Density functional theory beyond the Born-Oppenheimer approximation: Exact mapping onto an electronically non-interacting Kohn-Sham molecule

TL;DR

This paper introduces an exact extension of density functional theory that maps a molecular system onto a non-interacting Kohn-Sham molecule, capturing electron-nuclear interactions without assuming a specific wavefunction form.

Contribution

It presents a novel, exact framework for molecular DFT that extends beyond the Born-Oppenheimer approximation by mapping onto a fictitious non-interacting molecule.

Findings

Derived a self-consistent set of electronic and nuclear equations.

Established an exact adiabatic connection formula for the Hartree-exchange-correlation energy.

Proposed a practical density-functional approximation based on the new framework.

Abstract

This work presents an alternative, general, and in-principle exact extension of electronic Kohn-Sham density functional theory (KS-DFT) to the fully quantum-mechanical molecular problem. Unlike in existing multi-component or exact-factorization-based DFTs of electrons and nuclei, both nuclear and electronic densities are mapped onto a fictitious electronically non-interacting molecule (referred to as KS molecule), where the electrons still interact with the nuclei. Moreover, in the present molecular KS-DFT, no assumption is made about the mathematical form (exactly factorized or not) of the molecular wavefunction. By expanding the KS molecular wavefunction \`a la Born-Huang, we obtain a self-consistent set of "KS beyond Born-Oppenheimer" electronic equations coupled to nuclear equations that describe nuclei interacting among themselves and with non-interacting electrons. An exact adiabatic connection formula is derived for the Hartree-exchange-correlation energy of the electrons within the molecule and, on that basis, a practical adiabatic density-functional approximation is proposed and discussed.