Discrepancy Between DFT Ground-State Energies and ALAMODE Free Energy Calculations

[Zhennan-Lin]

Dear Dr. Tadano and Esteemed Colleagues

My first-principles DFT calculations clearly show that the low-temperature phase has lower energy (serving as the ground state at 0 K), which aligns with theoretical expectations. Using both harmonic and SCPH+harmonic approaches to describe the phonon spectrum for the low-temperature phase and the high-temperature phase, respectively. However, when computing thermodynamic properties within the quasi-harmonic approximation (QHA) framework - the high-temperature phase paradoxically exhibits lower free energy (F = U - TS) than the low-temperature phase even at 0 K.This free energy reversal contradicts both the DFT results and fundamental thermodynamic principles, I do not know how to find the potential problems; please give some suggestions.
Best regards,
Zhennan-Lin
East China Normal University
500 Dongchuan Road, Minhang District
Shanghai 200241, China

[ttadano]

Please describe how you computed the free energy more specifically.

[Zhennan-Lin]

Thank you for your response. After reviewing the paper (PHYSICAL REVIEW MATERIALS 3, 033601 (2019)), I noted that the Helmholtz free energy is computed as the sum of the static internal energy at 0 K, Eel(V) is obtained from DFT calculations, and the lattice vibrational contribution Fvib. The latter is determined using either the harmonic approximation or the more advanced self-consistent phonon (SCPH) method to account for anharmonic effects. Please correct me if my understanding is inaccurate.
Additionally, I would like to ask: For metallic systems, how the Helmholtz free energy ​ be properly evaluated?

[ttadano]

$F = U_{0} - TS_{el} + U_{ph} - TS_{ph} = U_{0} -TS_{el}+ F_{ph}$ where $U_0$ is the total energy of the system (e.g., total energy without entropy in VASP) and $S_{el}$ is entropy associated with the thermal broadening of electrons. $S_{el}$ can be computed numerically using the density of states, if an accurate determination is required.

Do you mean $U_{0}(A) < U_{0}(B)$ was obtained for systems A and B (high-T phase) but the free energy was $F(A) > F(B)$ at zero kelvin? If yes, that occurred probably because the zero-point energy was $U_{ph}(A) > U_{ph}(B)$.

It is important to use the same level of theory when comparing $F_{ph}$ of different systems. If you use SCPH for material A, you should also use SCPH for material B.

[Zhennan-Lin]

Thank you for your excellent response to my previous question.

[Zhennan-Lin]

Sorry, I noticed that the .thermo results do not include the Sph ( phonons entropy) values calculated by the SCPH , how can I get it ?

[ttadano]

We do not implement an option to print out $S_{ph}$ separately for an SCPH calculation. If you need it, you can compute it from the density of states by yourself using the expression for independent Boson.

[Zhennan-Lin]

Thanks for your answer and advice, I will try it.