On the consequences of the violation of the Hellmann–Feynman theorem in calculations of electric properties of molecules

The general relation between one-electron electric properties ε (dipole moment, polarizability, hyperpolarizability, etc.) of molecules calculated as energy derivatives (E) and as dipole expansion (D) is derived: εE=εD+εNHF, where εNHF represents the non-Hellmann–Feynman correction, which vanishes for wave functions satisfying the Hellmann–Feynman theorem. It is shown that in cases when the wave function does not satisfy the Hellmann–Feynman theorem (e.g., limited CI) not only the NHF correction may be very large, but what is more important, the elements of static polarizability tensors (α,β,γ, etc.) obtained via the dipole expansion are non-physical because they do not satisfy the Kleinman symmetry, and for example αijD≠αjiD, βijjD≠βjjiD or γiijjD=γijjiD≠γjjiiD (for i≠j, i=x,y,z). Finally, it is concluded that in the case of wave functions which do not satisfy the Hellmann–Feynman theorem only energy derivative methods are the correct way for calculations of one-electron electric properties of molecules.