Heilman-Feynman forces

The form of the basis set expansion is chosen to be convenient to model the system and to perform the integrals needed to solve the coupled Eqs. (13). Examples of convenient forms are Gaussians, numerical grids, and plane waves. Plane waves have often (but not exclusively) been chosen as the basis set expansion used in Car-Parrinello simulations for two reasons (i) It is generally desired to simulate extended systems, such as bulk materials, surfaces, and hquids, and plane waves provide a convenient way to model these systems using periodic boundary conditions (ii) forces on atomic nuclei can be calculated very efficiently if the electrons are described by plane waves by making use of the Heilman-Feynman theorem (yide infra). 

As noted above, when the orbitals are expanded in terms of plane waves, forces can be calculated accurately and efficiently using the Heilman-Feynman theorem. This theorem states that the sum of the last two terms of the derivative of the total energy with respect to a nuclear coordinate X, expressed in Dirac notation as follows. 

Thus the equilibrium value of Q is determined by the Heilman-Feynman theorem of balanced forces, and not by the maximum value of r/. This is not a violation of the PMH since neither v nor /x remains constant. 

Forces seem to be superficially simple since the Heilman-Feynman theorem tells us that for some atomic displacement, dZj, ... 

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