In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics. The theorem has been proven independently by many authors, including Paul Güttinger (1932), Wolfgang Pauli (1933), Hans Hellmann (1937) and Richard Feynman (1939). The theorem states
d E λ d λ = ⟨ ψ λ | d H ^ λ d λ | ψ λ ⟩ , {\displaystyle {\frac {\mathrm {d} E_{\lambda }}{\mathrm {d} {\lambda }}}={\bigg \langle }\psi _{\lambda }{\bigg |}{\frac {\mathrm {d} {\hat {H}}_{\lambda }}{\mathrm {d} \lambda }}{\bigg |}\psi _{\lambda }{\bigg \rangle },}
where
H ^ λ {\displaystyle {\hat {H}}_{\lambda }} is a Hermitian operator depending upon a continuous parameter λ , {\displaystyle \lambda ,}
| ψ λ ⟩ {\displaystyle |\psi _{\lambda }\rangle } , is an eigenstate (eigenfunction) of the Hamiltonian, depending implicitly upon λ , {\displaystyle \lambda ,}
E λ {\displaystyle E_{\lambda }\,} is the energy (eigenvalue) of the state | ψ λ ⟩ {\displaystyle |\psi _{\lambda }\rangle } , i.e. H ^ λ | ψ λ ⟩ = E λ | ψ λ ⟩ . {\displaystyle {\hat {H}}_{\lambda }|\psi _{\lambda }\rangle =E_{\lambda }|\psi _{\lambda }\rangle .}
Note that for systems with degenerate states, a refined version of the Hellmann–Feynman theorem is needed.
Proof 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
H ^ λ | ψ λ ⟩ = E λ | ψ λ ⟩ , {\displaystyle {\hat {H}}_{\lambda }|\psi _{\lambda }\rangle =E_{\lambda }|\psi _{\lambda }\rangle ,}
⟨ ψ λ | ψ λ ⟩ = 1 ⇒ d d λ ⟨ ψ λ | ψ λ ⟩ = 0. {\displaystyle \langle \psi _{\lambda }|\psi _{\lambda }\rangle =1\Rightarrow {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\langle \psi _{\lambda }|\psi _{\lambda }\rangle =0.}
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 {\displaystyle \lambda } :
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