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Seniority-zero Linear Canonical Transformation Theory

Daniel F. Calero-Osorio, Paul W. Ayers

TL;DR

This work develops Seniority-zero Linear Canonical Transformation (SZ-LCT), a unitary map that transforms the electronic Hamiltonian $\hat{H}$ into a seniority-zero form $\hat{H}_{SZ}$ via $\hat{H}_{SZ}=e^{\hat{A}}\hat{H}e^{-\hat{A}}$. The generator $\hat{A}$ combines one- and two-body excitations and is optimized by minimizing non-seniority-zero components in the BCH-expanded, two-body-truncated transformed Hamiltonian, enabling accurate treatment of dynamic and static correlation with a seniority-zero reference. Using spin-free operators and a seniority-zero reference (e.g., oo-DOCI), the method achieves sub-milliHartree accuracy across challenging benchmarks (H$_6$, BeH$_2$, BH) and scales effectively as $O(N^8/n_c)$ with available cores. This approach offers a promising route to efficient, high-accuracy treatments of strongly correlated electrons and potential mapping to quantum hardware via geminal-based seniority-zero representations.

Abstract

We propose a method to solve the electronic Schrödinger equation for strongly correlated systems by applying a unitary transformation to reduce the complexity of the physical Hamiltonian. In particular, we seek a transformation that maps the Hamiltonian into the seniority-zero space: seniority-zero wavefunctions are computationally simpler, but still capture strong correlation within electron pairs. The unitary rotation is evaluated using the Baker Campbell Hausdorff (BCH) expansion, truncated to two-body operators through the operator decomposition strategy of canonical transformation (CT) theory, which rewrites higher-rank terms approximately in terms of one- and two-body operators. Unlike conventional approaches to CT theory, the generator is chosen to minimize the size of non-seniority-zero elements of the transformed Hamiltonian. Numerical tests reveal that this Seniority-zero Linear Canonical Transformation (SZ-LCT) method delivers highly accurate results, usually with submilliHartree error. The effective computational scaling of SZ-LCT is $\mathcal{O}(N^8/n_c)$ , where $n_c$ is the number of cores available for the computation.

Seniority-zero Linear Canonical Transformation Theory

TL;DR

This work develops Seniority-zero Linear Canonical Transformation (SZ-LCT), a unitary map that transforms the electronic Hamiltonian into a seniority-zero form via . The generator combines one- and two-body excitations and is optimized by minimizing non-seniority-zero components in the BCH-expanded, two-body-truncated transformed Hamiltonian, enabling accurate treatment of dynamic and static correlation with a seniority-zero reference. Using spin-free operators and a seniority-zero reference (e.g., oo-DOCI), the method achieves sub-milliHartree accuracy across challenging benchmarks (H, BeH, BH) and scales effectively as with available cores. This approach offers a promising route to efficient, high-accuracy treatments of strongly correlated electrons and potential mapping to quantum hardware via geminal-based seniority-zero representations.

Abstract

We propose a method to solve the electronic Schrödinger equation for strongly correlated systems by applying a unitary transformation to reduce the complexity of the physical Hamiltonian. In particular, we seek a transformation that maps the Hamiltonian into the seniority-zero space: seniority-zero wavefunctions are computationally simpler, but still capture strong correlation within electron pairs. The unitary rotation is evaluated using the Baker Campbell Hausdorff (BCH) expansion, truncated to two-body operators through the operator decomposition strategy of canonical transformation (CT) theory, which rewrites higher-rank terms approximately in terms of one- and two-body operators. Unlike conventional approaches to CT theory, the generator is chosen to minimize the size of non-seniority-zero elements of the transformed Hamiltonian. Numerical tests reveal that this Seniority-zero Linear Canonical Transformation (SZ-LCT) method delivers highly accurate results, usually with submilliHartree error. The effective computational scaling of SZ-LCT is , where is the number of cores available for the computation.

Paper Structure

This paper contains 12 sections, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. The SZ-LCT method
  4. Spin-free operators
  5. Seniority-zero reference
  6. Implementation
  7. Cost and performance
  8. Results
  9. H_6
  10. BeH2
  11. BH
  12. Conclusions

Figures (3)

  • Figure 1: Results for the linear H6 chain (STO-6G): (a) dissociation curve; (b) energy differences.
  • Figure 2: Results for Be + H2 -> BeH2 insertion reaction. (a) dissociation curve; (b) energy differences.
  • Figure 3: Results for BH dissociation. (a) dissociation curve; (b) energy differences.