# Tracking excited states in wave function optimization using density matrices and variational principles

## TL;DR

This paper introduces a novel method for accurately tracking and optimizing individual excited states in quantum chemistry calculations, overcoming common issues like root flipping and near-degeneracies, and improving the performance of related perturbation theories.

## Contribution

The authors develop a state-specific optimization approach combining maximum overlap and variational principles, enhancing excited state tracking in complete active space methods.

## Key findings

- More effective than energy-based root selection and naive maximum overlap methods.
- Improves the performance of complete active space perturbation theory.
- Compatible with large active space methods and easy to implement.

## Abstract

We present a method for finding individual excited states' energy stationary points in complete active space self-consistent field theory that is compatible with standard optimization methods and highly effective at overcoming difficulties due to root flipping and near-degeneracies. Inspired by both the maximum overlap method and recent progress in excited state variational principles, our approach combines these ideas in order to track individual excited states throughout the orbital optimization process. In a series of tests involving root flipping, near-degeneracies, charge transfers, and double excitations, we show that this approach is more effective for state-specific optimization than either the naive selection of roots based on energy ordering or a more direct generalization of the maximum overlap method. Furthermore, we provide evidence that this state-specific approach improves the performance of complete active space perturbation theory. With a simple implementation, a low cost, and compatibility with large active space methods, the approach is designed to be useful in a wide range of excited state investigations.