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Analytical Nuclear Gradients of State-Averaged Configuration Interaction Singles Variants: Application to Conical Intersections

Takashi Tsuchimochi

TL;DR

This work addresses the need for reliable, low-cost treatment of conical intersections within CIS-based methods by deriving analytical nuclear gradients for state-averaged CIS (SACIS) and its spin-projected extension (SAECIS) using a Lagrangian formalism. A key advance is the explicit removal of null-space contributions in the coupled-perturbed equations, yielding stable gradients for geometry optimization and MECX searches. Benchmark results on ethylene and twelve MECXs show SACIS and SAECIS reproduce CX topology qualitatively and achieve mean RMSDs below 0.1 Å relative to high-level references, with SACIS offering the best cost-performance balance in general. Spin projection in SAECIS does not consistently improve CX descriptions and is computationally more demanding, though SAECIS can be advantageous when higher excited states with double-excitation character are important. Overall, SACIS provides a robust and efficient route to CX optimization within a mean-field framework, while SAECIS extends applicability to more strongly correlated situations at increased cost.

Abstract

We derive analytical nuclear gradients for state-averaged configuration interaction singles (SACIS) and its spin-projected extension (SAECIS), enabling efficient geometry optimization and minimum-energy conical intersection (MECX) searches within a low-cost CIS-based framework. The formulation employs a Lagrangian approach and explicitly removes null-space contributions in the coupled perturbed equations to ensure numerically stable gradients. For twisted-pyramidalized ethylene, both SACIS and SAECIS qualitatively reproduce the correct conical intersection topology, in sharp contrast to conventional CIS and ECIS. Benchmark calculations on twelve MECXs demonstrate that both methods reproduce geometries with mean RMSDs below 0.1~Å relative to high-level reference methods. SACIS captures the essential degeneracy through variational orbital relaxation, which alleviates ground-state Hartree--Fock (HF) orbital bias and effectively incorporates static correlation through localization effects; notably, spin projection is found to be non-essential for the qualitative description of these intersections. Overall, SACIS and SAECIS provide qualitatively reliable CX descriptions at mean-field computational cost in a black-box manner. Given their comparable accuracy and the additional overhead associated with spin projection, SACIS offers a more favorable cost-performance balance for general applications, whereas SAECIS may become advantageous when higher excited states with significant double-excitation character are involved.

Analytical Nuclear Gradients of State-Averaged Configuration Interaction Singles Variants: Application to Conical Intersections

TL;DR

This work addresses the need for reliable, low-cost treatment of conical intersections within CIS-based methods by deriving analytical nuclear gradients for state-averaged CIS (SACIS) and its spin-projected extension (SAECIS) using a Lagrangian formalism. A key advance is the explicit removal of null-space contributions in the coupled-perturbed equations, yielding stable gradients for geometry optimization and MECX searches. Benchmark results on ethylene and twelve MECXs show SACIS and SAECIS reproduce CX topology qualitatively and achieve mean RMSDs below 0.1 Å relative to high-level references, with SACIS offering the best cost-performance balance in general. Spin projection in SAECIS does not consistently improve CX descriptions and is computationally more demanding, though SAECIS can be advantageous when higher excited states with double-excitation character are important. Overall, SACIS provides a robust and efficient route to CX optimization within a mean-field framework, while SAECIS extends applicability to more strongly correlated situations at increased cost.

Abstract

We derive analytical nuclear gradients for state-averaged configuration interaction singles (SACIS) and its spin-projected extension (SAECIS), enabling efficient geometry optimization and minimum-energy conical intersection (MECX) searches within a low-cost CIS-based framework. The formulation employs a Lagrangian approach and explicitly removes null-space contributions in the coupled perturbed equations to ensure numerically stable gradients. For twisted-pyramidalized ethylene, both SACIS and SAECIS qualitatively reproduce the correct conical intersection topology, in sharp contrast to conventional CIS and ECIS. Benchmark calculations on twelve MECXs demonstrate that both methods reproduce geometries with mean RMSDs below 0.1~Å relative to high-level reference methods. SACIS captures the essential degeneracy through variational orbital relaxation, which alleviates ground-state Hartree--Fock (HF) orbital bias and effectively incorporates static correlation through localization effects; notably, spin projection is found to be non-essential for the qualitative description of these intersections. Overall, SACIS and SAECIS provide qualitatively reliable CX descriptions at mean-field computational cost in a black-box manner. Given their comparable accuracy and the additional overhead associated with spin projection, SACIS offers a more favorable cost-performance balance for general applications, whereas SAECIS may become advantageous when higher excited states with significant double-excitation character are involved.
Paper Structure (12 sections, 36 equations, 6 figures, 2 tables)

This paper contains 12 sections, 36 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. State-averaged CIS and ECIS
  4. SAECIS Lagrangian and its derivative
  5. Nuclear gradients
  6. Treatment of redundancies
  7. Computational details
  8. Results and discussions
  9. Ethylene
  10. Minimum Energy Conical Intersection
  11. Concluding remarks
  12. Influence of the null space in the Z-vector to the nuclear gradients

Figures (6)

  • Figure 1: Torsion and pyramidalization angles of C$_2$H$_4$.
  • Figure 2: Potential energy surfaces of ethylene as a function of bond torsion and pyramidalization angles $\varphi$ and $\theta$. (a) Description of conical intersection. (b) $S_0, S_1$ energy tomography along $\varphi$ at $\theta = 0^\circ$. (c) $S_0, S_1$ energy tomography along $\theta$ at $\varphi = 90^\circ$.
  • Figure 3: Configuration space accessible within ECIS.
  • Figure 4: Comparison of MECX structures. Gray: MRCI, Green: SF-TDDFT, Blue: SACIS, Red: SAECIS.
  • Figure 5: RMSD of optimized MECX geometry.
  • ...and 1 more figures