents

4.1.1 History . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Slater-type orbitals . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Gaussian-type orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3.1 Radial part . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3.2 Angular part .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3.3 Advantages and disadvantages of GTO’s . . . . . . . . . . . . . . . . . . . . . . . 15

4.4 Application of basis functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 CGTO (contracted Gaussians) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6 Basis set size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.7 Polarization functions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.8 Gaussian exponents . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.9 Basis-set superposition error . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Electron correlation 22

5.1 Introduction . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Non-dynamic correlation effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3 Dynamic correlation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3.1 How much of the electron correlation is described by a single determinant? . . . . . 25

5.3.2 Correlation energy . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3.3 How to account for electron correlation . . . . . . . . .. . . . . . . . . . . . . . . 26

6 Configuration Interaction 28

6.1 Properties of CI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.2 Matrix elements between Slater determinants . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 Properties of the CI expansion . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Review of Perturbation Theory 32

7.1 Auxiliary theorem . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2 How to calculate n-th order energy?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.3 Treating electron correlation via perturbation theory . . . . . . . . . . . . . . . . . . . . . . 35

7.4 Second-Order Energy (MBPT) . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Size Consistency 38

8.1 Size Consistency of HF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.2 Size Consistency of MBPT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.3 Trouble: Double CI Size Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Derivatives of the Energy 42

ii

Contents

9.1 The Potential energy surface . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.2 How to find stationary points? . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.3 Physical meaning of some derivatives of the potential energy . . . . . . . . . . .. . . . . . 44

9.4 The Hellmann-Feynman theorem . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 44

9.5 Gradients for RHF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Density functional theory (DFT) 49

10.1 An illustrative example: The Thomas Fermi model . . . . . . . . . . . . . . . . . . . . . . 49

10.2 The Hohenberg-Kohn theorems . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.3 The Kohn-Sham (KS) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.4 Exchange (and) correlation functionals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10.5 DFT in a Gaussian orbital basis . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 55

11 The Car-Parrinello Method 57

11.1 Plane wave basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.2 The Kohn-Sham equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.3 Pseudopotential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.4 CP Molecular dynamics . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

iii

1 Introduction

There have been many method developed to describe atoms and molecules in term of quantum mechanis.Due to the fact that it is nearly impossible to calculate the electron structure of molecules analytically correct,all those methods are approximations.

To the most known and used mehtods belong

� ab initio methods

– Hartree Fock

– electron correlation (configuration interaction, many body perturbation theory )

� semiempirical methods

� density functional theory

In the following the range of the practial applicability of the above methods will be discussed.

The choice of a basis sets as well as the chemist jargon (direct SCF, MP2, MNDO 6-31G**) will no longerbe a mysterium to us. The gained knowlegde can than be applied to critical reading of theoretical (andexperimental) literature.

In this lecture, the following topics have not been covered

� foundations of quantum mechanics (recommended)

� thorough derivation of approximations

� advanced theoretical methods

� latest developments

1

2 Quantum Mechanics

2.1 Some useful postulates

� Postulate 1

The state of the system is described by a wave function

��r�� r�� rn� t��

� is single valued, continuous, and quadratically integrable:Zj�j�dr �

Z��dr � finite

� Postulate 2

To every physical observableA there corresponds a linear Hermitian operator�A. To find the operator,expressA in cartesian coordinates and momenta, and substitute

r �� r

p �� p � �i�rExamples:

dipole moment� �P

i qiri, �� P

i qi �ri

kinetic energyK �P

i p�i ��mi, �K � �Pi �

�r�i ��mi

� Postulate 5

The expectation value of�A is obtained as

h �Ai � h�� A�ih��i

� Postulate 6

The time development of a system is given by the time dependent Schr¨odinger equation

i��t��t� � �H��t�

where �H is the Hamiltonian operator

�H � �V �r��

�mr�

of the stationary system.

The time independent Schr¨odinger equation is given by the famous eigenvalue equation

�H� � E��

� Postulate 7 (Spin)

Pauli exclusion (anti symmetry) principle for Fermions.

2

2.2 The Born-Oppenheimer approximation

2.2 The Born-Oppenheimer approximation

Quantum chemistry is the quantum mechanical treatment of molecules. We are dealing here with the BornOppenheimer approximation which separates electronic and nuclear degrees of freedom. WithVel � V �R�being the electronical potential andVnuc � Eel the nuclear one, the wave function separates within the BornOppenheimer approximation into two parts:

��R� r� � ��R� ��r�

R are the nuclear coordinates andr the electronic ones. This approximation makes sense because the nucleiare much heavier than the electrons, therefore the lighter electrons will instantaneously adapt to the nuclearconformation.

But the Born Oppenheimer Aprroximation has restrictions. It breaks down if

� �Eel � �Evib�rot

� photoionisation or electron-molecule scattering takes place,

� the temperature is high.

Within the Born-Oppenheimer approximation the nuclei are often described as classical objects.

2.3 Units, Symbols and Notations

2.3.1 Atomic units

length a� – bohr �� menergy EH – hartree �� J

EH � e�

��a�mass me – electron mass �� kgcharge e – (elementary charge) �� Cangular momentum � �� Jselectric dipole moment ea� �� Cmelectric dipole polarisability e�a��EH �� C�m�Jelectric field EH�ea� �� V�m

wave function q�� m��

2.3.2 Symbols

� wave function in general� Slater determinant� molecular orbital (MO)� spin MO, often�i�� i, �i�� i AO (basis function)�a vectorA matrix, tensor�ri position of electron i�i spin of electron i (� ��i�� i�)�xi spin coordinate of electron i, i.e.xi � ��ri� �i�

3

2 Quantum Mechanics

2.3.3 Notations

2.3.3.1 Dirac bras and kets:

j�ii = �i

h�ij = �i�

A j�ii = �A�i

h�ijA = �i� �A

h�ij�ji =R��i�j

h�ijA j�ji =R��i�A�j.

2.3.3.2 Two electron integrals notation:

� physicists’ notation

hijjkli �ZZ

�i��x��j��x��

r��k��x��l��x��d

�x�d�x�

�i� �x�� spin orbital i, occupied by electron 1.

� chemists’ notation

�ijjkl� �ZZ

�i��x��j��x��

r��k

��x��l��x��d�x�d�x�

�ijjkl� � hikjjli

� antisymmetrised 2-el. integrals

hijjjkli � hijjkli � hijjlki� �ikjjl� � �iljjk�

� spatial orbitals� chemists’ notation

�ijjkl� �ZZ

�i��x��j��x��

r��k

��x��l��x��d�x�d�x�

� index permutation symmetries

hijjkli � hjijlkihijjkli � hkljiji�

�complex orbitals

�ijjkl� � �kljij� real or complex�ijjkl� � �jijkl� � �ijjlk� � �jijlk� real

4

3 Hartree-Fock

The electronic Schr¨odinger Equation is given by

H� � E� (3.1)

whereH is the all electron Hamiltonian

H � H� �H� �H� (3.2)

H� is the one electron hamiltonian consisting of kinetic energy and nuclear attraction

H �NXi

�

�r� �

XA

ZAriA

(3.3)

H� is th 2-electron interaction (repulsion)

H� �

NXi

NXj�i

rij(3.4)

andH� is the repulsion of nuclei in molecules (mostly treated clasically)

H� �XA

XB�A

ZAZBRAB

(3.5)

� � ��r�� r�� r�� rN � is the all electron wave function andE denotes the total (electronic) energy.

3.1 Hartree approximation

In the one electron electron theory the, following approximation is made:

� � ��r�� r�� r�� rN � � ��r��r��r�� N ��rN � (3.6)

which is a separation ansatz and named ’The Hartree Approximation’.�i��ri� are the one electron wavefunctions, the orbital functions. The operatorh�i� can be defined to act only on the electron i and theproblem can be rewritten as a system of N one electron wave functions

hi�i��ri� � i�i��ri� (3.7)

5

3 Hartree-Fock

3.2 Pauli’s antisymmetry principle

When 2 electrons (which are of course Fermions) are exchanged,� changes sign. The aim is now to derivea approximation which is as easy to handle as the Hartree one and on the other side obeys Pauli’s principle.The solution is to use Slater determinants. An exchange of two electrons in a 2 spin orbital can be written as

� � �p��

(permutation of 1 and 2:) �p��

Written as a (Slater) determinant

� �p�

��

�� (3.8)

The exchange of two electrons is equal to the exchange of two rows in the determinant which results in achange of sign. In general:

� �p�

�� N �� N ��

...... � � �

...��N� ��N� � � � �N �N�

�� (3.9)

Shorthand:� � jj�� N jj

3.3 Hartree-Fock

1. restrict trial wave function to one Slater determinant

2. effective one electron Hamiltonian

(a) hi � ��r� �PA Za�riA� modifications� Huckel, extended H¨uckel �� Hartree-Fock

(b) mean-field approximation: one electron moves in the average field of all other electrons�Hartree-Fock / self-consitent field (SCF)

� HF is variational� HF energy� true energy

� HF gives the Slater determinant with the lowest possible energy (HF gives the best one determinantwave function)

� Derivation: see Szabo/Ostlund

� result fori � � � � � � N :f�i � i�i

Here,f is the Fock operator

f��

�r� �

XA

ZAriA

�

NXj�

�Jj�� Kj�� (3.10)

6

3.4 Remarks

�Jj�� is the Coulomb operator, effective one-electron operator defined by its action on�i��

h�i��j �Jj�� j�i��i �Z

��i ��i��

r��j��j��d�r�d�r�

� �iijjj� � hijjijiCoulomb integral�Kj�� exchange operator

h�i��j �Kj�� j�i��i �Z

��j��i��

r��i ��j��d�r�d�r� � �ijjij� � hijjjii

exchange integral

3.4 Remarks

� Coulomb integral can be interpreted as repulsion betweeen two charge densities� � ��

� Exchange integral has no obvious interpretation: Has nothing to do with physical exchange of elec-trons.

� Non local potential:

– Schrodinger equation� operator is known� finding of wave functions and energies.

– HF equations: operator dpends implucitly on wave functions� iterative solution, self consitentfield (SCF) method.

3.5 Hartree-Fock methods

There are different types of Hartree-Fock methods

� restricted Hartree-Fock (RHF) closed-shell singlett

i

x��

spatial MO’s doubly occupied (spin�, spin)

� restricted open-shell Hartree Fock (ROHF)

i

x��

��

spatial MO’s singly or doubly occupied

� unrestricted Hartree-Fock (UHF)

i

x��

��

different spatial MO’s for� and spins not E.F. of�S�

7

3 Hartree-Fock

Example: Li atom

ROHF:js��sj doublet

UHF: js�s��s�j lower energy, but not pure doublet

3.6 Hartree-Fock-Roothaan equations (for RHF)

HF equations are integro-differential equations which can be solved numerically for atoms (and diatomics)only. Here, a basis set expansion is introduced (mostly atomic orbitals)

RHF equations for closed-shell singlets

�F ��i�� i�i�� i � � � � � � Ne�� (3.11)

�i are the spatial MO’s.

�F �� h�� Xj

�� Jj�� Kj�� (3.12)

�i �Xq

Ciqq� �i� q � � � � � � N� (3.13)

�� C � (3.14)

�FXq

Ciqq � iXq

Ciqq (3.15)

Zp �F

Xq

Ciqq d� � i

ZpXq

Ciqq d� (3.16)

Xq

Ciq

Zp �Fq d� � i

Xq

Ciq

Zpq d� (3.17)

Xq

CiqFpq � iXq

CiqSpq (3.18)

F C � S C �diag� (3.19)

whereF is the Fock matrix,C the coefficient matrix,S the overlap matrix and �diag� the orbital energies.

C yF C � C yS C (3.20)

non-symmetric matrix eigenvalue problem. Transformation to a symmetric matrix� diagonalise� ,C

8

3.7 Matrix elements

3.6.1 Digression: How to solve the matrix equation?

� If S were the identity matrixE , then we would have

F C � C �diag�

C yF C � C yC �diag� � �diag� (3.21)

notice thatC yC � E . From a diagonalisation ofF one obtainsC and �diag�.

� Solution: makeS � E (in generalS �� E ) by choosing a new orthogonal basis: L¨odwin symmetricorthogonalisationS S�� E alsoS��S S �� E

How can we obtainS��?

1. diagonalizeS : U yS U � s (diag)

2. with sii � s��ii one obtainss��

3. ’undiagonalize’s�� with the back transformationU s��U y � S ��

Check:S ��S S �� U s��U yS U s��U y � U s��s s��U y � U U y � E

� transform RHF equationsC � � S ��C , C � S ��C �

F S ��C � � S S ��C � �diag�

S ��F S ��C � � S ��S S ��C � �diag�

F �C � � C � �diag�

3.7 Matrix elements

overlap matrix: Spq �R��p q � hpjqi

Fock matrix: Fpq �R��p

�Fq � hpjF jqi

Fpq �

Zp��h��

Xj

�� Jj�� Kj��q�� d�

� hpq � �

Zp��

��Xj

�j��

r��j��

��q�� Zp�� d�

��Xj

�j��

r��j��

��q�� d�� hpq � �

Xj

Zp��j��

r��j��q�� d� �

Xj

Zp��j��

r��j��q�� d�

Now �j �P

r cjrr, so

Fpq � hpq � �Xjrs

cjrcjs

Zp��r��

r��s��q�� d�

�Xjrs

cjrcjs

Zp��r��

r��s��q�� d�

We introduce now the denity matrixR � CyC ,Rrs �P

j cjrcjs

Fpq � hpq �Xrs

Rrs��pqjrs�� psjrq�� (3.22)

9

3 Hartree-Fock

3.8 Iterative scheme

initial guess�� Fdiagonalize�� C �� R� z

iterate

converged?�� stop

3.9 Hartree-Fock-Roothaan implementation

3.9.1 Classical SCF

Algorithm:

1. Calculate integralsSpq, hpq, �pq j rs� and store the results on disk

2. CalculateS

��.

3. Initial guess at MO coefficientsC

(semiempirical, only�h, smaller basis)

4. ConstructR� C

yC

5. Constuct the Fock matrixF

This is done integral-driven: a batch of integrals is read from disk and their contributions summed upinto F

. The storage for the matricesS

� S

�� C� R

andF

is ofO�N��.

6. Transform the Fock matrix:F

� � S

��FS

��

7. Diagonalize the transformed Fock matrix:C

�yFC

� �

diag �� C� S

��C

�

8. Convergence-Test:if energy change and density matrix change can be tolerated accept the actual matrices, else go backand repeat steps 4-8.

3.9.2 Direct SCF

The algorithm of the direct SCF implementation is slightly modified, specifically in the steps 1 and 5:

1. Calculate integralsSpq andhpq and keep them in the memory.

2. CalculateS

��

3. Initial guess at MO coefficientsC

4. ConstructR� C

yC

5. Constuct all �pq j rs� orbitals and sum them into F

6. Transform the Fock matrix:F

� � S

��FS

��

7. Diagonalize the transformed Fock matrix

10

3.10 Koopman’s theorem

8. Convergence-Test:if energly change and density matrix change can be tolerated , accept the actual matrices, else go backand repeat steps 4-8.

The changes have one minor disadvantage and several advantages. The disadvantage is, that one has tocalculate the 2-el integralsNiter times. Nowadays, one typically has value ofNiter � ��. The advantagesare: first, that there is no disk space neccesary for the�

� �N� � �N� � �N� � �N� integrals whereN is the

number of basis functions. (Example:N � �� corresponds to� � �� integrals� 1.6 GB). Second, theexecution time is sped up since the slow disk I/O is avoided. Third, the algorithm can be nicely parallelized.

To summarize the difference between the classical and direct SCF, one can state: direct SCF is especiallysuitable if the target problem involves large basis sets (withN � ��) and the target computer architectureprovides fast CPUs and slow I/Os. Since that is true in almost every case today, direct SCF is widely applied,in contrast to classical SCF.

3.9.3 Speed up Hartree-Fock

� Symmetry arguments:transform AO basis to symmetry-adapted basis. Thus,S

andF

factorize into blocks and many inte-

grals disappear. Observe however, that most molecules of interest are non-symmetric objects and onecannot exploit symmetry.

� Convergence accelerators:

1. Direct inversion of the iterative subspace DIIS.

2. TreatE as a function ofC

and extrapolate.

3. CalculateC

n�� fromC

n,C

n��,C

n��, ...

� Integral cutoff

1. Neglect�pq j rs� if �pq j rs� � �� EH

2. Ignore whole batches of integrals even before the begin of the calculations, e.g. ignore alld-orbitals� ��

� � �� integrals.

The consequences of these actions are advantageous in both methods: in classical SCF, the storage and I/Otime are reduced and in direct SCF, the cpu calculation time is reduced.

3.10 Koopman’s theorem

We assume an N-electron Hartree-Fock single determinant� with occupied spin orbitals of energies o andvirtual spin orbitals of energies v. Herein, the virtual orbitals are artifacts of the method. They are basisset dependent andnot convergent when applying greater basis sets since there is only the orthogonalityrequirement which is not sufficient for convergence.

However, Koopmans’ theorem gives us a way of calculating approximate ionization potentials and electronaffinities. He assumes (“frozen orbital approximation”) that the spin orbitals in the�N � �-electron states,i.e. the positive and negative ions are identical with those of theN -electron state.Clearly, this approximation neglects relaxation of the spin orbitals in the�N � �-electron states.

11

3 Hartree-Fock

The theorem states, that the energy level of the highest occupied state can be interpreted as a first approxi-mation of the ionization potential of theN -electron state and that the energy level of the lowest virtual statecan be interpreted as electron affinity.

Optimizing the spin orbitals in the�N � �-electron single determinants by performing a separate Hartree-Fock calculation on these states would be a more reasonable, but costly performance. Doing so, we wouldget lower energy levels, thus the neglect of relaxation in Koopmans’ theorem tends to produce too positveionization potentials and too negative electron affinities. Generally, the electron affinities obtained by Koop-mans’ theorem are worse than the ionization potentials since the virtual states do not converge.

12

4 Basis functions - basis sets

4.1 Introduction

Modern quantum chemistry uses the expansion theorem to represent the wave functionj�i of a molecularsystems in terms of basis functions. The following nomenclature is used:

j�i �PiCi j�ii (4.1)

j�ii � �A j��ni (4.2)

j�ki � j�k�ki (4.3)

j�ki �P�Ck� j�i (4.4)

where�i is the Slater determinant of the many-electron basis,�k are the spin orbitals,�k are the spatialmolecular orbitals,�k is the spin function (� or ) and� is the one-electron basis function (e.g. oftenatomic orbitals).

f�g denotes the basis set. Observe, that if the basis (both one-electron and many electron) is complete,the wave function would be exact. However, a complete basis involves an infinite number of basis functionswhich is impossible to realize. Thus, there is always a basis set truncation error.

4.1.1 History

In the theory of the hydrogen atom, the following variables and equations are of interest:

nlm � Rnl�r��lm��m�� (4.5)

�m��

��eim (4.6)

�lm��

��l � ��l � jmj�

��l � jmj��

Pjmjl �cos�� (4.7)

Rnl�r� � � �

�Z

na�

�� n� l � ��

�nf�n� l��g��

L�l��n�l ��

le�� (4.8)

13


ϑ

z

x

y

ϕ

Figure 4.1: Definition of spherical coordinates

where� � �Zna�

r anda� � h�

��e�(Bohr radius),P jmj

l the associate Legendre polynomials,L�l��n�l the asso-

ciate Laguerre polynomials. Hydrogen functions have the important property that they satisfy the followingorthonormality relation:

�nlmjn�l�m��

Z �

�

Z �

�

Z ��

��nlm�r� �� n� l�m��r� �� r� sin�d�d�dr

� �nn��ll��mm� (4.9)

The advantage of this method is thatS� E

due to the orthonormality relation, i.e. many integrals vanish.

But on the other hand, there are several severe disadvantages: first, the orthonormality relation only holdsfor one-atom systems and second, the integral evaluation is nearly impossible. Hence, the practical use islimited to atomic systems or to one-centre expansions which are exotic in molecular systems.

4.2 Slater-type orbitals

This method is a variation of hydrogen-like functions. The angular part�lm��m�� is still evaluated likehydrogen orbitals, hence we keep the angular orthogonality. What is new is a simplification of the radialpart: one applies some polynomial of simple power instead of the Laguerre polynomials:

Rn�r� ��n��p

��n��rn�� exp��r� (4.10)

where� � Z�sn� is the Slater exponent with screenings and the effective radial quantum numbern�. There

are empirical rules (“Slater rules”) to determine the concrete values fors andn�.As consequence, the radial functions are no longer orthogonal. On the other hand, the two-center integralsare now tractable and hence, the primary use of this method are calculations of diatomic molecules.

14

4.3 Gaussian-type orbitals


This method is one of the major inventions in Quantum Chemistry. It goes back to Boys in 1950.

4.3.1 Radial part

The main idea is to substitute polynomials or exponential functions ofr with a Gaussian function:

exp��r� �� exp��r��

It follows, that the functional form of all s-orbitals is the same. Specifically, the function for the 1s-orbitalreads as:

g�s�r� �

��

��

��

exp��r�� (4.11)

4.3.2 Angular part

Generally, there are two possibilities to calculate the angular part of the integral: either like in the hy-drogen atom by integrating over�lm��m�� or by executing the integral over cartesian coordinatesxlymznexp��r��. The latter would e.g. involve the following functions for ap- andd-orbital:

g�px�r� � ��

��x exp��r�� (4.12)

g�dxy�r� � ��

��xy exp��r�� (4.13)

In practice, one often proceeds as follows:

� Evaluate integrals over cartesian Gaussians, since that is fairly easy.

� Transform to “spherical Gaussians”Observe: Cartesiand and higher sets give rise to spurious functionsWe give an example:�d-orbitals:x� � y�� r� � z�� xy� xz� yz (5 functions).cartesian:x�� y�� z�� xy� xz� yz (6 functions).Hence, there is one redundant function in the Cartesian set. After symmetry adaption, that functioncan be identified with the 3s-orbital:x� � y� � z� � r� � “3s”. Redundant functions make the basisset unnecessarily big, and can cause linear-dependency problems.

4.3.3 Advantages and disadvantages of GTO’s

Figure 4.2 illustrates the major disadvantage of GTO’s:

Several GTO’s per STO are needed to describe the nuclear cusp and to mimic the long-range bahaviouraccurately. The major advantage of GTO’s is that the integral evaluation becomes easier, faster and worksalso for 3 or 4 centers. That is due to the following reasons:

15


nuclear "cusp"Ψ

r

GTO

STO

Figure 4.2: Sketch of an GTO and an STO

1. A product of two Gaussians is a Gaussian.Example: Unnormalized 1s GTO’s

g�s��r � �RA�g�s��r � �RB� � exp

��

��RA

��exp

��

��RB

��

� exp

��

��

��RA � �RB

��exp

��

��r � �RP

��(4.14)

with �RP � ��RA�� RB�� .

For general cartesian GTO’s:

�x�X lA��y � Y m

A ��z � ZnA� exp

��

��r � �RA

�� (4.15)

�x�X l�B��y � Y m�

B ��z � Zn�B � exp

��

��r � �RB

��

Pl�l� �x�Pm�m� �y�Pn�n� �z� exp

��

��x� �XA

��y � �YA

��z � �ZA

��

exp

��

��x� �XB

��y � �YB

��z � �ZB

��

i.e. the GTO’s factorize into x, y and z parts. We can take advantage of that, as can be seen in thefollowing example of�x orbitals:

16


β

RA

RP

RBα

α+β

Figure 4.3: Illustration of the product of two 1s Gaussians.

�x�XA� �x�XB� exp

��

��r � �RA

��exp

��

��r � �RB

�� (4.16)

�x�XA� �x�XB� exp

��

��

��RA � �RB

��exp

��

��r � �RP

��

exp

��

��

��RA � �RB

��x�XP �

� exp

��

��r � �RP

��xP � xA � xP � xB��x�XP � exp

��

��r � �RP

��xP � xA��xP � xB� exp

��

��r � �RP

��

i.e. the product of twopx orbitals becomes a sum of ans, apx, and adxx orbital at the center�P .

2. Laplace transform

The �r (Coulomb) operator can be transformed into a Gaussian.

r�

p�

Z �

�exp

��sr��psds (4.17)

e.g. nuclear attraction integral:ZZZexp

��

��r � �RA

��

j�r � �Rcjexp

��

��r � �RB

��d�r � (4.18)

C �ZZZZ

exp

��

��r � �RP

��ps exp

h�s �r �Rc�

�id�rds

This is similar for electron-repulsion integrals (2-electron-integrals):Z� � �

Zexp

��

��r� � �RA

��exp

��

��r� � �RB

�� (4.19)

j�r� � �r�j exp��

��r� � �RC

��exp

��

��r� � �RD

��d�r�d�r�

17


Thus, all integrals over GTO’s can be calculated analytically except for a non-analytical 1-dimensionalintegral

Fm�W � �

Z �

�t�m exp��Wt��dt (4.20)

that has to be calculated numerically (asymptotic series, continued function, power series, Gaussintegration, gamma function, table interpolation, Chebyshev polynomials,....)

4.4 Application of basis functions

In the following, we summarize the typical field of applications for the different methods.Atoms

� STO and numerical methods (finite differences inr)

Diatomics

� GTO

� (STO)

� (numerical methods: 2D finite differences, partial waves, 2D finite elements)

Polyatomics

� STO (semiempirical)

� GTO (ab-initio)

Continuum (and solid state)

� Plane wavesexp��i�r � �p� connected to STO’s

Exotic and historical

� One-center expansions (hydrids)

� lobe functions: s-GTO’s, but off-centre

� Floating spherical GTO’s: same, but makes position variational parameter.

4.5 CGTO (contracted Gaussians)

The GTOs have some deficiencies. They don’t give the nuclear cusp in a satisfactory way and they are shortrange. If one doesn’t want to use a huge basis it is not easy to overcome this. One possible solution is to fixa certain linear combination of Gaussians, i.e. “ contract “ them

lmn � xlymznXi

di exp��ir�� (4.21)

Thedi and�i are adjusted once and for all to satisfy

18

4.5 CGTO (contracted Gaussians)

Carbon Huzinaga DZ

� contraction coefficientS 6 1

4232.61 0.002029634.882 0.015535146.097 0.07541142.4974 0.25712114.1892 0.596551.9666 0.242517S 1 1

5.1477 1.0S 1 1

0.4962 1.0S 1 1

0.1533 1.0P 4 1

18.1577 0.0185343.9864 0.1154421.1429 0.3862060.3594 0.640089P 1 1

0.1146 1.0

Carbon 6-311G

� contraction coefficientS 6 1

4563.24 0.0019665682.024 0.015236154.973 0.076126944.44553 0.26080113.029 0.6164621.82773 0.221006SP 3 1

20.9642 0.11466 / 0.04024874.80331 0.919999 / 0.2375941.45933 -0.00303068 / 0.815854SP 1 1

0.483456 1.0 / 1.0SP 1 1

0.145585 1.0 / 1.0SP 1 1 diffuse function

6-311G+0.0438 1 / 1D 1 1 polarization function

6-311G*0.626 1.0

Table 4.1: Two different carbon basis sets as tabulated in quantum chemistry programs. On the left,the Huzinaga-Dunning double-zeta (DZ) set, consisting of 1 contracted ”s” (consisting of 6 GTO’s), 3uncontracteds, 1 contracted ”p” (consisting of 4 GTO’s) and 1 uncontracted basis functions. On the right,the 6-311G basis set, consisting of 1 contracted ”s” (consisting of 6 GTO’s), 1 contracted “sp” and 2uncontracted “sp” basis functions. Note that for the “sp”, � is the same fors andp, leading to efficientcomputation of the prefactors in products of two Gaussians. Additionally, the diffuse function and thepolarization function is given for the 6-311G case (cf. section 4.8)

� least-square STO (� STO-3G [3 Gaussians for a Slater], STO-4G, ...)

� atomic HF calculation (Dunning-Huzinaga)

� atomic correlated calculations (ANO). This is the most recent method.

Typical applications are highly contracted (lots of Gaussians fixed together) inner shells to model the nuclearcusp. These won’t change in the calculation. The valence shells need more flexibility so one uses not sostrong contraction or no contraction at all.

A typical example is 6-31G which means that the 1s orbital is modeled by a contraction of 6 primitive GTOs.For the 2s and 2p orbitals one uses 2 GTOs each one CGTO consisting of 3 primitive ones and one beinga bare one. CGTOs sometimes come with notations like (9s5p)�[4s2p] what means that for the s-orbital 9primitives are contracted to 4 CGTOs and 5 for 2 for the p-orbitals respectively.

19


4.6 Basis set size

1. There areminimum basis sets but except for few semi-empirical questions they are nowadays out ofuse. One has one basis function per occupied atomic orbital. However, shells have to be completed:In the boron atom, for example, only onep orbital is occupied. The other twop orbitals have to beincluded into the basis set, in order to complete the description of a boron atom in a molecule. Theabinitio names go with STO-3G, MINI or similar. But most use is in semi-empirical calculations.

2. The next (and probably minimal useful) case is thesplit valence case. For the inner shells one functionper orbital and for the outer two per orbital are used. So names like 3-21G mean 1 for the innermostand two for the next orbitals.

3. More accuracy providedouble-zeta, triple-zeta ... which means 2,3... functions per atomic orbital.The name is historically fundated cause in the days of STOs the zeta was in the exponentexp��r�.

4.7 Polarization functions

��

��

electric field, lower symmetryunperturbed spherical

Figure 4.4: Helium atom in spherical state or polarized

Polarization functions are, e.g., necessary in the following situations:

� The Helium atom in its ground state and in absence of external fields may well be described by a 1ssystem. But the lower (cylindrical symmetry) needs additional (2p) orbitals. Also to form bonds oneneeds p-orbitals as is seen in Li�.

� The Lithium atom lives in a 1s��s state which is not adequate for a bonded Li.

� In Be (s��s�) electron correlation plays a role and low lying p (and d) states mix into the groundstate.

� Also the calculation of excitation spectra for singly excited states with high transition moments needspolarization functions.

H� � ��g � ��

Usually one uses polarization functions only for the valence shell. The corresponding notation (e.g. inGaussian) reads in the case of one polarization function per atom:

� 6-31G* : 1 set ofd functions on atoms heavier than He,

� 6-31G** : like 6-31G* additionally 1 set ofp functions on H.

In the case of more than one polarization function per atom, you need to specify it explicitly. 6-311g(�df� �p)means in this context 3d functions and onef function on Li and higher additionally 2p functions onhydrogen.

20

4.8 Gaussian exponents

4.8 Gaussian exponents

The exponents of the Gaussians often form a geometric series:

�i�i��

� � const� (4.22)

Basis sets of this kind are often referred to as “even tempered basis sets”. This may be used if for somereason a basis set has to be “continued” i.e. an additional function is needed.

For very polarizable molecules, negative ions or Rydberg states so-called “diffuse functions” are used. Thismeans that additionally one s and one p functions are used. The respective Gaussian notation is 6-311+gor 6-311++g for heavy atoms only or for hydrogens also. An example for the resulting basis set is given intable 4.5

4.9 Basis-set superposition error

In the study of interacting atoms almost always one encounters the problem of basis-set superposition error(BSSE).

If the basis of atom A is incomplete (i.e. always) the basis of B will improve the wave function at A andtherefore lower the energy. This BSSE can come to the same order of magnitude as the interaction energyitself. As an example consider the water dimer:

�E � E�H�O�� E�H�O��

The different methods come with the following energies

method �E�SCF ��kJ mol�� BSSE/kJ mol��

STO 4G -26.4 -244-31G -32.2 �4�� -19.8 �3HF limit -15.4

If one wants to correct for the BSSE theCounterpoise correction may be appropriate. One calculates Awith its own basis and with B’s basis (but without B). This method has some rigor for HF. Little is provenfor higher approximations but it is used rather extensively.

��

��

BA

Figure 4.5: Illustration of basis set superposition error (BSSE)

21

5 Electron correlation

5.1 Introduction

Electrons interact via a �j�r��r�j �

�r��

potential. So each electron “feels” what all the others do instanta-neously (if we disregard relativistic delay). So the motion of the electrons iscorrelated.

In the Hartree-Fock approximation one electron only sees the average distribution of the others not theirinstantaneous position. So due to thismean-field approximation the treatment of electron correlation isincomplete.

5.2 Non-dynamic correlation effects

So correlated electron methods are in general more reliable than HF. One now distinguishes betweendy-namic andnon-dynamic correlation effects. The names are somewhat misguiding. Thenon-dynamic effectsare just everything for which single-determinant HF is not appropriate. But this has nothing to do withcorrelation effects. This deficiencies can be resolved typically by quite simple means. As an example welook on the dissociation of H� in minimum basis HF1. Let us write down the wave function

�RHF �p�� (5.1)

� ��sA � sBp

�� (5.2)

The overbar denotes the opposite spin. Now we have to perform some algebra

�RHF �

��sA�� sB��sA�� sB�� sA�� sB��sA�� sB�� (5.3)

�

��sA��sA�� sA��sB�� sB��sA�� sB��sB��

�sA��sA�� sA��sB�� sB��sA�� sB��sB��

�

��sA��sA�� sA��sA�� sB��sB�� sB��sB��

�sA��sB�� sA��sB�� sB��sA�� sB��sA��

Due to the symmetry of the problem with regard to exchanging nucleusA andB the first two terms and thelast two terms of the last equation are identical, leading to

�RHF � sA��sA�� sA��sA�� sA��sB�� sA��sB�� (5.4)

At bonding distance this makes no problem. But if we now look at the “molecule” at infinite separation. Inthis case the second term is what we expect, namely two separate nuclei carrying their respective electrons.

1The minimum basis is not causing trouble here. Better basis sets won’t resolve the problem.

22

5.3 Dynamic correlation

But the first term describes two ions an H� and an H�. The energy would be

ERHF �r �� E�H� �

��sAsAjsAsA�� (5.5)

which is not just the two hydrogens. This could be corrected for by 2 determinants (which would in thiscase mean full CI). Also the old valence bond idea could resolve this where one just uses

� �p��sA��sB�� sA��sB�� (5.6)

Also unrestricted Hartree-Fock gave no such problems.


Dynamic correlation manifests itself in real effects of instantaneous interaction of electronic charges. As anexample we discuss the dispersion interactions between atoms or molecules (London, van der Waals). Theyare not accounted for in the HF limit (even for a complete basis set). The interaction between two Argonatoms would be strictly repulsive, so Arliq� should not exist. The problem is that the mean-field center ofcharge does not coincide with the instantaneous center of charge. Due to fluctuations the dipole moment ofthe atom does not vanish for all times.

h��i � �� h�i� �� (5.7)

��t� �� t� �� (5.8)

If you put the two atoms together (R � overlap distance) they feel each other and polarize each othermutually. So they induce dipole moments (and higher multi-poles) upon each other. The interaction of theform

Einter ��A�t��B�ind�t�

R�(5.9)

leads to a lowering in total energy.This interaction is typically written in a series like

Edisper � �C�

R�� C�

R�� C��

R�� (5.10)

If we want to describe dispersive interaction there is no way out from electron correlation, i.e. a singledeterminant cannot do the job. And if it does it does it for the wrong reason, namely BSSE.

Dispersion interaction puts also some requirements to the function in the basis sets. If the unperturbedatom is spherically symmetric (e.g. rare gas) the fluctuating or induced dipole moments may not be de-scribable this way. So one has to use polarization functions. For dipolar interaction at least p functions, forquadrupolar interactions d functions are necessary. So Helium atoms have to be described by

�s�� s�� c��s�p� � ��

Never do electron correlation without polarization functions.

Dispersion usually falls off withR�� as leading term. The repulsion (Pauli’s principle) between two shellsgoes roughly exponentially, because the relevant quantity is the overlap of electron clouds. The wave func-tions fall off exponentially away from the core to leading order.

23


A dipole moment at B can of course be more easily induced if B is more polarizable. One expresses theenergy as Taylor expansion in electric dipole field (static case).

E��F � � E��F � �� perm� � �F �

��F��F � �� (5.11)

�� E

� �F� ��perm� � ��ind� � �� (5.12)

��ind� � ��F � �� (5.13)

Here� is a� �-tensor. So the dispersion goes proportional to the polarizability.

The exact relation (Casimir-Polder, 1947) for the leading coefficient in the dispersion series reads

C� �

Z �

��A�i��B�i��d�� (5.14)

Approximate relations come to

E�RAB� � �

R�AB

�

�

�A�B�IA

� �IB

� IA� IB � 1st ionization potentials (5.15)

(London relation)

� �

R�AB

�

�

�A�Bq�AnA

�q

�BnB

� nA� nB � “effective“ numbers of electrons (5.16)

(Slater-Kirkwood relation)

300.0

200.0

100.0

0.0

100.0

200.0

CH

C H

C H

C H

C H

C H

C HC H

4

2 6

3 8

4 10

5 12

6 14

7 16

8 18

He

Ne

Ar

Kr

Xe

-268.9-245.9

-185.7

-157.3

-107.1-164

-88.6

-42.1

-0.5

36.1

6998.4

125.7

boiling points

Figure 5.1: Boiling points increase with polarizability. The boiling point is a measure of dispersionenergy of non-polar fluids.

24


5.3.1 How much of the electron correlation is described by a single determinant?

An example for the treatment of electron correlations by a single determinant wave function is the following:

Let’s look at two electrons. Their wave function is given by

��x�� x�� k ��r��r�� k with x � ��r� ��

The probability density of finding electrons at the points (�r�� r�) in real space is

p��r�� r��

Zd��d��j��x�� x��j�

�

�

Zd��d��j�� j�

�

�

Zd��d��j��j�j��j�j��j�j��j�

�j��j�j��j�j��j�j��j�� c�c��

�

�

�j��j�j��j� � j��j�j��j��

�

�

��

��

Zd��d��

�� c�c�

�

If the electron spins are antiparallel (�� ) the integral in the last equation vanishes, as antiparallelspin eigenfunctions are orthogonal, so that

p��r�� r��

�

�j��j�j��j� � j��j�j��j��Since the joint probability is just the product of the independent (1-electron) probabilities, the positions ofthe electrons areuncorrelated. Let the electrons now be in the same orbital (�� ) and consider the point�r� � �r� �� r

p��r� �r� � j��r�j� ��

This means that there is a finite probability for the two electrons to be at the same place in space. In realitythis is impossible as there is the Coulomb interaction between the electrons with its singularity at zerodistance. This effect is an artifact of the mean-field approximation, as the electrons are treated as chargeclouds and not as point charges. Let’s now discuss the case of parallel spins (�� ). Here the spinintegral evaluates as 1, so that

p��r�� r��

��j��j�j��j� � j��j�j��j�

�� c�c�

In the special case�r� � �r� �� r this results in

p��r� �r� �

�

�j��j�j��j� � j��j�j��j� � �� c�c

��

So electrons with parallel spins cannot be in the same point in space even if they are in different orbitals(“Fermi hole”).

To put it together the single-determinant wave function

25


exact

EHF

Efull CI

EHF limit

Ecorr

E

complete basis

E

basis set size

small basis better basis

Figure 5.2: Illustration of basis size dependance ofEHF, and defintion ofEcorr

� (partially) correlates electrons with parallel spins.

� does not correlate electrons with opposite spins.

� does not correlate electrons within doubly occupied orbitals.

5.3.2 Correlation energy

A rigorous working definition of the correlation energy was given by L¨owdin. Considering the fact thatHF is based on a variational principle it should give the lowest energy if a complete basis set is used. ThisenergyEHF�limit is the lowest energy that can result from a single slater determinant wave function. If the“exact” energy (Eex) is known, for example from experimental values, subtracting relativistic contributions,the difference between them is defined to be the correlation energy

Ecorr �� EHF�limit �Eexact

� The difference betweenEHF andEfullCI is basis set dependent.Ecorr is only well defined for completebasis sets.

� Never calculate approximations usingEcorr andEHF obtained with different basis sets, for exampleEHF from a large basis andEcorr from a small basis.

� If EHF�limit is wrong because of inadequacy of a single RHF determinant (e.g.H� dissociation) thedefinition ofEcorr makes no physical sense.

5.3.3 How to account for electron correlation

There are several methods to deal with the effect of electron correlation:

26


1. Clever choice of wave functionsThe usual Slater determinant is the antisymmetrised product of one-electron wave functions. A betterbasis is formed by functions that are not orbitals but depend also on the distance between electrons:

�k � �k��r�� r�� r� � �r��

Functions of this type were used by Hylleraas in 1930 to calculate the He atom, by James and Coolidgein 1936 and by Kotos and Wolniewicz around 1960 to calculate theH� molecule. The latter is thebest calculation known.

2. Density functional theory (DFT)DFT is based on the Kohn-Sham equations

E � �

occXi

hii �

�

Zdr�dr�

��r��r��

j�r� � �r�j �Exc��r��r��r��

Exc is a functional of the electron density and its gradient. It is calledexchange correlation. TheKohn-Sham equation would be exact if this functional was known exactly. In real lifeEexc is knownonly approximately and there are many other different approximations. It gives the electron density�instead of the wave function.

3. Valence-bondIf the structure and the dissociation behavior of the molecule is known, the results can be improvedby a good choice of the wave function, but this method is difficult to extend to large systems, as thereare many spins.

4. More determinantsInstead of using only one slater determinant of the occupied orbitals, one can construct the wavefunction from several determinants, in which occupied orbitals are substituted by virtual ones.

� � c�jj��n��njj� c�jj��n��n��jj� ��

where�n�� is a virtual orbital. If only the coefficients of the matrices are optimized the method iscalledConfiguration Interaction (CI), if the orbitals within each determinant are optimized as wellone speaks ofMulti-Configuration SCF (MCSCF).

5. Perturbation theoryStarting from the HF determinant as 0th order wave function, the electron correlation is regarded as aperturbation. This is calledMany-body Perturbation Theory (MBPT).

27

6 Configuration Interaction

In the CI method the trial wave function is a linear combination of many Slater determinants�k, which arethemselves composed of molecular orbitals. These can be taken for example from HF calculations, whichhave the additional advantage of producing orthogonal orbitals. So the eigenvalue equation for the wavefunction

�CI �Xk

Ck�k

has to be solved.

6.1 Properties of CI

CI is trivial in theory:

1. Calculate an orthogonal set of MOsf�ig from a given set of AOs (using for example HF).

2. Construct all possible Slater determinantsf�kg of the MOs (taking into account the spin part as well).

3. Calculate the matrix elements of the Hamilton operatorHkl � h�kjH j�li.

4. Diagonalize H

As a result one obtains the energies and wave functions of all states that can be described by the basis,including excited states. In reality, however, things are a bit more complicated due to the following points:

ad 2. The number of determinants is gigantic. If the basis has gotN elements and there aren electrons,there are �

Nn

��

N �

n��N � n��

determinants. The problem is NP-complete, the number scales factorial withN . A practical exam-ple is Benzene in a 6-31g** standard basis. Here isN � ��, n � �, which gives about��

determinants.

ad 3. The calculation of the Hamilton matrix is even worse, as its dimension is the number of determinants.In the example, one would have to calculate a�� matrix, which needs�� TByte. Even a�� matrix needs almost 1 GByte.

ad 4. The Hamiltonian matrix has to be diagonalized. This is aN� problem.

How can these difficulties be overcome?

28

6.2 Matrix elements between Slater determinants

ad 2. Number of determinantsFull CI is only possible forvery small systems, e.g. double-zeta H�O. In all other cases the numberof determinants has to be restricted.

� Keeping in mind that the influence of higher order excitation determinants on the ground stateenergy is small, the set of determinants can be restricted to single excitation determinants (CIS)or to singles and doubles (CISD).

� The energy contribution of every determinant in a set is estimated (perturbation theory) and alldeterminants the contribution of which lies below a given threshold are discarded.

� It is very important to make use of every symmetry of the problem.

� If one constructs configuration state functions (CSF), which are spin eigenfunctions, the H ma-trix is block-diagonalized. This makes use of spin symmetry.

With those tricks one can get along with�� to �� determinants or CSFs.

ad 3. Storage of the matrixAs it is almost impossible to store H matrix, the matrix elements are recalculated every time they areneeded by the diagonalization algorithm (direct CI). Only the coefficient vectors are stored in memory.

ad 4. DiagonalizationUsually only the lowest eigenvalues and eigenvectors are of interest. To obtain them it is not necessaryto diagonalize the matrix fully. Instead a simple iteration algorithm may be used (Shavitt, Davidson).It can be improved by convergence accelerators.

Another problem is how to store the possible combinations of MOs for the determinants. A very elegantsolution has been found by Duch and Karwowski using graph theory. The electrons and MOs form verticesof a graph. The determinants are represented as paths in this graph.

6.2 Matrix elements between Slater determinants

The calculation of the matrix elements is simplified by some useful rules. The general matrix element is

Hkl � hKjH jLi �

where

H � H� �H� �Xi

hi �Xij

rij�

If more than two orbitals are different in�k and�l the matrix element vanishes. The other cases are:

Case 1: jKi � jLi hKjH� jKi �PN

m hmj h jmihKjH� jKi � �

�

PNm�n hmnj jmni

Case 2: jKi � j��m��i hKjH� jKi � hmjh jpijLi � j��p��i hKjH� jKi �

PNn hmnj jpni

Case 3: jKi � j��mn��i hKjH� jKi � �jLi � j��pq��i hKjH� jKi � hmnj jpqi

29

6 Configuration Interaction

6.3 Properties of the CI expansion

The CI wave function can be written as an expansion in the number of excited states in the determinant. Iflower indices denote occupied orbitals and upper indices mean virtual (=”excited”) orbitals, the CI ansatzup to second order can be written as

j��i � c� j��i�Xa�r

cra j�rai�

Xa�b�r�s

crsab j�rsabi� ��

where�� is the Hatree-Fock determinant. One expects that the singly excited determinants are closest inenergy and will contribute most to the ground state wave function, while higher orders contribute less andless. Consider now an expansion of only two terms

j��i � c� j��i� cra j�rai

The CI equation is given by the secular problem� h��jH j��i h��jH j�rai

h�rajH j��i h�r

ajH j�rai

��c�cra

�� E�

�c�cra

�The mixing between states is determined by the off-diagonal elements.

h�rajH j��i � hajh jri�

�Xb

harjbbi � habjbri � hajF jri � a�ar � ��

F denotes the Fock operator, which is of course diagonal in the basis of the HF orbitals, its eigenfunctions.The generalization of this result is called Brillouin’s theorem:

There is no direct interaction between the HF determinant and singly excited determinants.

This is also true for triply and higher excited determinants (cf. chapter 3). Even though, there are of courseindirect interactions between all determinants.Only doubly excited configurations do have direct interactionwith the HF state, so that they contribute most to the correlation energy. In spite of this, in most calculationsthe singly excited determinants are used as well, because

� is not very expensive,

� they contribute by indirect interaction via doubles,

� when calculating one electron properties the matrix element can be huge and so cancel out the small-ness of the coefficient (e.g. dipole moment).

An example: Dipole moment of CO

Method E �

SCF -112.788 -0.108SCF+138D -113.016 -0.068SCF+200D -113.034 -0.072SCF+138D+62S -113.018 +0.030Experiment +0.044

Another problem occurs when the symmetry of the problem changes in a reaction for example. Look at thereaction 2C�H� �C�H�.

The MOs are as follows:

30

6.3 Properties of the CI expansion

Symm. MO R �� R �shortAA: (1A-1B-2A+2B) �� A � ��B�AS: (1A-1B+2A-2B) �� A � �B�SA: (1A+1B-2A-2B) �� A � ��B�SS: (1A+1B+2A+2B) �� A � ��B�

The energy levels can be depicted in figure 6.1

σΑ + σΒ

R = oo R = short

π1∗ − π2∗

π1∗ + π2∗

π1 − π2

π1 + π2

σΑ∗ − σΒ∗

σΑ − σΒ

σΑ∗ + σΒ∗

Figure 6.1: Energy level crossing for the reaction 2C�H� � C�H�

31

7 Review of Perturbation Theory

The problem to be solved pertubativly is

H j�ii � Ei j�ii � �H� � V � j�ii (7.1)

We assume that the eigenvalue problem

H�

��i

E� E

��i

��i

E(7.2)

can be solved with reasonable accuracy (or even exactly).V is called the perturbation Hamiltonian, whichis known to be much smaller thanH�.We formally introduce a scaled perturbation

H � H� � � � V (7.3)

where we will later take�� .In the following we make use of the notation:

��i

E� jii

Now let us expand the eigenvectors and eigenvalues of our pertubation problem in a Taylor series in�

j�ii � jii� ��

i

E� ��

��i

E� � � � (7.4)

Ei � E��i � �E

��i � ��E

��i � � � � (7.5)

We call thecoefficients �n�i n-th order wave functions and theEn�i n-th order energies.

7.1 Auxiliary theorem

Let us assume that thejii form a properly normalized basis set

hijii � (7.6)

We chose the normalisation of�i such that

hij�ii � (7.7)

This intermediate normalisation is always possible, unlessjii and j�ii are orthogonal to each other (wewont discuss this case, it just complicates the formulas and can be looked up in standard text-books) Thisimplies:

hij�ii � hijii�z��Dij��

i

E� ��

Dij��

i

E� � � �� z

�

32

7.1 Auxiliary theorem

If this is to be true for allvalues of�, it follows

�Dij�n�

i

E� � for all n � �, (7.8)

i.e. higher-order wave functions are orthogonal to the zero-th-order wave function.

Let us plug in the Taylor expansion (7.4) into the full Schr¨odinger equation (7.1):

�H� � � � V �

�jii� �

��i

E� ��

��i

E� � � �

��

�E��i � �E

��i � ��E

��i � � � �

��jii� �

��i

E� ��

��i

E� � � �

��

If this expression is to be true for all values of� the coefficients of various orders of� on both sides mustbe equal:

� ��:H� jii � E

��i which is just the unperturbed Schr¨odinger equation.

� ��:H�

��i

E� V jii � E

��i

��i

E�E

��i jii

� ��:H�

��i

E� V

��i

E� E

��i

��i

E�E

��i

��i

E�E

��i jii

� � �

We multiply these conditions form the left byhij

� ��:hijH� jii � E

��i

� E��i hijii � E

��i (nothing new)

� ��:hijH�

��i

E� hij V jii � E

��i

Dij��

i

E�E

��i hijii

� E��i

Dij��

i

E� hijV jii � E

��i

� E��i � hij V jii since

Dij��

i

E� �.

� ��:hijH�

��i

E� hij V

��i

E� E

��i

Dij��

i

E�E

��i

Dij��

i

E�E

��i hijii

� E��i

Dij��

i

E� hijV

��i

E� E

��i

� E��i � hij V

��i

E� ��:� E

��i � hij V

��i

E...

33


7.2 How to calculate n-th order energy?

1. solve for (n-1)th-order wave function

2. calculateEn�i � hijV

��n��i

EExample: 2nd-order energy

H�

��i

E� V jii � E

��i

��i

E�E

��i jii

�E��i �H�

� ��i

E��V �E

��i

�jii � �V � hij V jii� jii

� multiply by a different eigenfunctionhnj �� hij of H� from the left:

hnj �E��i �H��

��i

E� hnjV jii � hij V jii hnjii

using orthogonality ofhnj andhij we get

E��i

Dnj��

i

E� hnjH�

��i

E� hnjV jii

�E��i �E

��i �

Dnj��

i

E� hnjV jii

Dnj��

i

E� hnjV jii

E��i �E��

i

(7.9)

� Expand��i in eigenfunction ofH�

��i �

Xn

c��n jni � (7.10)

where thec��n are simply given by (7.9):Dmj��

i

E�

Xn

c��n hmjni �Xn

c��n �mn � c��mDij��

i

E� c

��i � �

which finally gives:

��i �

Xn�i

c��n jni �Xn�i

Dnj��

i

Ejni �

Xn�i

jniDnj��

i

E(7.11)

� Now we can take this result for the��i and plug it into the formula for the second order energies:

Ei�� hij V��

i

E�Xn�i

hij V jniDnj��

i

E�

Xn�i

hijV jni hnjV jiiE��i �E

��n

�Xn�i

j hij V jni j�E��i �E

��n

(7.12)

34

7.3 Treating electron correlation via perturbation theory

Summary:What is needed for 2nd-order energy?

1. hij V jniThe integrals of perturbation Hamiltonian between all unpertubed eigenstates ofH�,

2. E��n

and all eigenvalues of the unperturbed Hamiltonian.

Similarly, but more complicated, this holds for higher orders:e.g. the 3rd-order energy:

E��i �

Xn �i

Xm�i

hij V jni hnjV jmi hmjV jii�E

��i �E

��n ��E

��i �E

��m �

�E��i

Xn�i

j hijV jni j��E

��i �E

��n ��

(7.13)

the effect is precalculatable� H � � �

7.3 Treating electron correlation via perturbation theory

Many-body perturbation theory (MBPT)Rayleigh-Schrdinger perturbation theory (RSPT)Moller-Plesset perturbation theory (MPPT,MP2,MP4,...)

� The unperturbed system is described by

H� j��i � E� j��i �whereH� is the Hartree-Fock Hamiltonian (not the Fock-Operator).

H� �

NXi

f�i�

E� �

NXi

i �

NXi

hijH jii�Xi

Xj

hijj jiji

�� EHF �Xa

haj h jai �

�

Xa

Xb

habj jabi

� The perturbation is described by

V � H �H�

� �

NXi

h�xi� �

NXi�

NXj�i

rij��

NXi

f�xi�

� �NXi

h�xi� �NXi�

NXj�i

rij��

NXi

h�xi� �NXi

VHF �xi��

�

NXi�

NXj�i

rij�

NXi

NXi

VHF �xi��

� (total electron-electron repulsion)—(Hartree-Fock electron repulsion)

35


VHF �xi��j�xi� �Xb

hbj

rijjbi�j�xi��

Xb

hbj

rijjji�b�xxi�

� coulomb — exchange

This is equivalently written in our stream-lined notation:

hij VHF jji �Xb

�hibjjbi � hibjbji� �Xb

hibj jjbi (7.14)

� Thus the 1st order energy becomes

E�� h��j V j��i

� h��jXi j

rijj��i � h��j

Xi

VHF �xi� j��i

�

�

Xab

habj jabi �Xa

hajVHF jai

� �

�

Xab

habj jabi

E�� E

��

Xa

a �

�

Xab

habj jabi � EHF

I.e. the first order pertubation correction restores the HF solution !

7.4 Second-Order Energy (MBPT)

Our general formula gives:

E��

Xn��

j h�jV jni j�E

�� E

��n

We knowj�i � j��i, but what are thejni ?

� singly excited states

h��jV j�rai � h��j �H �H�� j�rai� h��jH j�rai� z �h��jH� j�rai

� (Brillouin theorem)

� �h��j f j�rai � haj f jri � a�ar � �

sincejai � jri are eigenfunctions off .

36

7.4 Second-Order Energy (MBPT)

� doubly excited statesjni � j�rsabi

H� j�rsabi � E��abrs j�rsabi � �E

�� a � b � r � s� j�rsabi

E��

Xa b

Xr s

j h��jPi j

�rijj�rsabi j�

a � b � r � s

�Xa b

Xr s

j habj jrsi j� a � b � r � s

�

Xabrs


� triply- and higher-excited states do not interact directly withj��i

MBPT2 MBPT3 MBPT4

singles - - ES�

doubles ED� ED

� �ED� � ED

� �ED� �ED

�

triples - - ET� (expensive)

quadruples - EQ� (don’t need all)

Table 7.1: Contributions to correlation energy. Note: truncated schemes like MBPT4(SDQ) are also sizeconsistent.

CO� (63 basis fcts.) CO� (84 basis fcts.)AO integrals 20 62SCF 10 30integral trans. 27 173MBPT4 S 1 4MBPT4 D 7 88MBPT4 T 11 216MBPT4 Q 6 45total 82 618

Table 7.2: Time effort [CPU min.] (after S. Canuto 1989)

37

8 Size Consistency

One desirable property of quantum-chemical methods has been brought into QC by solid-state people. Insimple words, for a given calculated moleculeA with energyEA one constructs a “supermolecule”AAconsisting of twoA molecules, and treats them as one non-interacting molecule, i.e one runs a calculationwith two A molecules being largely separated. The method is said to besize-consistent if EAA � �EA.In the next three paragraphs, we will check the size consistency for HF, DCI and MBPT2 on the simpleexample ofA � H� in the minimum basis.

8.1 Size Consistency of HF

Remember that the wave function for one H� in the minimal basis HF solution is

�A � j�A�Ai � (8.1)

where� denotes the 1s-MO’s. The energy of the molecule A is then

EA �

�Xi�

h�Ajh j�Ai�

��h�A�Aj j�A�Ai� h�A�Aj j�A�Ai� (8.2)

� � h�Ajh j�Ai� h�A�Aj j�A�Ai �

For the supermoleculeAA one has for the wavefunction

�AA � j�A�A�B�Bi � (8.3)

and the energy

EAA ��Xi�

h�Ajh j�Ai��Xi�

h�B jh j�Bi�

�

�Xa�b�

habj jabi (8.4)

� h�Ajh j�Ai�

��

h�A�Aj j�A�Ai � h�A�Aj j�A�Ai� h�A�Bj j�A�Bi� h�A�B j j�A�Bi� h�A�Aj j�A�Ai� h�A�Aj j�A�Ai � h�A�Bj j�A�Bi� h�A�Bj j�A�Bi� h�B�Aj j�B�Ai� h�B�Aj j�B�Ai� h�B�Bj j�B�Bi � h�B�B j j�B�Bi� h�B�Aj j�B�Ai� h�B�Aj j�B�Ai� h�B�Bj j�B�Bi� h�B�B j j�B�Bi

iThe underlined terms are the self-interaction parts of the two electron integrals, which are all zero (Remem-ber: hijj jkli � hijjkli � hijjlki in physicist notation). The overlined terms are also zero, because we

38

8.2 Size Consistency of MBPT2

consider two H� with distance going to infinity, which means there is no overlap betweenA andB. theremaining non-zero contributions give:

EAA � h�Aj h j�Ai�

�� h��j j��i� (8.5)

� h�Aj h j�Ai� � h��j j��i

Thus we showed on this simple example that the HF method is size consistent. In fact, one can show thatthis is a general result.

8.2 Size Consistency of MBPT2

It is sufficient to consider the second order correction energy only, as we already know that the lower orderterms are equal to the HF terms, which is size consistent.

Remember: The second order energy correction in MBPT2 has the form

E��

Xa b

Xr s

j h��jP

r��ij j�rs

abi j� a � b � r � s

�Xa b

Xr s


� (8.6)

which means in our example for H� (�� j��i, �rsab �

��)E��

j h��j�� j�

� � � � �� (8.7)

Looking at the “H�”-supermolecule (�� j�A�A�B�Bi) we first notice there are two double excited states:

�rsab �

��A��A�B�B� or��A�A��B��B� (8.8)

The two other possible combinations (cross excitations) die on symmetry.

The corresponding second order energy correction is

E�� AA� �

j h�B�Bj��B��B� j�

�� B � ��

B

� �j h�A�Aj

��A��A� j�� A � ��

A

� (8.9)

� �j h��j �� j��

�

Again good news: In this example, MBPT2 is size consistent. Even better news: One can show rigorouslythat MPBT is size consistent

� in all orders

� in all contributions (singles, doubles, ...)

8.3 Trouble: Double CI Size Consistency

To perform our simple example for this case, we notice that for one H�, the determinants with one virtualMO do not contribute directly due to Brillouin’s theorem. But, the indirect contribution is also zero because

39

8 Size Consistency

of symmetry arguments (the HF soulution has�g whereas the singly excited terms have�u). Thereforeonly the doubly excited term�rsab �

�� survives in DCI. Remark: In this simple case, DCI is equivalentto full CI.

For the two non-interacting H�, the same symmetry reasoning yields zero contribution for the singly andtriply excited wave functions. Therefore we are left with the doubly excited states. But note that there alsoexists a quadruply excited state (j��A��A��B��Bi). Knowing that, it is easy to see that DCI cannot preservesize consistency: The quadruple excited state is not included in DCI for the “H�”, therefore we do not gettwice the energy of the DCI (=full CI) of H�. For a rigorous proof see Szabo/Ostlund, p.261.

In general, the message is:

� truncated CI is not size consistent,

� full CI is size consistent.

40

8.3 Trouble: Double CI Size Consistency

variationalsize consistentHellm

an-Feynmannon dynamical corr.

dynamical corr.excited states

HF

mea

n-fie

ldde

scrip

tion,

sing

lede

term

inan

t+

++

only

UH

F–

(Koo

pman

theo

rem

)lim

ited

CI

incl

ude

expl

icitl

yex

cite

dde

term

inan

tsse

lect

edac

cord

ing

toso

me

pres

crip

tion

+–

––

++

MC

SC

Fsm

allC

Iwith

orbi

tals

optim

ized

for

ever

yde

term

inan

t+

–+

+(–

)+

MB

PT

eval

uate

cont

ribut

ions

ofex

cite

dde

term

inan

tsco

nsis

tent

lyth

roug

hgi

ven

orde

rin

pert

urba

tion

theo

ry–

+–

only

ifba

sed

onU

HF

+–

CC

eval

uate

cont

ribut

ion

ofse

lect

edex

cita

tion

thro

ugh

infin

iteor

der

–+

–on

lyif

base

don

UH

F+

–

full

CI

incl

ude

allp

ossi

ble

dete

rmin

ants

,ex

actw

ithin

basi

sse

t++

++

++

Tab

le8.

1:P

rope

rtie

sof

diffe

rent

ab-in

itio

met

hods

41

9 Derivatives of the Energy

9.1 The Potential energy surface

If the Born-Oppenheimer approximation

�molecule

��R��r

�� nuclei

��R��electrons ��r� (9.1)

holds the effective potential energy for the nuclei as a function of their positions R is defined by

V��R�� h�eljHel��R� j�eli�

XA B

ZAZBRAB

(9.2)

This equation defines a potential energy surface PES (really a hypersurface) for the nuclei. The electronicHamiltonian depends parametrically of the nuclear coordinates

Hel

��R��Xi

�

�r�i �

XA

ZAriA

��Xi j

rij(9.3)

A schematic PES is shown in figure 9.1. Features of the PES can be related to molecular properties:

� geometries: A,C minimum (molecular structure)B saddlepoint (transition state)

� energetics: energy differences between minima (reaction enthalpies, stabilities)energy differences between minima and saddlepoints (activation energies)

� curvature: vibrational frequencies, normal modes (IR, Raman, thermochemistry)

� higher derivatives: (accurate) ro-vibrational spectra

� minimum-energy path between minima (A� B � C): reaction path?

9.2 How to find stationary points?

Minima on the PES represent molecular conformers, saddlepoints transition states. To locate them, onemakes use of optimisation techniques from numerical mathematics which use derivatives. The PES isTaylor-expanded around a stationary point�R�,

V��R�� V

��R�

��Xi

��V

�Ri

��R�

�Ri �R�

i

�(9.4)

�

�

Xi

Xj

��V

�Ri�Rj

��R�

�Ri �R�

i

� �Rj �R�

j

��

42

9.2 How to find stationary points?

C

A

B

Figure 9.1: Illustration of a potential energy surface (PES)

where the sums run over all relevant degrees of freedom. The first derivatives are components of the gradientvector

gradient �g � gi �

��V

�Ri

�(9.5)

(Note that the negative of the gradient is equal to the force acting along this degree of freedom.) The mixedsecond derivatives form the Hessian matrix

Hessian H � Hij ��V

�Ri�Rj(9.6)

The Taylor expansion can be written more compactly

V��R�� V

��R�

�� g �

��R� �R�

��

�

��R� �R�

�T�H �

��R� �R�

�(9.7)

At a stationary point, we have

�g��R�

��

If the stationary point is a minimum,H is positive definite: It has only positive eigenvalues after the onesbelonging to translation and rotation of the molecule as a whole have been removed. If the stationarypoint is ann-th order saddlepoint,H hasn negative eigenvalues. Physically most important are first-ordersaddlepoints.

Minima can be located by standard numerical techniques, such as simplex or Fletcher-Powell (uses onlyV ,no derivatives); steepest-descent, conjugate-gradient or variable-metric (uses gradients); Newton-Raphson(uses second derivatives). See a textbook on numerical methods for details (e.g. Chapter 10 of NumericalRecipes). Which method is being used depends not only on the specific molecular system, but also on thecomputer budget, the desired accuracy and — most of all — on the availability of analytical first and maybesecond derivatives. We will see that they can be quite complicated to calculate. Hence, not all derivativeshave been implemented for all electronic-structure methods.

43


The location of saddlepoints is not as straightforward. Methods include mode-walking (follow the softesteigenmode) and others.

9.3 Physical meaning of some derivatives of the potential energy

Some derivatives ofV with respect to an external perturbation, like the position of a nucleus�Ri or an electricfield �E, have a direct physical interpretation (to within a numerical factor):

�fi � � �V

� �Ri

� force on atomi

fij ��V

�Ri�Rj� harmonic force constant

�� V

� �E� dipole moment

�ij ��V

�Ei�Ej� static electric dipole polarisability (3x3 Cartesian tensor)

��

� �Ri

��V

� �E� �Ri

� dipole derivative (IR intensity)

��

� �Ri

��V

� �E� �E� �Ri

� polarisability derivative (Raman intensity)

9.4 The Hellmann-Feynman theorem

For the exact electronic wave function�, a useful relationship can be proven. The electronic energyE isgiven by

E � h�jH j�i � (9.8)

Its derivative with respect to some parameter� is obtained via the product rule

�E

�� h�j�H

��j�i� h �

��jHj�i� h�jHj �

��i (9.9)

SinceH is Hermitian,E can be extracted from the last two terms

�E

�� h�j�H

��j�i�E

�h ��

�j�i� h�j ��

�i�

(9.10)

Using the product rule backwards, we get

�E

�� h�j�H

��j�i�E

�

��h�j�i � (9.11)

Since� is normalised to some constant,h�j�i is independent of� and the last term is zero, i.e.

�E

�� h�j�H

��j�i (9.12)

which is called the Hellmann-Feynman theorem.

44

9.5 Gradients for RHF

Therefore, once the wave function of a system has been found, derivatives of the energy can be calculatedsimply by calculating expectation values of the corresponding derivative of the Hamiltonian. Derivatives ofthe wave functions with respect to lambda are not needed. This simplifies calculations. The perturbationlambda can be any external perturbation, for example the ones given in section 9.3.

Two remarks concerning the Hellmann-Feynman theorem:

1. It is always possible to determine derivatives numerically, for instance by calculatingE for � � ��,� � �� and� � �� . Then, one may use e.g. centered differences:�

�E

��

��

� E�� E��

��(9.13)�

��E

��

��

�

��E�� E�� E��

Finite differences are convenient (no derivatives necessary) but the number of energy calculationsquickly runs out of hand, if there are many degrees of freedom (or�’s): Imagine a calculation of aHessian matrix for a molecule of 30 atoms. If analytical second derivatives are available the completeHessian can be evaluated at a fraction of the cost.

2. The derivation above assumes that� is the EXACT wave function. Of the approximate wave functionsused in quantum chemistry some satisfy the Hellmann-Feynman theorem, others don’t. The ones thatdo satisfy the Hellmann-Feynman theorem are those that have been optimised with respect to allpossible parameters. The parameters have to include the coefficients of the basis functions (except forfull CI which is the exact solution within the given one-electron basis). Hence, the methods satisfyingHellmann-Feynman are:

� exact wave function

� Hartree-Fock (and self-consistent density functional)

� MCSCF

� full CI

The Hellmann-Feynman theorem is not strictly satisfied by, e.g.

� incomplete CI

� MBPT (except in infinite order!)

� coupled-cluster

� many others

For these wave functions, one may of course calculate analytic derivatives. However, the�� termhas to be properly taken into account.


Where the Hellmann-Feynman theorem is not applicable, the wavefunction derivatives must also be calcu-lated. Simplest example: Calculation of the energy function derivative in closed shell RHF.

45


The Hamiltonian is

ESCF � �

occXi

��hii � occXj

��Jij �Kij�

�A� z � i � energy of orbitali

(9.14)

where the one electron parthii, the Coulomb partJij and the exchange partKij are

hii �R�i

��

�r� �PA

ZArij

��j d�

�Jij �

R R�i��j��

�r��

�i��j�� d�� d�� iijjj��

Kij �R R

�i��j��r��

�i��j�� d�� d�� ijjij� �

(9.15)

To evaluate the energy derivative

�ESCF

��

occXi

�

��hii �

occXi

occXj

�

��Jij �Kij� (9.16)

we first consider the one electron part:

�

��hii � h �

��ijh j�ii� h�ij �h

��j�ii� h�ijhj �

��ii (9.17)

�

allXp�q

��

��Cip

�hpjh jqiCqi �

allXp�q

Cip

��

��hpj h jqi

�Cqi� z

def�� h�ii

�

allXp�q

Cip hpjh jqi��

��Ciq

�

Here the first and last term contain the�-dependence of the matrix elements over atomic orbitals (which canbe calculated once and forever), while the second term has to be determind in the calculation.

The derivative of the wave function can be written as linear combination ofall (occupied and unoccupied)molecular orbitals

��i��

�

allXk

U�ik�k� (9.18)

where the coefficientsU�ik express the admixture of other molecular orbitals (� perturbation theory). Thus

�occXi

�

��hii � �

occXi

h�ii � occXi

allXk

U�ikhik (9.19)

The derivative of the coulomb interaction and the exchange interaction can be rewritten in an analogousmanner, e.g.

�

��Jij �

�

��iijjj� � �iijjj�� z

def�� J�ij

�

allXk

U�ki�kijjj� � (9.20)

46


where – again – the derivative contained inJ�ij needs to be calculated only once:

J�ij �allX

p�q�r�s

CipCiqCjrCjs�

��hpqj

r��jrsi (9.21)

Thus

�

��ESCF � �

occXi

h�ii �

occXi

occXj

��J�ij �K�

ij

��

occXi

allXk

U�ikhik (9.22)

�

occXi

allXk

U�ik

occXj

��kijjj� � �kjjij��

�

occXi

��h�ii �

occXj

��J�ij �K�

ij

��A� z

� �h�ii � �i �

�

occXi

allXk

U�ik

��hik � occXj

��kijjj� � �kjjij��A

� z � hij �f�i� jki � i�ik

�

occXi

�h�ii � �i

��

occXi

U�ii i

It remains to express the coefficientsU�ik in terms of derivatives of atomic orbitals. This may be done byintroducing the projection operator for moelcular orbitals

S �allXi

j�ii h�ij (9.23)

and expand it in atomic orbitals, in complete analogy to eq. (9.17):

Sij � h�ijS j�ji � h�ij�ji � �ij

�allPp�q

CipCqj hpjqi (9.24)

The derivative yields

��

Sij � � � h ��

�ijS j�ji � h�ij �S�� j�ji � h�ijSj ��ji

� h ��

�ij �ji � S�ij � h�i j ��ji

� U�ji � S�ij � U�

ij

� U�ij � ��

�S�ij

(9.25)

where in the third step eq. (9.18) has been inserted, and

S�ijdef.�

allXp�q

CipCqj�

��hpjqi � (9.26)

47


Finally,

�

��ESCF �

occXi

�h�ii � �i

��

occXi

S�ii i � (9.27)

48

10 Density functional theory (DFT)

In the framework of density functional theory it can be shown, that the ground state properties of a quantummechanical system can already be obtained from the electron density��r� � j��r�j�. It is thus not necessaryto know the the complex wavefunction��r� itself. The ground state density��r� minimizes the Kohn-Shamenergy functional

EKS��

occXi

hij h jii�

�

Z Z��r��r��j�r � �r�j

d�rd�r� �Exc�� (10.1)

whereExc�� is the exchange-correlation functional.

Note the formal analogy to the HF energy equation

EHF � �occXi

hij h�i� jii� z 1-electron

��occXi

hijXj

Jj jii� z Coulomb

�occXi�j

�ijjij�� z exchange

� (10.2)

where the “best single slater determinant” for the ground state is obtained from the HF variational equations.However, the HF method gives anapproximation to the wavefunction, while the Kohn-Sham equationswould yield the exact ground state density, provided that the exact form exchange-correlation functionalwere known (for all interesting cases this it is unfortunately not known exactly).

10.1 An illustrative example: The Thomas Fermi model

Consider an ideal electron gas.�N non-interacting electrons occupy a cubic volume�V � �� as particlesin a box. The energy of the electrons is then discretized according to

E�nx� ny� nz� �h�

�ml��n�x � n�y � n�z� �

h�

�ml�R� (for sufficiently large n) (10.3)

The Pauli exclusion principle implies that the number of electrons with energy� (atT � �K) is given bythe volume of a sphere octant

��

�

�R�

��

�

�

��ml�

h�

��

(10.4)

The degeneracy factor 2 appears since two electrons with opposite spin are allowed to have the samenx,ny, nz. Thus, the density of states is

g� � � ��

�

�ml�

h� �� O�� (10.5)

49


Since forT � �K the sphere is completely filled up to the Fermi energy F , the total energy in�V is just

�E ��FR�

g� �d

� ��

��mh�

��

��F

(10.6)

Taking into account that�N � �� F � and introducing the electron density� � �N��V we find

�E ��

��N F (10.7)

��h�

�m

��

��

��

�V ��

��

�� z

def.� CF � ��

�V �� in atomic unitsm � � �

If the electron density is allowed to vary from point to point (inhomogeneous electron gas), the total energy(still equal to the total energy)is given by

E � TTF �� CF

Z��r�d�r (10.8)

Nuclear attraction and Coulomb repulsion are added for the more general case (but not exchange/correlation,which partially are accounted for inTTF ��):

ETF �� CF

Z��r�dr� � Z

Z��r�

rd�r �

�

Z Z��r��r��j�r � �r�j d

�rd�r� (10.9)

In order to determine the electron density it was thenassumed (already in 1927!) that the correct��r�minimizesETF �� with the constraint of constant number of electrons, i.e.

� � �

�ETF �� TF � �

Z��r�d�r �N�

�with N �

Z��r�d�r (10.10)

� �TF ��ETF ��

��r��

�

�CF��r�

�� Z

r�

Z��r��r � �r�

��d�rNote, however, that this variational principle – contrary to DFT – has not been derived rigorously from QMprinciples. It is rather based on physical intuition by noting the important role that variational principlesplay in physics. In general, Thomas Fermi theory gives in some sense “averaged” electron densities, i.e.radial oscillations showing up in more rigorous approaches (like HF) are missing.

Numerous extensions to Thomas Fermi theory have been derived, e.g. the von Weizs¨acker kinetic energyfunctional (1935)

TTFW �� CF

Z��r�d�r � �

��

�m

Z jr��r�j��r�

d�r (10.11)

10.2 The Hohenberg-Kohn theorems

A QM system is completely determined by the number of electronsN and the external potentialv��r� (i.e.electrostatic potential induced by the nuclei, external fields,� � � ). Then it holds, that

50

10.2 The Hohenberg-Kohn theorems

HK I: The external potential is determined, within a trivial additive constant, by the electron density ��r�.(uniqueness)

Proof: (for non-degenerate ground states)Assume, that the same density��r� results for two different external potentialsv andv�:

v��r�� H � � and v��r�� H � � �� (10.12)

The the variational principle for the ground state energyE� yields

E� � h��jH j��i � h��jH � j��i� h��jH �H � j��i� E�

� �R��r��v��r�� v��r��d�r �

(10.13)

Analogously

E�� h�jH � j�i � E� �

R��r��v��r�� v��r��d�r � (10.14)

Summation of eq. (10.13) and eq. (10.14) leads to the contradiction

E� �E�� E� �E�

� � (10.15)

which completes the proof.

Consequences of HK I: Since

� ��r� determines uniquelyv��r� (cf. HK I)

� ��r� determinesN �R��r�d�r

the system is completely described by��r�. Consequentlyall properties can be derived from��r�.

In order to determine��r�, an energy functional is needed:

E�� T �� Vee�� Vext��

� FHK �� z Hohenberg-Kohn

functional(universal)

�

Z��r�v��r�d�r� z

system dependentext. potential

(10.16)

with

FHK �� T �� J ��z�R R �r��r��

j�r��r�j d�rd�r�

Coulomb

�Exc�� (10.17)

whereExc�� is the non-classical exchange correlation functional, which is difficult to obtain and must beapproximated in all interesting cases.

The second Hohnberg Kohn theorem states

51


HK II: For a trial density e��r� with e��r� � � and N �R e��r�d�r, the true ground state energy E� is a

lower bound to the energy functional Ev�e��:E� � Ev�e�� FHK �e�� Z e��r�v��r�d�r � (10.18)

where the equality actually holds for the correct ground state density ��r�.

Proof:e��r�� ev��r� HK I�� eH � e�� Ev�e�� De��H ��e�E � h�jH j�i � Ev�� E� (10.19)

Consequence of HK II:The energy functional is variationally stationary for the correct ground state density, i.e:

�

�Ev��

Z��r�d�r �N�

�� (10.20)

or

� ��Ev ��

��r��

�FHK ��

��r�� v��r� � (10.21)

The latter equation tempts to interprete� as a “chemical potential” in the sense of Pauling’s electronegativity(= arithmetic average of the electron affinity and the ionization potential), although a rigorous relationshipcannot be established.

10.3 The Kohn-Sham (KS) method

We want a method to solve

E�� T �� J �� EXC ��

Z��r�v��r�d�r (10.22)

The three first terms of the l.h.s. areFHK �� where the second and third terms can be summarized asbefore asVee��. The contribution of the terms to the energy is very different in magnitude, in particularT �� EX � EC where the exchange termEX is typically ten times greater than the correlation termEC . Note that the virial theorem gives simply� T �� H �. The Thomas-Fermi approach consistsof solvingT �� directly from� as an approximation to the true kinetic energy functional. The Kohn-Shamapproach is to introduce an auxiliary wave function so that the largest part ofT �� can be evaluated exactlyand one worries about the rest later.

The exact kinetic energy is

T �

NXi

nih�ij �

�r�j�ii (10.23)

where the�’s are called natural orbitals and are no more than the eigenfunctions of the reduced 1-electrondensity matrix and contain the many-electron effects.ni is an effective occupation numbers that may befractional. The electron density is given thus by

��r� �NXi

niXs��

j�i�r� s�j� (10.24)

52

10.4 Exchange (and) correlation functionals

The following simplified kinetic termTS �� is introduced :

TS ��

NXi

h�ij �

�r�j�ii (10.25)

��r� �

NXi

Xs��

j�i�r� s�j�

The�’s are now the molecular orbitals of the determinantal wave function�S � �pNdet j��N j with

the occupation number�� . The�’s are eigenfunctions of the Hamiltonian withoutVee.

hS�i �

��

�r� � vs��r�

��i � i�i

TS �� h�sjXi

��

�r��j�si �

Xi

h�ij �

�r�j�ii

BecauseTS �� T �� the KS exchange-correlation functional contains not only the usual terms (ex-change,correlation,electronique repulsion) but also a kinetic energy correction term

E�� TS �� J �� Vee�� J �� T �� TS �� (10.26)

so that the terms in square brackets are the components ofEXC ��.

The Euler equation is given by

� � veff ��r� ��TS ��

��(10.27)

with

veff ��r� � v��r� ��J ��

��EXC ��

��

� v��r� �

Z Z��r��j�r � �r�j

d�rd�r� � vXC��r�

vXC��r� being the exchange correlation potential. The KS equations forN orbitals are��

�r� � veff ��r�

��i � i�i (10.28)

��r� �Xi

Xs

��i ��r� s��i��r� s�

with the constrainth�ij�ji � �ij . Hence, it is possible to recalculate in a self consistent procedureveff ��r� � f��r�� including the exchange-correlation potential from the electron density via the auxil-iary wave function.

10.4 Exchange (and) correlation functionals

We have seen that the DFT is exact and KS is exact, ifEXC �� can be calculated exactly.EXC �� can beestimated with the local-density approximation (LDA)

EXC �� ELDAXC �

Z��r� XC ��d�r (10.29)

53


(Note that exchange and correlation are not separated.) The philosophy is that the length scale of theexchange and correlation is smaller than the length scale of the variation of�. (This should be comparedwith the wave function theory : The HF exchange term is not local and neither is the CI correlation term.)The LDA is of course nearly exact for an uniform� and less accurate for a strongly varying�. vLDAXC , vXC

in the LDA approximation, is given by

vLDAXC ��ELDA

XC ��

��r�� XC ��r�� r�

� XC ��

��(10.30)

The KS equation with the LDA becomes �

�r� � v��r� �

Z��r��j�r � �r�j

d�r � vLDAXC ��r�

��i � i�i (10.31)

For the uniform electron gas XC � X � C . X is known exactly :

X � ��

��

��

�� r�. C is numerically well known. Today XC is nearly accurate for electron gas.

For atoms and molecules, the LDA is made using the Hartree-Fock-Slater (HFS) orX� introduced by Slaterin �. The idea is to replace the HF exchange operator by a local operator. In practice, the uniformelectron gas results is applied to infinitesimal portions of the system

vHFSXC ��r� � ��

��

�

��r��

�� (10.32)

For Slater� � while Kohn and Sham proposed thatX� is the exchange term (without correlation) if� � �

� . Empirically,� takes values between�� and��. In comparison with the Thomas-Fermi approach,the difference is mainly in the kinetic energy but there is a large improvement.The local spin density (LSD) approximation is similar in spirit to the unrestricted HF method.

N��r� � ��r� � ��r� (10.33)

N� �

Z��r�d�r N� �

Z��r�d�r

N � N� �N�

The KS equation becomes

��

�r� � v�eff ��r��i� � i��i� i � � �� N� � � � �� (10.34)

v�eff ��r� � v��r� �

Z��r��j�r � �r�j

d�r�EXC ��

��

��r�

Other terms could be added in the r.h.s. of the last equation like e.g.e��mcBz��r� to take into account the

presence of a magnetic field. We have as for the LDA case addition of the exchange and correlation terms

EXC �� EX ��

�� EC ��

For the exchange term

EX �� ELSD

X � ��

��

��

Z��

��

�� d�r (10.35)

� ��

��

��

Z��

��

��

�� d�r

54

10.5 DFT in a Gaussian orbital basis

with � ��

being the spin polarisation.ELSDC is an ugly and complicated term.

The exchange correlation terms can also been estimated by gradient correction. In this case the exchangecorrelation term is improperly called ”non-local” although it is in fact still mathematically local. Some ofthem are listed here:

� One example is thevon Weizsacker correction to Thomas-Fermi.

� Langreth and Mehl:

ELMXC �� ELDA

XC � �� Z jr�j�

��

�� exp��r��

� �

�d�r (10.36)

� Perdew and Yue have developped the generalised gradient approximation (GGA)

EGGAX ��

�

�

��Z

��F �s�d�r (10.37)

S �jr�j

�kF��r�� kF � ��

��

andF �S� � � � ��S� � S� � ��S��

��

� Becke: It is the most useful numerically.

EBX �� EHFS

X ��X��

A

Z jr��j��

��

� �Bjr��j��

��

��d�r

where A and B are parameters.

Combinations of gradient correction have been developped :BP, BLYP, BVWN ...There is also the possibility to combine exchange-correlation functionals with a HF exchange term (e.g.B3LYP):

EXC �� aEHFX �� a�EB

X �� bELY PC ��

10.5 DFT in a Gaussian orbital basis

The KS equation in orbital basis is �HKS

� iS

��ci � � (10.38)

�i �

NbasXi

Cipp��r� (10.39)

whereNbas is the number of basis functions. We have then

Spq � hpjqi (10.40)

HKS�pq � hpj �

�r� � v��r�jqi� hpjJ jqi� hpjXCjqi (10.41)

55


where the second and third term of the r.h.s. is the coulomb and exchange term respectively. The Coulombterm can be expressed in various ways. In the DFT language it is

hpjJ jqi �Z Z

�p��r�q��r��r��j�r � �r�j

d�rd�r� (10.42)

In the quantum chemistry language, the MO are explicitely introduced

hpjJ jqi �Z

�p��q��

r��

occXj

��j ��j��d� (10.43)

�

occXj

NbasXr�s

C�jrCjs

Z�p��

�r��

r��q��s��d�

The calculation are expensive : the Coulomb and HF exchange term are of the order ofNbas�. Hence,

instead charge fitting functionsgt��r� (Gaussian) are used :

��r� �

NfitXt

ctgt��r�hpjJ jqi �NfitXt

ct

Z�p��q��

r��gt��d�

This is computationally cheaper, the order being decreased toNbas�Nfit � Nbas

� The problem is now tofind the charge fitting coefficientsct The simplistic method (in practice not done) would use the followingminimisation :

min ��r�� fit��r�� min

Z �Xi

�i��r��i ��r��

Xtctgt��r�

��

d�r

A better way is to minimize the residual Coulomb energy

min

Z Z��r��

r��r��d�r�d�r�

where��r� � ��r��

Xtctgt��r�

with the constraint Z Xtctgt��r�d�r � N

We can also fit functions to the exchange-correlation potential

hpjXCjqi �Z Z

�p��q��

r��vXC��r��d�r�d�r�

vXC is local (even with gradient corrections) and is calculate on a grid

min

�� Xgridpoints

vXC��r��

Xu

Cugu��r�

��Agu��r� being a gaussian function

The exchange correlation scales also with the orderN�bas

KS HFh

N�bas N�

bas

J

N�bas N�

bas (N�bas possible)

XC

N�bas K

� N�

bas (reduction impossible)

56

11 The Car-Parrinello Method

The Car-Parrinello (CP) technique is the combination of

� electronic structure calculation, giving the energy and the forces

� and a molecular (dynamics) simulation.

It has the advantages, from the quantum side

� to be accurate

� to use no force-field

� the chemisty (reactions ...) is possible and from the molecular simulation side

� the environment

� temperature

� entropy

� dynamics

� statistical mechanics

The disadvantages are of course that it is less accurate than a quantum chemistry calculation and still expen-sive in computation time with regard to force-field simulation (good statistics only for a few picoseconds,and only few hundreds of atoms) The ingredients are

� DFT for electrons

� local density approximation or gradient correction

� plane wave basis set (faster than Gausssians)

� pseudopotentials in lieu of core electrons. (This is mandatory to use because of the preceding ”ingre-dient”)

� (classical) molecular dynamics for the nuclei

� fictious dynamics for electrons. This is the important trick of the method

� parallel implementation

57

11 The Car-Parrinello Method

11.1 Plane wave basis

The plan wave basis functions are given by

i��r� �X

�GCi��k� �G

exphi��k � �G��r

i(11.1)

where�G is the reciprocal lattice vector defined by�G ��l � ��m with �l a lattice vector andm an integer.

11.2 The Kohn-Sham equations

��e�

�mr� � Vion��r� � VH��r� � VXC��r�

��i��r� � i�i��r�

with Vion��r� � v��r� where the subscript ion means external potential due to the atomic cores (“ions”),

VH��r� � e�R �r��

j�r��r�jd�r� the ”Hartree” Coulomb potential andVXC��r� �

�EXC ��r��r� the exchange-correlation

potential.

The KS equation is rewritten in reciprocal space and with help of the plane waves

X�G�

��e�

�mj�k � �Gj�� G� �G�

� Vion� �G� �G�� VH� �G� �G�� VXC� �G� �G��Ci��k� �G

� iCi��k� �G(11.2)

The plane waves are orthonormal, i.e. there is no overlap matrix. The Laplace operatorr� is replaced by amultiplication operation in Fourier space. The energy cutoff�e�

�m j�k � �Gj� � Ecut � ��eV .

11.3 Pseudopotential

The valence orbitals are orthogonal to the inner(”core”)-shell orbitals of same symmetry. It implies ra-dial nodes close to the nucleus, rapidly oscillating function. They cannot be represented feasibly by planewaves. The inner electrons and smallr part of valence orbitals are not changed by chemical bonding. Thefirst solution is all electron calculations in plane-wave basis with sui table atom-centered basis functions(LMTO,LAPW,PAPW, ...). The second solution is the use of a pseudo-potentials. They leave the outer partof the valence orbitals unchanged. The radial nodes are removed and they replace the inner electrons. Thisshould be compared to atom-centered basis (e.g. Gaussians) in molecular quantum chemistry: The effectivecore potentials (ECPs) are used for speeding up the calculation with fewer electrons and keep the relativisticeffects (e.g. for heavy elements). ECPs are optional.

11.4 CP Molecular dynamics

The equation of motion for the nuclei is

MAd�

dt��RA�t� � � �

� �RA

EKS��R�t�� i�t��

XA�B

ZAZBRAB

�(11.3)

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11.4 CP Molecular dynamics

and for the electrons

�d�

dt��i�t� � � �

��iEKS��R�t�� i�t��

Xj�ij�j (11.4)

The last term in the r.h.s. is the orthonormality constraint while� is a fictious ”mass”, a parameter. It impliesthat the acceleration of the coefficient of the plane waveC

i��k� �Gis given by

�d�

dt�Ci��k� �G

� ��e�

�mj�k � �Gj� � �i�Ci��k� �G

�X�G�

�Vion� �G� �G�� VH� �G� �G��VXC� �G� �G��Ci��k� �G�

(11.5)

whereVion� �G � �G�� is the contribution of the nuclei and�i � �ii � h�ijHj�ii is an ”approximateconstraint” coming from the subsequent iterative orthogonalisation of the�i’s like, e.g.

��i � �i �

�

Xj �ih�j j�ii�j

Viewed from MD, the CP-MD is an ”extended-system” method. Viewed from DFT, CP-MD is avoiding adiagonalisation of the Hamiltonian.

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Practical QUantum

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