Spherical Cartesian

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates.

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

Pio. 2.1 The relationship between spherical polar and Cartesian coordinates. di pends on 6 and 0 ... 

The relationships among cartesian and spherical polar coordinates are given as... 

In performing integration over all space, it is necessary to convert the multiple integral from cartesian to spherical coordinates ... 

In many applications, derivative operators need to be expressed in spherical coordinates. In converting from cartesian to spherical coordinate derivatives, the chain rule is employed as follows ... 

Table 5-12 provides material balances for Cartesian, cylindrical, and spherical coordinates. The generic form applies over a unit cross-sectional area and constant volume ... 

The third quantum number m is called the magnetic quantum number for it is only in an applied magnetic field that it is possible to define a direction within the atom with respect to which the orbital can be directed. In general, the magnetic quantum number can take up 2/ + 1 values (i.e. 0, 1,. .., /) thus an s electron (which is spherically symmetrical and has zero orbital angular momentum) can have only one orientation, but a p electron can have three (frequently chosen to be the jc, y, and z directions in Cartesian coordinates). Likewise there are five possibilities for d orbitals and seven for f orbitals. 

The presence of a single polarization function (either a full set of the six Cartesian Gaussians dxx, d z, dyy, dyz and dzz, or five spherical harmonic ones) on each first row atom in a molecule is denoted by the addition of a. Thus, STO/3G means the STO/3G basis set with a set of six Cartesian Gaussians per heavy atom. A second star as in STO/3G implies the presence of 2p polarization functions on each hydrogen atom. Details of these polarization functions are usually stored internally within the software package. 

In addition to the Cartesian and normal/tangential coordinate systems, the cylindrical (Figure 2-11) and spherical (Figure 2-12) coordinate systems are often used. 

It does not make a significant difference that in practice one uses cartesian Gaussians rather than Gaussians with explicit inclusion of spherical harmonics. One... 

Inasmuch as this system is not used so often as Cartesian, cylindrical, or spherical coordinates, let us describe it in some detail. First of all, we find a condition when a family of non-intersecting surfaces can be a family of equipotential surfaces. Suppose that the equation of the surfaces is... 

In order to find the vertical component of the field we can apply the same approach as before, namely, the integration over the volume of the spheroid, only in this case the polar axis of the spherical system should be directed along the z-axis. However, we solve this problem differently and will proceed from the second equation of the gravitational field. In Cartesian system of coordinates we have... 

The integral can be solved by conversion from Cartesian to spherical coordinates. Then, the integration variable takes the convenient form dr = r drsinddddcj), which yields... 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

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... 

In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r, 6, cp. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. 

Another approximation to planar Couette conditions can be found in the cone-and-plate cell, shown in Figure 2.8.3. The angular speed of rotation of the cone is taken to be Q (in radians per second) while the angle of the cone is a (in radians) and is generally small, say 4-8°. A point in the fluid is defined by spherical polar (r, 0, ()>), cylindrical polar (q, z, cj)) or Cartesian (%, y, z) coordinates, where Q = y = rsin0 and z = rcos0. 

N is a normalization factor which ensures that = 1 (but note that the are not orthogonal, i. e., 0 lor p v). a represents the orbital exponent which determines how compact (large a) or diffuse (small a) the resulting function is. L = 1 + m + n is used to classify the GTO as s-functions (L = 0), p-functions (L = 1), d-functions (L = 2), etc. Note, however, that for L > 1 the number of cartesian GTO functions exceeds the number of (27+1) physical functions of angular momentum l. For example, among the six cartesian functions with L = 2, one is spherically symmetric and is therefore not a d-type, but an s-function. Similarly the ten cartesian L = 3 functions include an unwanted set of three p-type functions. 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

Equation (1) is the one-dimensional form of Fick s first law in Cartesian coordinates. In cylindrical and spherical coordinates, the form of Fick s first law for radial diffusion is... 

In this form the equation is rather cumbersome and not easily solved, so it is customary to express it in spherical polar coordinates r, 6, and, (p, where r is the distance from the nucleus and 6 and (p are angular coordinates, rather than in the Cartesian coordinates x, y, and z. The relationship of the polar coordinates to the Cartesian coordinates is shown in Figure 3.5. In this form V = e2/r, and the equation is easier to solve particularly because it can be expressed as the product R(r)Q(9)(dimensional functions R, the radial function, and 0 and , the angular functions. Corresponding to these three functions there are three quantum numbers, designated n, /, and m. 

Ionization surfaces calculated for several of the molecules listed in Table 1 are shown in Figures 10 and 11. The volume averaged cross sections determined from the volumes enclosed by these surfaces for nonspherical molecules such as C02 are in much better accord with experiment, consistent with the idea that the poor performance of Cartesian averaging for molecules such as C02 is due to the large departure from a spherical shape. Improved agreement with experiment is also... 

We have gone from Cartesian to spherical coordinates in the last member, as a convenience in imposing the limit on the integrations, and carried out the integration over angles, giving the factor of 4ir the division by 8 is because only positive values of nx, ny, and nz are to be considered. Now we have... 

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