Concerning analytical derivatives of kinetic and potential energies in the Hartree–Fock theory

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Two analytical procedures for obtaining the derivatives of the kinetic and potential energies in the Hartree–Fock theory are presented. They are the derivation from direct differentiation of the expectation values of the kinetic and potential energies, and that from the virial theorem. Since the molecular orbital wave function in the linear combination of atomic orbitals, in which the scale factor is not optimized, does not satisfy the virial theorem, the results by those two methods are different. The derivatives are compared in relation to the adopted basis sets, incorporation of the derivatives of atomic orbitals in the basis set, universal scaling, and the internuclear coordinate.