Solid-harmonic functions, definition

For r = 1, V = y so that y is the value of the solid harmonic in the surface of the unit sphere at points defined by the coordinates 6 and surface harmonics of degree /. The associated Legendre polynomials Pf(cosd) have l — m roots. Each of them defines a nodal cone that intersects a constant sphere in a circle. These nodes, as shown in Figure 20-5, are in the surface of the sphere and not at r = 0 as assumed in the definition of atomic orbitals. Surface harmonics are obviously undefined for r = 0. The linear combinations i/ i i/i i define one real and one imaginary function directed along the X and Y Cartesian axes respectively, but these functions (denoted and ipy) are no longer eigenfunctions of L, but of or Ly instead. 

Consider Eq. (6.84). This result was obtained for a harmonic system of identical and equivalent atoms. We could however reverse our reasoning and define a vibrational spectrum for a dense atomic system from the velocity autocorrelation function according to Eq. (6.84). Since this function can be computed for all systems, including liquids and disordered solids, we may use (6.84) as a definition of a spectrum that may be interpreted as density of modes fimction for such media. We can then use it in expressions such as (4.33), and (6.92). Is this approach to dynamics in condensed phases any good ... 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

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