Paradoxical Role of the Kinetic-Energy Operator in the Formation of the Covalent Bond

The origin of chemical binding is analyzed with the help of variational reasoning for the ground state of the hydrogen molecule–ion. The bond-parallel component of the kinetic-energy integral is shown to be the critical term. The effect of electron sharing on this term is such that the variation process yields, at all internuclear distances, a lower energy for the molecule than for the separated atoms. This appears to be paradoxical in as much as, at the equilibrium distance, the potential part of the binding energy is negative and the kinetic part is positive. The paradox is resolved by showing that the actual values of the potential and kinetic binding energy are the result of a scale variation which is similar to that found in the isolated atom and contracts the wavefunction towards the two nuclei. The variational origin of this redistribution of the kinetic- and potential-energy integrals, which establishes the virial relationship, is analyzed. It is concluded that the variational facts are inadequately described by the view that the chemical bond is formed “because the potential energy can be lowered through accumulation of charge in the bond.” Rather, the accumulation of charge in the bond is essential for chemical binding because it profoundly influences the behavior of the kinetic-energy integral under scale variations.