# Numerical Calculations of Electric Response Properties Using the Bubbles and Cube Framework

## TL;DR

This paper introduces a numerical method for calculating how Hartree–Fock orbitals respond to electric fields, using a combination of bubble and cube expansions.

## Contribution

A new fully numerical method for electric response calculations using bubble and cube expansions is introduced.

## Key findings

- The method uses Green’s function and Helmholtz kernel convolutions for orbital optimization.
- Polarizabilities for He, H2, and NH3 were calculated and match literature values.
- The approach combines bubble expansions with cube-based tensorial basis functions on a 3D grid.

## Abstract

We have developed
a fully numerical method for calculating the
response of the Hartree–Fock orbitals to an external electric
field. The Hartree–Fock orbitals are optimized using Green’s
function methods by iterative numerical integration of the convolution
with the Helmholtz kernel. The orbital response is obtained analogously
by iterative numerical integration of the convolution with the Helmholtz
kernel of the Sternheimer equation. The orbitals are expanded in atom-centered
functions (bubbles), consisting of numerical radial functions multiplied
by spherical harmonics. The remainder, i.e., the difference between
the bubble expansion and the exact orbitals, is expanded in numerical
tensorial local basis functions on a three-dimensional grid (cube).
The methods have been tested by calculating polarizabilities for He,
H2, and NH3, which are compared to the literature
values.