Spectroscopy of polyatomic molecules

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

In electronic spectroscopy of polyatomic molecules the system used for labelling vibronic transitions employs N, to indicate a transition in which vibration N is excited with v" quanta in the lower state and v quanta in the upper state. The pure electronic transition is labelled Og. The system is very similar to the rather less often used system for pure vibrational transitions described in Section 6.2.3.1. 

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

The second problem relates to the inclusion, or otherwise, of molecular symmetry arguments. There is no avoiding the fact that an understanding of molecular symmetry presents a hurdle (although I think it is a low one) which must be surmounted if selection rules in vibrational and electronic spectroscopy of polyatomic molecules are to be understood. This book surmounts the hurdle in Chapter 4, which is devoted to molecular symmetry but which treats the subject in a non-mathematical way. For those lecturers and students who wish to leave out this chapter much of the subsequent material can be understood but, in some areas, in a less satisfying way. 

The algebraic approach begins with the notion of a zeroth-order description based on a dynamical symmetry, a concept which is a generalization of the usual definition of the symmetry group of the Hamiltonian. What a dynamical symmetry means in practice is that one constructs a zeroth-order Hamiltonian for which there is a full set of quantum numbers for labeling the eigenstates and that the energy is an analytical function of these quantum numbers. In the infrared or Raman spectroscopy of polyatomic molecules (1) one knows that to zeroth order it is practical to represent the spectrum by a Dunham-type formula... 

The symmetry of nuclear displacements arises most commonly in connection with vibrational spectroscopy of polyatomic molecules [7]. Let us compare the nuclear displacements of a symmetric linear triatomic, XYX, illustrated in Fig. 4.2, with those shown in Fig. 3.9 for a homonuclear diatomic molecule, which also has cylindrical symmetry. It was pointed out that in the latter case there is no way of reducing the symmetry of the potential energy of X2 below Doo/i by nuclear motion in the case of the triatomic molecule, there is. 

---