Excited state orbital optimization via minimizing the square of the gradient: General approach and application to singly and doubly excited states via density functional theory

TL;DR

This paper introduces a general Square Gradient Minimization approach for reliably converging excited state solutions in quantum chemistry orbital optimization, outperforming existing methods in challenging cases and enabling accurate excited state predictions.

Contribution

The paper presents a novel SGM method that converges excited states without collapse, applicable to various DFT approaches, and demonstrates its effectiveness over traditional techniques.

Findings

SGM converges challenging excited states where MOM fails.

Accurately predicts doubly excited state energies.

Improves excited state calculations beyond TDDFT limitations.

Abstract

We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting Square Gradient Minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant $\Delta$SCF and spin-purified Restricted Open-Shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It is found that SGM can converge challenging states where the Maximum Overlap Method (MOM) or analogues either collapse to the ground state or fail to converge. We also report that $\Delta$SCF/ROKS predict highly accurate excitation energies for doubly excited states (which are inaccessible via TDDFT). Singly excited states obtained via ROKS are also found to be quite accurate, especially for Rydberg states that frustrate (semi)local TDDFT. Our results suggest that orbital optimized excited state DFT methods can be used to push past the limitations of TDDFT to doubly excited, charge-transfer or Rydberg states, making them a useful tool for the practical quantum chemist's toolbox for studying excited states in large systems.