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

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 ΔSCF
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 ΔSCF/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.

本研究提出了一种通用的方法，旨在将激发态解收敛至任何量子化学轨道优化过程，且避免了变分塌陷的风险。由此产生的平方梯度最小化（SGM）方法仅需解析能量/拉格朗日轨道梯度，且在有限差分法实现时，仅比基态轨道优化多耗费三倍的迭代成本。SGM方法被应用于单行列式ΔSCF和自旋纯化受限开放壳Kohn-Sham（ROKS）两种途径，以探究轨道优化DFT激发态的准确性。研究发现，SGM能够收敛那些最大重叠法（MOM）或其类似方法要么塌陷至基态要么无法收敛的难题状态。此外，我们还报告了ΔSCF/ROKS能够预测双激发态的高精度激发能量（这些状态通过TDDFT无法访问）。通过ROKS获得的单激发态也被发现具有很高的准确性，尤其是对于那些令（半）局部TDDFT感到挫败的里德伯态。我们的结果表明，轨道优化激发态DFT方法可以突破TDDFT的局限性，扩展至双激发、电荷转移或里德伯态，从而成为研究大型系统中激发态的实用量子化学工具箱中的有效工具。