Majorana operators

It has become customary to call the Casimir operator of U12(2), Majorana operator since it was introduced by Majorana in the 1930 s within the context of other problems,... 

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... 

When converted to the local vibrational quantum numbers of Eq. 4.53, the selection rules (4.69) imply that the Majorana operator conserves the quantity... 

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-... 

Figure 4.13 Schematic representation of the effects of the Majorana operator Mn, which removes the degeneracy of the local modes and of the Fermi operator hi, which splits the degenerate (when Xn = - 1) normal multiplets.

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

Terms involving Majorana operators are nondiagonal, but their matrix elements can be simply constructed using the formulas discussed in the preceding sections. The total number of parameters to this order is 15 in addition to the vibron numbers, N and N2- This has to be compared with 4 for the first-order Hamiltonian (4.91). For XY2 molecules, some of the parameters are equal, Xi,i = X2,2 XU2 = X2,12, Y112 = Y2 U, A] = A2, reducing the total number to 11 plus the vibron number N = Aj = N2. Calculation of vibrational spectra of linear triatomic molecules with second-order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.8. 

A more general expression can be obtained by adding the Majorana operators of Sections 4.13 and 4.21 and/or higher-order terms. 

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... 

As a result of the introduction of Majorana operators one moves away from the local limit. In order to emphasize this point it is convenient to relabel the vibrational quantum numbers and to introduce the usual labeling IV v2 v3 V44 V55 >. This labeling is shown in Figure 5.5 and used in Table 5.2. 

The second term is the Majorana operator, My. This operator has both diagonal and off-diagonal matrix elements... 

The Majorana operators My annihilate one quantum of vibration in bond / and create one in bond j, or vice versa. 

It is instructive to analyze the effect of the interaction terms (Majorana operators) in Eq. (6.24). These terms split the degeneracies of the multiplets of Figure 6.1, as shown in Figure 6.3. Thus, the Majorana terms remove the degeneracies of the local modes and bring the behavior of the molecule towards the normal limit, precisely in the same way as in tri- or tetratomic molecules. 

These matrix elements can also be written in the local basis v v ). For this purpose it is convenient to introduce a slightly different form of the interaction term, often referred to as the Majorana operator, Mi2-> which is related to Cu (2) h 2 by means of several conventions. We choose to define Mjj such that in place of (2.45), we obtain the following matrix elements ... 

Figure 11. Block structure of the Hamiltonian matrix caused by the Majorana operator.

The secular problem is solved easily by using the local eigenvalues [Eq. (3.72)] and the matrix elements of Majorana operators [Eq. (3.46)]. Let us consider the explicit form of the secular determinant for the first polyad, p = l. We obtain... 

Figure 20. Splitting of the degenerate CH stretches in C2H4 by the Majorana operators.

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... 

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