The vibrational equation

The point of changing from Cartesian displacement coordinates to normal coordinates is that it brings about a great simplification of the vibrational equation. Furthermore, we will see that the normal coordinates provide a basis for a representation of the point group to which molecule belongs. 

Eqn (9-3.1) can then be separated into three eigenvalue equations for the three types of motion. The three eigenvalues will be Wu (translational energy , W 1 (rotational energy) and Wyih (vibrational energy) 

Of the three eigenvalue equations, the one of interest to us is the vibrational equation. It has a particularly simple form when normal coordinates are employed because the classical kinetic and potential energies then have no cross terms (see eqns (9-2.17) and (9-2.18)) and this fact leads to a simple form for their quantum mechanical analogues (the kinetic energy and potential energy operators). The vibrational equation is thus 

Each term of this sum is a function of just one normal coordinate and is 

Eqn (9-3.3) is the same as the well known one-dimensional harmonic oscillator equation and has as its solutions 

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The vibration problem is strictly analogous to the buckling problem. Accordingly, because of similar complexity, the derivation of the vibration equations is not attempted. The vibration boundary conditions are identical to the buckling boundary conditions. 

The general dispersion formula obtained for the coupling of the vibrational equations with the Maxwell field can be brought into the form of Fresnel s wellknown equation for the wave normal from crystal optics. It is usually written in the form... 

Normal coordinates The vibrational equation The r (or representation The reduction of T ... 

The vibration equations of the self-anchored suspension bridges can be obtained from following operation ... 

Sinee the last term is a eonstant, this immediately gives the separation of the rotations from the vibrational equation (6.24)... 

Based on the vibration equation of the piezoelectric vibrator, the elecnomechanical coupling factor, k, can be calculated ... 

The equation shows that the function Ki(0) terms of the second order, indeed plays the role of the potential energy of the nuclei. For the existence of a stable molecule we now obtain the further requirement that the extreme value of Vn(0 determined through (40) be a minimum since the quadratic form must be positive definite, so that all degrees of freedom are stable and vibrations around the equilibrium position axe possible. As is known, the vibrational equation (46) is separable into normal coordinates rn by a linear transformation. If (T siO the... 

The left hand side ag iin corresponds, because of (66), to the vibrational equation (45). The right hand side therefore must be normal to o ,. Substituting the values of Xn) and Xn from (47b) and (57) and using the symbol... 

In general terms, since an exact solution of the vibrational equation in terms of anharmonic wave function is not possible, use is made of the fact at for finite displacements at each step of the potential energy or transitional dipole moment expansion, the higher terms are much smaller than the respective lower terms. A perturbation theoiy treatment becomes, therefore, feasible. The potential energy may be expressed in the form [3]... 

In molecular crystals or in crystals composed of complex ions it is necessary to take into account intramolecular vibrations in addition to the vibrations of the molecules with respect to each other. If both modes are approximately independent, the former can be treated using the Einstein model. In the case of covalent molecules specifically, it is necessary to pay attention to internal rotations. The behaviour is especially complicated in the case of the compounds discussed in Section 2.2.6. The pure lattice vibrations are also more complex than has been described so far . In addition to (transverse and longitudinal) acoustical phonons, i.e. vibrations by which the constituents are moved coherently in the same direction without charge separation, there are so-called optical phonons. The name is based on the fact that the latter lattice vibrations are — in polar compounds — now associated with a change in the dipole moment and, hence, with optical effects. The inset to Fig. 3.1 illustrates a real phonon spectrum for a very simple ionic crystal. A detailed treatment of the lattice dynamics lies outside the scope of this book. The formal treatment of phonons (cf. e(k), D(e)) is very similar to that of crystal electrons. (Observe the similarity of the vibration equation to the Schrodinger equation.) However, they obey Bose rather than Fermi statistics (cf. page 119). 

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