The Heilman-Feynman Theorem

S referred to as the Heilman-Feynman theorem. It was widely used to investi- te isoelectronic processes such as isomerizations X —> Y, barriers to internal Otation, and bond extensions where the only changes in the energy are due to banges in the positions of the nuclei and so the energy change can be calculated Nn one-electron integrals. 

This set of equations is known as the Heilman-Feynman theorem (see, e.g., Hirschfelder et al. (1954)). It is a set of simple first order differential equations for the energy levels En- But (4.1.53) is not a closed system of differential equations since we do not know the behaviour of the perturbation matrix elements as a function of e. In an attempt to close the system (4.1.53) we compute... 

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. 

Note Pulay s mention of the Heilman-Feynman theorem. We have moved on since 1969, especially in the development and application of analytical methods for evaluating the gradients. 

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

It should be noted that the Heilman— Feynman theorem applies to all molecular interactions, not just Coulombic ionic. Therefore, electrostatic equation (38) applies to all types of solutes (ions, dipoles, induced dipoles, hydrogen bonded, etc.) interacting in a solution or at a solid—solution interface. 

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