Applying Generalized Variational Principles to Excited-State-Specific Complete Active Space Self-consistent Field Theory

Abstract

We employ a generalized variational principle to improve the stability, reliability, and precision of fully excited-state-specific complete active space self-consistent field theory. Compared to previous approaches that similarly seek to tailor this ansatz’s orbitals and configuration interaction expansion for an individual excited state, we find the present approach to be more resistant to root flipping and better at achieving tight convergence to an energy stationary point. Unlike state-averaging, this approach allows orbital shapes to be optimal for individual excited states, which is especially important for charge-transfer states and some doubly excited states. We demonstrate the convergence and state-targeting abilities of this method in LiH, ozone, and MgO, showing in the latter that it is capable of finding three excited-state energy stationary points that no previous method has been able to locate.

Energy Landscape of State-Specific Electronic Structure Theory

“…4,125,126 The combined Lagrangian may include functionals that target a particular energy (such as eq 41), the square-magnitude of the local gradient, orthogonality to a nearby state, or a desirable dipole moment. 126 This approach is particularly suited to systems where there is a good initial guess for the excited state or its properties, and a good choice of objective functions can significantly accelerate numerical convergence. 125 These constraints may also help to prevent the drift of stochastic variance optimization algorithms by increasing the effective barrier heights between minima with the correct target properties.…”

“…Finally, although not considered in this work, the generalized variational principle developed by Neuscamman and co-workers includes Lagrange multipliers to simultaneously optimize multiple objective functions that target a particular excited eigenstate. ,, The combined Lagrangian may include functionals that target a particular energy (such as eq ), the square-magnitude of the local gradient, orthogonality to a nearby state, or a desirable dipole moment . This approach is particularly suited to systems where there is a good initial guess for the excited state or its properties, and a good choice of objective functions can significantly accelerate numerical convergence .…”

Optimization Stability in Excited-State-Specific Variational Monte Carlo

“…58,59 However, as we show in our results, orbital optimization can still be challenging for the LM, and we also consider VMC optimization with the orbitals left at shapes obtained from a recent state-specific CASSCF approach. 38,60,61 This approach of combining state-specific quantum chemistry with VMC has recently been shown to provide accurate excitation energies across a range of different types of excited states 35 and we provide another example of this here for the excited state in CN5. However, the key thrust of the present study is optimization stability, and so in many of our results we will be optimizing orbitals within VMC to test stability in that context.…”

“…Throughout our results, we employ and optimize one- and two-body Jastrow factors, which are constructed with splines for the functions χ k and u kl . For our stability analysis, we also optimize the orbitals of our Slater expansions, benefiting from recent methodological improvements with the table method. , However, as we show in our results, orbital optimization can still be challenging for the LM, and we also consider VMC optimization with the orbitals left at shapes obtained from a recent state-specific CASSCF approach. ,, This approach of combining state-specific quantum chemistry with VMC has recently been shown to provide accurate excitation energies across a range of different types of excited states and we provide another example of this here for the excited state in CN5. However, the key thrust of the present study is optimization stability, and so in many of our results we will be optimizing orbitals within VMC to test stability in that context.…”

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory

“…Instead, the limitations of current excited-state CASSCF formalisms and the development of non-ground-state SCF optimization algorithms have inspired several new investigations into state-specific CASSCF excited states. In particular, Neuscamman and co-workers have developed generalized variational principles , and the W Γ approach inspired by MOM-SCF, demonstrating that the issues of root flipping and variational collapse to the ground state can be successfully avoided. Despite these advances, we still do not have a complete understanding of the multiple stationary points on the SS-CASSCF energy landscape, and several practical questions remain.…”