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Hellmann–Feynman theorem

The Hellmann–Feynman theorem states that once the spatial distribution of the charged particles (usually the electron clouds) has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using the classical electrostatics. It is named for its independent provers Hans Hellmann (1936) and Richard Feynman (1939).

This quantum mechanics theorem relates the energy eigenvalues of a time-independent Hamiltonian operator to the parameters composing it. In general, the theorem states that, $\frac{\partial {E_n}}{\partial {\lambda}}=\int{\psi_n^*\frac{\partial{\hat{H}}}{\partial{\lambda}}\psi_nd\tau}$

where $\hat{H}$ is the parameterized Hamiltonian operator,

En is the nth Hamiltonian eigenvalue,

ψn is the nth Hamiltonian eigenvector,

λ is a continuous parameter of interest,

and dτ implies an integration over the complete domain of the eigenvectors.

The proof

Using the Dirac's bra-ket notation, we can write $\frac{\partial E}{\partial\lambda}$ $= \frac{\partial}{\partial\lambda}\langle\psi|\hat{H}|\psi\rangle$ $= \langle\frac{\partial\psi}{\partial\lambda}|\hat{H}|\psi\rangle + \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle + \langle\psi|\hat{H}|\frac{\partial\psi}{\partial\lambda}\rangle$ $= E \langle\frac{\partial\psi}{\partial\lambda}|\psi\rangle + \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle + E \langle\psi|\frac{\partial\psi}{\partial\lambda}\rangle$ $= E \frac{\partial}{\partial\lambda}\langle\psi|\psi\rangle + \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle$ $= \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle.$

Note that the last step is only valid for the exact wave function, not (necessarily) for approximate ones. Hence, for instance, the Hellmann-Feynman force of a Hartree-Fock wave function differs quite significantly from the true energy gradient.