Majorana operators 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. 

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. 

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... 

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... 

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 ... 

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. 

In this operator the Majorana term originates the off-diagonal elements in the basis (4.124). Such elements are very important for a linear symmetric molecule such as HCCH or DCCD. The purely local part of the Hamiltonian (4.126) has eigenvalues given by... 

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