The results of a calculation of phonon spectra can be used to compute energy (E), entropy (S), free energy (F), and
lattice heat capacity (C_{v}) as functions of temperature. The CASTEP total energy yields the total electronic energy at 0 K. The
vibrational contributions to the thermodynamic properties are evaluated to compute E, S, F, and C_{v} at finite temperatures as
discussed below.

Thermodynamic calculations can be performed only if the system is in the ground state, that is, geometry optimization is fully converged. This means that all the phonon eigenfrequencies must be real and non-negative.

When you perform a vibrational analysis with CASTEP the results of the thermodynamic calculations can be visualized using the thermodynamic analysis tools.

The formulas below are based on work by Baroni et al. (2001).

The temperature dependence of the energy is given by:

where E_{zp} if the zero point vibrational energy, k is Boltzmann's constant, ħ is Planck's constant and
F(ω) is the phonon density of states. E_{zp} can be evaluated as:

The vibrational contribution to the free energy, F, is given by:

The vibrational contribution to the entropy, S, can be obtained using:

The lattice contribution to the heat capacity, C_{v}, is:

A popular representation of the experimental data on heat capacity is based on the comparison of the actual heat capacity to that predicted
by the Debye model. This leads to the concept of the temperature dependent Debye temperature, Θ_{D}(T). Heat capacity in
Debye model is given by (Ashcroft and Mermin, 1976):

where N is the number of atoms per cell. Thus, the value of the Debye temperature, Θ_{D}, at a given
temperature, T, is obtained by calculating the actual heat capacity, Eq. CASTEP 84, then inverting
Eq. CASTEP 85 to obtain Θ_{D}.

Accelrys Materials Studio 8.0 Help: Wednesday, December 17, 2014