Coxeter's Quaternions Reflections PDF

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Zhilinskin& / Physics Reports 341 (2001) 85}171 135 Fig. 19. erent symmetry types as a function of the total polyad quantum number N . Numbers of states are given for four one-dimensional representations of D group. k I> , k"1, 2,2 , (1! )(1! )(1! )(1! )(1! k I>) , k"1, 2,2, g (2k#1, , )"  (1! )(1! )(1! )(1! )(1! ) "u, g . (79) (80) Fig. erent symmetry types as a function of N . erent symmetry types scale in fact by a constant factor in the high energy region (Quack, 1977). That is why the most important is the number of states of the (k"0, g,#) symmetry within the polyads.

State. erent one-dimensional B representations and as was initially introduced by Zhilinskii and Brodersen (1994) any vibrational component can be labeled by D (> ? with "g, u due to equivalence A ;D0,D0.  E S For tetrahedral ¹ molecules the list of possible decompositions of a rotational structure of B degenerate vibrational states into isolated vibrational components was given by Zhilinskii and Brodersen (1994). The general solution for the E vibrational state is particularly simple. There is an in"nite number of solutions E;D("D(>B#D(\B E E S (44) with "$(6k#2),$(6k#4), k"0, 1, 2,2 .

The multiplet structure. . Within each multiplet, levels with the same J are distinguished by J so that the  X  "rst order splitting is a linear function of J (Landau and Lifshitz, 1965). X The transformation of the eigenfunctions of the Hamiltonian in Eq. (45) from the limit "0 to the limit "1 is a well-known transformation from the uncoupled to a coupled basis for two angular momenta, "NN SS 2P"NSJJ 2. erent. Consequently, a number of levels is redistributed among the multiplets at intermediate values of the control parameter .

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Quaternions Reflections by Coxeter


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