Majorana operators couplings

The off-diagonal elements of the Majorana operator in Eq. (4.29) illustrate, for the first time, the appearance of a nonlinear resonance (Section 3.4) within the algebraic approach. For two identical modes, an m = (1,-1) resonance is expected to be very important. Mx2 indeed couples such [nearly degenerate, cf. Figure 4.1 or Eq. (3.28)] states whose quantum numbers differ by m. Note furthermore that the 0 (2) quantum number is conserved, va + vb = const, as expected for a 1,-1 resonance because m = (1,1) is orthogonal to m = (1,-1), cf. Eq. (3.31). 

We have already discussed in Section 4.5 the local-to-normal transition for two coupled oscillators. The situation is quite analogous for two coupled rovibrators. The local-to-normal transition can be described by combining the operators of the local chain with those of the normal chain. It is convenient to introduce the Majorana operator... 

Figure 4.12 Representation of the couplings induced by the Majorana operator in the first multiplet, n = 1. N and give the order of magnitude of the couplings.

For the first multiplet, n = 1, and similar expressions for the other multiplets. The states of the second multiplet, n = 2, which are coupled by the operator Mn, are (04°0), (12°0), (02° 1) (20°0), (10°1), (00°2). From the structure of the matrices, one can see that the Majorana operator does two things simultaneously. It produces the local couplings that are needed to go from local to normal situations, and it introduces, when viewed from the normal-mode basis, Dar-ling-Dennison (1940) couplings of the type < v v 2, v3IVIv( -F 2, v , v3 2 >. 

Note that there is a duality that stems from the two different ways one can view the Hamiltonian (4.67) (Lehmann, 1983 Levine and Kinsey, 1986). As written, the Majorana operator serves to couple the local-mode states. But the Majorana operator is [cf. Eq. (4.66)] the Casimir operator of U(4) and is a leading contributor to the Hamiltonian, Eq. (4.56) describing the exact normal-... 

We consider next Majorana-type couplings. These are introduced, as in the previous case of triatomic molecules (cf. Section 4.17), by the operators... 

In the same way as discussed in the preceding sections, one can include Majorana operators, M13 and M23. Since these are in the wrong coupling scheme, one must use the recoupling techniques of Section 2. The matrix elements of Mn and M23 are given by... 

These matrix elements are equivalent to those of Eq. (3.124), apart from anharmonic contributions of the order of v/N. So we see that the extended Majorana operator has the required effect on the states involved in the resonance mechanism. At the same time > SB does not preserve the coupled 65 (2) symmetry in other words, + Vg is not conserved anymore. Consequently, the block-diagonal structure of the Hamiltonian operator is destroyed and the numerical diagonalization of... 

It is, of course, possible to extend this calculation to obtain, in closed analytical form, the first excited polyad, Vj = 2. The result is shown schematically in Fig. 34. In particular, we notice the direct coupling between pairs of stretching modes in light of the selection rule (for Ub = const) A(u -I-Uj) = 0 and Au, Au = 0, 1. This means that the (initially degenerate) stretches 100), 010) now mix and split under the effect of Mj2. Due to the symmetry under bond exchange, we obtain either symmetric or antisymmetric wavefunctions, as discussed for the one-dimensional case. The difference here is the presence of the bending mode, which is also involved in the coupling scheme induced by the Majorana operator. We can see in both Fig. 34 and Eq. (4.45) that... 

As discussed previously for bent molecules, the local model (4.54) is a poor approximation when intermode coupling occurs, so we now need to introduce the Majorana operator. The explicit analysis of this problem is perfectly analogous to the previous one, apart from the different conversion law between algebraic and vibrational quantum numbers. Moreover, in a linear molecule we expect to obtain vibrational wavefunctions... 

Although we will not discuss in detail this particular aspect of anharmonic resonances, it is important to note that Darling-Dennison couplings are automatically included by the action of the Majorana operator. A practical way to convince ourselves of this inclusion is to diagonalize (either numerically or in closed form) the Hamiltonian matrix explicitly for the first two polyads of levels and then to convert, in normal-mode notation, the vibrational states obtained. As discussed in Ref. 11, the Hamiltonian (4.38) can also be written (neglecting Cj2 and Cj2 interactions) as... 

With the local basis, we are ready to construct a triatomic-like Hamiltonian operator where most of the physically relevant interactions should be either diagonal or in the nondiagonal form of the Majorana operator. This is a direct consequence of our choice for the coupling scheme (1 -I- 2) -f- 3, which is, in fact, done to favor interactions of the type (H-2). So for a linear tetratomic molecule we write the following Hamiltonian operator ... 

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