Vibron number

Here N is the vibron number (related to the number of bound states, as shown in the sections to follow). The U(3) representations are characterized in general by three quantum numbers. However, in the reduction of totally symmetric states of U(4), only totally symmetric states of U(3) appear, characterized by a single quantum number, nK, which can take the values... 

These equations provide an explicit relationship between the parameters appearing in the Morse potential, re, 3, V0, and the reduced mass (i and the algebraic parameters E 0, A, B, and N. Particularly important is the relation N + 2 = 2a, which shows explicitly how the vibron number N is related to the number of bound states in the potential of Figure 2.8. 

The numbers N, N2 are the vibron numbers of each bond. As discussed in Chapter 2 they are related to the number of bound states for bonds 1 and 2, respectively. For Morse rovibrators they are given by Eq. (2.111) that is, they are related to the depth of the potentials. They are fixed numbers for a given molecule. The numbers (0], co, X], X2 are related to the vibrational quantum numbers, as discussed explicitly in the following sections. We have written the O] (4) representations as (C0i, 0) and not simply as aq, since for coupled systems one can have representations of 0(4) in which the second quantum number is not zero. The values of (iq and (02 are given by the rule (2.102),... 

The two most useful sets are the bond displacements themselves, and the symmetry coordinates. The use of the latter leads naturally to a scheme in which the Hamiltonian for bent molecules is no longer diagonal in the total 0(4) quantum numbers (ti, x2), and thus one loses the simple form of the secular equation (Figure 4.11). The secular equation must be now diagonalized in the full space with dimensions that become rapidly larger. This scheme, developed by Leviatan and Kirson (1988), can be implemented only if the vibron numbers N are relatively small, N < 10. 

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. 

In these formulas the vibron numbers Nl, N2, N3 have been omitted, as well as the quantum numbers that are zero, /, and M. 

In addition, since all bonds are equivalent, the vibron numbers Nj must be all equal, IV, = 1VH. Thus, the symmetry of the molecule imposes the following conditions on the coefficients in Eq. (6.17) ... 

There are several ways in which the Hamiltonian Hct(p, q) can be converted to an equivalent Hamiltonian where the kinetic energy has the simple form with the mass being independent of coordinates and momenta.4 These ways differ in order 1 /N, where N is the vibron number. We present first a particularly simple construction. 

CC bends). In practice, such unwanted modes are moved at energies 10 (or even more) times than those of the true vibrational levels. Removal of the spurious mode could be accomplished by giving the parameter A a large value. However, we proceed in a slightly different way, which involves giving a value to N (the vibron number) that is very large. 

Let US consider a practical example. We want to use the algebraic model for describing the vibrational spectrum of HCN, a linear nonsymmetric molecule (Fig. 35). As per custom, we first determine the vibron numbers Aj and N2 of the CN and HC bonds, respectively, by using Eq. (4.114). We obtain Aj = 156 and Aj = 43. As in Section IV.B.l [Eq. (4.41)] we then recover from the purely local model, the initial guesses of the algebraic parameters... 

After the assignment of the three vibron numbers A, i = 1, 2, 3, the Hilbert model space of the physical problem is given by the basic states of the symmetric irreducible representations [NJ0[N2]0[N3] of the SGA (4.121). To specify these basic states unambiguously, we choose a complete subalgebra chain of (4.121). The first step is the usual local (bond) assignment of Morse rovibrating units, U,(4) D 0,(4) (i = 1, 2, 3), leading to... 

In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

Figure C3.5.5. Vibronic relaxation time constants for B- and C-state emitting sites of XeF in solid Ar for different vibrational quantum numbers v, from [25]. Vibronic energy relaxation is complicated by electronic crossings caused by energy transfer between sites.

The vibronic coupling model has been applied to a number of molecular systems, and used to evaluate the behavior of wavepackets over coupled surfaces [191]. Recent examples are the radical cation of allene [192,193], and benzene [194] (for further examples see references cited therein). It has also been used to explain the lack of structure in the S2 band of the pyrazine absoiption spectrum [109,173,174,195], and recently to study the photoisomerization of retina] [196],... 

Apparently, the most natural choice for the electronic basis functions consist of the adiabatic functions / and tli defined in the molecule-bound frame. By making use of the assumption that A" is a good quantum number, we can write the complete vibronic basis in the form... 

In this section, we briefly discuss spectroscopic consequences of the R-T coupling in tiiatomic molecules. We shall restrict ourselves to an analysis of the vibronic and spin-orbit structure, detennined by the bending vibrational quantum number o (in the usual spectroscopic notation 02) and the vibronic quantum numbers K, P. 

Figure 3. Low-energy vibronic spectrum in a. 11 electronic state of a linear triatomic molecule, computed for various values of the Renner parameter e and spin-orbit constant Aso (in cm ). The spectrum shown in the center of figure (e = —0.17, A o = —37cm ) corresponds to the A TT state of NCN [28,29]. The zero on the energy scale represents the minimum of the potential energy surface. Solid lines A = 0 vibronic levels dashed lines K = levels dash-dotted lines K = 1 levels dotted lines = 3 levels. Spin-vibronic levels are denoted by the value of the corresponding quantum number P P = Af - - E note that E is in this case spin quantum number),...

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

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