Many-Body Algebraic Theory

For molecules with many atoms, the simultaneous treatment of rotations and vibrations in terms of vector coordinates r1, r2, r3, . . . , quantized through the algebra . . .G ≡ U1(4) ⊗ U2(4) ⊗ U3(4). . . . . . .(6.1). . . becomes very cumbersome. Each time a U(4) algebra is added one must go through the recoupling procedure using Racah algebra, which, although feasible, is in practice very time consuming. An alternative treatment, which can be carried out for molecules with any number of atoms, is that of separating vibrations and rotations as already discussed in Sections 4.2-4.5 for triatomic molecules. For nonlinear molecules, there are three rotational degrees of freedom, described by the Euler angles α, β, γ of Figure 3.1, and thus there remain 3n − 6 independent vibrational degrees of freedom, where n is the number of atoms in the molecule. For linear molecules, there are two rotational degrees of freedom, described by the angles α, β, and thus there remain 3n − 5 independent vibrational degrees of freedom, some of which (the bending vibrations) are doubly degenerate. In this alternative treatment, the algebraic theory of polyatomic molecules consists in the separate quantization of rotations and vibrations. Each bond coordinate is then a scalar, and the corresponding algebra is that of U(2). In polyatomic molecules, the geometric symmetry of the molecule also plays a very important role. For example, the benzene molecule, which is the example we discuss in this book has the point group symmetry D6h. A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates.