Resolution of the Identity and Local Correlation Approximations

Even within the context of a given target level of theory, e.g., MP2/aug-cc-pVTZ, there exist various approximations that allow one to estimate the desired 

In the RI/DF approximation, computing the necessary two-electron integrals is sped up by introducing an auxiliary basis set and representing products of orbitals as  

If one evaluates this product on a grid instead of using atom-centered Gaussian functions for the auxiliary functions Pi r), one then obtains the very similar pseudospectral approximation.Minimizing the self-interaction error, the four-index electron repulsion integrals, 

This RI approximation can be applied to the four-index integrals needed during a Hartree-Fock computation, and also to the integrals needed for post-Hartree-Fock procedures such as 

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 O(N f) (with Nbf the 

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