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Orthogonally Constrained CASSCF Framework: Newton-Raphson Orbital Optimization and Nuclear Gradients

Loris Delafosse, Vincent Robert, Saad Yalouz

TL;DR

The paper tackles the challenge of describing both ground and excited states in strongly correlated systems where state-averaged orbitals hinder state-specific relaxation. It introduces the orthogonally constrained CASSCF (OC-CASSCF) framework and a practical two-step implementation combining OC-CASCI and Newton-Raphson orbital optimization, plus analytical nuclear gradients. Benchmarks on LiH (and supplementary H2O) show OC-CASSCF delivers energies and wavefunctions in near-FCI accuracy with high state fidelity and accurate transition dipoles, outperforming SA-CASSCF, and enables reliable geometry optimization of excited states. The approach promises a robust zeroth-order description for multi-state problems and could be integrated with perturbative corrections such as CASPT2 or NEVPT2 for larger systems.

Abstract

In a recent work, we introduced the foundations of an orthogonally constrained complete active space self-consistent field (OC-CASSCF) framework that produces state-specific molecular orbitals for mutually orthogonal multiconfigurational electronic states. In the present study, we extend this approach by incorporating a Newton-Raphson orbital-optimization scheme, for which we derive analytical expressions of the orbital gradient and Hessian. Furthermore, we outline a practical route toward the evaluation of analytical nuclear gradients, enabling geometry optimizations within the OC-CASSCF formalism. Benchmark calculations on the three lowest singlet states of LiH and H$_2$O molecules demonstrate a systematic improvement as compared to conventional state-averaged CASSCF, even when using modestly sized active spaces.

Orthogonally Constrained CASSCF Framework: Newton-Raphson Orbital Optimization and Nuclear Gradients

TL;DR

The paper tackles the challenge of describing both ground and excited states in strongly correlated systems where state-averaged orbitals hinder state-specific relaxation. It introduces the orthogonally constrained CASSCF (OC-CASSCF) framework and a practical two-step implementation combining OC-CASCI and Newton-Raphson orbital optimization, plus analytical nuclear gradients. Benchmarks on LiH (and supplementary H2O) show OC-CASSCF delivers energies and wavefunctions in near-FCI accuracy with high state fidelity and accurate transition dipoles, outperforming SA-CASSCF, and enables reliable geometry optimization of excited states. The approach promises a robust zeroth-order description for multi-state problems and could be integrated with perturbative corrections such as CASPT2 or NEVPT2 for larger systems.

Abstract

In a recent work, we introduced the foundations of an orthogonally constrained complete active space self-consistent field (OC-CASSCF) framework that produces state-specific molecular orbitals for mutually orthogonal multiconfigurational electronic states. In the present study, we extend this approach by incorporating a Newton-Raphson orbital-optimization scheme, for which we derive analytical expressions of the orbital gradient and Hessian. Furthermore, we outline a practical route toward the evaluation of analytical nuclear gradients, enabling geometry optimizations within the OC-CASSCF formalism. Benchmark calculations on the three lowest singlet states of LiH and HO molecules demonstrate a systematic improvement as compared to conventional state-averaged CASSCF, even when using modestly sized active spaces.
Paper Structure (9 sections, 15 equations, 2 figures)

This paper contains 9 sections, 15 equations, 2 figures.

Table of Contents

  1. Theory
  2. Orthogonally Constrained-CASSCF (OC-CASSCF)
  3. Practical Implementation: Two-Step Approach
  4. Orthogonally Constrained Configuration-Interaction:
  5. Orthogonally Constrained Orbital-Optimization:
  6. Analytical OC-CASSCF Nuclear Gradients
  7. Numerical results
  8. Energies and Wavefunctions Properties
  9. Nuclear Gradients and Geometry Optimizations

Figures (2)

  • Figure 1: Evolution of energies and wavefunction-based properties for LiH (with a minimal STO-6G basis set and a penalty term $\Delta = 1$ Ha). Left panel: Dissociation curves for the three low-lying singlet states of the molecule. The solid black curves show the FCI reference results. The blue curves with markers represent the ground and first excited states obtained using the SA-CASSCF method. Similarly, the orange curves with markers correspond to the ground, first and second excited states obtained with the OC-CASSCF method. Right top panel: evolution of the state fidelity (as defined in Eq. (\ref{['eq:Fidelity']})) of the SA-CASSCF and OC-CASSCF states relative to the FCI reference. Right bottom panel: evolution of the transition dipole magnitude betwene the ground and first excited states (as defined in Eq. (\ref{['eq:trans_di']})) obtained with SA-, OC-CASSCF and FCI.
  • Figure 2: Illustration of the nuclear gradients computation for the LiH molecule (using minimal STO-6G basis set with $\Delta = 1$ Ha). Left panels: Gradient amplitudes across the internuclear distance for the ground, first and second excited states. Analytical gradients are shown with orange curves, while black full lines are used for numerical gradients. Right panels: Geometry optimization for the two excited states using a gradient descent approach with $\eta = 1$ (see Eq. (\ref{['eq:gradient_descent']})). Evolution of internuclear distance $x$ at each optimization step (orange full line). FCI equilibrium positions are shown with black dashed lines.