# How can we calculate the Gibbs free energy from first principles?

I am trying to calculate the Gibbs free energy for a bulk material. Since DFT ground state calculations give the energy at 0K, a temperature dependent part has to be added in this to obtain correct result.

Can someone refer me to the correct procedure to calculate this quantity, for example using VASP?

To extend your 0K energies to finite temperature, you essentially need to add up the different sources of entropy. Some examples as reviewed by Hickel et al. :

• Vibrational entropy (usually split into harmonic and anharmonic contributions)
• Electronic entropy
• Magnetic entropy
• Configurational entropy

Each of these contributions has some underlying theory or competing theories. For example, electronic entropy is quite straightfoward and is effectively a post-processing step on the electronic density of states from a typical DFT calculation. Harmonic vibrational contributions could be approximated by one of the flavors of the Debye model or treated explictly via phonons. Magnetic entropy is relatively difficult to compute and there are multiple approaches with different performance/accuracy tradeoffs.

Most of the formulations involve computing the individual contributions as a Helmoltz energy on a grid of volumes: $$F(T,V) = E_0(V) + F_\textrm{vib}(T,V) + F_\textrm{el}(T,V) + F_\textrm{mag}(T,V) + \dots \tag{1}$$ and computing the Gibbs energy by a Legendre transform $$G = F + PV \tag{2}$$

The exact procedures depend on what physics is energetically relevant for your application and how much compute you can afford to capture that physics.

 T. Hickel, B. Grabowski, F. Körmann, J. Neugebauer, Advancing density functional theory to finite temperatures: Methods and applications in steel design, J. Phys. Condens. Matter. 24 (2012). doi:10.1088/0953-8984/24/5/053202.