Zero vibronic

As we see, the complexity of (4) is already reduced by symmetry rules, which allow us to identify the non zero vibronic coupling constants. Moreover, the application of the Wigner-Eckart theorem [33, 81] yields a further reduction of the complexity for degenerate irreducible representations. 

The synnnetry selection rules discussed above tell us whether a particular vibronic transition is allowed or forbidden, but they give no mfonnation about the intensity of allowed bands. That is detennined by equation (Bl.1.9) for absorption or (Bl.1.13) for emission. That usually means by the Franck-Condon principle if only the zero-order tenn in equation (B 1.1.7) is needed. So we take note of some general principles for Franck-Condon factors (FCFs). 

If the solution of the zero-order Schiodinger equation [i.e., all teiins in (17) except V(r,Ro) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling tenns. If F) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

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),...

Figure 5, Low-eriergy vibronic spectrum in a electronic state of a linear triatomic molecule. The parameter c determines the magnitude of splitting of adiabatic bending potential curves, is the spin-orbit coupling constant, which is assumed to be positive. The zero on the...

Figure 6. Bending potential curves for the X Ai, A B electronic system of BH2 [33,34], Full hotizontal lines K —Q vibronic levels dashed lines /f — I levels dash-dotted lines K — 2 levels dotted lines K — 3 levels. Vibronic levels of the lower electronic state are assigned in benf notation, those of the upper state in linear notation (see text). Zero on the energy scale corresponds to the energy of the lowest vibronic level.

The first term on the right-hand side is the same as in Equation (7.128). Herzberg and Teller suggested that the second term, in particular (dRg/dQj), may be non-zero for certain non-totally symmetric vibrations. As the intensity is proportional to Rgy this term is the source of intensity of such vibronic transitions. 

A further advantage is the higher efficiency of the alexandrite laser because of its being a four-level laser. In the illustration in Figure 9.2(c), level 4 is a vibronic level and level 3 the zero-point level of the T2 state. Level 2 is a vibronic level of the 2 state and level 1 the zero-point level. Because of the excited nature of level 2 it is almost depopulated at room... 

Nevertheless, 1,4-difluorobenzene has a rich two-photon fluorescence excitation spectrum, shown in Figure 9.29. The position of the forbidden Og (labelled 0-0) band is shown. All the vibronic transitions observed in the band system are induced by non-totally symmetric vibrations, rather like the one-photon case of benzene discussed in Section 7.3.4.2(b). The two-photon transition moment may become non-zero when certain vibrations are excited. 

Figure 9.33 Single vibronic level fluorescence spectra obtained by collision-lree emission Irom the zero-point level of the state of (a) pyrazine and (b) perdeuteropyrazine. (Reproduced, with permission, Ifom Udagawa, Y., Ito, M. and Suzuka, I., Chem. Phys., 46, 237, 1980)...

Figure 6.1. Jablonski-type diagram for pyrazine. The zero-field splittings (between tx, tV) t2) are not drawn to scale. Spin polarization ( x x x) resulting from the most probable intersystem crossing routes and part of the emission spectrum where different vibronic bands (v = /,/, k) have different zf origins are schematically indicated. (After El-Sayed.(17))...

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 term III scattering (equation 8) is the weakest in the three scattering mechanisms, as shown by two derivative terms (M ) in the electronic transition integrals. Clearly, for a dipole forbidden transition (M° = 0) the only non-zero term is term III. The term in scattering results in binary overtone and combination transitions of vibronically active modes. It is noted that no fundamental transition survives. 

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),... 

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

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