Potential energy, anharmonic terms symmetry

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

Secondly, and most seriously, the validity even of the harmonic frequencies of Table 1 may be questioned 45). The observed binary and ternary bonds are all of symmetry class T(in thehexacarbonyls) or 41 or (in the case of Mn(CO)5Br), and these symmetry classes are repeated several times both in the fundamental and in the ternary region. Thus we have satisfied the conditions for Fermi resonance. Of course, to show that Fermi resonance is symmetry-allowed is not the same as showing that it occurs, but there is every reason to suspect it in the present case. The physical origin of anharmonicity lies in the existence of direct or crossed cubic and quartic terms in the potential energy expression ). 

This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

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