8.4: Quantum forces - the Hellmann-Feynman Theorem
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For many systems one is often interested in forces as well as energies. If we can write the energy of a in state ϕ as E=⟨ϕ|ˆH|ϕ⟩ and differentiate with respect to some quantity α then
dEdα=⟨dϕdα|ˆH|ϕ⟩+⟨ϕ|dˆHdα|ϕ⟩+⟨ϕ|ˆH|ϕ⟩
But since ˆH|ϕ⟩=E|ϕ⟩ and ⟨ϕ|ϕ⟩ is 1 for normalisation:
dEdα=⟨ϕ|dˆHdα|ϕ⟩+Eddα⟨ϕ|ϕ⟩+⟨ϕ|dˆHdα|ϕ⟩
This result is called the Hellmann-Feynman theorem: the first differential of the expectation value of the Hamiltonian with respect to any quantity does not involve differentials of the wavefunction.
For example, if α represents the position of a nucleus in a solid, then the force on that nucleus is the expectation value of the force operator dˆHdα. It can be applied to any quantity which is a differential of the Hamiltonian provided the basis set does not change.
Caveat: if we use an incomplete basis set which depends explicitly the positions of the atoms, then we have |ϕ⟩=∑n,i|un,i(r)⟩. This give spurious so-called “Pulay” forces if ϕ is not an exact eigenstate.