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Physics LibreTexts

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+Eddαϕ|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.


This page titled 8.4: Quantum forces - the Hellmann-Feynman Theorem is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.

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