SDS: the ‘static–dynamic–static’ framework for strongly correlated electrons

Abstract

A genetic ‘static–dynamic–static’ (SDS) framework is proposed for describing strongly correlated electrons. It permits both simple and sophisticated parameterizations of many-electron wave functions. One particularly simple realization amounts to constructing and diagonalizing the Hamiltonian matrix in the same number of many-electron basis functions in the primary (static), external (dynamic) and secondary (static) subspaces of the full Hilbert space. It combines the merits of both internally and externally contracted configuration interaction as well as intermediate Hamiltonian approaches. When the Hamiltonian matrix elements between the contracted external functions, with the coefficients determined by first order perturbation, are approximated as the diagonal elements of the zeroth-order Hamiltonian $H_0$, we obtain a multi-state multi-reference second-order perturbation theory (denoted as SDS-MS-MRPT2) that scales computationally with the fifth power of the molecular size. Depending on how $H_0$ is defined, various variants of SDS-MS-MRPT2 can be obtained. For simplicity, we here choose $H_0$ as a multi-partitioned Møller–Plesset-like diagonal operator. Further combined with the string-based macroconfiguration technique, an efficient implementation of SDS-MS-MRPT2 is realized and tested for prototypical systems of variable near-degeneracies. The results reveal that SDS-MS-MRPT2 can well describe not only standard benchmark systems but also problematic systems. Taking SDS-MS-MRPT2 as a start, the accuracy may steadily be increased by relaxing the contraction of the external functions and/or iterating the diagonalization–perturbation–diagonalization procedure. As such, the SDS framework offers a very powerful scenario for handling strongly correlated systems.

Appendix: String-based macroconfigurations

Since their introduction into quantum chemistry by Knowles and Handy [41], representation of determinants as separate strings of $m_s=\frac{1}{2}$ (alpha) occupancies and $m_s=-\frac{1}{2}$ (beta) occupancies have become well entrenched [42–44]. This representation has several distinct advantages, e.g., the simple coupling coefficients (either −1, 0 or 1) and the compact independent indexing of alpha and beta strings (i.e., pruned binomial trees). However, extension of this concept, originally designed for FCI, to more complicated calculations that have multiple groups of orbitals (e.g., MRCISD with core, valence and virtual orbitals) has generally been accompanied by a reduction in some of the advantages of the original method. In the SDS framework introduced in this work, including its MRPT2 realization, there is a special need to limit excitations between particular groups of orbitals.

The partitioning of orbitals into arbitrary groups together with assigning fixed number of electrons for each group, i.e., a macroconfiguration $\kappa _i=G_1^{N_1^{{\kappa }_1}}G_2^{N_2^{{\kappa }_2}}\ldots G_g^{N_g^{{\kappa }_g}}$, has been shown to lead to both reliable means of generating complete and incomplete model spaces and computational efficiencies [40]. The advantages of articulating parts of the Hilbert space with macroconfigurations and the advantages of representing determinants with strings would seem to be incompatible. However, we demonstrate that this is not the case.

Fig. 6

Representation of a double excitation from a ket determinant involving groups 2 and 7 to a bra determinant involving groups 4 and 6

The critical insight that allows full use of both the concepts of macroconfigurations and strings is the notion of spectator groups. Suppose that a class of excitations from a set of ket determinants to another set of bra determinants can be characterized as a one-electron excitation from group $G_j$ to $G_i$ and another from $G_l$ to $G_k$. A concrete example is depicted in Fig. 6. In this example, there are four orbital groups, corresponding to eight spin-orbital groups. The indices of the generated strings within groups $G_k$ and $G_i$ can be calculated straightforwardly as walks on a (universal) pruned binary tree, as in the original work of Knowles and Handy. The indices of the strings within $G_l$ and $G_i$ are even simpler, as they can be generated sequentially. The key to efficiency is that the product of the resulting coupling coefficient multiplied by the two-electron integral [i.e., $(ij|kl)$] is identical for every bra/ket pair that shares identical walks in the spectator groups (i.e., groups 1, 3, 5 and 8 in the considered example). It can be appreciated that the number of bras and kets sharing this matrix element could be very large (e.g., if group 8 is the beta excited orbitals and there are two electrons in this group, then its contribution alone can be on the order of 10,000 for a modest molecule using a triple split valence one-electron basis set).

Sharing the single matrix element by a truly large number of bra/ket pairs can only occur if the indexing of the bra and ket determinants can be done efficiently. This is accomplished by precomputing the base index of each group of the bra and ket macroconfigurations. For example, each walk within group 3 is offset from the previous one by the product of the total number of walks in group 2 multiplied by the total number of walks in group 1. Then, fragments of walks up to and including group 3 can be computed as the contribution from groups 1 and 3 added to the contribution from group 2. Since the contributions from groups 1 and 3 do not change as a result of the class of excitations connecting the ket and bra macroconfigurations, the sequence of contributions to the overall indices from the spectator groups can be precomputed at the macroconfiguration level.

The described algorithm is clearly not dependent on the number of spin-orbital groups. This allows one further efficiency in that Abelian molecular point group symmetry restrictions can be imposed at the macroconfiguration level, provided that each spin-orbital group contain only spin orbitals that transform as the same irreducible representation. In practice, this requires a sequestering of spin orbitals for each group of occupation-restricted orbitals. For example, if the CASSCF active orbitals transform as, say, two of the irreducible representations of the molecule, then four groups of spin orbitals would be generated.

The described algorithm is applicable to any partitioning of the Hilbert space that can be represented as direct sum of determinants generated from macroconfigurations. This includes all finite-order PT, including multi-reference variants, of present interest, as well as (multi-reference) CI and CEPA methods. Consequently, the computational realization was straightforwardly validated against well-established computer programs.