This proof of the Hellmann–Feynman theorem requires that the wavefunction be an eigenfunction of the Hamiltonian under consideration; however, one can also prove more generally that the theorem holds for non-eigenfunction wavefunctions which are stationary for all relevant variables. 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 wavefunction 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 λ:
Alternate proof
The Hellmann–Feynman theorem is actually a direct, and to some extent trivial, consequence of the variational principle from which the Schrödinger equation may be derived. This is why the Hellmann–Feynman theorem holds for wave-functions that, though not eigenfunctions of the Hamiltonian, do derive from a variational principle. This is also why it holds, e.g., in density functional theory, which is not wave-function based and for which the standard derivation does not apply. According to the Rayleigh–Ritz variational principle, the eigenfunctions of the Schrödinger equation are stationary points of the functional : The eigenvalues are the values that the Schrödinger functional takes at the stationary points: where satisfies the variational condition: Let us differentiate Eq. using the chain rule: Due to the variational condition, Eq., the second term in Eq. vanishes. In one sentence, the Hellmann–Feynman theorem states that the derivative of the stationary values of a function with respect to a parameter on which it may depend, can be computed from the explicit dependence only, disregarding the implicit one. On account of the fact that the Schrödinger functional can only depend explicitly on an external parameter through the Hamiltonian, Eq. trivially follows.
Example applications
Molecular forces
The most common application of the Hellmann–Feynman theorem is to the calculation of intramolecular forces in molecules. This allows for the calculation of equilibrium geometries – the nuclear coordinates where the forces acting upon the nuclei, due to the electrons and other nuclei, vanish. The parameter λ corresponds to the coordinates of the nuclei. For a molecule with 1 ≤ i ≤ N electrons with coordinates, and 1 ≤ α ≤ M nuclei, each located at a specified point
Expectation values
An alternative approach for applying the Hellmann–Feynman theorem is to promote a fixed or discrete parameter which appears in a Hamiltonian to be a continuous variable solely for the mathematical purpose of taking a derivative. Possible parameters are physical constants or discrete quantum numbers. As an example, the radial Schrödinger equation for a hydrogen-like atom is which depends upon the discrete azimuthal quantum numberl. Promoting l to be a continuous parameter allows for the derivative of the Hamiltonian to be taken: The Hellmann–Feynman theorem then allows for the determination of the expectation value of for hydrogen-like atoms: In computing the energy derivative we need to know how depends on. We usually think of these quantum numbers as being independent, but here we must vary the solutions so as to keep the number of nodes in the wavefunction fixed. The number of nodes is, so .
In the end of Feynman's paper, he states that, "Van der Waals's forces can also be interpreted as arising from charge distributions with higher concentration between the nuclei. The Schrödinger perturbation theory for two interacting atoms at a separationR, large compared to the radii of the atoms, leads to the result that the charge distribution of each is distorted from central symmetry, a dipole moment of order 1/R7 being induced in each atom. The negative charge distribution of each atom has its center of gravity moved slightly toward the other. It is not the interaction of these dipoles which leads to van der Waals's force, but rather the attraction of each nucleus for the distorted charge distribution of its own electrons that gives the attractive 1/R7 force."
Hellmann–Feynman theorem for time-dependent wavefunctions
For a general time-dependent wavefunction satisfying the time-dependent Schrödinger equation, the Hellmann–Feynman theorem is not valid. However, the following identity holds: For
Proof
The proof only relies on the Schrödinger equation and the assumption that partial derivatives with respect to λ and t can be interchanged.
