# Explicitly correlated Helium wave function and hyperspherical coordinates

## TL;DR

This paper introduces a new class of explicitly correlated wave functions for Helium-like atoms using hyperspherical coordinates, achieving highly accurate ground state energies and satisfying cusp conditions.

## Contribution

It proposes a novel functional form of wave functions explicitly depending on interelectronic and hyperspherical coordinates, improving accuracy for Helium atom calculations.

## Key findings

- Achieved the best ground state energy for Helium with hyperspherical coordinates.
- Wave functions satisfy Kato cusp conditions within 6.10^{-4} accuracy.
- Included negative powers of R in wave functions as a necessary condition for Schrödinger equation solutions.

## Abstract

Wave functions of a new functional kind have been proposed for Helium-like atoms in this work . These functions explicitly depend on interelectronic and hyperspherical coordinates. The best ground state energy for the Helium atom $ -2.903724376677 a.u.$ has been calculated with variational method with basis set of simple functions with a single exponential parameter. To the author's knowledge, this is the best result with use of hyperspherical coordinates so far. Comparable result has been obtained for the hydrogen anion. For Helium atom, our best wave functions matched the Kato cusp conditions within the accuracy below $6.10^{-4} $. An important feature of proposed wave functions is the inclusion of negative powers of $R=\sqrt(r^{2}_{1}+r^{2}_{2})$ in combination with positive powers of $r_{12}$ into the wave function. We showed that this is necessary condition for proposed wave function to be a formal solution of Schr\"odinger equation.