[Paper Review] Imaginary shift in CASPT2 nuclear gradient and derivative coupling theory

Jae Woo Park, Rachael Al-Saadon+2 more|arXiv (Cornell University)|Apr 14, 2019

Spectroscopy and Quantum Chemical Studies4 citations

TL;DR

This paper presents an analytical nuclear gradient theory for CASPT2 with imaginary shift regularization, extending prior real-shift implementations. The method improves accuracy in excited-state properties and conical intersection calculations with only a minor computational overhead, primarily from evaluating imaginary shift-specific density matrix terms, as demonstrated on p-HBDI and FeP models.

ABSTRACT

We report the analytical nuclear gradient theory for complete active space second-order perturbation theory (CASPT2) with imaginary shift, which is commonly used to avoid divergence of the perturbation expression. Our formulation is based on the Lagrangian approach and is an extension of the algorithm for CASPT2 nuclear gradients with real shift. The working equations are derived and implemented into an efficient parallel program. Numerical examples are presented for the ground- and excited-state geometries and conical intersections of a green fluorescent protein model chromophore, $p$-HBDI$^-$. We also report timing benchmarks with adenine, $p$-HBDI$^-$, and iron porphyrin. It is demonstrated that the energies and geometries obtained with the imaginary shift improve accuracy at a minor additional cost which is mainly associated with evaluating the effective density matrix elements for the imaginary shift term.

Motivation & Objective

• Develop an analytical nuclear gradient theory for CASPT2 using imaginary shift regularization to avoid intruder state divergences.
• Address the limitations of real shift in CASPT2 by implementing a more robust, singularity-free regularization scheme.
• Ensure accurate computation of excited-state geometries, conical intersections, and vertical/excited-state energies.
• Minimize computational overhead by efficiently integrating imaginary shift terms into existing CASPT2 gradient algorithms.
• Validate the method on benchmark systems including p-HBDI and iron porphyrin with varying active space sizes.

Proposed method

• Adapt the Lagrangian approach to derive analytical nuclear gradients for CASPT2 with imaginary shift, extending prior real-shift formulations.
• Derive working equations for the imaginary shift terms, including $\mathbf{d}^{(2)}_{\text{shift}}$ and $\tilde{y}_{I,M}^{\text{shift}}$, which account for the regularization in the energy denominator.
• Implement the theory within the BAGEL quantum chemistry package, leveraging parallel computing for efficiency.
• Integrate the imaginary shift terms into the solution of the $\lambda$-equation and $Z$-vector equations with minimal modification to existing code.
• Use a complex shift $1/\Delta \to \Re[1/(\Delta + i\epsilon)]$ to ensure singularity-free behavior, especially critical near conical intersections.
• Optimize computational cost by combining new terms with conventional CASPT2 contributions, reducing overhead to ~15% for large active spaces.

Experimental results

Research questions

• RQ1 How does the imaginary shift in CASPT2 nuclear gradients improve accuracy compared to the real shift, particularly near conical intersections?
• RQ2 What is the computational cost overhead of including imaginary shift terms in CASPT2 nuclear gradient calculations?
• RQ3 How does the performance of the imaginary shift method scale with increasing active space size?
• RQ4 To what extent does the imaginary shift reduce sensitivity to the shift parameter $\epsilon$ compared to the real shift?
• RQ5 Can the imaginary shift formulation maintain numerical stability and accuracy in excited-state geometry optimizations and conical intersection calculations?

Key findings

• The imaginary shift formulation yields less sensitive results to variations in $\epsilon$ compared to the real shift, enhancing robustness in excited-state calculations.
• For small active spaces (e.g., adenine, p-HBDI), the computational cost of imaginary shift gradients is nearly identical to real shift, with only 2–3% additional time.
• For larger active spaces (e.g., FeP with CAS(10e,9o)), the wall time increased by ~15% (2947 s vs. 3387 s), primarily due to $O(N_{\text{act}}^9)$ scaling of imaginary shift term evaluation.
• The imaginary shift term $\mathbf{d}^{(2)}_{\text{shift}}$ accounted for 52% of the correlated density matrix computation time in FeP, reaching 383 seconds, while being negligible (<1 second) in smaller systems.
• The additional cost for solving the $\lambda$-equation increased by only 11 seconds for FeP, indicating minimal impact on iterative solvers.
• The method was successfully implemented and integrated into the public BAGEL quantum chemistry package, enabling broader use in multiconfigurational quantum chemistry applications.