# Application of the Hylleraas-$B$-spline basis set: Nonrelativistic Bethe logarithm of helium

## TL;DR

This paper introduces a new Hylleraas-$B$-spline basis set approach for highly precise nonrelativistic Bethe logarithm calculations of helium, significantly improving accuracy and developing a novel solver for numerical challenges.

## Contribution

It applies the Hylleraas-$B$-spline basis set to helium Bethe logarithm calculations and develops a multiple-precision eigenvalue solver for enhanced numerical stability.

## Key findings

- Achieved 7-9 digit precision in Bethe logarithm for helium states up to n=10.
- Improved accuracy over traditional $B$-spline basis set methods.
- Developed a multiple-precision generalized eigenvalue solver for precision calculations.

## Abstract

In this work, we report an application of Hylleraas-$B$-spline basis set to the nonrelativistic Bethe logarithm calculation of helium. The Bethe logarithm for $n\ ^1S$, $n$ up to 10, states of helium are calculated with a precision of 7-9 significant digits in two gauges, which greatly improves the accuracy of the traditional $B$-spline basis set. In addition, to deal with the numerical linear correlation problem in Bethe logarithm calculation, we developed a multiple-precision generalized symmetric eigenvalue problem solver (MGSEPS). This program may be very useful to precision calculations.