Question

1. (a) Describe in your own words and mathematically the harmonic oscillator approximation to molecular vibration. (b) What does this approximation lack in terms of representing real molecular bonds? (c) When is the approximation most valid and when are higher order approximations necessary?

Answer 1

(a) Actually a large number of systems are governed (at least approximately) by the ahrmonic oscillator equations. Whenevr we studies the behavior of a physical system in the neighborhood of a stable equilibrium position, one arrives at the equations which , in the limit of small oscillations, are those of a harmonic oscillator.For example, the vibrations of the atoms of a molecule about their equilibrium position can be approximated by a harmonic oscillator potential because atoms are modelled like attached to each other by spring like forces.

Cosider an arbitray potential V(x) whcich has a minimum at . Expanding the function V(x) in a taylor's series in the neighborhood of x_o , we obtaion:

  

The coefficients of this expansion are given by :

And the linear term (x-x_o) is zero since x_o corresponds to a minimum of V(x). The force derived from the potential V(x) is, in the neighborhood of x_o:

Since x=x_o represents a minimum, the coefficient b is positive.

The point x=x_o corresponds to a stable equilibrium position for the particle: F_x is zero for x=x_o. Moreover , for (x-x_o) suffuciently small, F_x and (x-x_o) have opposite signs since b is positive.

If the amplitude of the motion of the particle about x_o is sufficiently small for the term (x-x_o)³ to be negligible compared to the preceding ones, we have the harmonic oscillator since the dynamical equation can be approximated by:

  

The corresponding angular frequaency is given by

So we can approximate the molecular vibrations to harmonic potential.

(b)We negelcted the higher rder terms in this expression which is not actually the case, so some deviations maybe reported from experimental data.

(c) When the potential representing the molecule is shallow enough that we can neglect higher order derivatives, the approximation is valid. But if the potential is not shallow or dont even have a global minimum, we could lend up in trouble. In this case we use perturbation theory to get closer to exact solution of the problem.