text: The most common application of the Hellmann–Feynman theorem is 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  corresponds to the coordinates of the nuclei. For a molecule with  electrons with coordinates , and  nuclei, each located at a specified point  and with nuclear charge , the clamped nucleus Hamiltonian is
  The -component of the force acting on a given nucleus is equal to the negative of the derivative of the total energy with respect to that coordinate. Employing the Hellmann–Feynman theorem this is equal to
  Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms. Differentiating the Hamiltonian yields
  Insertion of this in to the Hellmann–Feynman theorem returns the -component of the force on the given nucleus in terms of the electronic density  and the atomic coordinates and nuclear charges: