Half-Projected σ Self-Consistent Field For Electronic Excited States

Abstract

Fully self-consistent mean-field solutions of electronic excited states have been much less accessible compared to ground state solutions (e.g., Hartree-Fock). The main reason is that most excited states are energy saddle points, and hence energy-based optimization methods such as ∆-SCF often collapse to the ground state. Recently, our research group has developed a new method, σ-SCF [J. Chem. Phys., 147, 214104 (2017)], that successfully solves the "variational collapse" problem of energy-based methods. Despite the success, σ-SCF solutions are often spin-contaminated for open-shell states due to the single-determinant nature; unphysical behaviors such as disappearing solutions and discontinuous first-order energy derivatives are also observed along with the spontaneous breaking of spin or spatial symmetries. In this work, we tackle these problems by partially restoring the broken spin-symmetry of a σ-SCF solution through an approximate spin-projection scheme called half-projection. Orbitals of the projected wave function are optimized in a variation-after-projection (VAP) manner. The resulting theory, which we term half-projected (HP) σ-SCF, brings substantial improvement to the description of singlet and triplet excitations of the original σ-SCF method. Numerical simulations on small molecules suggest that HP σ-SCF delivers high-quality

Improving half-projected spin-contaminated wave functions by multi-configuration perturbation theory

“…Therefore, in a previous paper, we investigated the performance of the so-called half-projection (HP) technique advocated by Smeyers 40 . This early method has recently been revisited and successfully applied within a mean-field theory 41 as well as to geminal-based wave functions 35 . As shown in Ref.…”

Using SCF metadynamics to extend density matrix embedding theory to excited states

“…These methods include ∆-SCF, 14 σ -SCF, 15 and half-projected σ -SCF. 16 These methods should be explored as alternatives for more complicated systems.…”

“…It is difficult to study excited states using cheaper Hartree-Fock (HF) theory, 11 but higher lying self-consistent field (SCF) solutions 12 to the HF equations have been compared to excited states. These include Delta SCF (∆-SCF), 13,14 σ -SCF, 15,16 and SCF metadynamics. 17 An attractive compromise between the cheap HF and accurate high level methods is the method of embedding, where an impurity, treated at a high level of theory, is embedded within a bath, treated at a low level of theory.…”

On the variational principle for the non-linear Schrödinger equation

“…It is worth mentioning that the above procedure may be, in principle applied to any self-consistent eigenstate of a nonlinear Hamiltonian, though optimization for excited states may not be easy. Numerically stable procedures for determining excited states variationally, that make use of second centralized moments, were proposed recently [31,32] for linear Hamiltonians.…”