Advances in Generalized Valence Bond-Coupled Cluster Methods for Electronic Structure Theory
The electron-electron correlation term in the electronic energy of a molecule is the most difficult term to compute, yet it is of both qualitative and quantitative importance for a diverse range of chemical applications of computational quantum chemistry. Generalized Valence Bond-Coupled Cluster (GVB-CC) methods are computationally efficient, size-consistent wavefunction based methods to capture the most important static (valence) contributions to the correlation energy. Despite these advantages early GVB-CC methods suffer four major short-comings: over-localization leading to symmetry breaking, poor behavior with spin-unrestriction, neglect of the dynamic (or residual) correlation energy, and multiple orbital minima. The work presented here is directed at rectifying these major short-comings of the GVB-CC methods.
The GVB-CC methods suffer from symmetry-breaking (SB) in systems with multiple resonance structures, which arises due to neglected correlations. We show how these problems can be significantly removed by using 2nd order perturbation theory (PT) for weak correlations coupling 3 different electron pairs, and (infinite order) CC theory for stronger correlations involving electrons in only 1 or 2 different pairs. The resulting Three-pair corrected Imperfect Pairing (TIP) method works quite well in removing SB from aromatic hydrocarbons, but it is shown that to robustly combine CC and PT it is necessary to modify several aspects of the basic method. A penalty function term needs to be introduced to regularize the PT amplitudes and ensure they remain small. When TIP is compared side-by-side with CC treatments of the 3-pair correlations, the results suggest that the penalty function is beneficial for any hybrid CC/PT method that includes orbital optimization. TIP greatly reduces SB and recovers a significantly higher fraction of the valence electron correlation energy than earlier double excitation based GVB-CC methods.
Spin-unrestriction is typically defined as a free variation of the molecular orbitals of different spins in order to lower the molecular energy. In GVB-CC methods, this approach often leads to undesirable artifacts, so therefore we develop an alternate method for spin-unrestriction that transfers concepts from corresponding orbitals in Hartree-Fock theory to active space electron correlation methods. The Unrestriction-in-Active-Pairs (UAP) procedure forces an orbital to only mix with its corresponding virtual orbital to spin-unrestrict, thus limiting the number of degrees of freedom that the molecular orbitals can use to spin-unrestrict and thereby eliminating many of the undesirable artifacts of spin-unrestriction. It can be shown that in the unrestricted limit of ROHF fragments (the UAP dissociation orbitals) the CC equations are singular if only the strongly correlated electrons are considered. The CC equations can be regularized to alleviate this problem. UAP when combined with the GVB-CC model chemistries we have developed makes a powerful tool for predicting potential energy surfaces including appropriate orbitals on the molecular fragments at dissociation.
Finally, we consider the extension of the standard single-determinant Kohn-Sham (KS) method to the case of a multi-configuration auxiliary wave function, such as the GVB-CC methods. By applying the rigorous Kohn-Sham method to this case, we construct the proper interacting and auxiliary energy functionals. Following the Hohenberg-Kohn theorem for both energy functionals, we derive the corresponding multi-configuration Kohn-Sham equations, based on a local effective potential. At the end of the analysis we show that, at the ground state, the auxiliary wavefunction must collapse into a single-determinant wave function, equal to the regular KS wavefunction. We also discuss the stability of the wave function in multi-configuration density functional theory methods where the auxiliary system is partially interacting, and the residual correlation is evaluated as a functional of the density. We also discuss improvements to the definition of the residual correlation energy of the partially interacting system.