General framework for calculating spin–orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions

Abstract

Standard implementations of nonrelativistic excited-state calculations compute only one component of spin multiplets (i.e., Ms = 0 triplets); however, matrix elements for all components are necessary for deriving spin-dependent experimental observables. Wigner-Eckart's theorem allows one to circumvent explicit calculations of all multiplet components. We generate all other spin-orbit matrix elements by applying Wigner-Eckart's theorem to a reduced one-particle transition density matrix computed for a single multiplet component. In addition to computational efficiency, this approach also resolves the phase issue arising within Born-Oppenheimer's separation of nuclear and electronic degrees of freedom. A general formalism and its application to the calculation of spin-orbit couplings using equation-of-motion coupledcluster wave functions are presented. The two-electron contributions are included via the mean-field spin-orbit treatment. Intrinsic issues of constructing spin-orbit mean-field operators for open-shell references are discussed, and a resolution is proposed. The method is benchmarked by using several radicals and diradicals. The merits of the approach are illustrated by a calculation of the barrier for spin inversion in a high-spin tris(pyrrolylmethyl)amine Fe(II) complex.

How to stay out of trouble in RIXS calculations within equation-of-motion coupled-cluster damped response theory? Safe hitchhiking in the excitation manifold by means of core–valence separation

“…In contrast to multi-reference approaches, EOM-CC does not involve systemspecific parameterization (e.g., active-space selection), thus satisfying Pople's requirements of theoretical model chemistry 40 that can be used for systematic studies and comparisons between different systems. The EOM-CC framework yields reliable lowerorder properties such as solvatochromic shifts 41 , transition dipole moments 35 , spin-orbit [42][43][44][45] and non-adiabatic couplings [46][47][48] , as well as higher-order properties 49 such as two-photon absorption cross sections [50][51][52][53][54][55] , static and dynamical polarizabilities [56][57][58][59] . Whereas the bulk of prior developments and applications of the EOM-CC methods as well as of the closely related coupled-cluster response theory [60][61][62] were in the VUV regime, these methods are now being extended to the X-ray regime and their performance is being explored for computing, for example, XAS [15][16][17]19,[63][64][65] , XES 24,66 , and RIXS 24,66 spectra.…”

Calculation of Spin–orbit Couplings Using RASCI Spinless One-Particle Density Matrices: Theory and Applications

“…We construct u following the prescription given by Pokhilko and collaborators, [19] through the spin-conserving αα and ββ parts of the 1TDM (γ) between states with the same spin projection,…”

“…We implemented the computation of SOCs within the RASCI framework in a development version of the Q-Chem electronic structure package, [24] using general libraries developed for SOC calculations within EOM-CC. [19] The required integrals are evaluated using the King-Furlani algorithm, [66] as implemented by Epifanovsky et al [62] Spin-orbit NTOs [29] are computed and analyzed using the libwfa library. [67] RASCI calculations were carried out with RAS1 and RAS3 subspaces including all occupied and virtual orbitals, respectively.…”

“…Effective computational algorithms [17][18][19] for the calculation of SOC matrix elements often exploit Wigner-Eckart's theorem, [20,21] which enables the evaluation of the entire SOC matrix between two multiplets from just one multiplet component M (for example, the M = 0 component of a triplet state). Recently, Pokhilko and collaborators [19] have reported an algorithm for evaluating SOCs by application of Wigner-Eckart's theorem to the reduced spinless density matrix generated for one specific transition. Formulated in the spin-orbital form, [22] this approach is ansatz agnostic and is applicable to transitions between any types of open-shell states, as long as respective transition density matrices can be obtained.…”

“…[26] This algorithm has been also adopted by Meitei et al [27] in a pilot RASCI (restricted active space configuration interaction) code within Psi4. Here we extend the implementation of Pokhilko [19] to production-level implementation of the RASCI family of methods [28] within Q-Chem, which enables calculations of a large variety of open-shell and electronically excited states. In addition to the calculation of SOCs, we also enabled calculation of natural transition orbitals (NTOs) of the spinless transition density matrix (1TDM), which provides insight into the mechanism of the spin-orbit interactions.…”