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On the Coupled Cluster Doubles Truncation Variety of Four Electrons

Fabian M. Faulstich, Vincenzo Galgano, Elke Neuhaus, Irem Portakal

Abstract

We extend recent algebro-geometric results for coupled cluster theory of quantum many-body systems to the truncation varieties arising from the doubles approximation (CCD), focusing on the first genuinely nonlinear doubles regime of four electrons. Since this doubles truncation variety does not coincide with previously studied varieties, we initiate a systematic investigation of its basic algebro-geometric invariants. Combining theoretical and numerical results, we show that for $4$ electrons on $n\leq 12$ orbitals, the CCD truncation variety is a complete intersection of degree $2^{\binom{n-4}{4}}$. Using representation-theoretic arguments, we uncover a Pfaffian structure governing the quadratic relations that define the truncation variety for any $n$, and show that an exact tensor product factorization holds in a distinguished limit of disconnected doubles. We connect these structural results to the computation of the beryllium insertion into molecular hydrogen ({Be$\cdots$H$_2$ $\to$ H--Be--H}), a small but challenging bond formation process where multiconfigurational effects become pronounced.

On the Coupled Cluster Doubles Truncation Variety of Four Electrons

Abstract

We extend recent algebro-geometric results for coupled cluster theory of quantum many-body systems to the truncation varieties arising from the doubles approximation (CCD), focusing on the first genuinely nonlinear doubles regime of four electrons. Since this doubles truncation variety does not coincide with previously studied varieties, we initiate a systematic investigation of its basic algebro-geometric invariants. Combining theoretical and numerical results, we show that for electrons on orbitals, the CCD truncation variety is a complete intersection of degree . Using representation-theoretic arguments, we uncover a Pfaffian structure governing the quadratic relations that define the truncation variety for any , and show that an exact tensor product factorization holds in a distinguished limit of disconnected doubles. We connect these structural results to the computation of the beryllium insertion into molecular hydrogen ({BeH H--Be--H}), a small but challenging bond formation process where multiconfigurational effects become pronounced.
Paper Structure (14 sections, 9 theorems, 80 equations, 6 figures, 1 table)

This paper contains 14 sections, 9 theorems, 80 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous works and perspective
  3. The electronic structure problem and coupled-cluster theory
  4. The coupled cluster doubles truncation variety
  5. Construction for four electrons
  6. Complete intersections
  7. A numerical proof for the complete intersection
  8. Quadrics $P_J$ and Pfaffians
  9. Preliminaries on Pfaffians, $\mathop{\mathrm{SL}}\nolimits$-modules and $\mathfrak{S}_4$-modules
  10. Decomposition of the second symmetric power
  11. Example: $4$ electrons in $8$ spin-orbitals
  12. The tensor product of Pfaffians
  13. The Coupled Cluster Doubles Degree
  14. Quantum Chemistry Simulation

Key Result

Proposition 2.1

The affine truncation variety $V_{\{2\}}^{\mathbb A} \subset \mathcal{H}\simeq \mathbb C^{{n \choose 4}}$ is isomorphic to the affine graph of the map

Figures (6)

  • Figure 1: Depiction of the C$_{2v}$ insertion pathway for BeH$_2$. From left to right: X-position equal to 0, 3.7, and 5.0.
  • Figure 2: Molecular orbitals along the Be insertion coordinate in BeH$_2$. The top row shows the highest occupied molecular orbital (HOMO), the bottom row shows the lowest unoccupied molecular orbital (LUMO). From left to right: fully inserted geometry at $X=0$, intermediate geometry at $X=3.7$, and separated geometry at $X=5.0$.
  • Figure 3: Molecular orbital energies along the beryllium insertion pathway into molecular hydrogen.
  • Figure 4: (a) Full energy spectrum of the Hamiltonian in molecular orbital basis (b) Real-valued CCD energy solutions
  • Figure 5: Different CC solution counts along the beryllium insertion pathway into molecular hydrogen.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Example 1.1: $d=2,n\geq4$
  • Example 1.2: $d=2,n=5$
  • Remark 1.3
  • Definition 1.4
  • Proposition 2.1
  • Example 2.2: $8$ and $9$ spin orbitals
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 20 more