Laplacian operator in Cartesian coordinates

Here, /u and e are the permeabihty and permittivity of the dielectric media, respectively, and k = 2n f. .JJie is the wave number of the dielectric. The operator v is called transverse Laplacian operator. In Cartesian coordinates... 

In the above equation, A and B run over the M nuclei while i and J are assigned for the N electrons in the system. The first two terms describe the kinetic energy of the electrons and the nuclei, respectively. The symbol represents the Laplacian operators in Cartesian coordinates and Ma is the mass of nucleus A in multiples of the mass of an electron. The last three terms describe the potential part contributed by attractive electrostatic interactions between the nuclei and the electrons, the repulsive potential due to the electron-electron interactions, and the repukive potential due to the nucleus-nucleus interactions, respectively. The distance between particles p and q is denoted by rpq (and in a similar case Rpq)- the wave function of the fill state of the... 

Here, A and B run over the M nuclei while i and j denote the N electrons in the system. The first two terms describe the kinetic energy of the electrons and nuclei respectively, where the Laplacian operator V2 is defined as a sum of differential operators (in cartesian coordinates)... 

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 operator V2, which is known as the Laplacian, takes on a particularly simple form in Cartesian coordinates, namely,... 

The explicit dependence on R is not shown for equation (A.6) and will not be given in subsequent equations, it being understood that unless stated otherwise, we are working within the BO approximation. The Laplacian operator V(/)2 in Cartesian coordinates for the 7th electron is given by... 

It is worth noting at this point that the Laplacian operator V2 = V V plays an important role in this text. In Cartesian coordinates,... 

The term V2 is the laplacian operator, which in cartesian coordinates (given by x, y, and z) is... 

The operator is called the Laplacian. In Cartesian coordinates the Laplacian of (j) is given by... 

Niven established the connections between the ellipsoidal harmonics, expressed in cartesian coordinates, and the spheroconal harmonics, expressed in spheroconal coordinates, in the respective factors of Eq. (18), by requiring that the eigenfunctions h satisfy the Laplace equation [18]. The application of the Laplace operator on the eigenfunctions with the condition of vanishing leads to the zeros 0, of the respective polynomials, which are real and different in their respective domains ccartesian coordinates leads to the corresponding condition for its being harmonic... 

The validity of Eq. (4.110) can also be shown by transforming p, i.e., the Laplacian A, given in Cartesian coordinates to spherical coordinates. It then turns out that the components of the angular momentum operator read... 

The first term in Equation (A9.2) is the kinetic energy operator, which contains the Laplacian ( del squared ). This is the three-dimensional equivalent to the onedimensional kinetic energy operator we met for the harmonic oscillator in Appendix 6. In Cartesian coordinates, the Laplacian operator is defined as... 

The relative Schrodinger equation cannot be solved in Cartesian coordinates. We transform to spherical polar coordinates in order to have an expression for the potential energy that contains only one coordinate. Spherical polar coordinates are depicted in Figure 17.3. The expression for the Laplacian operator in spherical polar coordinates is found in Eq. (B-47) of Appendix B. The relative Schrodinger equation is now... 

This operator is related to the Laplacian operator, (del squared), in spherical polar coordinates. From the definition of in Cartesian coordinates, and from the coordinate transformation given previously. 

The Laplacian operator has its simplest form in Cartesian coordinates... 

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. 

X = thermal conductivity which is assumed temp independent, cal/cm-°K-sec Q = heat of reaction, cal/g e = fraction of expl reacted, dimensionless V2=Laplacian operator, which in Cartesian space coordinates x, y, z is... 

The particles are numbered from 1 to N with Mi the mass of particle i, R, = [A, Yt Zi) a column vector of Cartesian coordinates for particle i in the external, laboratory fixed, frame, Vr the Laplacian in the coordinates of R, and Ri — Rj the distance between particles i and j. The total Hamiltonian, eqn.(l), is, of course, separable into an operator describing the translational motion of the center of mass and an operator describing the internal energy. This separation is realized by a transformation to center-of-mass and internal (relative) coordinates. Let R be the vector of particle coordinates in the laboratory fixed reference frame. 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

This form of the equation is not easily applied to rotational motion because the Cartesian coordinates used do not reflect the centro-symmetric nature of the problem. It is better to express the Schrodinger equation in terms of the spherical polar coordinates r, 6 and 0, which are shown in Figure 5.7. Their mathematical relationship to x. y and z is given on the left of the diagram. In terms of these coordinates the Laplacian operator becomes ... 

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