The Alchemical Integral Transform revisited

TL;DR

This paper rigorously derives the kernel of the Alchemical Integral Transform (AIT), discusses parametrization in multiple dimensions, and provides analytical expressions for energy and density changes in various quantum systems.

Contribution

It offers a rigorous mathematical foundation for AIT, including kernel derivation and parametrization strategies, expanding its applicability to complex quantum systems.

Findings

Derived the kernel $\\mathcal{K}$ of AIT.

Provided analytical expressions for energy and density changes.

Applied the method to multiple quantum systems including harmonic oscillator and hydrogen-like atoms.

Abstract

We recently introduced the Alchemical Integral Transform (AIT) enabling the prediction of energy differences, and guessed an Ansatz to parametrize space $\pmb{r}$ in some alchemical change $\lambda$. Here, we present a rigorous derivation of AIT's kernel $\mathcal{K}$ and discuss the parametrization $\pmb{r}(\lambda)$ in $n$ dimensions, i.e. necessary conditions, mathematical freedoms and additional constraints when obtaining it. Analytical expressions for changes in energy spectra and densities are given for a number of systems. Examples include homogeneous potentials like the quantum harmonic oscillator, Hydrogen-like atom, and Dirac well, both for one- and multiparticle cases, and a multiparticle system beyond coordinate scaling for harmonic potentials.