Lecture 9: Molecular integralevaluationevaluation

Integrals of the Hamiltonian matrixover Gaussian-type orbitals

Gaussian-type orbitals

• The de-facto standard for electronic-structurecalculations is to use Gaussian-type orbitals withvariable exponents– This is because they lead to much more efficient

evaluation of two-electron integralsevaluation of two-electron integrals

Gaussian-type orbitals

• The usefulness of the GTOs is based on the Gaussianproduct rule

GTO basis sets

• Primitive set• Contraction• Valence & core polarization• Further augmentation• Further augmentation• Various established generation philosophies

– Correlation-consistent & polarization-consistent sets– ANO sets– Pople-style segmented contraction– Completeness-optimized

• We will now consider two types of integrals:– One-electron integrals

Molecular integrals

– Two electron integrals

– Compare with the integrals related to the molecularHamiltonian

Cartesian Gaussians

• The orbitals β will be real-valued Cartesian GTOs

– Here i+j+k=l (”Cartesianquantum numbers”)

• From the integrals over the CGTOs we will obtain the integrals in (contracted) spherical-harmonic GTO basisas linear combinations

Gaussian overlap distributions

• The Cartesian GTOs can be factored in Cartesiandirections

• Consequently, the Gaussian overlap distribution• Consequently, the Gaussian overlap distribution

will factorize in Cartesian directions

Overlap integrals

• Let’s begin with overlap integrals

that also factorize as

( , , ) ( , , )ab ikm a jln b

S G a G b d< ò r R r R r

• Employing the Gaussian product rule, we obtain

where

aba b

p a b

m <∗

< ∗

A Bp

AB A B

aX bXx

a bX X X

∗<

∗< ,

Overlap integrals

• We can similarly write the in each Cartesiandirection

• Then, by invoking the following (Obara-Saika) recurrence relations we can obtain the overlap

00S

recurrence relations we can obtain the overlapintegrals up to arbitrary Cartesian quantum numberin each Cartesian direction

– The final overlap integral is obtained as a product of different Cartesian components

Multipole-moment integrals

• Integrals of the form

are obtained through relations

efg e f g e f gab a C C C b ij kl mn

S G x y z G S S S< <

1< ∗ ∗ ∗

• Special cases: overlap, dipole and quadrupoleintegrals

11, 1, , 1

1, 1 1, , 1

1 11, , 1

1( )

21

( )21

( )2

e e e e ei j PA ij i j i j ij

e e e e ei j PB ij i j i j ij

e e e e eij PC ij i j i j ij

S X S iS jS eSp

S X S iS jS eSp

S X S iS jS eSp

,∗ , ,

,∗ , ,

∗ ,, ,

< ∗ ∗ ∗

< ∗ ∗ ∗

< ∗ ∗ ∗

One-electron integrals with differentialoperators

• We will need e.g. in evaluation of the kinetic energyoperator in the one-electron Hamiltonian integrals of kind

• These will again factorize as• The Obara-Saika relations are

with 0ij ij

D S<

efg e f gab ij kl mn

D D D D<

One-electron integrals with differentialoperators

• Now we can obtain for

212

ab a b

ab a b

ab a b

P i G G

L i G G

T G G

< , Ñ

< , ´Ñ

< , Ñ

r

r

r

the following expressions (taking z-component onlyfor the momentum integrals)

2ab a bT G G< , Ñ

∋ (

0 0 1

1 1 0 1 1 0

2 0 0 0 2 0 0 0 2

( )

12

zab ij kl mnzab ij kl mn ij kl mn

ab ij kl mn ij kl mn ij kl mn

P iS S D

L i S D S D S S

T D S S S D S S S D

< ,

< , ,

< , ∗ ∗

One-electron Coulombic integrals

• Let us set up an analogous scheme for one-electronCoulombic integrals

• The Obara-Saika relations are written for auxillaryΠ

• The Obara-Saika relations are written for auxillaryintegrals Π as

One-electron Coulombic integrals

• The auxillary integrals have the special cases

where and F is the Boys function

Boys function

• For integrals featuring the 1/r singularity, i.e. Coulombic integrals, we will need a special functioncalled the Boys function

Two-electron Coulombic integrals

• Let us finally set up the Obara-Saika (like) scheme for two-electron Coulombic integrals

• Employ again the auxillary integrals that feature the special cases

1. Generate Boys functions2. Vertical recursion

3. Electron-transfer recursion

Two-electron Coulombic integrals

3. Electron-transfer recursion

4. Horizontal recursion

Computational requirements of two-electron integrals

Step Floating pointoperations

Memoryrequirement

Boys functions Lp4 Lp4

Vertical recursion L4p4 L3p4

Transfer recursion L6p4 L6p4

Primitive contraction L6p4 L6Primitive contraction L6p4 L6

Horizontal recursion I L9 L7

Harmonics conversion I L8 L5

Horizontal recursion II L8 L6

Harmonics conversion II L7 L4

L is the angular momentum (1-4)p is the number of primitives in the shell (~1-20)

The multipole method

• Let us consider a 2D system decomposed three times

Level-2 Level-3

The multipole method

• The fast multipole method (FMM): Only the NN interactions are treated explicitly; while the rest areobtained as multipole expansions of each box– The number of NN contributions scales linearly with the

size of the systemsize of the system

• With continuous charge distributions (Gaussians) the FMM has to be generalized (CFMM)– Linearly scaling number of nonclassical integrals– The rest can be computed from multipole expansions