S-uncoupling operator

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,... 

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,... 

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. 

The L-uncoupling operator, —(l/2//ii2)(J+L-+J-L+), which is responsible for the evolution as J increases from Hund s case (a) to case (d), causes numerous perturbations between states that differ by AO = AA = 1 and with AS = 0. This specific type of rotational perturbation is often called a gyroscopic perturbation. 

If the rotational structure is resolved between the two 2n3/2 and 2n1/2 ionization thresholds, it is possible to assign definite ion-core rotational J+ values to each autoionized resonance, and the resonances are then described by Hund s case (e). Their wavefunctions are explicitly known as linear combinations of the case (a) wave functions, due to the mixing by the rotational operator (the j-uncoupling operator, see Section 8.7, Eqs. (8.7.9) - (8.7.14)). Consequently, the resonances no longer have a well-defined A-value [for example (AB+2n)d<5 1nj can be mixed with (AB+2n)d7r xEj)] and the value of A cannot be predicted without calculations. Such a study has been performed for HBr (Irrgang, et al., 1998). 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

When we uncouple the commutator on the left-hand side of Eq. (331), nondiagonal corrections to the operator ak electric field S and the interaction with phonons can be defined from the equation of motion... 

We have assumed that no angular momentum contribution assists. Then the basis set of spin functions consists of the uncoupled set Sa,Msa) Sb, Msb), or the coupled set SA,SB,S,Ms), its size is N = (2Sa + l)(2Ss + 1). Additionally, the orbital angular momentum can be added and then the basis set becomes a direct product of all orbital and spin functions. In a special case, spin delocalisation (double exchange) operates. 

Equations (36 to 38) provide a convenient starting point for most treatments of currents through molecular junchons where the couphng between the two metal electrodes is weak. In this case, it is convenient to write the system s Hamiltonian as the sum, H = Ho + V, o( a part Ho that represents the uncoupled electrodes and spacer and the couphng V between them. In the weak coupling, limit the T operator... 

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