Testing: The Hellmann-Feynman Theorem

The implementation of any self-consistent exchange-correlation functional can be checked by comparing numerical and analytic forces arising from microscopic displacements of nuclei within an arbitrary system. The agreement between the two forces demonstrates that the XC energy functional is consistent with the potential.

The analytic forces are obtained from a theorem due to Hellmann [105] and Feynman [106], which states that when the valence electron wavefunctions are variationally optimised, the physical force on an ion is simply the classical electrostatic force due to the electrons and nuclei. Due to the extensive basis sets used in plane-wave calculations, the energy is essentially variational and so the force should be given by the same Hellmann-Feynman expression. In other basis set calculations, such as those using localised Gaussian functions, the basis size is smaller and so the Hellmann-Feynman theorem does not hold. An additional term known as the Pulay force [107] must be calculated in these cases.

The HCTH implementation was checked by calculating forces within the ${\hbox{H}}_2$ molecule, using a bond length of $0.800$Å. The numerical force ${\bf F}^{\hbox{{num}}}$ in the molecule was determined from the displacement of one of the ions through a distance $\delta {\bf r}$, from its equilibrium position ${\bf R}_0$,

$${\bf F}^{\hbox{{num}}} = - \frac {E({\bf R}_0 + \delta {\bf r}) - E({\bf R}_0 - \delta {\bf r})}{2 \delta {\bf r}} \, .$$ (3.21)

Since the Hellmann-Feynman theorem requires the wavefunctions to be variationally optimised, the best agreement between the numerical and Hellmann-Feynman forces is dictated by the completeness of the basis set. Therefore the kinetic energy cutoff, $E_{\hbox{{cut}}}$, of the plane-wave expansion must be sufficiently large in order to make accurate comparisons. The test system consisted of placing an H$_2$ molecule in a supercell of size $4\times2\times2$Å, with the bond axis along the larger cell dimension. The supercell approach allows finite systems such as isolated atoms and molecules to be treated within a periodic representation, by placing the species in the centre of a periodic cell that has a sufficient amount of vacant space in order to minimise the interactions from the equivalent species in neighbouring cells. Ordinarily, a total energy convergence test is performed with respect to the size of the supercell to ensure that these neighbouring interactions are diminished, however this is not necessary for the purpose of these tests.

Using a well converged cutoff energy, the numerical and analytic Hellmann-Feynman forces agreed to within $0.002\%$, for a displacement of $\delta {\bf r} = 0.01$Å. The fact that this is a very small discrepancy and that the same degree of accuracy was also obtained with the previously established PW91 functional in CASTEP, demonstrates the correct implementation of the HCTH functional. The spin-polarised version of HCTH was also checked using the open shell ${\hbox{H}}_2^{-}$ molecule.