Energy Landscape of State-Specific Electronic Structure Theory

Abstract

State-specific approximations can provide a more accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere, and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated, and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree–Fock and excited-state mean-field representations of singlet H 2 (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape, and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimization landscape and elucidate the challenges faced by variance optimization algorithms, including the presence of unphysical saddle points or maxima of the variance.

Applying Generalized Variational Principles to Excited-State-Specific Complete Active Space Self-consistent Field Theory

“…To understand how these advantages come about, let us turn to discussing recent progress in the use of quasi-Newton methods to minimize energy-gradient-based objective functions, which have proven effective in the context of both the excited-state mean field (ESMF) ansatz − and Kohn–Sham ΔSCF . Essentially, the idea is to search for energy saddle pointswhich in full CI (FCI) would be the exact excited statesby minimizing the norm of the energy gradient with respect to the variational parameters.…”

Variational Density Functional Calculations of Excited States: Conical Intersection and Avoided Crossing in Ethylene Bond Twisting

“…Symmetry breaking is expected to be similarly important in calculations of excited states. Excited states can be found as higher-energy self-consistent field (SCF) solutions of the KS equations, corresponding to stationary points on the electronic energy surface other than the ground state minimum. ,− Unlike TDDFT, such calculations are variational (i.e., based on the calculus of variation), and they can provide better approximations to long-range charge-transfer, ,, Rydberg, , core-level, , and other excitations , characterized by significant change in the electron density. However, while excited state DFT has been applied in a variety of fields, − only a few studies of electronic near-degeneracies have been reported, ,, and the results appear to be contradictory.…”

Variational Density Functional Calculations of Excited States: Conical Intersection and Avoided Crossing in Ethylene Bond Twisting

“…Excited states can be found as higher-energy self-consistent field (SCF) solutions of the KS equations, corresponding to stationary points on the electronic energy surface other than the ground state minimum. 26,[55][56][57][58][59][60][61][62][63][64][65] Unlike TDDFT, such calculations are variational (i. e. based on the calculus of variation), and can provide better approximations to long-range charge-transfer, 26,59,66 Rydberg, 67,68 core-level, 69,70 and other excitations 59,61 characterized by significant change in the electron density. However, while excited state DFT has been applied in a variety of fields, [71][72][73][74][75][76][77] only a few studies of electronic near-degeneracies have been reported 26,78,79 and the results appear contradictory.…”