Coupled cluster theory: Towards an algebraic geometry formulation

Abstract

Coupled cluster theory produced arguably the most widely used high-accuracy computational quantum chemistry methods. Despite the approach's overall great computational success, its mathematical understanding is so far limited to results within the realm of functional analysis. The coupled cluster amplitudes, which are the targeted objects in coupled cluster theory, correspond to solutions to the coupled cluster equations, which is a system of polynomial equations of at most degree four. The high dimensionality of the electronic Schrödinger equation and the non-linearity of the coupled cluster ansatz have so far stalled a formal analysis of this polynomial system. In this article, we present algebraic investigations that shed light on the coupled cluster equations and the root structure of this ansatz. This is of importance for the a posteriori evaluation of coupled cluster calculations. To that end, we investigate the root structure by means of Newton polytopes. We derive a general v-description, which is subsequently turned into an h-description for explicit examples. This perspective reveals an apparent connection between Pauli's exclusion principle and the geometrical structure of the Newton polytopes. We also propose an alternative characterization of the coupled cluster equations projected onto singles and doubles as cubic polynomials on an algebraic variety with certain sparsity patterns. Moreover, we provide numerical simulations of two computationally tractable systems, namely, the two electrons in four spin-orbitals system and the three electrons in six spin-orbitals system. These simulations provide novel insight into the root structure of the coupled cluster solutions when the coupled cluster ansatz is truncated.

Figures (4)

• Figure 1: Graph that shows the dimensionality of the intersection of facets of the third Newton polytope. If two facets have a four-dimensional intersection they are connected with a solid blue line, and if two facets have a three-dimensional intersection they are connected with a dotted red line.
• Figure 1: Number of Newton iterations for randomly perturbed initial guess. The imposed perturbations are of size $0.1$, $0.6$, $1.0$, $2.0$ in \ref{['fig:newtonfit_01']}, \ref{['fig:newtonfit_06']}, \ref{['fig:newtonfit_1']}, \ref{['fig:newtonfit_12']}, respectively.
• Figure 2: The $\varepsilon$-energy trajectory of CCS, where $\varepsilon$ was varied between zero and eight. The value of $\varepsilon$ is indicated through different colors, the six FCI energies are marked with red vertical lines.
• Figure 2: Success rate of Newton's method as a function of the norm of the perturbation $\Vert t\Vert$.

Theorems & Definitions (20)

• Lemma 3.1: CAR
• Remark 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• Lemma 3.6
• Lemma 3.7
• Theorem 3.8
• Remark 3.9
• Remark 4.1