Harmonics, general tensor

The various mechanisms of mixing are thoroughly discussed by Wybourne (2). One of the most important mechanisms responsible for the mixing is the coupling of states of opposite parity by way of the odd terms in the crystal field expansion of the perturbation potential V, provided by the crystal environment about the ion of interest. The expansion is done in terms of spherical harmonics or tensor operators that transform like spherical harmonics. This can be formulated in a general Eq. (1)... 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The problem is similar to that involved in harmonic force field calculations, but more difficult in almost all respects. In simple cases one may attempt to solve directly, or graphically, for some of the anharmonic 0 values using the observed values of the spectroscopic constants in equations like (61) and (62). These may then be related to / values through the L tensor as described on pp. 124—132. However, such methods are of only limited value. The more general method of calculation is to attempt an anharmonic force field refinement, in which a trial force field is refined, usually in a large non-linear least-squares calculation, to give the best agreement between the observed and... 

From the measurement of the intensity of the generated second harmonic for a dipolar molecule and with knowledge of the value of the dipole moment, an experimental value for / can be obtained. The relation of this value to the hyperpolarizability tensor components, in the completely general case... 

The total-intensity hyper-Rayleigh scattering experiment basically consists of measuring the amount of incoherently scattered optical second harmonic (parallel and perpendicular output polarizations) versus a reference. When only this single measurement is done, only one independent tensor component can be obtained. For charge transfer molecules, this is generally approximated by P. The depolariza-... 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

In this product the strain tensor components in the crystallite reference system are used for the SODFs. With this choice the calculation of the macroscopic strains and stresses e, and x,- requires only the harmonic coefficients of /=0 and 1 = 2 (see Section 12.2.6.3). Similar to the ODF [Equation (23)], the WSODFs are expanded in a series of generalized spherical harmonics ... 

To illustrate a case to be revisited in detail later, we explicitly derive the rate for coherent second-harmonic generation in a system containing M molecules. Using the general expression Eq. (52) for the radiation tensor together with the Golden Rule, and retaining a sum over the emitted harmonic in the matrix elements, we first obtain an expression of the form... 

Let us now consider more specifically the case of a medium possessing an excited state u) close in energy to that of the emitted harmonic, 2h( >. For practical application, this condition is generally more useful than resonance at the fundamental frequency, since the latter condition is likely to result in a substantial loss of pump power through conventional single-photon absorption. In view of its denominator structure, it is clearly the first term in Eq. (85) that will provide the major contribution to the nonlinear response tensor... 

Since the transformation properties of spherical harmonics are well known, the spherical-tensor notation has some advantages, particularly in the derivation of general theorems however, the reality of cartesian tensors also has its attractions, especially for small values of /. Normally, the moments of a particular three-dimensional molecule are most conveniently given in an x,y,z frame. 

These components transform under rotation like the spherical harmonic functions T]m. In general, an irreducible spherical tensor Tim transforms like the function Ylm. The spherical components of a second-rank tensor are again collected in Table 1.13. 

Although we have introduced irreducible spherical tensors, we do not yet have a formalism which admits ready generalization to D dimensions. This can be accomplished by transforming the spherical tensor to a Cartesian tensor. The spherical components of a tensor are related by a unitary transformation to Cartesian components [11,12,13]. For example consider a spherical harmonic of / = 1 (a spherical tensor of rank 1) written as a three component vector... 

Stone [35] showed that for deltahedral clusters the L" MOs are generally bonding. The L" MOs are, therefore, generally antibonding, since the parity inversion operation leads to a reversal of the bonding properties of L" and L . He used the methodology of Tensor Surface Harmonic theory to derive the n + 1 skeletal electron pair rule for... 

One application of the stress theorem is the study of elastic properties of solids, which becomes straightforward when a suitable finite macroscopic strain is applied to the solid. When the wavefunctions of the distorted solid are known, the stress tensor is evaluated with the stress theorem. In the harmonic approximation elastic constants are defined as the ratio of stress to strain, and it is furthermore possible to go to large strains to obtain all nonlinear elastic properties. In general it is necessary to be concerned with internal strains that may appear microscopically owing to the lower symmetry of the strained solid. In section 6 we show in detail how this problem is solved by combining the stress and force theorems. 

The familiar expressions for other permanent electrostatic interactions in terms of T-tensors [28] and spherical harmonics can be found elsewhere [3]. Early simulation studies placed the permanent moments within a spherical LJ-core [29]. This generalized Stockmayer potential is now frequently used as a basic model to examine the effects of boundary conditions in simulations of polar fluids [30]. In the simulation of "realistic" liquids it is now more usual to graft the multipole moment onto an ISM core. The first simulations of this type of model involved a diatomic LJ-core and a permanent quadrupole to model Na and Bra [8, 12]. The point dipole interaction has also been used with a non-spherical core to model HF [31]. This particular simulation also included the quadrupole-dipole (0 j) and quadrupole-quadrupole (66) interactions. Point moments are readily included in an m.d. simulation. For linear molecules acting with a central anisotropic potential the force on molecule i is given by [32]... 

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