Laplacian of the charge density

Carroll, M.T., Chang, C. and Bader, R.F.W. (1988) Prediction ofthe structures of hydrogen-bonded complexes using the Laplacian of the charge density, Mol. Phys., 63, 387 105. 

FIGURE 1. Laplacian of the charge density contour map of CH3CI in the HCC1 plane. Reprinted with permission from Z. Shi and R. J. Boyd,./. Phys. Chem., 95, 4698. Copyright (1991) American Chemical Society... 

C) Relief map of the negative Laplacian of the charge density, -V2p, of the chlorine molecule. One can notice three depletions of the electron charge density in the region perpendicular to the bond, and along the interatomic axis, for the colinear approach at the both ends. 

Figure 5. A) Relief map of the negative Laplacian of the charge density, -V2p, of the carbon dioxide molecule.

The full usefulness of the classification using V Pb must await the development of the quantum mechanical aspects of the theory. The Laplacian of the charge density appears in the local expression of the virial theorem and it is shown that its sign determines the relative importance of the local contributions of the potential and kinetic energies to the total energy of the system, A full discussion of this topic is given in Section 7.4. 

Unless demanded by symmetry, the value of will not be an extremum at a critical point in p. Thus, the critical points in V and p will not, in general, coincide and the distribution of eleetronie eharge, even in a one-electron system, is not determined entirely by the external foree — V V t). Bohm (1952) ascribed the stability of a stationary state in a quantum system to the balance between the classical force —W and the quantum mechanical force which is given by the gradient of the quantum potential . For a one-electron system at a critical point in p(r), Bohm s quantum mechanical force is just the right-hand side of eqn (3.8). The Laplacian of the charge density, the quantity appearing in the quantum potential, is an important local property of a system and is the subject of Chapter 7. 

Integration of the right-hand side of eqn (5.82) in the manner indicated in eqn (5.80) transforms it into an integral of the Laplacian of the charge density. A typical term in this integration can be transformed using Gauss s theorem to yield... 

Because of eqn (6.73) and the vanishing of the Laplacian of the charge density over an atomic basin, the following identities hold,... 

A check on the accuracy of the numerical integrations of the atomic properties is provided by the evaluation of 1,(12). This quantity, defined in eqn (5.50), is proportional to the integral of the Laplacian of the charge density over the basin of the atom or, equivalently, to the flux in the gradient vector field of p through the surface of the atom,... 

The terms in the expansion derived from derivatives with respect to X are identical to those obtained by taking the corresponding derivatives of the charge density p(X) itself. The dyadic Wp is the Hessian matrix of p, whose eigenvectors and eigenvalues determine the properties of the critical points in the charge distribution. The trace of this term is the Laplacian of the charge density, V p. 

It is shown in this chapter that, while the Lewis model of the electron pair does not correspond to the existence of spatially localized pairs of electrons, it does find a more abstract but no less real physical expression in the topological properties of the Laplacian of the charge density, the quantity V p(r) as defined in eqn (2.3) (Bader and Ess6n 1984 Bader et al. 1984). The... 

Laplacian of the charge density V p(r), along with p(r) and Vp(r), serve to define the conceptual models of chemistry and they provide the necessary basis for the theoretical description of these models. The Laplacian of the charge density V p(r), as demonstrated in the two preceding chapters, plays a dominant role throughout the theory of atoms in molecules. It is shown here that the Laplacian provides a link between theory and the chemical models of geometry and reactivity that are based upon the Lewis model. 

Where then to look for the Lewis model, a model which in the light of its ubiquitous and constant use throughout chemistry must most certainly be rooted in the physics governing a molecular system If one reads the introductory chapter on fields in Morse and Feshbach s book Methods of theoretical physics (1953), one finds a statement to the effect that the Laplacian of a scalar field is a very important property, for it determines where the field is locally concentrated and depleted. The Laplacian of the charge density at a point r in space, the quantity V p(r), is defined in eqn (2.3). This property of the Laplacian of determining where electronic charge is locally concentrated and depleted follows from its definition as the limiting difference between the two first derivatives which bracket the point in question as defined in eqn (2.2) and illustrated in Fig. 2.2. 

The Laplacian of the charge density plays a central role in the theory of atoms in molecules where it appears as an energy density, that is. 

Electrostatic potential maps have been used to make predictions similar to these (Scrocco and Tomasi 1978). Such maps, however, do not in general reveal the location of the sites of nucleophilic attack (Politzer et al. 1982), as the maps are determined by only the classical part of the potential. The local virial theorem, eqn (7.4), determines the sign of the Laplacian of the charge density. The potential energy density -f (r) (eqn (6.30)) appearing in eqn (7.4) involves the full quantum potential. It contains the virial of the Ehrenfest force (eqn (6.29)), the force exerted on the electronic charge at a point in space (eqns (6.16) and (6.17)). The classical electrostatic force is one component of this total force. 

The Laplacian of the charge density as it appears in the local expression of the virial theorem, eqn (7.4), compares twice the kinetic energy density C(r) not with contributions to the potential F, but with F, the electronic potential energy. This is an important distinction from the point of view of whether or not a system is bound. Consider an interaction where V p is predominantly negative over the binding region and net forces of attraction act on the nuclei. In this case the local contribution to the virial of the Hellmann-Feynman forces exerted on the electrons, which, as explained following eqn (6.60), is... 

It is important to emphasize that the correlation which exists between the properties of the Fermi hole and the VSEPR model is made most evident through the use of the properties of the Laplacian of the charge density. The correspondence is greatest when the position of the reference electron... 

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