Lecture 9: Molecular integral
evaluation
Integrals of the Hamiltonian matrix
over Gaussian-type orbitals
Gaussian-type orbitals
• The de-facto standard for electronic-structure
calculations is to use Gaussian-type orbitals with
variable exponents
– This is because they lead to much more efficient
evaluation of two-electron integrals
Gaussian-type orbitals
• The usefulness of the GTOs is based on the Gaussian
product rule
GTO basis sets
•
•
•
•
•
Primitive set
Contraction
Valence & core polarization
Further augmentation
Various established generation philosophies
–
–
–
–
Correlation-consistent & polarization-consistent sets
ANO sets
Pople-style segmented contraction
Completeness-optimized
Molecular integrals
• We will now consider two types of integrals:
– One-electron integrals
– Two electron integrals
– Compare with the integrals related to the molecular
Hamiltonian
Cartesian Gaussians
• The orbitals β will be realvalued Cartesian GTOs
– Here i+j+k=l (”Cartesian
quantum numbers”)
• From the integrals over the
CGTOs we will obtain the
integrals in (contracted)
spherical-harmonic GTO basis
as linear combinations
Gaussian overlap distributions
• The Cartesian GTOs can be factored in Cartesian
directions
• Consequently, the Gaussian overlap distribution
will factorize in Cartesian directions
Overlap integrals
• Let’s begin with overlap integrals
Sab <
òG
ikm
(r, a, Ra )G jln (r,b, Rb )dr
that also factorize as
• Employing the Gaussian product rule, we obtain
where
ab
m<
a ∗b
p < a ∗b
aX A ∗ bX B
xp <
a ∗b
X AB < X A , X B
Overlap integrals
• We can similarly write the S 00 in each Cartesian
direction
• Then, by invoking the following (Obara-Saika)
recurrence relations we can obtain the overlap
integrals up to arbitrary Cartesian quantum number
in each Cartesian direction
– The final overlap integral is obtained as a product of
different Cartesian components
Multipole-moment integrals
• Integrals of the form
g
Sabefg < Ga xCe yCf zCg Gb < S ije Sklf Smn
are obtained through relations
1
(iSie,1, j ∗ jSie, j ,1 ∗ eSije ,1 )
2p
1
Sie, j ∗1 < X PBSije ∗ (iS ie,1, j ∗ jS ie, j ,1 ∗ eSije ,1 )
2p
1
Sije ∗1 < X PC S ije ∗ (iS ie,1, j ∗ jSie, j ,1 ∗ eSije ,1 )
2p
Sie∗1, j < X PAS ije ∗
• Special cases: overlap, dipole and quadrupole
integrals
One-electron integrals with differential
operators
• We will need e.g. in evaluation of the kinetic energy
operator in the one-electron Hamiltonian integrals of
kind
• These will again factorize as Dabefg
• The Obara-Saika relations are
with
Dij0 < S ij
g
< Dije Dklf Dmn
One-electron integrals with differential
operators
• Now we can obtain for
r
Pab < ,i Ga Ñ Gb
r
Lab < ,i Ga r ´Ñ Gb
1
Tab < , Ga Ñ2 Gb
2
the following expressions (taking z-component only
for the momentum integrals)
1
Pabz < ,iSij0Skl0 Dmn
0
0
Lzab < ,i(Sij1 Dkl1 Smn
, Dij1Skl1 Smn
)
∋
1 2 0 0
0
2
Tab < , Dij Skl Smn ∗ Sij0Dkl2 Smn
∗ Sij0Skl0 Dmn
2
(
One-electron Coulombic integrals
• Let us set up an analogous scheme for one-electron
Coulombic integrals
• The Obara-Saika relations are written for auxillary
integrals 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 function
called 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
Two-electron Coulombic integrals
1. Generate Boys functions
2. Vertical recursion
3. Electron-transfer recursion
4. Horizontal recursion
Computational requirements of twoelectron integrals
Step
Floating point
operations
Memory
requirement
Boys functions
Lp4
Lp4
Vertical recursion
L4p4
L3p4
Transfer recursion
L6p4
L6p4
Primitive 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 are
obtained as multipole expansions of each box
– The number of NN contributions scales linearly with the
size 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