Tetrahedral rotation functions

It may happen that for certain values of / more than one totally symmetric combination (10) may exist. In that case, the / index should be understood as a composite label. As an example, we list in Table I, for values of l up to 10, the tetrahedral rotation functions that transform according to the totally symmetric representation of the tetrahedral group T. These functions are normalized such that... 

Tetrahedral Rotation Functions Adapted to the Site Group O [See Eq. (33)] ... 

Fig. 11a and b. Decay of the alignment echo height as a function of the mixing time x2 for different motional mechanisms, a Tetrahedral jumps as a model for conformational changes b Diffusive motion, the solid lines correspond to unrestricted rotational diffusion, the dashed lines to diffusion restricted to an angular region of 8°. Note the strong dependence of the decay curves on the evolution time t, in case of diffusive motion... 

Expansion (7.58) becomes obvious if we consider the lattice symmetry. Two nearest neighbors belonging to the same sublattice are situated on the diagonals of the opposite vertices and their contributions to the crystalline field should be equal, so that the odd harmonics vanish. Coefficients /34 and j86 can be calculated using the AAP method. Yamamoto et al. [1977] found /34 = 5.43, p6 = 7.18 these numbers were later revised by Smith [1990] so as to give the best fit to the libration spectrum (/34 = 6.50, /36 = -5.33). The intermolecular potential is expanded in a series in rotational tetrahedral functions uv, which can be expressed in terms of Wigner functions at 7 = 3 ... 

Caswell and Schmir (1979) that PLNM/ALPH effects will be observed only in conformationally restricted systems. These authors studied products of hydrolysis of -imidate [92] and an equimolar mixture of this and the Z-imidate [93] as a function of pH. The percentage of ester in both cases was quantitatively described by the same titration curve, the yield of 101% ester at low pH falling to 65% at high pH, with the change governed by an apparent p/fa-value of 10.2. Rotation about the C—O or C—N bonds of the tetrahedral intermediate [94] is thus fast compared with its decomposition across the whole pH range. 

As one of the special features, MOMEC has a plane twist function. This has been included to limit the tetrahedral twist in four-coordinate compounds, where 1,3-nonbonded interactions lead to a preference for a tetrahedral arrangement (see Section 3.6). That is, the plane twist potential can be used to induce a square-planar arrangement or, using constraints, any intermediate structure can be enforced. The same potential can in principle be used for other structural features (see Fig. 17.14.1), such as the Bailar twist of six-coordinate complexes or for computing the rotational barrier of metallocenes. However, at present it has only been implemented in MOMEC for the tetrahedral twist and no parameters have been included as yet. 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

As in the Rayleigh case, the pair polarizability results as well from nonlinear light scattering mechanisms (induced by hyperpolarizabilities and permanent multipole moments). For tetrahedral molecules nonlinear mechanisms contribute to some correlation functions listed in Table V—only those related to the depolarized spectrum and governed by double rotational transitions (QQ, QO, and 00). The nonlinear origin corrections Atp j7"1 j2 which must be added to the linear origin terms cp 1 °f Table V for a tetrahedral molecule are successively [17]... 

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