Arbitrarily accurate quantum alchemy.

Guido Falk Von Rudorff

doi:10.1063/5.0073941

Abstract

Doping compounds can be considered a perturbation to the nuclear charges in a molecular Hamiltonian. Expansions of this perturbation in a Taylor series, i.e., quantum alchemy, have been used in the literature to assess millions of derivative compounds at once rather than enumerating them in costly quantum chemistry calculations. So far, it was unclear whether this series even converges for small molecules, whether it can be used for geometry relaxation, and how strong this perturbation may be to still obtain convergent numbers. This work provides numerical evidence that this expansion converges and recovers the self-consistent energy of Hartree-Fock calculations. The convergence radius of this expansion is quantified for dimer examples and systematically evaluated for different basis sets, allowing for estimates of the chemical space that can be covered by perturbing one reference calculation alone. Besides electronic energy, convergence is shown for density matrix elements, molecular orbital energies, and density profiles, even for large changes in electronic structure, e.g., transforming He3 into H6. Subsequently, mixed alchemical and spatial derivatives are used to relax H2 from the electronic structure of He alone, highlighting a path to spatially relaxed quantum alchemy. Finally, the underlying code that allows for arbitrarily accurate evaluation of restricted Hartree-Fock energies and arbitrary order derivatives is made available to support future method development.

Highlights

At the core of most quantum chemistry methods is the search for sufficiently good approximations of the total energy E of a system, given the positions RI and nuclear charges ZI of all N nuclei, the net charge Q, and the spin σ
Several formulations have been developed, all based on the derivatives of energies or electron densities ρ with respect to either the nuclear charges or the external potential: a Taylor expansion of the energy,[4] derivatives from conceptual Density Functional Theory (DFT),[5] and Alchemical Perturbation Density Functional Theory (APDFT) that is based on density derivatives.[2]
Panel (a) demonstrates that for the first few orders, automatic differentiation and finite differences agree, while for higher orders, automatic differentiation diverges, which is expected due to the finite precision used in the corresponding code

Summary

Introduction

At the core of most quantum chemistry methods is the search for sufficiently good approximations of the total energy E of a system, given the positions RI and nuclear charges ZI of all N nuclei, the net charge Q, and the spin σ. In the realm of quantum alchemy, non-integer nuclear charges are considered as a conceptual tool to interpolate between compounds, as, e.g., done by Wilson in 1962.1 This allows us to employ well-established methods such as perturbation theory in order to describe similar compounds if the electronic structure of one compound is already available. While some formulations even allow for a change in numbers of electrons,[6,7] this work considers isoelectronic changes only

Objectives

Methods

Results

Conclusion