Solid harmonics normalization

Table 15. Solid Harmonics normalized to A7ij(2l- -1) over the Unit Sphere. When normalized to unity they are the usual real hydrogen angular functions... 

Figure 7.7 Cumulative reaction probability of the reaction H + H2 as a function of energy in (a) log and (b) linear scales. Dotted harmonic normal mode approximation. Solid result of the normal form calculation up to quartic...

The angular dependence of the coefficients C R, < a. < b) can be expressed in a closed form. The relevant formulae are obtained by asymptotic expansion of the polarization series truncated at some finite order. In practice such an asymptotic expansion is best performed by evaluating the polarization energies (as given by equations 9, 18, and 21) using the multipole expansion of the electrostatic potential l/ ri — r2. The latter expansion can be written in terms of either the Cartesian or the spherical tensors. The spherical formulation appears to be more popular because it leads much more easily to closed formulae and only this formulation will be considered in this article. Denoting by (r) the regular solid harmonic r Cim(0,), where Cim 0,) is the spherical harmonic in the Racah normalization and with the Condon and Shortley phase, one can write ... 

To remove the constant fraction in front of each term, the solid harmonics in Racah s normalization are used ... 

In this normalization, Coo(, spherical harmonics in Table 6.1. 6.4.2 THE SOLID HARMONICS... 

Except for the normalization constant (see Exercises 6.3 and 6.4), this step completes our derivation of the solid harmonics (6.4.33) from Laplace s equation. Our results are summarized in (6.4.14)-(6.4.16). 

Box 6.1 One-electron basis functions expressed in terms of the generalized L uerre polynomials (6.5.1) and the solid harmonics (6.4.13). The functions are not normalized... 

The purpose of the present exercise is to scale the complex solid harmonics yi in such a way that the angular part is normalized to 1, is a real function and the components are related as... 

It should be realized that the real regular solid harmonics Rj O ) and f (r) introduced here differ from the real solid harmonics Si r) of Section 6.4.2. Thus, whereas the real functions in Section 6.4.2 were obtained by a unitary transformation of the complex ones (in Racah s normalization), the transformation in (9.13.45) and (9.13.46) is not unitary. It is adopted here since it leads to simpler expressions for the multipole expansions. 

Using these relations, it is trivial to set up an explicit expression for their evaluation analogous to (6.4.47) for S/m(r). The recurrence relations may be obtained from (6.4.55), (6.4.56) and (6.4.69). Taking into account the different normalizations (9.13.76) and (9.13.77) and also the symmetry relations (9.13.47) and (9.13.48), we obtain after some algebra the following set of recurrence relations for the scaled regular solid harmonics ... 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

Figure 9.9 Simulated normalized line shapes of -polarized (a-c) and p-polarized (if-/) second-harmonic signals for quarter waveplate measurements (a) and (if) hypothetical achiral surface (hs = 0.5 fp = 0.75, gp = —0.5), (b) and (if) hypothetical chiral surface with in-phase chiral coefficient (fs = 0.75, hs = 0.5 fp = 0.75, gp = —0.5, hp = 0.25), (c) and (/) hypothetical chiral surface with out-of-phase chiral coefficient ( fs = 0.75 0.25i, hs = 0.5 fp = 0.75, gp = —0.5, hp = 0.25z). Upper (solid line) and lower (dashed line) sign in expansion coefficients correspond to two enantiomers. Rotation angles of 45° and 225° (135° and 315°) correspond to right-hand (left-hand) circularly polarized light and are indicated for one of enantiomers with open and filled circles, respectively.

An example of an otherwise fairly normal reactive intermediate where the harmonic approximation fails is phenylcarbene (see Fig. 17.8) Although the match beween calculated and observed IR bands is quite satisfactory across most of the spectmm, it breaks down completely in the region of 750-950 cm, where there is not even a remote similarity between the pattern of three bands predicted by B3LYP calculations (dashed line) and those found experimentally (solid line, downward pointing peaks). 

Lattice vibrations are fundamental for the understanding of several phenomena in solids, such as heat capacity, heat conduction, thermal expansion, and the Debye-Waller factor. To mathematically deal with lattice vibrations, the following procedure will be undertaken [7] the solid will be considered as a crystal lattice of atoms, behaving as a system of coupled harmonic oscillators. Thereafter, the normal oscillations of this system can be found, where the normal modes behave as uncoupled harmonic oscillators, and the number of normal vibration modes will be equal to the degrees of freedom of the crystal, that is, 3nM, where n is the number of atoms in the unit cell and M is the number of units cell in the crystal [8],... 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

Normally, in solid-state problems a harmonic approximation is used enabling one to introduce normal coordinates for the intermolecular subsystem. The density matrix in these coordinates equals the product of the density matrices of independent oscillators, the expressions for which are well-known [158],... 

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