Spherical Laplacian

In fact, eigenfunctions for the spherical Laplacian can be written down explicitly in terms of Legendre polynomials. The Legendre polynomials are defined in terms of two indices as follows ... 

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 charge density study of benzoylacetone [8] revealed that the Laplacian at the bond critical points between the enol hydrogen and the oxygens has a negative value. This means that the bonds between that hydrogen and both the oxygens have covalent character. Furthermore the populations of the spherical valence parts of the multipole... 

To obtain the Laplacian in spherical coordinates appropriate second derivatives. Again, as an exan (16) can be written as... 

Electrons in the core of an atom are fully localized into spherical shells but not into opposite-spin pairs. In an isolated atom the valence shell electrons are similarly localized into a spherical shell. The Laplacian shows that in each of these spherical shells there is a spherical region of charge concentration and a spherical region of charge depletion. But in these regions there is no localization of electrons of opposite spin into pairs. There are no Lewis pairs or electron pair domains in an inner shell. The domain of each electron is spherical and fully delocalized through the shell. 

In addition to these new two functions, the Laplacian operator V2 is written in spherical coordinates as... 

In the case of flux with spherical symmetry, i.e., with no dependence on the latitude and longitude, gradient and Laplacian operators must be expressed as a function of the radial distance r to the origin... 

The wave function, V /(r), is a function of the vector position variable r. To determine it at every point in space it is convenient to take advantage of the fact that the potential V(r) depends only on the scalar interatomic distance r. In spherical coordinates (Figure 1.2), the Laplacian operator V2 has the form... 

Here, ma is the mass of the nucleus a, Zae2 is its charge, and Va2 is the Laplacian with respect to the three cartesian coordinates of this nucleus (this operator Va2 is given in spherical polar coordinates in Appendix A) rj a is the distance between the jth electron and the a1 1 nucleus, rj k is the distance between the j and k electrons, me is the electron s mass, and Ra>b is the distance from nucleus a to nucleus b. 

In our discussion of spherical harmonics we will use an expression of the three-dimensional Laplacian in spherical coordinates. For this we need spherical coordinates not just on but on all of three-space. The third coordinate is r, the distance of a point from the origin. We have, for arbitrary (x, y, zY e... 

Exercise 3.26 (Used in Proposition A.3) Consider the Laplacian in spherical coordinates (see Exercise 1.12) ... 

The following proposition justifies the rehance on spherical harmonics in spherically symmetric problems involving the Laplacian. To state it succinctly, we introduce the vector space C2 C 2(R3) continuous functions whose first and second partial derivatives are all continuous. 

The right-hand side of eqn. (9), which is the diffusion equation or Fick s second law, involves two spherically symmetric derivatives of p(r, t). In the general case of three-dimensional space, lacking any symmetry, it can be shown that the Laplacian operator... 

In this equation, V2 = d2/dx2 + d2/dy2 + d2/dz2 denotes the Laplacian operator of cartesian second derivatives, p(r) is the charge density in a spherical shell of radius r and infinitesimal thickness dr centered at the particle of interest (see diagram), k is the effective dielectric constant, and e0 is the permittivity of free space (8.854 x 10 12 in SI units). The energy of interaction / , of ions of charge z,c with their surroundings,... 

If one uses the chain rule to transform the Laplacian V2= d2/dx2 + 92/ 9y2+ 92/9z2 into spherical polar coordinates, the result is... 

X = r sin0 cos< , Y = r sin0sin0, Z = rcos9, so that, in spherical polar coordinates, the Laplacian is given by... 

If we consider spherical vessels of radius ro such that T will depend only on the distance from the center r, then T = T r,t)y and the Laplacian " can be written in spherical coordinates... 

Summarizing, the far and near field differ in three respects. First they do so in range. Common double layer fields extend over distances of order x" in the absence of an external field such fields are radial for a spherical double layer, as shown In fig. 3.86,bl. On the other hand, the range of the far fields is of the order of the particle radius a, which for the case considered, means that they extend far beyond the double layer. In the second place they differ In magnitude, as already stated. Thirdly, the difference is that in the near field there exist local excess charges, whereas in the far field each volume element is electro-neutral. In mathematical language, p [r,0) = 0, where r and 6 are defined in fig. 3.87. Consequently, the Laplacian of the potential is also zero in the far field. 

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