Vibronic levels

For molecules we can use Bom-Oppenlieimer wavefimctions and talk about emission from one vibronic level to another. Equation (B1.1.5T equation tb 1.1.6) and equation tb 1.1.7) can be used just as they were for absorption. If we have an emission from vibronic state iih to the lower state a, the rate constant for emission would be given by... 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

Introduction of the vibronic coupling (s / 0) causes removal of the above degeneracy and leads to the general vibronic-spin-orbit pattern presented in the central part of Figure 3. Each vibronic level is characterized by a particular K... 

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 4. Spin-orbit splitting in AT — 1 and 2 vibronic levels of the state of NCN. Solid lines connect the results of calculations thar employ ab initio computed potential curves [28], For comparison the results obtained by employing experimentally derived potential curves (dashed lines) [30,31] are also given. Full points represent energy differences between P — K — and P — K spin levels, and crosses are differences between P — K + I and P — K levels.

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 excellent agreement of the results of HCR ab initio studies with the corresponding experimental findings clearly shows that the strongest influence on the numerical accuracy of the vibronic levels have effects outside of the R-T effect, that is, primarly the replacement of the effective bending approaches employed in previous works by a full 3D treatment of the vibrational motions (for an analysis of this matter see, e.g., [17]). Let us note, however, that such a... 

It follows that the only possible values for la + Ip are S A and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number I = la + Ip. The off-diagonal elements of [second and third of Eqs. (60)] connect the basis functions with I — la + Ip and I — l + l — l 2A. 

In other cases, the zeroth-order vibronic levels are generally more than twofold degenerate and the perturbative handling is much more complicated. An exception is the case 07= , U( =li K = 0 with the twofold degenerate zeroth-order level. The basis functions are 1 1 1 1 —) = 1) and 1 —1 1 —l- -)s 2). The zeroth-order energy is... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

If emission is from only one vibrational level of the upper electronic state it is referred to as single vibronic level fluorescence (or phosphorescence). 

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... 

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)...

Single vibronic level, or dispersed, fluorescence spectroscopy... 

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