Majorana operators triatomic molecules

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. 

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

The matrix elements of the operator Ml2 are easy to construct since they are identical to those already encountered in triatomic molecules [Eq. (4.70)]. The corresponding secular equation can be diagonalized, yielding the results shown in Figure 5.5. The main effect of the Majorana term is splitting of the degenerate C-H stretching modes into g and u species, as in the previous triatomic case, Section 4.5. 

Let us consider the explicit effect of the Majorana operator on a symmetric triatomic molecule. In light of the symmetry under bond exchange, the Hamiltonian operator (4.23) can be written 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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