Direct unconstrained optimization of excited states in density functional theory

TL;DR

This paper introduces VM TIDFT, a novel unconstrained optimization method for excited states in DFT that avoids variational collapse and accurately describes complex excitations.

Contribution

The paper develops VM TIDFT, a new approach allowing nonorthogonal excited state optimization with continuous orthogonality enforcement, enabling the use of molecular orbital coefficients and unconstrained algorithms.

Findings

Robust convergence with preconditioned conjugate gradient algorithm.

Accurate energies for charge-transfer and double-electron excitations.

Effective alternative to TDDFT for difficult excited states.

Abstract

Orbital-optimized density functional theory (DFT) has emerged as an alternative to time-dependent (TD) DFT capable of describing difficult excited states with significant electron density redistribution, such as charge-transfer, Rydberg, and double-electron excitations. Here, a simple method is developed to solve the main problem of the excited-state optimization -- the variational collapse of the excited states onto the ground state. In this method, called variable-metric time-independent DFT (VM TIDFT), the electronic states are allowed to be nonorthogonal during the optimization but their orthogonality is gradually enforced with a continuous penalty function. With nonorthogonal electronic states, VM TIDFT can use molecular orbital coefficients as independent variables, which results in a closed-form analytical expression for the gradient and allows to employ any of the multiple unconstrained optimization algorithms that guarantees convergence of the excited-state optimization. Numerical tests on multiple molecular systems show that the variable-metric optimization of excited states performed with a preconditioned conjugate gradient algorithm is robust and produces accurate energies for well-behaved excitations and, unlike TDDFT, for more challenging charge-transfer and double-electron excitations.