Algebraic Theory of Molecules (Topics in Physical Chemistry by F. Iachello, R. D. Levine

By F. Iachello, R. D. Levine

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Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term EQ. 31) with eigenvalues The spectrum of Eq. 1. This is the spectrum of the one-dimensional truncated harmonic oscillator with a maximum vibrational quantum number equal to N. Thus N + I represents the number of bound states. When N —> oo one recovers the full oscillator spectrum.

46 Chapter 2 These equations provide an explicit relationship between the parameters appearing in the Morse potential, re, p\ V0, and the reduced mass (i and the algebraic parameters E'0, A, B, and N. 8. An alternative form of Eq. 1, The two chains (I) and (II) span the entire set of analytically solvable problems in three dimensions. Any solution for a generic potential can be expanded in terms of either of the two, since both provide an orthonormal set in three dimensions. An interesting quantity that characterizes the properties of the two solutions is the nonrigidity parameter R introduced by Berry (1980) where Erot and Evib are the energies of the first rotational and vibrational energy level.

40 Chapter 2 The spherical components of n^ are related to the Cartesian components by Sometimes the operators o1',^ are denoted by With these definitions the creation operators o f , TtJ transform as spherical tensors under rotation. The annihilation operators do not. However, it is easy to construct operators that do transform as spherical tensors [Eq. 23)]. 1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. 25). Note that each tensor operator of multipolarity A, has 2X, + 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form.

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