Interaction tensor dipole

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)... 

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 screened dipole interaction tensor (taking r = for clarity) then becomes... 

III. Electronic dipole-dipole interaction tensor D Fine structure or zero-field parameters D,E... 

In general, the anisotropic dipole-dipole interaction between the two unpaired spins in the triplet state (Table 3) gives rise to a very broad EPR spectrum, with in the derivative representation characteristic peaks for directions of B parallel to the principal axes of the dipole-dipole interaction tensor D. For reaction centers the spectrum shows a peculiar distribution of lines that are in emission or in ab-... 

This result is accurate in general to within the square of the dipole interaction tensor T. 

Figure 8 Various relaxation mechanisms contributing to the relaxation rate, dipole-dipole interaction tensor.

In the case of dipole-dipole interactions between nuclear spins, the Hamiltonian can be separated into an uncorrelated product of a spin part, and the dipole-dipole interaction tensor. 

In intermolecular dipole-dipole relaxation, due to translational diffusion, the nuclear spins are not interacting long enough to create any spin correlation. Hence, each spin constitutes its own separate spin system and the expressions for the relaxation time only depends on the correlation function of the dipole-dipole interaction tensor. 

Furthermore, the lack of correlation of the different spins results in a decoupling of the different spin dipole-dipole interactions experience by the nuclear spin, Thus, even though there in general exists a correlation of the different dipole-dipole interaction tensors, the final expression for intermolecular relaxation time only consists of a sum of auto correlation functions of the individual interactions. 

The relative ease by which the expressions can be derived is, however, not to be mistaken for conceptual simplicity. The dynamics in the dipole-dipole interaction tensor depends in a complex manner on many body interactions and the intrinsic relative motions of the molecules. Hence, it is a non-trivial task to relate it to the motion of individual molecules. 

As mentioned above, the source of paramagnetic relaxation is simply a dipole-dipole interaction, which as such is well understood. The complexity of paramagnetic relaxation stems from the electron spin relaxation, and the fact that the dynamics of the electron spin might be correlated with the dipole-dipole interaction tensor. 

This contains an TCP of the TpaL tensor, which is derived from the electron spin and dipole-dipole interaction tensor(See equation (11)). Hence, the first question we confront is whether those tensors are correlated or not. In case they are not the total TCP can be decomposed into a product of auto correlations for the the electron spin and dipole-dipole interaction tensor, respectively. In case they are, however, it is necessary to consider the whole TCP and the electron spin has to be correlated with the dipole-dipole interaction tensor. The time dependence in the electron spin tensor can be obtained by integrating the time dependent Schrbdinger equation for the electron spin under the electron spin Hamiltonian. The electron spin is just like the nuclear spin precessing around the external magnetic field and influenced by molecular dynamics. 

The dynamics of the dipole-dipole interaction tensor was averaged over each proton of the complex, and had a correlation time around 50 ps. Due to the rigidity of the complex, the decay in the TCP was caused by reorientation of the whole complex and the wagging motion of the water molecules. The fluctuations of symmetry was studied both from the individual water molecules distortion from their ideal symmetry positions and from the symmetry modes of the complex. The symmetry modes were well defined for the oxygen atoms in the water molecule, which show small distortions. The orientations of the water molecules, on the other hand, were too widely distributed from such an analysis to be meaningful. The time scale of the symmetry modes was in the sub-picosecond regime, much too fast to be correlated to the dipole-dipole interaction tensor. Hence, the decomposition of the total TCP into a spin part and a space part is well motivated. 

Here f is the interatomic axis, and Vzz is the component of the retarded dipole-dipole interaction tensor generated by the linearly z-polarized laser light... 

Once the summation over virtual photon wave-vectors and polarizations in Eq. (5.10) is performed, the result can be cast in terms of a retarded resonance electric dipole-electric dipole interaction tensor V, (a>, R) (Power and Thirunamachandran 1983 Andrews and Sherborne 1987), using the identity... 

The susceptibility formula (2.20) can readily be used for calculating the attraction between atoms or molecules, if these particles are replaced by electric dipole oscillators. Let us consider two electric dipole oscillators i and j with elongations u,- and Uj at positions and rj. The electrostatic force exerted on dipole j by a unit moment of dipole i is given by the respective component of the dipole interaction tensor... 

The semiclassical substitution of electric dipole oscillators for molecules requires that one three-dimensional dipole is attached to each allowed electron transition. Each molecule has to be replaced by an ensemble k of independent three-dimensional dipole oscillators whose eigen-frequencies correspond to the allowed one-electron excitation energies. The energy of interaction of molecules i and j is derived by summing Eq. (2.22) over all dipoles k representing molecule i and over all dipoles / representing molecule j. These summations do not affect the dipole interaction tensors T. Tyj, which depend exclusively on the separation and mutual orientation of molecules i and J. We are left with the summation over the susceptibilities x (c(J) of dipoles k at molecule i... 

It is possible to find the interaction energy between two molecules by integrating their polarizabilities multiplied by the dipole interaction tensor along the imaginary frequency axis. Since in Eq. (2.24) we use the electrostatic interaction tensor T,- the frequency integration actually affects only the polarizabilities. These quantities are strictly real on the imaginary frequency axis. 

The dipole interaction tensor (2.21) is obviously proportional to the inverse cube of the separation r,j = r, —of molecules i and j. The interaction energy AE is found to be proportional to the inverse sixth power of Tij. In the simplest case of spherical atoms, we obtain... 

The dipole interaction tensors T,y(o)), Tj-,((u) can be found from Maxwell s equations. In this chapter we restrict ourselves to using the nonretarded interaction tensor (2.21). The case of retardation is considered in chapter 5. 

---