What is the Exchange Repulsion Energy? Insight by Partitioning into Physically Meaningful Contributions

It is shown that the exchange repulsion energy, E_xr, can be rationalized by partitioning the respective energy expression for two systems with Hartree-Fock orbitals into physically meaningful contributions. A division of E_xr into a positive kinetic and a negative potential part is possible, but these contributions correlate only poorly with the actual exchange repulsion energy. A more meaningful partitioning is derived, where all kinetic energy contributions are collected in a term that vanishes for exact Hartree-Fock orbitals due to their stationarity conditions. The remaining terms can be distinguished into an exchange integral contribution as well as contributions to the repulsion energy with two, three and four orbital indices. The forms, relationships and absolute sizes of these terms suggest an intuitive partitioning of the exchange repulsion energy into Molecular Orbital Pair Contributions to the Exchange repulsion energy (MOPCE). Insight into the analytic form and quantitative size of these contributions is provided by considering the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msup><mml:mrow></mml:mrow> <mml:mn>3</mml:mn></mml:msup> <mml:msubsup><mml:mi>Σ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo></mml:msubsup> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:msub><mml:mi>σ</mml:mi> <mml:mi>g</mml:mi></mml:msub> <mml:mn>1</mml:mn> <mml:msub><mml:mi>σ</mml:mi> <mml:mi>u</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:annotation>${^3 \Sigma _u^+ (1\sigma _g 1\sigma _u )}$</mml:annotation> </mml:semantics> </mml:math> state of the H₂ molecule, the water dimer, as well as an argon atom interacting with Cl₂ and N₂.