The alchemical integral transform revisited

We recently introduced the Alchemical Integral Transform (AIT), enabling the prediction of energy differences, and guessed an ansatz to parameterize space r in some alchemical change λ. Here, we present a rigorous derivation of AIT's kernel K and discuss the parameterization r(λ) 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 such as 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.