Configuration interaction extension of AGP for incorporating inter-geminal correlations

Abstract: In this paper, we develop a class of antisymmetrized geminal power configuration interaction (AGP-CI) wave functions that extend the AGP framework by incorporating inter-geminal correlations through a CI expansion. To make these wavefunctions computationally tractable, we evaluate them by rewriting the AGP-CI ansatz as a linear combination of AGPs (LC-AGP), for which overlaps and Hamiltonian matrix elements can be computed with standard AGP machinery. Motivated by border-rank decompositions, we further reorganize this ansatz into a compact linear combination of AGPs depending on a small deformation parameter $τ$, which controls how closely the truncated expansion approximates the full AGP-CI state. Benchmark applications to the Hubbard model and to the small molecules H$_2$O and N$_2$ demonstrate that the proposed wavefunctions achieve consistently high accuracy and outperform the LC-AGP, particularly for systems with more electrons and in strongly correlated regimes.

• The paper presents a configuration interaction extension of AGP that replaces geminal operators with independent pair-creation operators.
• The methodology utilizes border-rank decomposition and a deformation parameter to achieve polynomial scaling for efficient strong correlation treatment.
• Numerical benchmarks on the Hubbard model and small molecules demonstrate improved accuracy and computational robustness compared to LC-AGP.

AGP-CI Wavefunctions: Incorporating Inter-Geminal Correlations via Border-Rank Decomposition

Overview of Geminal-Based Electronic Structure Methods

The antisymmetrized geminal power (AGP) ansatz has been widely employed in quantum chemistry for capturing intra-pair electron correlations. AGP relies on a power of a single geminal operator applied to the vacuum, effectively treating all electron pairs in an identical manner. This construction is inherently limited, as it treats inter-geminal (pair-pair) correlations only at the mean-field level. Multiple strategies have been proposed to address this deficiency, including LC-AGP, which forms linear combinations of distinct AGP states, and APG, which antisymmetrizes products of distinct geminals. However, APG decompositions scale exponentially in computational cost with respect to particle count, and LC-AGP displays optimization challenges and scaling limitations for strongly correlated and large electron systems.

Configuration Interaction Extension of AGP: AGP-CI Ansatz

This work introduces the AGP-CI family of wavefunctions, directly extending the AGP reference through a CI-like expansion that systematically replaces geminal operators with independent pair-creation operators. The AGP-CI($k$) wavefunctions retain up to $k$ independent pair replacements and can be viewed as homogeneous polynomials in the commuting geminal operators. The key technical novelty is the explicit exploitation of the Waring rank structure of the associated polynomials, enabling AGP-CI decomposition into a manageable number of AGP-like terms.

The border-rank approximation is then invoked, introducing a small deformation parameter $\tau$. This parameter regulates the convergence of a finite-difference expansion toward the target AGP-CI polynomial form. Crucially, the number of AGP terms required for AGP-CI$\tau$ grows only polynomially, and in many cases is reduced to $\mathcal{O}(1)$ for finite $k$. This allows efficient evaluation using AGP machinery and provides a significant computational speedup, particularly for strongly-correlated and large systems.

Numerical Results: Hubbard Model and Molecular Benchmarks

Benchmarking was performed on both the periodic one-dimensional Hubbard model and small molecules (H$_2$O, N$_2$) using the STO-6G basis. The AGP-CI wavefunctions, especially AGP-CID and AGP-CIT, consistently demonstrated higher numerical accuracy compared to LC-AGP at fixed numbers of AGP terms. This advantage was most pronounced in the strongly correlated regime ($U/t=10$ for the Hubbard model) and for systems with larger electron counts, where LC-AGP exhibited substantial degradation in energy accuracy and exhibited non-monotonic convergence under increased expansion order.

The introduction of the border-rank AGP-CI$\tau$ form resulted in aggressive reduction in computation cost (scaling as $k$0 relative to exact Waring decomposition), with controlled approximation errors. For the 12-electron Hubbard system, AGP-CID's error remained stable while LC-AGP's error increased rapidly. AGP-CI$k$1 proved robust to electron count and correlation strength, with systematic improvement as excitation level in the CI expansion was increased.

For molecular systems, AGP-CI$k$2 delivered comparable or superior accuracy relative to LC-AGP in strongly correlated N$k$3, and produced smooth potential energy surfaces, avoiding the optimization irregularities inherent to LC-AGP. In the evaluation of double occupancy, AGP-CI$k$4 consistently yielded lower root-mean-squared error compared to LC-AGP, confirming its utility for both energetic and observable predictions.

Theoretical and Practical Implications

By leveraging border-rank decomposition, AGP-CI$k$5 enables systematic and polynomially-scaling parameterization of geminal-based wavefunctions, opening the door to tractable, high-accuracy treatments of strong correlation in larger many-electron systems. AGP-CI$k$6 provides a flexible bridge between the physics-oriented CI expansion and the computational machinery of AGP overlap/Hamiltonian evaluation, supporting direct variational optimization and structured initialization.

The polynomial scaling in AGP term count avoids the exponential blowup of APG, while eliminating the optimization and scaling bottlenecks endemic to LC-AGP. The border-rank AGP-CI$k$7 form offers practical speedups, numerical robustness, and monotonic accuracy improvement with expansion order, making it a promising ansatz for quantum chemistry and condensed matter applications.

Further methodological refinement can be anticipated by improving optimization protocols for both AGP-CI$k$8 and LC-AGP, and by adapting the approach to larger and more chemically realistic basis sets. Analysis of border-rank parameterization in connection with tensor decomposition theory may yield additional compression and expressive power. In broader context, the AGP-CI$k$9 approach is highly relevant for the ongoing development of quantum many-body wavefunctions in both ab initio and model Hamiltonian studies, with implications for both algorithm design and conceptual understanding of electron correlation.

Conclusion

The AGP-CI framework, as formulated in this study, achieves robust incorporation of inter-geminal correlations via a configuration interaction expansion and a border-rank decomposition. The resulting AGP-CI$\tau$0 ansatz enables polynomial scaling, high accuracy, and practical evaluation in strongly correlated and large electron systems. Numerical benchmarks confirm the systematic improvement and stability of AGP-CI$\tau$1 relative to LC-AGP, with significant implications for computational many-body electronic structure theory. Future research will focus on optimization refinement, expansion to larger basis sets, and further exploration of border-rank in quantum wavefunction compression and parameterization.