The Anharmonic Diatomic Molecule

The potential energy of a diatomic molecule is illustrated in Fig.5.1. A mathematical expression that actually represents a potential curve of this form and which is often used for molecules has been proposed by MORSE [5.1], It is 

For very small amplitudes of vibration (T = 0), the mean separation between the atoms is aQ for the classical case. For large amplitudes (high temperatures), however, the mean separation will be a aQ due to the deviations of cp(r) from the parabolic curve. This is the qualitative explanation of thermal expansion. Let 

We note that gQ is, in general, a negative constant. Neglecting the quartic enharmonic term in (5.3), the equation of motion can be written in the form 

7) we have included the first three terms of the Fourier expansion of the motion. If we retained the term proportional to Wq in (5.4), the expansion (5.7) would also contain a term proportional to cos3o)t. Physically it is clear that the Fourier decomposition of the motion of a sphere in a potential as illustrated in Fig.5.1 contains not only the frequency w (harmonic term), but also the overtones 2o), 3o), etc. Substituting (5.7) in (5.4), 

Here we have used (5.7) and the fact that cosu)t = cos2o)t = 0. Equation (5.11) means that the mean distance a is larger than ag. The force constant f (curvature of (p(r) at r = a) is smaller than the force constant fg (curvature of cp(r) at r = ag) this gives rise to a decrease of the frequency as expressed by (5.9). 

---

As an introduction, the chapter begins with the anharmonic diatomic molecule. Then we study the thermal properties (free energy, equation of state, thermal expansion and specific heat) of the classical anharnx)nic linear chain. Two important concepts are introduced the Gvuneisen pavametev and the quasiharmonic approximation. In this approximation, the temperature dependence of the force constants and phonon frequencies is only due to the... 

---