Spherical harmonics 59

The basis sets that we have considered thus far are sufficient for most calculations. However, for some high-level calculations a basis set that effectively enables the basis set limit to be achieved is required. The even-tempered basis set is designed to achieve this each function m this basis set is the product of a spherical harmonic and a Gaussian function multiplied... 

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

When the above analysis is applied to a diatomic species such as HCl, only k = 0 is present since the only vibration present in such a molecule is the bond stretching vibration, which has a symmetry. Moreover, the rotational functions are spherical harmonics (which can be viewed as D l, m, K (Q,< >,X) functions with K = 0), so the K and K quantum numbers are identically zero. As a result, the product of 3-j symbols... 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

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. 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

For practical purposes two different approaches have been used. If the nuclear framework has a center with high degree of symmetry, it may be convenient to expand the Hartree-Fock functions fk(r) in terms of spherical harmonics Ylm(6, q>) around this center ... 

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

In atomic physics, it is sometimes necessary to calculate integrals over products of three spherical harmonics. These can be reduced conveniently, with the help of the foregoing expression, to integrals over the product of two spherical harmonics, which are known.13 Thus... 

Campbell s Theorem, 174 Cartwright, M. L., 388 Caywood, T. E., 313 C-coefficients, 404 formulas for, 406 recursion relations, 406 relation to spherical harmonics, 408 tabulations of, 408 Wigner s formula, 408 Central field Dirac equation in, 629 Central force law... 

The distribution of orientation of the structural units can be described by a function N(0, solid angle sin 0 d0 dtp d Jt. It is most appropriate to expand this distribution function in a series of generalised spherical harmonic functions. 

Using the summation theorem for spherical harmonics, these correlation functions may be represented as scalar products... 

The model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

The radial functions, R depend only upon the distance, r, of the electron from the nucleus while the angular functions, (6,(p) called spherical harmonics, depend only upon the polar coordinates, 6 and Examples of these purely angular functions are shown in Fig. 3-11. 

The orbitals d. and dy can be expressed in terms of the complex forms di and whose angular parts are given by the spherical harmonics and respectively. The matrix of orbital angular momentum about the z axis in the complex basis is... 

Here, Yx m( j) denotes a spherical harmonic, coj represents the spherical polar angles made by the symmetry axis of molecule i in a frame containing the intermolecular vector as the z axis. The choice of the x and y axes is arbitrary because the product of the functions being averaged depends on the difference of the azimuthal angles for the two molecules which are separated by distance r. At the second rank level the independent correlation coefficients are... 

If we expand the value of the displacement field c[) in terms of spherical harmonics according to = Jt/cj) t/(cos 0) /(r =0), it is then possible to write down equations of motion for the (/, m) components of both ripplon and phonon displacements ... 

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