On the description of conical intersections—A continuous representation of the local topography of seams of conical intersection of three or more electronic states: A generalization of the two state result

Abstract

For conical intersections of two states (I,J = I + 1) the vectors defining the branching or g-h plane, the energy difference gradient vector g(I,J), and the interstate coupling vector h(I,J), can be made orthogonal by a one parameter rotation of the degenerate electronic eigenstates. The representation obtained from this rotation is used to construct the parameters that describe the vicinity of the conical intersection seam, the conical parameters, s(x)(I,J)(R), s(y)(I,J)(R), g(I,J)(R), and h(I,J)(R). As a result of the orthogonalization these parameters can be made continuous functions of R, the internuclear coordinates. In this work we generalize this notion to construct continuous parametrizations of conical intersection seams of three or more states. The generalization derives from a recently introduced procedure for using non-degenerate electronic states to construct coupled diabatic states that represent adiabatic states coupled by conical intersections. The procedure is illustrated using the seam of conical intersections of three states in parazolyl as an example.

The Representation and Parametrization of Orthogonal Matrices

“…That is, the Stiefel manifold itself is parametrized in two steps as the product QZ . Examples of this situation are the determination of one or more adiabatic electronic states within a diabatic basis, , “perturb-then-diagonalize” formulations of wave functions, ,, and other similar separations that occur when slowly varying “weaker” correlations are separated from more abrupt, or even discontinuous, “stronger” correlations.…”

On the Construction of Property Based Diabatizations: Diabolical Singular Points

“…The equivalence of criteria and , is somewhat more complicated to establish. We sketch the demonstration below and refer the reader to ref or refs and for details. Introduce the [ N state ] independent rotations equivalently which are to be applied to Ψ ( a ) for infinitesimal θ i , j .…”

“…Introduce the [ N state ] independent rotations equivalently which are to be applied to Ψ ( a ) for infinitesimal θ i , j . We have previously shown , that the pairwise rotation in eq 11 results in the following equations From eqs and , we obtain that condition [] is equivalent to .…”

Unitary coupled-cluster based self-consistent polarization propagator theory: a quadratic unitary coupled-cluster singles and doubles scheme

“…[36][37][38] Available calculations of same symmetry conical intersections have used hermitian excitedstate formulations such as the algebraic diagrammatic construction (ADC) methods, 31,39 time-dependent density functional theory (TDDFT)-based techniques, 40,41 constrained density functional theory-configuration interaction (CDFT-CI) method, 42 and multireference techniques including complete active space self-consistent-field (CASSCF) method, 43 CAS second-order perturbation theory (CASPT2), 44 multi-reference perturbation theory (MRPT), 45,46 and multi-reference configuration interaction (MRCI) methods. [47][48][49][50][51] The MRCI method as a non-perturbative wavefunction based approach has exhibited robust performance. However, the lack of size-extensivity in MRCI often poses difficulties in obtaining accurate electronic energies.…”