Tensor interaction

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The expansion of Fourier components of the dipole interaction tensor in the vicinity of the minimum point at the boundary of the first Brillouin zone, with the Cartesian axes Ox and Oy respectively chosen along bi and b2 (see Fig. 2.9b), has the form... 

Fig. 2.10. Eigenvalues of the Fourier component of the dipole-dipole interaction tensor in two-dimensional infinite lattices. The solid lines are for a triangular lattice, the dashed lines are for an analytical approximation (2.2.9), and the dotted lines are for a square lattice.

An inference of fundamental importance follows from Eqs. (2.3.9) and (2.3.11) When long axes of nonpolar molecules deviate from the surface-normal direction slightly enough, their azimuthal orientational behavior is accounted for by much the same Hamiltonian as that for a two-dimensional dipole system. Indeed, at sin<9 1 the main nonlocal contribution to Eq. (2.3.9) is provided by a term quadratic in which contains the interaction tensor V 2 (r) of much the same structure as dipole-dipole interaction tensor 2B3 > 0, B4 < 0, only differing in values 2B3 and B4. For dipole-dipole interactions, 2B3 = D = flic (p is the dipole moment) and B4 = -3D, whereas, e.g., purely quadrupole-quadrupole interactions are characterized by 2B3 = 3U, B4 = - SU (see Table 2.2). Evidently, it is for this reason that the dipole model applied to the system CO/NaCl(100), with rather small values 0(6 25°), provided an adequate picture for the ground-state orientational structure.81 A contradiction arose only in the estimation of the temperature Tc of the observed orientational phase transition For the experimental value Tc = 25 K to be reproduced, the dipole moment should have been set n = 1.3D, which is ten times as large as the corresponding value n in a gas phase. Section 2.4 will be devoted to a detailed consideration of orientational states and excitation spectra of a model system on a square lattice described by relations (2.3.9)-(2.3.11). 

In an effort to understand the mechanisms involved in formation of complex orientational structures of adsorbed molecules and to describe orientational, vibrational, and electronic excitations in systems of this kind, a new approach to solid surface theory has been developed which treats the properties of two-dimensional dipole systems.61,109,121 In adsorbed layers, dipole forces are the main contributors to lateral interactions both of dynamic dipole moments of vibrational or electronic molecular excitations and of static dipole moments (for polar molecules). In the previous chapter, we demonstrated that all the information on lateral interactions within a system is carried by the Fourier components of the dipole-dipole interaction tensors. In this chapter, we consider basic spectral parameters for two-dimensional lattice systems in which the unit cells contain several inequivalent molecules. As seen from Sec. 2.1, such structures are intrinsic in many systems of adsorbed molecules. For the Fourier components in question, the lattice-sublattice relations will be derived which enable, in particular, various parameters of orientational structures on a complex lattice to be expressed in terms of known characteristics of its Bravais sublattices. In the framework of such a treatment, the ground state of the system concerned as well as the infrared-active spectral frequencies of valence dipole vibrations will be elucidated. 

By introducing the Fourier components K (k) of the dipole-dipole interaction tensor for the dense Bravais lattice, the desired quantities V°jP (k) for the complex lattice can be determined as122... 

Values of the force matrix components On and 0)2 are specified by the corresponding Fourier components of the dipole-dipole interaction tensors F P(k)... 

The potential of mean torque, parameterized by the local interaction tensors Xa and Xc (assume to be cylindrical) for the aromatic core and the C C segment, respectively, can be mapped out at each temperature by fitting the observed quadrupolar splittings in the mesophase. Furthermore, the order matrix of an average conformer of the molecule can also be calculated.17 Now pn is needed to describe the internal dynamics. 

The applicability of this selection technique depends on the anisotropy and the relative orientation of the various interaction tensors. Frequently occurring situations in transition metal complexes suitable for treatment by this technique are16,76) ... 

Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y.

RDC data can allow the five averages (i.e., the elements of r1- in Eq. (23)) describing each dipolar interaction tensor to be determined, and subsequently interpreted in terms of mean internuclear vector orientations and an associated description of dynamics. To date, there are a couple of different approaches which have been introduced for the analysis and interpretation of RDCs acquired in multiple alignment media. 

A probably more correct treatment has recently been given by Fixman (91) and by Pyun and Fixman (92). These authors avoid the preaveraging of the Oseen-interaction tensor [eq. (3.20)]. A comparison of results will be given at the occasion of the discussion of eigen values. Fortunately, Zimm s results appear to be only slightly different. 

By contrast, when the active space is restricted to the spin-only kets, the influence of all attainable excited states manifests itself in the filling of the MPs (tensors). In such a case the g-tensor deviates considerably from the free-electron value, the TIP appears substantial, and the spin-spin interaction tensor transforms to high values of the ZFS parameters (D and E). 

We note that the values of the hydrodynamic interaction tensor (2.6) averaged beforehand with the aid of some kind of distribution function, are frequently used to estimate the influence of the hydrodynamic interaction, as was suggested by Kirkwood and Riseman (1948).4 For example, after averaging with respect to the equilibrium distribution function for the ideal coil and taking the relation (1.23) into account, the hydrodynamic interaction tensor (2.6) assumes the following form... 

The nuclear spin interaction tensor is most readily expressed in its principal axis frame where only the M = 0, 2 terms are nonzero (and only the M = 0 term is nonzero for axial symmetry). It can then be expressed in the laboratory frame via... 

Since for each pair of dipoles we have many possible directions of transition moments, it is useful to rewrite (1.34) in a tensorial form using the dipole-dipole interaction tensor < >(m — n) ... 

The problem is to discuss the generalized polarizability ae(1.49) with the matrix a 1 not commuting with that of the dipolar interactions, 0. To show that the pure retarded interactions may be discarded in the dynamics of mixed crystals, we assume here that the coulombic interactions are suppressed in (ft. The interaction tensor is then reduced to its retarded term (1.74). Then the dispersion is given by (1.35) ... 

D dynamic-angle correlation spectroscopy (DACSY) (Fig. 11) was proposed by Medek et al.,202 which enables one to obtain the correlation of the Hamiltonians of various ranks. Clearly, this method offers more information than most conventional correlation spectra, so more precise determination of the interaction tensors and their relative orientations can be achieved. 

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