lecture 9: molecular integral evaluation - chem.helsinki.fimanninen/aqc2012/session180412.pdf ·...
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Lecture 9: Molecular integralevaluationevaluation
Integrals of the Hamiltonian matrixover Gaussian-type orbitals
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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
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Gaussian-type orbitals
• The usefulness of the GTOs is based on the Gaussianproduct rule
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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
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• We will now consider two types of integrals:– One-electron integrals
Molecular integrals
– Two electron integrals
– Compare with the integrals related to the molecularHamiltonian
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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
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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
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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
∗<
∗< ,
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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
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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
,∗ , ,
,∗ , ,
∗ ,, ,
< ∗ ∗ ∗
< ∗ ∗ ∗
< ∗ ∗ ∗
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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<
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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
< ,
< , ,
< , ∗ ∗
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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
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One-electron Coulombic integrals
• The auxillary integrals have the special cases
where and F is the Boys function
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Boys function
• For integrals featuring the 1/r singularity, i.e. Coulombic integrals, we will need a special functioncalled the Boys function
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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
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1. Generate Boys functions2. Vertical recursion
3. Electron-transfer recursion
Two-electron Coulombic integrals
3. Electron-transfer recursion
4. Horizontal recursion
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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)
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The multipole method
• Let us consider a 2D system decomposed three times
Level-2 Level-3
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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