Symmetric resonance states

Figure 9. Transition probability amplitude, to- o( ) assisted by the 3-rd effective complex resonance potential energy surface, (a) phase, (b) absolute value, (c) trajectory. The phase changes by tt in resonances. The phase drops by tt between resonances, when t E) intersects the origin. The position of resonance states is plotted in squares (open squares -symmetric resonance states, full squares - antisymmetric resonance states). Note that due to the symmetry properties of the problem, transition is not assisted by antisymmetric resonances.

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. 

As shown, ia the case of chlotination of aEyl chloride, the resonance states of the chloroaEyl radical iatermediates are not symmetrical and their propagation reactions lead to the two different dichloropropene isomers ia an approximate 10 90 ratio (26). In addition, similar reactions result ia further substitution and addition with products such as trichloropropanes, trichloropropenes, tetrachloropropanes, etc ia diminisbing amounts. Propylene dimerization products such as 1,5-hexadiene, benzene, 1-chloropropane, 2-chloropropane, high boiling tars, and coke are also produced ia smaE amounts. 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

Even if the equations arrived at seem familiar there are some obvious fundamental differences. First of all, the ansatz singles out two mirror spaces, where the particle and its mirror image may be located, respectively. Second, the expanded complex symmetric representation takes into account complex resonance states. Note also, as said above, that the complex symmetry can be obtained from the hermitean representation via a non-positive definite metric, i.e. 

The method of perturbed stationary state (PSS) was applied to the heavy particle collision of symmetrical resonance by Buckingham and Dalgarno, Matsuzawa and Nakamura, and Kolker and Michels. Although the essential idea of our method is the same as in these papers, except for the inclusion of four levels, we include a brief description of the PSS method for later discussion. 

The basic idea of the PSS method is to start from the molecular wave-function and to take the nonadiabaticity as a perturbation. However, in the case of symmetrical resonance the energy splitting of a and corresponding u state gives the transition probability even when the nonadiabatic interaction is negligibly small. The Schrodinger equation describing the whole system composed of two atoms is... 

M. V. Basilevsky and V. M. Ryaboy, Direct calculation of resonant states in reactive scattering—application to linear triatomic systems. Int. J. Quantum Chem. 19 611 (1981) M. V. Basilevsky and V. M. Ryaboy, Decay dynamics of triatomic molecules. Quantum calculations for non-symmetric linear systems. Chem. Phys. 86 61 (1984). 

It is reasonable to deduce that the observed UV absorption at 205 nm (about 49000 cm ) involves a totally symmetric excited state in which all four Ti-0 bonds are symmetrically stretched with respect to the ground state (due to the Ti-O antibonding character of the LUMO). Thus the symmetric Ti04 stretching meets both selection rules a) and b) (see the introduction) for the resonance enhancement of its Raman intensity. We therefore conclude that the observed band at 1125 cm (calculated at 1100 cm in the TiSi4 cluster) can be attributed to this kind of vibration. Based on the discussion above, this vibration can be also described as the in-phase combination of the four Ti-O-Si stretching modes, or as the in-phase combination of the four Si-O " bonds perturbed by Ti. 

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