The Hellmann-Feynman constraint

The electrostatic Hellmann-Feynman theorem states that for an exact electron wave function, and also of the Hartree-Fock wave function, the total quantum-mechanical force on an atomic nucleus is the same as that exerted classically by the electron density and the other nuclei in the system (Feynman 1939, Levine 1983). The theorem thus implies that the forces on the nuclei are fully determined once the charge distribution is known. As the forces on the nuclei must vanish for a nuclear configuration which is in equilibrium, a constraint may be introduced in the X-ray refinement procedure to ensure that the Hellmann-Feynman force balance is obeyed (Schwarzenbach and Lewis 1982). 

The Hellmann-Feynman constraint has been applied successfully to the exocyclic fluorine, carbon, and nitrogen atoms in tetrafluoroterephthalonitrile (1,4-dicyano-2,3,5,6-tetrafluorobenzene). Charge balance is achieved without deterioration of the least-squares agreement factors, though the resulting changes in the density maps are very small (Hirshfeld 1984) (see chapter 5). 

We note that the constraint applies to the static density. Its application therefore requires adequate deconvolution of thermal motion and electron density effects. 

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