Solid harmonics table

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... 

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

Fig. 6.1. The real solid harmonics S/ ,(r) for Z < 4. The functions have been arranged as in Table 6.3.

Note that all terms in the solid harmonics of order / in (6.4.47) are of degree / exactly. The reader may wish to verify that (6.4.47) reproduces the real solid harmonics in Table 6.3. 

The real solid harmonics may be generated from (6.4.47) the first few are illustrated in Figure 6.1 and listed in Table 6.3. We also recall that the STOs are given by the expressions... 

Figure 3. Time dependence of the second harmonic coefficient, d33, for corona-poled (PS)O-NPP films. A. Simultaneously poled (180°C) and cross-linked with 0.50 equiv. 1,2,7,8-diepoxyoctane/phenol OH B. Poled at 180°C C. Poled at 150°C. The solid lines are least-squares fits to equation 1, yielding the decay parameters in Table II.

Table 5.8.4. Raman and infrared bands (in cm ) of Nj" (in solid N AsFg") and their assignments based on calculated harmonic frequencies (in cm- ) of gaseous Nj"...

Based on the optimized structure, the harmonic vibrational frequencies of Nf have been calculated at the CCSD(T)/6-311+G(2d) level, and these results are listed in Table 5.8.4, along with the experimental data. Comparing the experimental and calculated results, we can see that there is fairly good agreement between them, bearing in mind that the calculations are done on individual cations and experimental data are measured in the solid state. Such an agreement lends credence to the structure optimized by theoretical methods. 

Table 7. The two sets of solid spherical harmonics (/ = 2 and 1 = 4), appearing in the integrals d>, are normalized to 4nj(21+1) over the unit sphere...

Figure 8. Semiclassical (solid bars) and quantum (dithered bars) Fano factors versus order N of harmonic generation for (a) fundamental and (b) Ahh-harmonic modes in the quasistationary noenergy-transfer regime. Panels (a) and (b), for N = 1 — 5, correspond to Tables I and II, respectively. It is seen that the quantum results are well fitted by the semiclassical Fano factors. According to both analyses, the third-harmonic mode has the most suppressed photocount noise.

Figure 13. Absorption and resonance Raman profiles for azulene in CS2 at 300 K 44 The solid lines are theoretical curves computed using a seven-mode stochastic harmonic model without Dushinsky rotation (Table I). The absorption curve (upper panel) is co

Figure 8. Vibration-rotation contributions to the entropy for H20 as a function of temperature. The lower curve (O, solid line) shows the entropy predicted by the classical rigid rotator-quantum harmonic oscillator (RR/HO) model, the intermediate curve ( , dashed line) shows data taken from the JANAF Thermochemical Tables, and the upper curve (0> solid line) is obtained from differentiating the fourth-order fit to the AOSS-U free energy calculations shown in Figure 6. Note that (as expected) all three curves coincide at low temperatures and that the JANAF and AOSS-U results are in dose agreement throughout the entire temperature range.

We now examine actual instances. In Table I we have listed the parameters pertinent to the preceding discussion for a number of solids the lowest two librational frequencies are considered. We see that COj, NjO, Clg, C2N3, OCS, and benzene all have k larger than 10(X), and many of them several times that value. We conclude therefore that the harmonic oscillator result is probably quite accurate for the librations in these solids. On the other hand, a-Ng, CO, CgH, and the hydrogen halides (HCl, HBr) have low values of k and therefore the harmonic oscillator results may be inaccurate. It is difficult, on the other hand, to assess the errors... 

The zero-field oe-o type SHG, in our opinion, may be accounted for by a multipolar (the quadrupolar) mechanism known for the solid crystals. The effective second-order susceptibilities calculated from the experimental data on the oe-o synchronism are shown in the Table. Their order of magnitude is typical of the quadrupolar SHG mechanism. So, the phenomenon of the zero-field second harmonic generation in nematic and... 

German DIN standards on the characterization of dispersed or porous solids are collected in ref. [6]. The most comprehensive description of the adsorption method is found in an lUPAC recommendation (7). Nowadays national standards are being harmonized either in the framework of the European Communities or at the international level. A list of standardization committees working in this field is appended (Table 4). Different methods of particle counting and characterization are collected in a VDI manual [8j. 

It is important to note that also nonchiral molecules are capable of forming chiral mesophases. In particular, molecules with a bent core ( bananashaped molecules) can build polar, and even chiral liquid crystal structures [75]-[78]. Bent-core molecules form a variety of new phases (B1-B7, Table 1.3) which differ from the usual smectic and columnar phases (see also Chapter 8). As a consequence of the polar arrangement, antiferroelectric-like switching was observed in the B2 phase formed by bent-core molecules, and second harmonic generation was found in both the B2 phase and the B4 phase. The latter phase is probably a solid crystal. It consists of two domains showing selective reflection with opposite handedness. In the liquid crystalline B2 phase, the effective nonlinear susceptibility can be modulated by an external dc field [79] (Figure 1.15). 

Use of this potential can then provide a quantitative measure of cohesion in these solids. One measure of the strength of these potentials is the vibrational frequency that would correspond to a harmonic oscillator potential with the same curvature at the minimum this is indicative of the stiffness of the bond between atoms. In Table 1.1 we present the frequencies corresponding to the Lennard-Jones potentials of the common noble gas elements (see following discussion and Table 1.2 for the relation between this frequency and the Lennard-Jones potential parameters). For comparison, the vibrational frequency of the H2 molecule, the simplest type of covalent bond between two atoms, is about 500 meV, more than two orders of magnitude larger the Lennard-Jones potentials for the noble gases correspond to very soft bonds indeed ... 

The spherical harmonics are conveniently normalized so that on squaring, multiplying by sin 6 dd d (an element of solid angle) and integrating over 6 and the result is unity. The first few spherical harmonics are listed in Table A. 1.1. 

Below we will come back to the reptation model in context with the dynamics of polymers confined in tube-like pores formed by a solid matrix. For a system of this sort the predictions for limits (II)de and (III)de (see Table 1) could be verified with the aid of NMR experiments [11, 95] as well as with an analytical formalism for a harmonic radial potential and a Monte Carlo simulation for hard-pore walls [70]. The latter also revealed the crossover from Rouse to reptation dynamics when the pore diameter is decreased from infinity to values below the Flory radius. 

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