Field dependence of the energy of a molecule in a magnetic field

By combining the variational principle and the Hellmann-Feynman theorem, various forms of virial equations are obtained: for a many-electron system in arbitrary scalar and vector potential fields, for a neutral or charged molecule in an arbitrary vector potential field, for a molecule and an atom in a homogeneous magnetic field. These virial equations and their integrals represent exact results for the field-dependence of the ground-state and excited-state energies of the considered systems. Their validity is demonstrated not only for the exact eigenenergies, but also for a wide class of approximate energies, including the Hartree-Fock approximation and the intense-field limit of the statistical Thomas-Fermi model.