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Abstract

The energy difference between two iso-electronic systems can be accurately approximated by the alchemical first order Hellmann-Feynmann derivative for the 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) [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 [J. Chem. Phys. 70, 1573 (1979)]. Numerical estimates accurately reflect trends in the charge-neutral iso-electronic diatomic molecule series with 14 protons (N$_2$, 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.