Pseudospectral Local MP2 Techniques

Second order Møller-Plesset perturbation theory (MP2) is perhaps the most widely used ab initio electron correlation methodology, recovering a large fraction of the correlation energy at a relatively low computational cost. The method greatly improves Hartree-Fock treatments of properties such as transition states, dispersion interactions, hydrogen bonding, and conformational energies. However, the scaling of conventional MP2 algorithms with system size is formally nN⁴, where N is the number of basis functions and n the number of occupied orbitals, due to the necessity of carrying out a four index transformation from atomic basis functions to molecular orbitals. In principle, it is possible to reduce this scaling by using integral cutoffs, as for Hartree-Fock calculations. However, the reduction is noticeably less effective in MP2, particularly for the large, correlation-consistent basis sets that are required for accurate correlation effects on observable quantities. Thus, MP2 techniques have traditionally been used primarily for small molecules.

Several years ago, Pulay and coworkers [94, 95] formulated a version of MP2 in which the occupied orbitals are first localized (e.g., via Boys localization [97]) and the virtual space correlating such orbitals are then truncated to a local space, built from the atomic basis functions on the local atomic centers orthogonalized to the occupied space. Another critical advantage of LMP2 is that one can very precisely control which region of the molecule is correlated, reducing CPU costs enormously. The method has been shown to yield an accuracy for relative energies that is, if anything, superior to conventional MP2, due to elimination of basis set superposition error [96]. However, localized MP2 implementations in conventional electronic structure codes have not yet led to substantial reductions in CPU time, since the first few steps of the necessary four-index transformation are unaffected by localization of the occupied orbitals, and the localized orbitals have tails that extend throughout the molecule.

We have carried out extensive tests demonstrating the accuracy and computational efficiency of the pseudospectral implementation of LMP2, as detailed in ref. [16]. In the pseudospectral approach, we assemble two-electron integrals over molecular orbitals directly and are thus able to fully profit from the huge reduction in the size of the virtual space in Pulay’s theory. Formally, the PS implementation of LMP2 scales as nN³; however, various types of cutoffs and multigrid procedures can reduce this to ~N². In fact, for calculations involving both the 6‑31G** and Dunning cc‑pVTZ basis sets, we find a scaling ~N^2.7 with system size.

The physical idea behind the LMP2 method is that if the molecular orbitals are transformed so that they are localized on bonds or electron pairs, correlation among the occupied pairs can be described by the local orbital pairs and their respective local pair virtual spaces defined from the atomic orbitals on the relevant atom or pair of atoms. The localized orbitals can be generated by any unitary transformation of the canonical orbitals. For LMP2, we use Boys-localized [97] orbitals, for which the term is maximized. The local virtual space for each atom is defined by orthogonalizing its atomic basis functions against the localized molecular orbitals. The correlating orbitals included in the local virtual space are thus mostly near the atom itself, but because of the orthogonalization procedure, they are not particularly well localized.

The Jaguar LMP2 program uses Pulay’s method [94, 95, 96] to expand the first order wave function correction as a linear combination of determinants formed by exciting electrons from localized orbitals i and j to local virtual space correlation orbitals p and q:

(1)

For local MP2, we must iteratively solve the following equation, which has been derived in detail by Pulay and Sæbo, for the coefficients :

(2)

Here F is the Fock matrix, S is the overlap matrix, and T is the residual matrix defined by this equation. The exchange matrix K_ij is restricted to the dimensions of the virtual space corresponding to the occupied localized molecular orbitals i and j. The simplest updating scheme for the coefficients is to obtain updated coefficients iteratively from the equation:

(3)

where ε_i and ε_j are the matrix elements F_ii and F_jj in the localized molecular orbital basis and ε_p and ε_q are the eigenvalues of the Fock matrix in the local virtual basis.

From the C_ij coefficients and the exchange matrices K_ij, Jaguar computes the second order energy correction from the equations:

(4)

(5)

where the bracket in Eq. (4) denotes a trace and δ_ij is 1 if i = j and 0 otherwise. Computing the exchange matrix elements for Eq. (4) is approximately 80% of the work for an energy correction computation, while generating the C_ij coefficients comprises about 20% of the work.

Jaguar performs localized MP2 calculations using pseudospectral methods, evaluating integrals over grid points in physical space in a manner similar to that described for HF calculations in The Pseudospectral Method. The two-electron exchange integrals needed for Eq. (4) are evaluated over grid points g as follows:

(6)

where Q_ig is the least squares fitting operator for molecular orbital i on grid point g, R_pg is the physical space representation of virtual orbital p, and A_jqg is the three-center, one-electron integral over the occupied molecular orbital j and the local virtual orbital q. The last term is related to the three-center, one-electron integrals in atomic orbital space, A_klg, described in Eq. (2) in The Pseudospectral Method, by

(7)

The summation is performed in two steps, first summing over k to form intermediates A_jlg,

(8)

then summing over l to yield the integrals in molecular orbital space

(9)

Jaguar’s local MP2 module also includes analytical corrections similar to those described earlier for Hartree-Fock calculations, and a length scales algorithm, both of which are explained in reference [13].