Computations of Noncovalent Interactions - Reviews in Computational Chemistry - page 15

value at a significantly reduced computational cost. One such approach is the
resolution of the identity (RI) approximation, 68,69 also referred to as the density
fitting (DF) approximation. 70 Applied with a well-chosen auxiliary basis, this
approach introduces errors that are tiny compared to the error made by using
a noninfinite basis set. 71,72
In the RI/DF approximation, computing the necessary two-electron inte-
grals is sped up by introducing an auxiliary basis set and representing products
of orbitals as 71

X

c P
nm

n ð r Þ m ð r Þ P i ð r Þ ½ 12

i

If one evaluates this product on a grid instead of using atom-centered Gaussian
functions for the auxiliary functions P i ð r Þ , one then obtains the very similar

pseudospectral approximation. 73,74 Minimizing the self-interaction error, the
four-index electron repulsion integrals,

ð

1
r 12 r ð r 2 Þ s ð r 2 Þ

d 3 r 1 d 3 r 2 ð mn j rs Þ¼ m ð r 1 Þ n ð r 1 Þ ½ 13

may be approximated as 71

X
PQ ð mn j

Þ 1 ð mn j rs Þ P Þð P j Q ð Q j rs Þ ½ 14

where ð P j Q Þ is a two-index electron repulsion integral and ð mn j P Þ and ð Q j rs Þ

are three-index electron repulsion integrals:

ð P

1
r 12 Q

d 3 r 1 d 3 r 2

ð P j Q Þ¼ ð r 1 Þ ð r 2 Þ ½ 15

and

ð

1
r 12 P

d 3 r 1 d 3 r 2 ð mn j P Þ¼ m ð r 1 Þ n ð r 1 Þ ð r 2 Þ ½ 16

This RI approximation can be applied to the four-index integrals needed dur-
ing a Hartree-Fock computation, 68,71 and also to the integrals needed for
post-Hartree-Fock procedures such as MP2. 69,70
Although the above equations show that the RI approximation makes
the formalism somewhat more complex, this speeds up the computation
because it can be broken down into less costly steps. Formally, computing
all of the possible four-index integrals would scale as

N bf Þ

Oð (with N bf the