Symmetry dynamic

Contents Lattice Dynamics. - Symmetry. - Inter-molecular Potentials. - Anharmonic Interactions. - Two-Phonon Spectra of Molecular Crystals. -Infrared and Raman Intensities in Molecular Crystals. 

Dynamical symmetries for one-dimensional problems can be studied by considering all the possible subalgebras of U(2). There are two cases... 

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 E(). The algebra U(l) has a linear invariant... 

Thus, to lowest order, a Hamiltonian with this dynamic symmetry is Ha) = E0 + enx. 

Since nx is an invariant, so is n. One can thus write down the most general bilinear algebraic Hamiltonian with dynamic symmetry U(l) as... 

Dynamic symmetries for chain (II) correspond to an expansion of the Hamiltonian in terms of invariant operators of 0(2). The linear invariant is... 

The most general Hamiltonian with dynamic symmetry (II) again has a form similar to Eq. (2.43), with both linear and quadratic terms. This is a peculiar feature of one-dimensional problems. In order to simplify the discussion of three-dimensional problems, we prefer to consider a Hamiltonian with only quadratic terms... 

A general potential V(r) corresponds to a generic algebraic Hamiltonian (2.29). In the most general case the solution cannot be obtained in explicit form but requires the diagonalization of a matrix. The matrix is (N + 1) dimensional. An alternative approach, useful in the case in which the potential does not deviate too much from a case with dynamical symmetry, is to expand it in terms of the limiting potential. For the Morse potential, this implies an expansion of the type (1.7)... 

Dynamical symmetries for three-dimensional problems can be studied by the usual method of considering all the possible subalgebras of U(4). In the present case, since one wants states to have good angular momentum quantum numbers, one must always include the rotation algebra, 0(3), as a subalgebra. One can show then that there are only two possibilities, corresponding to the chains... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

The more familiar rigid rovibrator has the dynamical symmetry associated with chain (II). The Hamiltonian up to quadratic terms is... 

Figure 2.6 The potential V(r) that corresponds to the dynamical symmetry (I). The potential is nonrigid because [cf. Eq. (2.113)] the rotational spacings are comparable to the vibrational ones. Tn the harmonic limit V(r) is the potential of an isotropic harmonic oscillator.

Figure 2.8 The potential V(r) that corresponds to the dynamic symmetry (II).

The Morse potential results derived on the basis of the dynamic symmetry derived for chain (II) of U(4) provide a reasonably good description of spectra... 

The point made in Eq. (3.31), namely, that the coupled, old, n action variables can be transformed to new, uncoupled, n - 1 conserved action variables is one to which we shall repeatedly return, in the quantum-algebraic context, in Chapters 4—6. Of course, we shall first discuss H0, which has n good quantum numbers, and which we shall call a Hamiltonian with a dynamical symmetry. At the next order of refinement we shall introduce coupling terms that will break the full symmetry but that will still retain some symmetry so that new, good, but fewer quantum numbers can still be exactly defined. In particular, we shall see that this can be done in a very systematic and sequential fashion, thereby establishing a hierarchy of sets of good quantum numbers, each successive set having fewer members. 

Consider first chain (I). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of the chain,... 

We have called the vibrational quantum numbers here Vj, v2, v3 in order to distinguish them from the local quantum numbers, va, vl , vc. Note that, in view of the presence of the missing label, %, the normal basis is not very convenient for calculations. The spectrum corresponding to Eq. (4.59) is shown in Figure 4.8. There are fewer examples of molecules for which the dynamical symmetry of the normal chain II, provides a realistic zeroth-order approximation. The normal behavior arises when the masses of the three atoms are comparable, as, for example in XY2 molecules with mx = mY. More examples are discussed in the following sections. 

An important problem of molecular spectroscopy is the assignment of quantum numbers. Quantum numbers are related to conserved quantities, and a full set of such numbers is possible only in the case of dynamical symmetries. For the problem at hand this means that three vibrational quantum numbers can be strictly assigned only for local molecules (f = 0) and normal molecules ( , = 1). Most molecules have locality parameters that are in between. Near the two limits the use of local and normal quantum numbers is still meaningful. The most difficult molecules to describe are those in the intermediate regime. For these molecules the only conserved vibrational quantum number is the multiplet number n of Eq. (4.71). A possible notation is thus that in which the quantum number n and the order of the level within each multiplet are given. Thus levels of a linear triatomic molecules can be characterized by... 

As mentioned already in Chapter 2, the algebras U(l) and 0(2) are isomorphic (and Abelian). A consequence of this statement is that in one-dimension there is a large number of potentials that correspond exactly to an algebraic structure with a dynamical symmetry. Of particular interest in molecular physics are ... 

Situations in which the Hamiltonian does not have a dynamic symmetry (i.e., it contains Casimir operators of both chains), as, for example,... 

In the previous sections the correspondence between the Schrodinger picture and the algebraic picture was briefly reviewed for some special cases (dynamical symmetries). In general the situation is much more complex, and one needs more elaborate methods to construct the potential functions. These methods are particularly important in the case of coupled problems. This leads to the general question of what is the geometric interpretation of algebraic models. 

According to Eq. (2.70), one has in this case two dynamic symmetry chains... 

Cooper, I. L., and Levine, R. D. (1991), Computed Overtone Spectra of Linear Triatomic Molecules by Dynamical Symmetry, /. Mol. Spectr. 148, 391. 

Kellman, M. E. (1983), Dynamical Symmetries in a Unitary Algebraic Model of Coupled Local Modes of Benzene, Chem. Phys. Lett. 103,40. 

In this model, idealized skeletons are assumed, deformations of the skeleton by ligands being neglected. The ligands have a static or dynamic symmetry about their skeletal bond axes which is compatible with the skeletal symmetry. Here it should be noted that the conceptual dissection of molecules into skeleton and ligands has been a standard procedure developed by stereochemists quite some time ago. 

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