# Required number of states increases only moderately with the problem size for antisymmetrized geminal powers

## TL;DR

This paper introduces an efficient algorithm using antisymmetrized geminal powers to accurately compute ground-state energies of many-electron systems, demonstrating moderate growth in required states with system size.

## Contribution

The authors develop an optimized variational algorithm for antisymmetrized geminal powers, reducing computational bottlenecks and enabling accurate energy calculations for larger systems.

## Key findings

- Achieved sub-milihartree accuracy for water molecule energy.
- Captured the correct ground state tendency in 1D Hubbard model.
- Energy convergence slows in larger 2D Hubbard models.

## Abstract

We propose an algorithm to obtain the ground-state energy of a many-electron system using the variational wave function of a linear combination of antisymmetrized geminal powers. We optimized this algorithm to obtain the energy and the other parameters of a many-electron system. Also we clarified the bottleneck of the total calculation in the tensor contraction and successfully reduced the computational time. As a result, we can use an extended number of geminal states to obtain the ground state of the water molecule and Hubbard models. The result for the water molecule with the Dunning double-zeta basis is of the sub-milihartree order above the energy of exact diagonalization. Further, we observe that the result for the one-dimensional Hubbard model with 14 sites shows good tendency to capture the right ground state and that for the two-dimensional Hubbard model still lacks some part of the energy reflecting the large size of the Hilbert space. We conclude that the required number of terms for geminal states for sufficiently accurate energy is only moderately affected by the problem size. We further show other technical details for the numerical algorithms of geminal states in the variation process. It is expected that with the use of more extended computing resources and larger sizes of electronic systems, our algorithm can provide improved results.