The Electrostatic Theorem

Hellmann and Feynman independently applied Eq. (14.68) to molecules, taking A as a nuclear Cartesian coordinate. We now consider their results. 

As usual, we are using the Born-Oppenheimer approximation, solving the electronic Schrodinger equation for a fixed nuclear configuration [Eq. (13.4)]  

Tne kinetic-energy part of H is independent of the nuclear Cartesian coordinates, as can be seen from (1420). Hence 

The variable rg is the distance between nucleus d and point (x, y, z) in space  

What is the significance of (14.76) In the Born-Oppenheimer approximation, U xa, y , Za,. ..) is the potential-energy function for nuclear motion, the nuclear Schrodinger equation being 

Equation (14.132) has a simple physical interpretation. Let us imagine the electrons smeared out into a charge distribution whose density in atomic units is p(x, y,z). The force on nucleus 8 exerted by the infinitesimal element of electronic charge -p dx dy dz is [Eq. (6.56)] 

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The requirements of binding, as viewed through the electrostatic theorem, emphasize the existence of an atomic interaction line as a necessary condition for a state to be bound, whether it be at the shared or closed-shell limit of interaction. The differing properties associated with the distributions of electronic charge at the shared and closed-shell limits of interaction are reflected in the differing mechanisms by which the forces on the nuclei are balanced to achieve electrostatic equilibrium in the two cases. 

Equation (40) is usually called the electrostatic theorem. This states that the force acting on any nucleus can be calculated by classical electrostatics from the quantum-mechanical charge distribution p of the electrons and the point charges Z of the other nuclei. 

In the preceding sections we have studied diatomic interactions via U(R). However, the study of diatomic interactions can also be carried out in terms of the force F(R) instead of the energy U(R), where R denotes the internuclear separation. Though there are several methods for the calculation of the force, the electrostatic theorem of Hellmann (1937) and Feynman (1939) is of particular interest in this section, since the theorem provides a simple and pictorial method for the analysis and interpretation of interatomic interactions based on the three-dimensional distribution of the electron density p(r). An important property of the Hellmann-Feyn-man (HF) theorem is that underlying concepts are common to both the exact and approximate electron densities (Epstein et al., 1967, and references therein). The force analysis of diatomic interactions is a useful semiclassical and therefore intuitively clear approach. And this results in the analysis of diatomic interactions via force functions instead of potential ones (Clinton and Hamilton, 1960 Goodisman, 1963). At the same time, in the authors opinion, it serves as a powerful additional instrument to reexamine model diatomic potential functions. 

In his paper on the electrostatic theorem, Feynman (1939) conjectured that the inward polarization of the atomic densities may be the origin of... 

From the chemical point of view, we must say these equations are not tractable and provide no useful information. In common, the study carried out by many authors (Salem, 1963b Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Bader, 1960b Murrell, 1960 Berlin, 1951 Ben-ston and Kirtman, 1966 Davidson, 1962 Benston, 1966 Bader and Bandrauk, 1968b Kern and Karplus, 1964 Cade et al., 1966 Clinton, 1960 Phillipson, 1963 Empedocles, 1967 Schwendeman, 1966) on the force constants is based on the application of the virial and the Hellmann-Feynman or the electrostatic theorems. In particular, the Hellmann-Feynman theorem provides the expression for ki which relates the harmonic force constant to the properties of molecular charge distribution p(r), i.e., it follows (Salem, 1963b) that... 

SHAPE-ENERGY RELATIONS EOR THE COMPUTATION OF FORCES AND GEOMETRY OPTIMIZATION BASED ON MACROMOLECULAR ELECTRONIC DENSITIES AND THE ELECTROSTATIC THEOREM... 

If reasonably accurate electronic densities are available, then the forces acting on the nuclei can be approximately determined by a simple application of the electrostatic theorem, an important variant of the Hellmann-Feynman theorem. In turn, these forces can be used for geometry optimization. 

The electrostatic theorem, as a special case of the Hellmann-Feynman theorem, has some limitations. These limitations are briefly reviewed helow, following the discussion of the connections between the Hellmann-Feynman theorem and general properties of potential energy hypersurfaces. ... 

It is quite reasonable that the electrostatic theorem follows from the Bom-Oppenheimer approximation, since the rapid motion of the electrons allows the electronic wave function and probability density to adjust immediately to changes in nuclear configuration. The rapid motion of the electrons causes the sluggish nuclei to see the electrons as a charge cloud, rather than as discrete particles. The fact that the effective forces on the nuclei are electrostatic affirms that there are no mysterious quantum-mechanical forces acting in molecules. 

Let us consider the implications of the electrostatic theorem for chemical bond-... 

Let us consider the implications of the electrostatic theorem for chemical bonding in diatomic molecules. We take the internuclear axis as the z axis (Fig. 14.4). By symmetry the X and y components of the effective forces on the two nuclei are zero. [Also, one can show that the z force components on nuclei a and b are related by = P z,b (Prob. 14.40). The effective forces on nuclei a and b are equal in magnitude and opposite in direction.]... 

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