Symmetry, Invariants, And Topology In Molecular Models by ZHILINSKII B. I.

By ZHILINSKII B. I.

Physics reviews, quantity 341, no 1, February 2001 , pp. 85-171(87)

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It can be always written in the form  Q b Q\ Q\ BG hS (u, N mod+lcm(d ),) . (67) N? G S ? In fact for each particular value of in Eq. (67) the real period can be shorter than the lcm(d ) but it G is always a divisor of it. 4. Two polyad quantum numbers. , 1999). Moreover, this example shows the natural way of the generalization of the generating function approach to more complicated examples with several polyad quantum numbers. We remind that C H is a linear molecule with seven vibrational degrees of freedom and the   D point group symmetry of the equilibrium con"guration in the ground electronic state.

31) Equivalently Eq. S ¸ ) . (32) > \ \ > X X This is a particular case of the model operator studied by Pavlov-Verevkin et al. (1988) in order to demonstrate the dynamical meaning of the formation of the conical intersection points. Let us remark that the classical Hamiltonian symbol (31) corresponds to two rotational energy surfaces E "$"L""sin ". I. Zhilinskin& / Physics Reports 341 (2001) 85}171 115 An exact quantum solution of Eq. (32) or Eq. (30) may be easily found by noting that the projection of the formally constructed total angular momentum J "¸ #S is an integral of X X X motion.

We are interested in molecular applications and therefore, we only analyze model vibrational Hamiltonians which can be initially approximated by a harmonic oscillator (small vibrations near the equilibrium). Resonances between the vibrational modes may be approximate or exact (due to symmetry). Vibrational structure of molecules provides a great number of examples of both kinds. Table 15 summarizes molecular examples with typical and quite interesting resonance conditions. In each case we consider K vibrational modes with frequencies , i"1,2, K, and suppose a resonance G condition : : 2 : +n : n : 2 : n .

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