Quantum numbers triatomic molecules

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),...

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

Figure 4.5 Local vibrational quantum numbers of a linear triatomic molecule. The arrows indicate the corresponding displacements.

An important problem of molecular spectroscopy is the assignment of quantum numbers. Quantum numbers are related to conserved quantities, and a full set of such numbers is possible only in the case of dynamical symmetries. For the problem at hand this means that three vibrational quantum numbers can be strictly assigned only for local molecules (f = 0) and normal molecules ( , = 1). Most molecules have locality parameters that are in between. Near the two limits the use of local and normal quantum numbers is still meaningful. The most difficult molecules to describe are those in the intermediate regime. For these molecules the only conserved vibrational quantum number is the multiplet number n of Eq. (4.71). A possible notation is thus that in which the quantum number n and the order of the level within each multiplet are given. Thus levels of a linear triatomic molecules can be characterized by... 

Figure 4.14 Local vibrational quantum numbers of bent triatomic molecules. Also shown are the relative displacements of the atoms in the different modes.

Figure 4.19 Normal-mode vibrational quantum numbers for a bent triatomic molecule. Contrast the results for water, which is (cf. Table 4.6) near the local-mode limit with that for S02, which is near the normal-mode limit.

Fig. 1.13 The Pettifor structure map for sp-valent AB2 triatomic molecules with N 16 where N is the total number of valence electrons. The full triangles and circles correspond to bent and linear molecules respectively whose shape is well established from experiment or self-consistent quantum mechanical calculations. The open symbols correspond to ambiguous evidence. The data base has been taken from Andreoni et a/. (1985).

Since we have abandoned the individual quantum numbers of the degenerate modes Qx and Qy, we will replace the summation over the 3N — 6 normal modes in energy expressions by a summation over the distinct vibrational frequencies. For a linear triatomic molecule there are four normal modes, but only three distinct vibrational frequencies. In the harmonic-oscillator approximation, the energy contribution of the doubly degenerate modes is... 

The possible dissociation channels for the fragmentation of a triatomic molecule were discussed in Section 1.4. The linear ABC molecule can fragment into three chemical channels, A+B+C, A+BC(n), and AB(n )+C with the diatoms being produced in particular vibrational states denoted by quantum numbers n and n, respectively. Furthermore, each of the fragment atoms and molecules can be created in different electronic states. The total energy Ef = Ei + hu is the same in all cases and therefore the different channels are simultaneously excited by the monochromatic light pulse. The dissociation channels differ merely in the products and in the way the total energy partitions between translation and vibration. 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

The theory outlined above can be used to calculate the exact bound-state energies and wavefunctions for any triatomic molecule and for any value J of the total angular momentum quantum number. We can solve the set of coupled equations (11.7) subject to the boundary conditions Xjfi (R Jp) —> 0 in the limits R —> 0 and R — oo (Shapiro and Balint-Kurti 1979). Alternatively we may expand the radial wavefunctions in a suitable set of one-dimensional oscillator wavefunctions ipm(R),... 

Let us for simplicity discuss a triatomic molecule, for example H2O, with fi perpendicular to the plane defined by the three atoms. In that case, the photon will mainly excite molecules that are perpendicularly aligned to the Eo vector, i.e., that lie in a plane perpendicular to Eo- If the dissociation time is small compared to the rotational period of the parent molecule, the rotational vector of OH will be preferentially directed parallel to the laboratory 2-axis because the recoil of H and OH proceeds in-plane. This would lead to a distribution in the projection quantum number mj which is strongly peaked near mj j. For a parallel transition, on the other hand, we would expect the opposite situation, i.e, j would be aligned perpendicularly to the 2-axis and P(mj) would peak near mj - 0. 

Figure 2.2 presents a schematic diagram of the possible vibrational modes for diatomic, linear triatomic, and bent triatomic molecules. The vibrational term symbol provides an ordered list of quantum numbers... 

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