The Laplacian

Let us analyse p near an interface. The Laplacian in tire curvilinear coordinates (ir )can be written such that (A3.3.71) becomes (near the interface)... 

Before we start to calculate the Laplacian matrix we define the diagonal matrix DEG of a graph G. The non-diagonal elements are equal to zero. The matrix element in row i and column i is equal to the degree of vertex v/. 

As both the diagonal matrix DEG and the adjacency matrix A are symmetric it follows that the Laplacian matrix (Eq. (12)) is also a symmetric one. 

FIGURE 13.5 Isosurface plots, (a) Region of negative electrostatic potential around the water molecule. (A) Region where the Laplacian of the electron density is negative. Both of these plots have been proposed as descriptors of the lone-pair electrons. This example is typical in that the shapes of these regions are similar, but the Laplacian region tends to be closer to the nucleus. 

Wave functions can be visualized as the total electron density, orbital densities, electrostatic potential, atomic densities, or the Laplacian of the electron density. The program computes the data from the basis functions and molecular orbital coefficients. Thus, it does not need a large amount of disk space to store data, but the computation can be time-consuming. Molden can also compute electrostatic charges from the wave function. Several visualization modes are available, including contour plots, three-dimensional isosurfaces, and data slices. 

The symbol V is called del and in cartesian coordinates V, known as the laplacian, is given by... 

It also contains the laplacian which is here defined as... 

Equation 2.3-1 expresses the Laplacian meaning of probability. It is applicable when the number of result,s are countable and... 

The theory of atoms in molecules defines chemical properties such as bonds between atoms and atomic charges on the basis of the topology of the electron density p, characterized in terms of p itself, its gradient Vp, and the Laplacian of the electron density V p. The theory defines an atom as the region of space enclosed by a zero-/lMx surface the surface such that Vp n=0, indicating that there is no component of the gradient of the electron density perpendicular to the surface (n is a normal vector). The nucleus within the atom is a local maximum of the electron density. 

In the Atoms In Molecules approach (Section 9.3), the Laplacian (trace of the second derivative matrix with respect to the coordinates) of the electron density measures the local increase or decrease of electrons. Specifically, if is negative, it marks an area where the electron density is locally concentrated, and therefore susceptible to attack by an electrophile. Similarly, if is positive, it marks an area where the electron density is locally depleted, and therefore susceptible to attack by a... 

A natural question is how does the local period-doubling behavior cf the logistic map translate to its CML-version incarnation Without loss of generality, let us consider the Laplacian-coupled version of the logistic-driven CML ... 

By definition, the Laplacian of U represents the divergence of the attraction field, and, correspondingly, its value characterizes the density of masses at same point. Now the following question arises. What does the Laplacian tells us about the behavior of the potential To answer this question we first consider the simplest case, when U depends on one argument, x, Fig. 1.7a. Then, we can represent the derivatives as ... 

Taking into account the fact that Ax — Ay — Az — h and substituting Equation (1.62) into the expression for the Laplacian ... 

Of course, both statements can be proved from the theorem of uniqueness for the attraction field. In addition, it is appropriate to comment a linear function reaches its maximum at terminal points of the interval. The same behavior is observed in the case of harmonic functions, which cannot have their extreme inside the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point, and, correspondingly, the Laplacian would differ from zero. At the same time, saddle points may exist. 

Since the right hand side contains the sum of the second derivatives, we will make use of the Laplacian inside the earth and express the second derivative of the potential along the vertical through the mean curvature, density, and angular velocity. Then, we have... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

The symbols and are, respectively, the laplacian operators for a single nucleus and a single electron. The variable is the distance between nuclei a and / , Vai the distance between nucleus a and electron i, and the distance between electrons i and j. The summations are taken over each pair of particles. The quantity e is equal to the magnitude of the electronic charge e in CGS units and to e/(47reo) / in SI units, where eo is the permittivity of free space. 

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