We present a time-dependent density functional theory (TDDFT) based approach to compute the light-matter couplings between two different manifolds of excited states relative to a common ground state. These quantities are the necessary ingredients to
solve the Kramers--Heisenberg equation for resonant inelastic X-ray scattering (RIXS) and several other types of two-photon spectroscopies. The procedure is based on the pseudo-wavefunction approach, where TDDFT eigenstates are treated as a configuration interaction wavefunction with single excitations, and on the restricted energy window approach, where a manifold of excited states can be rigorously defined based on the energies of the occupied molecular orbitals involved in the excitation process. We illustrate the applicability of the method by calculating the 2p4d RIXS maps of three representative Ruthenium complexes and comparing them to experimental results. The method is able to accurately capture all the experimental features in all three complexes, with relative energies correct to within 0.6 eV at the cost of two independent TDDFT calculations.
The Hartree-Fock problem was recently recast as a semidefinite optimization over the space of rank-constrained two-body reduced-density matrices (RDMs) [Phys. Rev. A 89, 010502(R) (2014)]. This formulation of the problem transfers the non-convexity o
f the Hartree-Fock energy functional to the rank constraint on the two-body RDM. We consider an equivalent optimization over the space of positive semidefinite one-electron RDMs (1-RDMs) that retains the non-convexity of the Hartree-Fock energy expression. The optimized 1-RDM satisfies ensemble $N$-representability conditions, and ensemble spin-state conditions may be imposed as well. The spin-state conditions place additional linear and nonlinear constraints on the 1-RDM. We apply this RDM-based approach to several molecular systems and explore its spatial (point group) and spin ($S^2$ and $S_3$) symmetry breaking properties. When imposing $S^2$ and $S_3$ symmetry but relaxing point group symmetry, the procedure often locates spatial-symmetry-broken solutions that are difficult to identify using standard algorithms. For example, the RDM-based approach yields a smooth, spatial-symmetry-broken potential energy curve for the well-known Be--H$_2$ insertion pathway. We also demonstrate numerically that, upon relaxation of $S^2$ and $S_3$ symmetry constraints, the RDM-based approach is equivalent to real-valued generalized Hartree-Fock theory.
