By F. Iachello, R. D. Levine
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Extra resources for Algebraic Theory of Molecules (Topics in Physical Chemistry Series)
Using Eq. 76), see also Eq. 123), one can account for deviations from linearity. 10 37 Transitions in one-dimensional problems In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by \N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here F x, Fy, Fz).
Count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 4 Basis states (representations) The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A.
An algebraic treatment of this problem thus requires the use of the algebra corresponding to the unitary group U(4) (lachello, 1981; lachello and Levine, 1982). , 4), operators. The 4x4=16 bilinear products b^b^ span the algebra of U(4) [Eq. 21)]. This form of the algebra, called the uncoupled form, is not well suited for the analysis of the problem since one wants states of good angular momentum. The corresponding operators must then have definite transformation properties under rotations. This is equivalent to saying that Cartesian coordinates are not particularly useful to solve the Schrodinger equation with a spherically symmetric potential and that one prefers to use spherical coordinates.