# The He atom revisited

## TL;DR

This paper improves the efficiency of Feynman-Kac path integral calculations for the helium atom by applying a Generalized Feynman-Kac method, achieving faster convergence and more precise energy estimates.

## Contribution

It introduces a Generalized Feynman-Kac approach with importance sampling to accelerate convergence in quantum path integral calculations for helium.

## Key findings

- Faster convergence of path integrals for helium atom energies.
- More accurate ground and excited state energy estimates than previous variational methods.
- Application of space averaging enhances computational efficiency.

## Abstract

In the Feynman-Kac[1] path integral approach the eigenvalues of a quantum system can be computed using Wiener measure which uses Brownian particle motion. In our previous work[2-3] on such systems we have observed that the Wiener process numerically converges slowly for dimensions greater than two because almost all trajectories will escape to infinity[4]. One can speed up this process by using a Generalized Feynman-Kac (GFK) method[5] in which the new measure associated with the trial function is stationary, so that the convergence rate becomes much faster. We thus achieve an example of Importance Sampling and in the present work we apply it to the Feynman-Kac(FK) path integrals for the ground and first few excited state energies for He to speed up the convergence rate. We calculate the path integrals using space averaging rather than the time averaging as done in the past. The best previous calculations from Variational computations report precisions of Hartrees, whereas in most cases our path integral results obtained for the ground and first excited states of He are lower than these results by about Hartrees or more.

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Source: https://tomesphere.com/paper/1703.08977