Canonical-ensemble state-averaged complete active space self-consistent field (SA-CASSCF) strategy for problems with more diabatic than adiabatic states: charge-bond resonance in monomethine cyanines.

Abstract

This paper reviews basic results from a theory of the a priori classical probabilities (weights) in state-averaged complete active space self-consistent field (SA-CASSCF) models. It addresses how the classical probabilities limit the invariance of the self-consistency condition to transformations of the complete active space configuration interaction (CAS-CI) problem. Such transformations are of interest for choosing representations of the SA-CASSCF solution that are diabatic with respect to some interaction. I achieve the known result that a SA-CASSCF can be self-consistently transformed only within degenerate subspaces of the CAS-CI ensemble density matrix. For uniformly distributed ("microcanonical") SA-CASSCF ensembles, self-consistency is invariant to any unitary CAS-CI transformation that acts locally on the ensemble support. Most SA-CASSCF applications in current literature are microcanonical. A problem with microcanonical SA-CASSCF models for problems with "more diabatic than adiabatic" states is described. The problem is that not all diabatic energies and couplings are self-consistently resolvable. A canonical-ensemble SA-CASSCF strategy is proposed to solve the problem. For canonical-ensemble SA-CASSCF, the equilibrated ensemble is a Boltzmann density matrix parametrized by its own CAS-CI Hamiltonian and a Lagrange multiplier acting as an inverse "temperature," unrelated to the physical temperature. Like the convergence criterion for microcanonical-ensemble SA-CASSCF, the equilibration condition for canonical-ensemble SA-CASSCF is invariant to transformations that act locally on the ensemble CAS-CI density matrix. The advantage of a canonical-ensemble description is that more adiabatic states can be included in the support of the ensemble without running into convergence problems. The constraint on the dimensionality of the problem is relieved by the introduction of an energy constraint. The method is illustrated with a complete active space valence-bond (CASVB) analysis of the charge/bond resonance electronic structure of a monomethine cyanine: Michler's hydrol blue. The diabatic CASVB representation is shown to vary weakly for "temperatures" corresponding to visible photon energies. Canonical-ensemble SA-CASSCF enables the resolution of energies and couplings for all covalent and ionic CASVB structures contributing to the SA-CASSCF ensemble. The CASVB solution describes resonance of charge- and bond-localized electronic structures interacting via bridge resonance superexchange. The resonance couplings can be separated into channels associated with either covalent charge delocalization or chemical bonding interactions, with the latter significantly stronger than the former.