Determination of the exchange interaction energy from the polarization expansion of the wave function

TL;DR

This paper compares three formulas for calculating exchange energy in a hydrogen-proton system, showing that the variational formula converges fastest and highlighting the advantages of polarization expansion and variational principles in molecular interaction calculations.

Contribution

The paper introduces and compares three methods for calculating exchange energy, demonstrating the superior convergence of the variational formula and emphasizing the effectiveness of polarization expansion.

Findings

All three formulas yield the correct asymptotic expression at large distances.

The SAPT formula converges geometrically with error ~3^{-K}.

The variational formula exhibits the fastest convergence with error ~K^{1/2} [a^{K}/(K+1)!]^2.

Abstract

The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surface-integral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances $R$, all three formulas yield the correct expression $-(2/e)Re^{-R}$, but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error falling as $3^{-K}$, where $K$ is the order of the applied polarization expansion. The error of the surface-integral formula decreases exponentially as $a^K/(K+1)!$, where $a=\ln2 -\tfrac{1}{2}$. The variational formula performs best, its error decays as $K^{1/2} [a^{ K}/(K+1)!]^2$. These convergence rates are much faster than those resulting from approximating the wave function through the multipole expansion. This shows the efficiency of the partial resummation of the multipole series effected by the polarization expansion. Our results demonstrate also the benefits of incorporating the variational principle into the perturbation theory of molecular interactions.