Vibronic states, coupling

The application of OODR in molecular beams to the analysis of complex perturbed spectra is illustrated by Fig. 5.21, which shows a section of the visible NO2 spectrum around X = 488 nm. In spite of the small residual Doppler width of 15 MHz, not all lines are fully resolved and the analysis of the spectrum turns out to be very difficult, because the upper state is heavily perturbed. If in the OODR experiment the pump laser Li is kept on line 1, one obtains the two OODR signals as shown in the upper left part of Fig. 5.21, which proves that lines 1 and 4 in the lower spectrum share a common lower level. The right part of Fig. 5.21 also shows that the lines 2 and 5 start from a common lower level as well as the lines 3 and 6. The whole lower spectrum consists of two rovibronic transitions 7", K" = (10,5) (/, K ) = (11,5) (each with three hfs components), which end in two closely spaced rotational levels with equal quantum numbers (/, K ) that belong to two different vibronic states coupled by a mutual interaction [547]. 

In condensed phases, spectra are commonly measured in absorption. Three main types of transitions are observed in the absorption spectra of the actinide ions (1) Laparte-forbidden f to f transitions, (2) orbitally allowed 5/ to 6d transitions, and (3) metal to ligand charge transfer. Of these, study of internal f to f transitions has found wide use in the investigation of actinide chemistry. These band usually in the visible and ultraviolet regions, can be easily identified because of their sharpness, and are sensitive to the metal environment. As discussed earlier, the 5/ orbitals of the actinide elements are more exposed than the lanthanide 4/ orbitals, and therefore, crystal field effects are larger in the 5/ series. The f to f transitions for actinide elements may be up to 10 times more intense and twice as broad as those observed for the lanthanides, due to the action of crystal fields. In addition, extra lines resulting from vibronic states coupled to / / states have been observed. 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

The lowest adiabatic state is completely dissociative. The second and third are bound, but an efficient predissociation of the associated vibronic states can be predicted, on the basis of the strong couplings and small energy gaps in the region of the minima. 

Figure 10. Low-energy vibronic levels in the X2II state of HCCS computed in various approximations [152]. Hq zeroth-order approximation (both vibronic and spin-orbit couplings neglected). Hi. vibronic coupling taken into account, spin-orbit interaction neglected. Hi + Hs0 both vibronic and spin-orbit couplings taken into account. Solid horizontal lines K = 0 vibronic levels dashed line K — 1 dash-dotted lines K = 2 dotted lines K — 3. Values of the quantum numbers V4, N of the basis functions dominating the vibronic wave function of the level in question are indicated. Approximate correlation of vibronic states computed in various approximations is indicated by thin lines. In all cases the stretching quantum numbers are assumed to be zero.

The electronic spectra of these species have been extensively investigated in the past using both theory and experiment (64,65,80,81). The experimental spectra are generally poorly resolved and contain a high density of coupled vibronic states (34), which severely complicates the assignments (82). For example, there has been considerable debate about the nature of the initially populated state in Cr(C0)6 photodissociation (83-86). It was until recently believed to be of ligand field (LF) nature, but... 

In what follows we study the coupling matrix elements between vibronic states following the treatment of Lin,88 and of Bixon and Jortner.8 To begin, we display the coupling matrix elements in terms of intramolecular normal coordinates (see eq. (4-10)) ... 

Their system consists of a dilute crystal such that solute-solute interactions are negligible. Interactions between solvent and solute are important only insofar as the lattice vibrations are coupled with the zero-order nonstationary levels of the final state. The initial state is considered to be an equilibrated vibronic state, which implies v = 0 state for temperatures around 77°K. 

The C k are the Clebsch-Gordan (CG) coupling coefficients for the icosahedral system [28]. There is more than one quadratic term produced by the G and H modes. These quadratic terms can be derived from the CG coefficients. We add one of these terms to the Hamiltonian to show that the quadratic G0(g h) system can have a non-degenerate ground vibronic state with a realistic choice of coupling constants. 

The energies of the vibronic states have been calculated analytically by the shift transformation method developed initially by Bates et al. [24]. Consider now the quadratic coupling term as a perturbation on the linear system. The resultant energies have similar forms to those of the linear system namely ... 

Fig. 3. The tunneling splitting 8 (vertical axis) between the G and A states as a function of the linear coupling constants KG and KH for states associated with D3d wells. The two curves on the surface mark the region where the G vibronic state crosses the A vibronic state and the region where the D3d extrema are wells on the lowest APES, respectively. The region marked Overlap is where 8 has negative values and the ground state has A symmetry and the results are physically acceptable.

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