Alchemical diastereomers from antisymmetric alchemical perturbations
The energy difference between two iso-electronic systems can be approximated by the first order Hellmann–Feynman derivative with respect to the linear alchemical coupling parameter, evaluated using the electron density of the corresponding averaged Hamiltonian. This approximation is exact up to third order because even-order contributions cancel out. This finding holds for any iso-electronic compound pair (dubbed “alchemical diastereomers”), regardless of differences in configuration, composition, or energy, and consequently, relative energy estimates for all possible iso-electronic alchemical diastereomer pairs require only O(1) self-consistent field cycles for any given averaging reference Hamiltonian. We discuss the relation to the Verlet algorithm, alchemical harmonic approximation (AHA) [Krug et al., J. Chem. Phys. 162, 044101 (2025)], relative properties such as forces, ionization potential or electron affinities, and Levy’s formula for relative energies among iso-electronic systems that uses the averaged electron density of the two systems [Levy, J. Chem. Phys. 70, 1573 (1979)]. Density functional theory based estimates accurately reflect trends in the charge-neutral iso-electronic diatomic molecule series with 14 protons (N2, CO, BF, BeNe, LiNa, HeMg, HAl), with systematically increasing errors. Using alchemical Hellmann–Feynman derivatives for toluene, we demonstrate the concept’s broader applicability by estimating relative energies for all 36 possible alchemical diastereomer pairs from vertical iso-electronic charge-neutral antisymmetric BN doping of toluene’s aromatic ring, with mean absolute errors of a few milli-Hartrees.