Laplacian of the density

Proynov, E. I., Vela, A., Salahub, D. R., 1994, Nonlocal Correlation Functional Involving the Laplacian of the Density , Chem. Phys. Lett., 230, 419. 

BR Becke and Roussel s 1989 MGGA exchange functional that includes a dependence on the Laplacian of the density in addition to its gradient. Becke, A. D. and Roussel, M. R. 1989. Phys. Rev. A, 39, 3761. 

The Laplacian of the electron density plays a dominant role throughout the theory.191 In addition, Bader has shown that the topology of the Laplacian recovers the Lewis model of the electron pair, a model that is not evident in the topology of the electron density itself. The Laplacian of the density thus provides a physical valence-shell electron pair repulsion (VSEPR) basis for the model of molecular geometry and for the prediction of the reaction sites and their relative alignment in acid-base reactions. This work is closely tied to earlier studies by Bader of the electron pair density, demonstrating that the spatial localization of electrons is a result of a corresponding localization of the Fermi correlation hole. 

Although X-ray crystallography has mainly been used for structure determination in the past, the diffraction data, especially if measured carefully and to high orders, contain information about the total electron density distribution in the crystal. This may be analyzed to provide essential information about the chemical properties of molecules, in particular the characterization of covalent and hydrogen bonds, both from the point of view of the valence electron density, the Laplacian of the density and derived energy density distribution. In addition, calculation of the molecular electrostatic potential indicates direction of chemical attack as well as how molecules can interact with their environment. 

Proynov El, Vela A, Salahub DR (1994) Nonlocal correlation functional involving the Laplacian of the density, Chem Phys Lett, 230 419-428... 

This real matrix may be diagonalized to give the principal axis of curvature and the trace of the hessian matrix i.e., the laplacian of the density, is an invariant. 

In order to make improvements over the LSDA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not only the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA) and is the second rung on Jacob s Ladder. The third rung, meta-GGA functional, includes a dependence on the Laplacian of the density (V p) or on the orbital kinetic energy density (t). The fourth row, the hyper-GGA or hybrid functionals, includes a dependence on exact (HF) exchange. Finally, the fifth row incorporates the unoccupied Kohn-Sham orbitals. This is most widely accomplished within the so-called double hybrid functionals. 

As noted in Chapter 2, deformation densities suffer from ambiguities in definition of reference state and are not (even in theory) directly related to other bond properties. Inspection of the total molecular density or its derivative may ultimately prove more valuable (Bader et al., 1979 Bader, 1985). A start has been made in this endeavor by Swope and Downs (1988), who found features in the Laplacian of the density for B-O bonds... 

FIGURE 5. Summary of electronic distribution in aniline, (a) Bond distances (A), NBO charges [bracket, in au] and Wiberg indices (parentheses, in au). (b) Topology of the electron density determined from atom-in-molecule calculations p(r) = electron density, L = Laplacian of the density defined as L(r) = —V2p(r) and e = ellipticity of the bond critical point, (c) Laplacian map of the density, (d) Iso-surfaces of the electron localization function, ELF = 0.87 the values are the populations of the valence basins... 

The atoms in molecules (AIM) model of a molecule proposed by Bader14 is completely based on the properties of the electron density. The maxima and minima of the density are used to define a volume in the space, which can be associated with a particular atom in a molecule. Notice that such atoms in a molecule are clearly not spherical, and they extend over all the space with sharp boundaries between two atoms. The information about the zones of depletion or accumulation of charge is extracted from the Laplacian of the density, V2p(r). The Laplacian of a function determines where the function will locally augment its value, V2p(r) < 0, and where it will decrease its value, V2p(r) > 0. This cannot be made using just the density because the density is a monotonically decreasing function. Hence, the regions of the space where V2p(r) < 0... 

The LYP functional as given by Eq. (157) is a meta-GGA, because it depends on the Laplacian of the density. Miehlich et al. [193] pointed out that V p(r) can be eliminated from the LYP formula by partial integration. Their transformed version of LYP,... 

Inclusion of either the Lapladan or orbital kinetic energy density as a variable leads to the so-called meta-GGA functionals, and functionals which in general use orbital information may also be placed in this category. Calculation of the orbital kinetic energy density is numerically more stable than calculation of the Laplacian of the density, and the two t functions in eq. (6.44) are common components of meta-GGA functionals. 

For slowly varying densities, the kinetic energy functional can be represented by one of its gradient expansions. The gradient expansion of the kinetic energy density is not unique since it relies upon different derivations techniques [35], which yield or not a contibution of the laplacian of the density in the second order correction. In the following we will consider the expansion expression which does not involve V /i(r) ... 

The oxygen net charge increases with the oxygen coordination and the ionic character is 35% for coordination 2, 55% for coordination 3 and 75 % for 4. The values provided by the Bader s analysis for quartz and cristobalite are significantly lower those that assumed fi om electronegativity. In both cases the Laplacian of the density is positive and the... 

According to Perdew s ladder (Fig. 2.2), the next rung corresponds to the meta-GGA functionals [35]. The difference between this third rung and the previous one (i.e. the GGA rung) is that this type of exchange-correlation functionals expands GGA to include further the kinetic energy density t (i.e. the Laplacian of the occupied orbitals) and/or the Laplacian of the density V p(r). 

Fig. 6.6 Additive, Effective, Hamiltonian kinetic energy densities, together with the laplacian of the density along the C-C (a), and C-H (b) intemuclear lines for the C2H4 molecule computed at the HF//TZV(3d,p)++ level...

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