# Difference between potential energy, free energy and Coulomb energy in solid state physics

I often encounter terms such as (Helmholtz, Gibbs) free energy, potential energy and total energy when describing the energy of a physical system at atomic level. Sometimes I stumble upon Coulomb energy, which adds more to the confusion. My (vague) understanding is that potential energy of a cell should be equal to Coulomb energy of that cell.

Could someone explain the differences/similarities between these terms, including what contributes to the "total energy" found in a DFT (VASP) calculation?

I can't answer in the context of DFT/VASP, where these terms might have specific uses, but I can offer some general comments.

These terms are not necessarily mutually exclusive:

• Total energy probably refers to the combined kinetic and interaction energy $$E=K+P...$$ resulting from summing/integrating over the whole Hamiltonian. In DFT this would be the combination of all functionals.
• [Helmholtz/Gibbs] Free Energy refers to the thermodynamic energy available to do work (at constant T). Basically, the total internal energy of the system subtracting out the energy that is 'trapped in entropy.'
• Potential energy (in this context) most likely refers to the particle-particle interactions. This could include the energy from the pseudopotentials and the electron-electron (Coulomb) interactions.
• Coulomb energy: in the most general sense, you are correct: the only source of potential energy is the Coulomb force. However, it's possible that this term is referring to a specific contribution such as just the potential energy from the psuedopotentials. (Someone better-versed in DFT can provide a more intelligent answer here). In the specific context of the reference you cited, the Coulomb energy appears to be calculated from the Coulomb interactions of the nuclei (in the lattice) with the electronic charge uniformly smeared out between them.
• +1. I thing most of these terms mean the same thing whether using DFT/VASP or not. For "total energy", often in ab-initio codes the "total energy" really means "total electronic energy" after assuming the nuclei are clamped in a Born-Oppenheimer type spirit, but VASP is not just electronic structure, so let's allow a VASP user to comment further on that part! Jul 10 '20 at 3:03
• Thanks taciteloquence and @NikeDattani for your input. It does clarify these terms. I will wait for someone to comment in terms of DFT/VASP. Jul 10 '20 at 9:56
• The potential energy is different from the Coulomb energy. The first one deal only with the geometrical/spacial distribution of particles. The second one deal only with charges. On the other hand, the Helmholtz (A) & Gibbs (G) energy are the so called chemical potentials. Being A = E-TS and G = H -TS where E: internal energy, T: temperature, S: entropy and H: enthalpy (= E + PV, P: pressure and V: volume).
– Camps
Jul 10 '20 at 11:27
• @Camps thanks for the clarification. In that case I have a question where, in order for a material to be potentially stable, should it have low Coulomb energy or high Coulomb energy? In terms of potential energy, we know that a material sitting at a local minimum of potential energy surface (PEC) could be stable. Could a similar analogy be drawn for Coulomb energy as well? Jul 10 '20 at 12:35
• In general, a negative (lower) value of energy indicates more stability. In the case of Coulomb energy, as it is defined as the energy due to the interaction of a positive point charge with the system charges, the negative value indicate an attractive interaction whereas a positive value indicate repulsive interaction.
– Camps
Jul 10 '20 at 16:39