Polarization moments, classical

The multipole (or polarization) moments introduced according to (2.14) present a classical analogue of quantum mechanical polarization moments [6, 73, 96,133, 304]. They are obtained by expanding the quantum density matrix [73, 139] over irreducible tensor operators [136, 140, 379] and will be discussed in Chapters 3 and 5. 

Following the general approach, as presented in Chapter 2, let us expand the solution (3.5) over spherical functions (2.14) in order to pass from pa(9,(p) to classic ground state polarization moments aPq (2.16). It is important to stress that, since absorption is non-linear with respect to the exciting light, here, unlike in Section 2.3, we obtain polarization moments aPq of rank k > 2 in the ground state. We will denote the rank and the projection by k and q respectively (unlike K and Q for the excited state). We can, however, state that all the produced polarization... 

Table 3.1. Analytical expressions for classic polarization moments apQ (k < 4) describing optical polarization of angular momenta in the ground (initial) state via light absorption...

It is again very fruitful here to expand the probability densities pa(9, spherical functions (2.14) and to pass to equations for classical polarization moments ap, tPg, as determined by (2.16). The decisive simplification of equations consists of the fact that the derivative of the spherical function Ykq 9, [Pg.107]

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

Table 5.1. Comparison between classical pq and quantum mechanical fq polarization moments...

In just the same way as in the case of fluorescence intensity, the asymptotic equations of motion of polarization moments (5.54), (5.55) and (5.87), (5.88) must coincide with the corresponding equation of motion of classical multiple moments, as introduced by Eq. (2.16). We will show that this is indeed so in the following section. 

Equations of motion of probability density and of classical polarization moments... 

In order to obtain the equations of motion of classical polarization moments, we must base our methods on the system of equations of motion of the probability density pa(6, angular momentum vector 3(6, optical pumping. For a number of maximally simplified situations, where the probability density in the ground state pa(6,(p) does not depend on that of the excited state pb(6,(p), we have already encountered such equations in preceding chapters see e.g., (3.4), or (4.5) and (4.6). 

Summing up the above, we may conclude that the classical system of equations (5.93) and (5.94), together with the above given additional terms, coincides perfectly with the asymptotic system of equations of motion of quantum mechanical polarization moments (5.87) and (5.88). This result was actually to be expected from correspondence principle considerations. 

Using the explicit form of spherical functions (Appendix B), it is also possible to obtain an interpretation, analogous to (D.47) and (D.48) for classical polarization moments cpg, 9Pg with other values of K, Q. It may be pointed out that, sometimes, for instance in [19,95], normalization W = 1 is used. 

In the present book we have used the cogredient expansion form (2.14), where, as distinct from the standard form, an additional normalizing factor has been introduced, namely (—l)< v/(2K + l)/4n. Our expansion of the classical probability density p(0, differs from the standard one in exactly the same way as the expansion of the quantum mechanical density matrix p over 2Tq differs from the expansion over lTg. In Section 5.3 we present a comparison between the physical meaning of the classical polarization moments pg, as used in the present book, and the quantum mechanical polarization moments fg, as determined by the cogredient method using normalization (D.ll). 

Thus, we have attempted to give, in the present appendix, an idea of the various methods of determining classical and quantum mechanical polarization moments and some related coefficients. We have considered only those methods which are most frequently used in atomic, molecular and chemical physics. An analysis of a great variety of different approaches creates the impression that sometimes the authors of one or other investigation find it easier to introduce new definitions of their own multipole moments, rather than find a way in the rather muddled system of previously used ones. This situation complicates comparison between the results obtained by various authors considerably. We hope that the material contained in the present appendix might, to some extent, simplify such a comparison. 

Molecules do not consist of rigid arrays of point charges, and on application of an external electrostatic field the electrons and protons will rearrange themselves until the interaction energy is a minimum. In classical electrostatics, where we deal with macroscopic samples, the phenomenon is referred to as the induced polarization. I dealt with this in Chapter 15, when we discussed the Onsager model of solvation. The nuclei and the electrons will tend to move in opposite directions when a field is applied, and so the electric dipole moment will change. Again, in classical electrostatics we study the induced dipole moment per unit volume. 

The charge distribution of neutral polar molecules is characterized by a dipole moment which is defined classically by jx = E, , , where the molecular charge distribution is defined in terms of the residual charges (qt) at the position r,. The observed molecular dipole moment provides useful information about the charge distribution of the ground state and its ionic character. 

Both species exhibit the expected linear geometry that maximizes the dominant n- - a interaction. However, these isomers are rather perplexing from a dipole-dipole viewpoint. The dipole moment of CO is known to be rather small (calculated Fco = 0.072 D), with relative polarity C- 0+. 40 While the linear equilibrium struc-ture(s) may appear to suggest a dipole-dipole complex, robust H-bonds are formed regardless of which end of the CO dipole moment points toward HF This isomeric indifference to dipole directionality shows clearly that classical dipole-dipole interactions have at most a secondary influence on the formation of a hydrogen bond. 

Classical relaxors [22,23] are perovskite soUd solutions like PbMgi/3Nb2/303 (PMN), which exhibit both site and charge disorder resulting in random fields in addition to random bonds. In contrast to dipolar glasses where the elementary dipole moments exist on the atomic scale, the relaxor state is characterized by the presence of polar clusters of nanometric size. The dynamical properties of relaxor ferroelectrics are determined by the presence of these polar nanoclusters [24]. PMN remains cubic to the lowest temperatures measured. One expects that the disorder -type dynamics found in the cubic phase of BaTiOs, characterized by two timescales, is somehow translated into the... 

Since a molecule with a center of symmetry, such as one belonging to point groups D h (n even). Cntl (n even). Dtlll (n odd), O, and //, cannot have a dipole moment, no matter how polar the individual bonds (Chapter 3), dipole moments have proved to be useful in distinguishing between two structures Much of the classic chemistry of square planar coordination compounds of the type MA2B2 was elucidated on the basis of cis isomers having dipole moments and traps isomers having none (see Chapter 12). ... 

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