Singlet Field Equation

To present the results in the clearest possible fashion, we give only the kinetic equation for the irreversible reaction 

Our description of reactions in terms of fluctuations about equilibrium is consistent with this irreversible case under some conditions. We consider a small fluctuation from complete equilibrium. In the subsequent decay of 

The equations in this form are especially suitable for comparison with diffusion equation approaches, which are most often applied to the irreversible reaction case. The reaction scheme we have selected is also convenient because B clearly plays the role of the sink in the diffusion equation approaches, and a rather direct comparison with these methods is possible. 

The kinetic equation for the A species phase-space correlation function 

The contributions to the collision operator A (l z) describe the following types of dynamic event the A operators are Enskog collision operators and describe uncorrelated binary collision events describes uncorrelated elastic collisions of A with solvent molecules 

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To examine the relation between the pair kinetic equation (7.32) and the corresponding propagator for the doublet field that enters into the singlet field equation derived in Appendix C, consider (C.12). The static memory kernel ab,ab defined in (C.l 1) may be written in a form closely related to that in (7.32) by using the static hierarchy. For a hard-sphere system, the static hierarchy takes the form" ... 

ROKS is based on the sum method by Ziegler, Rauk, and Baerends [45], and allows for MD simulations in the first excited singlet state (Sj) [24]. The theoretical framework has been generalized to arbitrary spin states [25], and a new algorithm for solving the self-consistent field equations has been introduced recently [46]. 

In Appendix B we show that the action of the pseudo-Liouville operator on the singlet field generates a coupling to the doublet field, and so on. One may show that if the phase-space density fields are defined as in (7.5), if density fields up to the th order are included explicitly in the description, the random forces corresponding to all fields lower than the nth are zero. Hence only the damping matrix corresponding to this /ith-order field is nonzero. As an example, consider the case of singlet and doublet fields that are explicitly treated. In this case, (7.14) reduces to two coupled equations of the form... 

The pseudo-Liouville operator does couple these doublet fields to triplet fields such as 8 abs cds involving the solvent molecules. Thus one of the simplest forms for the pair kinetic equation can be obtained by explicitly including doublet and triplet fields in the generalized Langevin equation. This procedure yields a treatment of the effects of solvent dynamics on the motion of the reactive pair that is much more sophisticated than that given in the singlet kinetic equation discussed in the preceding... 

Since the derivation of the pair equation exactly parallels that for the singlet kinetic equation, the details are sketched in Appendix D and not given here. It is quite easy to derive a kinetic equation for the general reversible reaction case the calculations need only be carried out in matrix form. To avoid this more complex notation and to present the results in simple form, however, we again give only the results for the irreversible decay of the AB pair field. 

Consider first the kinetic equation for the singlet field in a nonreactive system. It has the general form given in (7.2a), which now reduces (suppressing species labels) to... 

An equation with the form of the macroscopic law in (2.16) can be obtained from the singlet field kinetic equation by projecting out the velocity dependence of the phase-space correlation functions. A comparison of the resulting equation with this macroscopic law can then yield a microscopic correlation function expression for the rate kernel. 

The operator on this correlation function, involving the doublet field 5ajab(12), may now be compared directly with the operator in the pair kinetic equation (7.32). There, of course, the possibility of soft forces between the solute species was also taken into account. The ring operator in (7.33) and (7.34) takes the place of < >ab,ab above. In the singlet kinetic equation that we used in Section VII.C, we ignored fl t... 

For example from optical absorption measurements the temperature dependence of the ground-state splitting of TmV04 was measured (see fig. 17.45). Thereby optical transitions are taking place between the (split) ground-state doublet and an excited singlet state. The results can be compared with the molecular field equation... 

Field-induced exciton-breaking rate. To check the validity of this hypothesis, we have to demonstrate that we can reproduce the measured singlet exciton population quenching using the following equation ... 

Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

Fig. 7.5. Plot of the measured A(p), as defined by equation (7.7), in N2-isobutane mixtures. Data were taken at magnetic fields of 0.375 T (x) and 0.425 T (o). Reprinted from Physical Review Letters 72, Al-Ramadhan and Gidley, New precision measurement of the decay rate of singlet positronium, 1632-1635, copyright 1994 by the American Physical Society.

In equation (4) A is the parameter of the trigonal crystal field. This crystal field splits the 7 2-state into an orbital singlet and a doublet, the parameter A is defined in such a way that for positive A the ground state is the orbital singlet. 

While the singlet ground state will be unaffected by an external magnetic field, the S 1 state will become Zeeman split into the Ms =—1,0, 1 sublevels. Thus, the energies of the four possible 15, Ms) states are known (10, 0)) = Jex (ll, + 1)) = —g/rB77 (ll, 0)) = 0 (ll, —1))= I /ab//. When these four expressions are introduced into the Van Vleck equation, the Bleaney-Bowers equation7 is obtained. This equation describes the temperature dependence of the susceptibility (for the zero-field limit), independent of the sign of Jcx ... 

A mean-field treatment gives the following gap equation [71] for singlet superconductivity ... 

It is noteworthy that the S-To conversion rate is given by Qn for a radical pair with 7-0 J as shown in Chapter 3. When 7 - 0 J,Ag = 0.01, B = 1 T, and A/g tB = Ai/gfis = 0 T, the rate becomes 4.4 x 10 s from problem 3-5. If such S-To conversion rate is comparable to the escape rate of two radicals from a solvent cage, appreciable MFEs and MIEs can be observed. In some cases, this condition can be satisfied in homogeneous solvents. If two radicals are confined with membranes, micelles, or chemical bonds, the escape rate of the two radicals becomes much smaller than the S-To conversion rate. In this case, the S and To states attain equilibrium and the T i-To and T i-S relaxations become important under sufficiently high fields as shown in Fig. 7-4(b). In 1984, the author s group proposed the relaxation mechanism (RM) [2] in order to explain MFEs and MIEs on chemical reactions in confined systems. When st(0 T), stCB) fcp in the RM as shown in Fig. 7-4, the rate equations of the populations of the singlet and triplet radical pair ([S] and [T ] for n = +1,0, and -1) produced from a triplet precursor can be represented as follows [2] ... 

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