This proof of the Hellmann–Feynman theorem requires that the wave function be an eigenfunction of the Hamiltonian under consideration; however, it is also possible to prove more generally that the theorem holds for non-eigenfunction wave functions which are stationary (partial derivative is zero) for all relevant variables (such as orbital rotations). The Hartree–Fock wavefunction is an important example of an approximate eigenfunction that still satisfies the Hellmann–Feynman theorem. Notable example of where the Hellmann–Feynman is not applicable is for example finite-order Møller–Plesset perturbation theory, which is not variational.
The proof also employs an identity of normalized wavefunctions – that derivatives of the overlap of a wave function with itself must be zero. Using Dirac's bra–ket notation these two conditions are written as
The proof then follows through an application of the derivative product rule to the expectation value of the Hamiltonian viewed as a function of :