When a hydrogen molecule forms, 4.52 eV of energy is released, while for $\text{H}_2^+$ the binding energy is 2.77 eV. Such a binding energy is the difference of energies that have four terms in them: (1) the kinetic energy of the electron(s), (2) the potential energy of the electron(s) interacting with the nuclei, (3) the electron-electron interaction, and (4) the proton-proton interaction.
Explanations of $\sigma$ bonding in freshman chemistry texts tend to focus on #2. However, if we want to explain the difference in energy between bonding and anti-bonding orbitals, then it seems plausible that there should be a large difference in kinetic energy, #1. This is because the KE of the bonding orbital is low compared to that of the antibonding orbital because the bonding orbital basically a particle in a long box with wavelength in the long direction equal to twice the length of the box. In the antibonding case, this component of the wave-vector should be basically doubled.
All four of these energy terms can be represented by quantum-mechanical observables, and they can therefore be defined numerically, and calculated for a given set of trial wavefunctions. How much of the truth is captured by explanations that only mention #2?
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