Pseudo-mass parameterized alchemical equation: a generalisation of the molecular Schr\"odinger equation

TL;DR

This paper introduces a novel alchemical Schrödinger-like equation with nuclear charges as variables, enabling a unified treatment of nuclear and electronic coordinates, and explores its mathematical and physical properties.

Contribution

It presents a new pseudo-mass parameterized alchemical equation, defines the alchemical function space, and extends the Hellmann-Feynman theorem to non-stationary states.

Findings

Derivation of the geometric phase of alchemical dynamics

Simplification of the equation for hydrogen-like ions revealing conformal invariance

Extension of the Hellmann-Feynman theorem to time-dependent alchemical dynamics

Abstract

We introduce a pseudo-mass parameterized Schr\"odinger-like alchemical equation which contains nuclear charges as variables, treating nuclear charges, nuclear coordinates and electronic coordinates on the equal footing. The eigenfunctions of the alchemical equation are the wave functions of nuclear charges, just like conventional wave functions are of coordinates. A mathematical definition of alchemical function space is given to hold the ''nuclear charge wave function''. The geometric phase of alchemical dynamics and alchemical phase space are also derived. For hydrogen-like ion, the alchemical equation can be simplified into a strong repulsive inverse square potential equation, which refers to the quantum anomaly and keeps conformal invariance in non-relativistic quantum mechanics. An extension of the Hellmann-Feynman theorem, which applies to the non-stationary state in time-dependent clamped-nuclear-charge alchemical dynamics, is also proved.