Pseudospectral Resolution-of-the-identity MP2 Techniques

Second order Møller-Plesset perturbation theory (MP2) [93] recovers a large fraction of correlation energy at a relatively low computation cost, and as a result is widely used in ab initio electron correlation methodology. The MP2 correlation energy is calculated as

,

(1)

where indices i, j label the occupied molecular orbitals (MO), a, b label the virtual molecular orbitals, and ϵp is the p-th spin orbital energy.

The conventional MP2 algorithm formally scales as N_occN⁴, where N is the number of atomic orbital (AO) basis functions and N_occ is the number of occupied orbitals. This is largely due to the necessary four index integral transformations from AO space to MO space:

(2)

where μ, ν, ρ, σ are AO indices and C_μp represents the MO coefficients of spin orbital p.

The product of orbital pairs, for example, i and a, may be treated as electron densities. In resolution-of-the-identity MP2 (also known as density-fitted MP2), we approximate these electron densities with the use of auxiliary basis functions. For example, for a general four index integral,

(3)

where P and Q are auxiliary basis functions. From this approximation, any four index integral transformations become effectively a three index integral transformation, which is much less computationally demanding. By means of the RI method, the RI-MP2 calculation formally scales as N_occ²N_vir²N_aux, where N_occ is the number of occupied orbitals, N_vir is the number of virtual orbitals, and N_aux is the number of RI auxiliary basis functions. The details of RI-MP2 can found in the references [305–308].

Note, the calculation of the RI-MP2 energy itself does not use the pseudospectral method. However, by default, the SCF wavefunction and RI-MP2 gradients are obtained with the pseudospectral method.