A Diabatic Representation Including Both Valence Nonadiabatic Interactions and Spin−Orbit Effects for Reaction Dynamics

Abstract

A diabatic representation is convenient in the study of electronically nonadiabatic chemical reactions because the diabatic energies and couplings are smooth functions of the nuclear coordinates and the couplings are scalar quantities. A method called the fourfold way was devised in our group to generate diabatic representations for spin-free electronic states. One drawback of diabatic states computed from the spin-free Hamiltonian, called a valence diabatic representation, for systems in which spin-orbit coupling cannot be ignored is that the couplings between the states are not zero in asymptotic regions, leading to difficulties in the calculation of reaction probabilities and other properties by semiclassical dynamics methods. Here we report an extension of the fourfold way to construct diabatic representations suitable for spincoupled systems. In this article we formulate the method for the case of even-electron systems that yield pairs of fragments with doublet spin multiplicity. For this type of system, we introduce the further simplification of calculating the triplet diabatic energies in terms of the singlet diabatic energies via Slater's rules and assuming constant ratios of Coulomb to exchange integrals. Furthermore, the valence diabatic couplings in the triplet manifold are taken equal to the singlet ones. An important feature of the method is the introduction of scaling functions, as they allow one to deal with multibond reactions without having to include Comparison of the spin-coupled adiabatic energies obtained from the spin-valence diabatics with those obtained by ab initio calculations with geometry-dependent spin-orbit matrix elements shows that the new method is sufficiently accurate for practical purposes. The method formulated here should be most useful for systems with a large number of atoms, especially heavy atoms, and/or a large number of spin-coupled electronic states.

Model space diabatization for quantum photochemistry

“…In the original paper on applying the fourfold way with MC-QDPT, 19 we pointed out the dependence of MC-QDPT energies on orbital rotations, and we resolved the ambiguity by defining the adiabatic energies as those calculated using the DMOs rather than the canonical molecular orbitals (CMOs). In the original work and subsequent work 26,[43][44][45][46][47][48] we found, when we checked, only small differences between the two sets of adiabatic energies; however, for a current application to thioanisole, we found differences of up to 0.8 eV when using DMOs obtained by the fourfold way and up to 0.05 eV when using DMOs obtained by the threefold way. Therefore, we developed the scheme presented here to give a diabatic potential energy matrix that, when diagonalized, gives precisely the adiabatic energies of standard QDPT with CMOs.…”

“…The fourfold way was first formulated at the CASSCF level 18 and then extended to the MC-QDPT level 19 and to include spin-orbit coupling. 45 At the MC-QDPT level, it can be used with diabatic molecular orbitals obtained at either the MC-QDPT level 19 or the CASSCF level. 29 It has been successfully applied to a variety of systems at both the CASSCF and MC-QDPT levels.…”

“…29 It has been successfully applied to a variety of systems at both the CASSCF and MC-QDPT levels. 26,[43][44][45][46][47][48] However, as mentioned in the Introduction, for some cases, the original fourfold way at the MC-QDPT level may give results that are quite different from the standard MC-QDPT method. These problematic cases are the motivation for developing the present MSD strategy.…”

Comment on “Fe2: As simple as a Herculean labour. Neutral (Fe2), cationic (Fe2+), and anionic (Fe2−) species” [J. Chem. Phys. 142, 244304 (2015)]

“…A widely used approximation in treating nonadiabatic dynamics is to treat the adiabatic states in the absence of SOC as diabatic states for the full Hamiltonian including spin-orbit coupling. 4,5 For example, it has been emphasized that an "important effect of SOC is that it causes spin-forbidden processes to become partially allowed through interaction and mixing of states of different spin multiplicity." 5 In this sense, the adiabatic states in the absence of spin-orbit coupling (states with well defined Λ-S labeling) become diabatic states for the full problem.…”

“…4,5 For example, it has been emphasized that an "important effect of SOC is that it causes spin-forbidden processes to become partially allowed through interaction and mixing of states of different spin multiplicity." 5 In this sense, the adiabatic states in the absence of spin-orbit coupling (states with well defined Λ-S labeling) become diabatic states for the full problem. For Fe 2 , the full-problem ground state has contributions from three such diabatic states, resulting in two avoided crossings, each due to two states, when spin-orbit coupling is included.…”