6 Wave Function-Based Correlation Methods

6.16 Simplified Coupled-Cluster Methods Based on a Perfect-Pairing Active Space

The methods described below are related to valence bond theory and are handled by the GVBMAN module. The following models are available:

CORRELATION
       Specifies the correlation level in GVB models handled by GVBMAN.
TYPE:
       STRING
DEFAULT:
       None No Correlation
OPTIONS:
       PP CCVB GVB_IP GVB_SIP GVB_DIP OP NP 2P
RECOMMENDATION:
       As a rough guide, use PP for biradicaloids, and CCVB for polyradicaloids involving strong spin correlations. Consult the literature for further guidance.

Molecules where electron correlation is strong are characterized by small energy gaps between the nominally occupied orbitals (that would comprise the Hartree-Fock wave function, for example) and nominally empty orbitals. Examples include so-called diradicaloid molecules,441 or molecules with partly broken chemical bonds (as in some transition-state structures). Because the energy gap is small, electron configurations other than the reference determinant contribute to the molecular wave function with considerable amplitude, and omitting them leads to a significant error.

Including all possible configurations however, is a vast overkill. It is common to restrict the configurations that one generates to be constructed not from all molecular orbitals, but just from orbitals that are either “core” or “active”. In this section, we consider just one type of active space, which is composed of two orbitals to represent each electron pair: one nominally occupied (bonding or lone pair in character) and the other nominally empty, or correlating (it is typically anti-bonding in character). This is usually called the perfect pairing active space, and it clearly is well-suited to represent the bonding/anti-bonding correlations that are associated with bond-breaking.

The quantum chemistry within this (or any other) active space is given by a Complete Active Space SCF (CASSCF) calculation, whose exponential cost growth with molecule size makes it prohibitive for systems with more than about 14 active orbitals. One well-defined coupled cluster (CC) approximation based on CASSCF is to include only double substitutions in the valence space whose orbitals are then optimized. In the framework of conventional CC theory, this defines the valence optimized doubles (VOD) model,495 which scales as 𝒪(N6) (see Section 6.10.2). This is still too expensive to be readily applied to large molecules.

The methods described in this section bridge the gap between sophisticated but expensive coupled cluster methods and inexpensive methods such as DFT, HF and MP2 theory that may be (and indeed often are) inadequate for describing molecules that exhibit strong electron correlations such as diradicals. The coupled cluster perfect pairing (PP),199, 79 imperfect pairing932 (IP) and restricted coupled cluster935 (RCC) models are local approximations to VOD that include only a linear and quadratic number of double substitution amplitudes respectively. They are close in spirit to generalized valence bond (GVB)-type wave functions,310 because in fact they are all coupled cluster models for GVB that share the same perfect pairing active space. The most powerful method in the family, the Coupled Cluster Valence Bond (CCVB) method,847, 848, 849 is a valence bond approach that goes well beyond the power of GVB-PP and related methods, as discussed below in Sec. 6.16.2.