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Algebraic Geometry for Spin-Adapted Coupled Cluster Theory

Fabian M. Faulstich, Svala Sverrisdóttir

Abstract

We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride.

Algebraic Geometry for Spin-Adapted Coupled Cluster Theory

Abstract

We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride.
Paper Structure (18 sections, 9 theorems, 102 equations, 4 figures, 1 table)

This paper contains 18 sections, 9 theorems, 102 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous Works and Perspective
  3. Notational Preamble
  4. The Electronic Structure Problem
  5. Second Quantization and Fermi Dirac Algebra
  6. Representations of the Fermi--Dirac Algebra
  7. Spin of a Single Electron
  8. Spin of $d$-Electron States and Spin-Singlet $d$-Electron States
  9. Spin of Fock States and Spin-Singlet Fock States
  10. Spin-Adapted Coupled Cluster Theory
  11. Spin Singlet Truncation Varieties
  12. Spin-Adapted Coupled Cluster Equations
  13. Numerical Simulations
  14. Scaling of Generic Spin Restricted Systems
  15. Lithium Hydride
  16. ...and 3 more sections

Key Result

Lemma 3.5

The character of the representation $\mathcal{H}_d$ of ${\rm SU}(2)$ is the Laurent polynomial If $d = 2k$ then $\mathcal{H}_d$ decomposes into a sum of integer irreducible representations $V_j$ where $0 \le j \le k$, with multiplicity If $d = 2k + 1$ then $\mathcal{H}_d$ decomposes into a sum of half-integer irreducible representations $V_{j + \frac{1}{2}}$ where $0 \le j \le k$ with multiplici

Figures (4)

  • Figure 1: Comparison of the number of roots between spin restricted and spin generalized CC equations for CCS (left) and CCD (right) at $k=1$. The reduction in degree translates into a corresponding reduction in the number of solution paths that must be tracked.
  • Figure 2: LiH dissociation in a minimal basis ($k=2$, $m=4$): comparison of RCCSD solution branches with the exact eigenvalue curves. (a) All RCCSD solution branches compared to the exact spectrum. (b) RCCSD solutions lying near an eigenvalue (the physically relevant).
  • Figure 3: LiH dissociation in a full basis ($k=2$, $m=6$): comparison of RCCD solutions with the exact eigenvalues. (a) All RCCSD solutions compared to the exact spectrum. (b) RCCD solutions lying near an eigenvalue (physically relevant).
  • Figure 4: H$_2$O dissociation in a minimal basis ($k=4$, $m=6$): comparison of RCCD solutions with the exact eigenvalues. (a) All RCCD solutions compared to the exact spectrum. (b) RCCD solutions lying near an eigenvalue (physically relevant).

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4: Proper embeddings of $\mathcal{H}_d^{{\rm SU}(2)}$
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • Remark 3.7
  • ...and 17 more