S-uncoupling

This is an approach for the calculation of the microcanonical rate constant k(E) for indirect unimolecular reactions that is based on several approximations. The molecule is represented by a collection of s uncoupled harmonic oscillators. According to Appendix E, such a representation is exact close to a stationary point on the potential energy surface. Furthermore, the dynamics is described by classical mechanics. 

The probability is calculated according to classical statistical mechanics (Appendix A.2). According to Eq. (A.49), the density of states (number of states per unit energy) for s uncoupled harmonic oscillators with frequencies z/j is... 

From Appendix A.2, we have classical expressions for the sum and density of states of s uncoupled harmonic oscillators. Thus, the sum of states is... 

That is, values of the off-diagonal matrix elements of the BJ S, S-uncoupling operator, within a 2S+1An multiplet state, must be much smaller than the energy separation between multiplet components that result from differences between diagonal matrix elements of Hso. Since matrix elements of the S-uncoupling operator are proportional to J, at sufficiently high-J case (a) ceases to be as good an approximation as case (b). 

This off-diagonal matrix element connects the same basis states as a term in HROT (S-uncoupling operator), but its sign is opposite to that of the Bv contribution. The off-diagonal Bv term appears with a negative sign while 7 appears with a positive sign because of the phase convention [Eq. (3.2.85c) and Eq. (3.2.86)],... 

When the S-uncoupling operator acts between two components of a multiplet state that belong to the same vibrational quantum number, then the vibrational part of the B(R)L S matrix element is... 

However, if by chance there is a near degeneracy between the O spin-component of the nth level and the fi = fi 1 components of the (v + l)th level of the same electronic state, then the S-uncoupling operator can cause a perturbation between these levels. In the harmonic approximation and using the phase choice that all vibrational wavefunctions are positive at the inner turning point,... 

Figure 3.17 Anomalous A-doubling in the X2nr state of 15N80Se, due to S-uncoupling perturbations (From Jenouvrier, et al., 1973.)...

The only nonzero off-diagonal matrix element between these two substates, which differ by AO = AE = 1, is given by the S-uncoupling part of the rotational Hamiltonian. Using Table 3.2, one obtains... 

It is necessary to assume that the two substates, 2IIi/2 and 2II3/2, have identical potential curves, and thus the same vibrational wavefunctions. The other part of the S-uncoupling operator, J+S-, acts between 2II 3/2 (0 = — ) and 2n -1/2 (0 = — 5), and gives the same matrix element. There are no off-diagonal matrix elements of the S-uncoupling operator between O > 0 and 0 < 0 2II basis functions. Thus the e and / 2II basis functions, which are linear combinations of fi and — fi basis functions, have the same matrix elements of the rotational operator as the separate signed-fi functions. 

Starting from a 2II matrix expressed in terms of case (a) basis functions, case (b) energy level expressions have been derived for BJ A. This means that the eigenfunctions are almost exactly the case (b) basis functions. Alternatively, in the case (b) basis the S-uncoupling operator, — 2B(R)J S, can be replaced by — B (N2 — J2 — S2) because... 

Thus, it is evident that the only nonzero case (b) matrix elements of the S-uncoupling operator are AN = A J = AS = 0 and that all off-diagonal matrix elements of this operator between Fi and F2 basis functions vanish. The departure from case (b) at low J is caused by AN = 1 matrix elements of the Hso and Hss operators. 

These two components differ by Afi = +1 and interact via a nonzero matrix element of the S-uncoupling operator. This interaction can never be neglected, since the two interacting components have the same energy. The matrix element,... 

Is the perturbation matrix element J-dependent (heterogeneous perturbation or S-uncoupling in either the perturbed or perturbing state) Figure 5.8 contrasts the level shifts resulting from J-independent perturbation matrix elements with those from matrix elements proportional to J. 

Both Ok and fl. - flj. provide useful insights into the causes and rates of specific dynamical processes. Sections 9.4.9 and 9.4.10 provide analyses of the dynamics of the S-uncoupling operator in a 25+1A state and the 1 2 anharmonic coupling operator that contributes to Intramolecular Vibrational Redistribution (IVR) in a polyatomic molecule and illustrate the diagnostic power of the flk + fij. and lk — resonance and rate operators. 

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