Which Energy Value to Use for Binding Energy Calculations in VASP?

[hszhao.cn@gmail.com]

Dear VASP Forum,

I'm calculating binding energies for my system and I'm uncertain which energy value from the OUTCAR file should be used. In the output, I see these lines:

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free energy    TOTEN  =       -24.22450672 eV

energy without entropy =      -24.22361411  energy(sigma->0) =      -24.22428357

I've heard that when calculating binding energies, one should use the "energy without entropy" value rather than the TOTEN or energy(sigma->0) values. Is this correct? Should I use -24.22361411 eV for binding energy calculations, or should I use the free energy (TOTEN) value of -24.22450672 eV, or perhaps the energy(sigma->0) value of -24.22428357 eV?

Could you please explain the differences between these energy values and which one is most appropriate for binding energy calculations?

Thank you for your help.

Regards,
Zhao

[jonathan_lahnsteiner2]

Dear hszhao,

The different energy values in VASP can be interpreted as follows:
Total Free Energy (TOTEN):
Includes the total electronic energy plus the entropy contribution from the finite electronic temperature (set by the SIGMA parameter).
Energy(sigma->0):
This is an extrapolated value of TOTEN to the limit where sigma = 0. So an extrapolation to zero temperature for the electrons.
Energy without Entropy:
Is the total free energy without the entropy term due to electrons.So the difference between TOTEN and Energy without Entropy gives the entropy contribution due to the electrons. You can verify this in your OUTCAR file and compare to "entropy T*S EENTRO".

In your case I would recommend to use TOTEN for the computation of the binding energies because it will involve not only potential and kinetic energies but also the entropy contributions due to the electrons.

All the best Jonathan

[hszhao.cn@gmail.com]

Dear Jonathan,

So the difference between TOTEN and Energy without Entropy gives the entropy contribution due to the electrons. You can verify this in your OUTCAR file and compare to "entropy T*S EENTRO".

Yes, you are right, as shown below:

In my example:

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Free energy TOTEN = -24.22450672 eV
Energy without entropy = -24.22361411 eV
The difference is: -24.22450672 - (-24.22361411) = -0.00089261 eV

This can be verified by the corresponding result in my OUTCAR, as shown below:

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(datasci) werner@x13dai-t:~/Desktop/std_vasp_slurm/b1/fe/cd/b1fecd84-2a91-41fa-b7e0-f63215ca6329_1$ ug EENTRO | tail -1
OUTCAR:  entropy T*S    EENTRO =        -0.00089262

In your case I would recommend to use TOTEN for the computation of the binding energies because it will involve not only potential and kinetic energies but also the entropy contributions due to the electrons.

1. Does TOTEN in VASP correspond to Helmholtz free energy?
2. Do you also mean that, for the computations such as cohesive energy, formation energy, and formation enthalpy, TOTEN still should be used?
3. By the way, I've also been investigating which energy value is most appropriate for an equation of state (EOS) fitting, and I noticed that the Equation of State Workflow in atomate2 utilizes the "energy(sigma->0)" value for E-V fitting. Could you please explain why "energy(sigma->0)" might be preferred over "energy without entropy" or TOTEN when performing equation of state calculations? Is there a theoretical justification that makes this extrapolated value more suitable for obtaining accurate equilibrium volume, bulk modulus, and other EOS parameters?

Thank you for your insights on this matter.

Regards,
Zhao

[hszhao.cn@gmail.com]

Dear Jonathan.

After further reflection, I would like to add the following to my current understanding. Based on your explanation, I want to confirm my understanding and share my derivation as described below:

1. Regarding TOTEN: If I understand correctly, TOTEN corresponds to the electronic Helmholtz free energy, aka, F_elec = U_elec - T_elec*S_elec, calculated by VASP for the electronic subsystem. Is this interpretation correct?

2. By using the following derivation process, we can get the Gibbs free energy at a given 𝑇 and 𝑝: G(T,p) = E₀(V) + F_vib(T,V) + pV.

Starting from the standard thermodynamic definition:
G = U + pV - TS

We can separate the electronic and the thermally excited lattice dynamics, aka, phonon, contributions:
U = U_elec + U_vib
S = S_elec + S_vib

Substituting:
G = U_elec + U_vib + pV - T(S_elec + S_vib)
G = (U_elec - T*S_elec) + (U_vib - T*S_vib) + pV

Since TOTEN = F_elec = U_elec - T*S_elec, and defining F_vib = U_vib - T*S_vib:
G(T,p) = TOTEN(V) + F_vib(T,V) + pV

Using this paper's notation where E₀(V) = TOTEN(V):
G(T,p) = E₀(V) + F_vib(T,V) + pV

This ensures we properly account for electronic entropy in the total Gibbs free energy calculation.

Best regards,
Zhao

[hszhao.cn@gmail.com]

Dear Jonathan.

After further reflection, I would like to add the following to my current understanding. Based on your explanation, I want to confirm my understanding and share my derivation as described below:

1. Regarding TOTEN: If I understand correctly, TOTEN corresponds to the electronic Helmholtz free energy, aka, F_elec = U_elec - T_elec*S_elec, calculated by VASP for the electronic subsystem. Is this interpretation correct?

2. By using the following derivation process, we can get the Gibbs free energy at a given 𝑇 and 𝑝: G(T,p) = E₀(V) + F_vib(T,V) + pV.

Starting from the standard thermodynamic definition:
G = U + pV - TS

We can separate the electronic and the thermally excited lattice dynamics, aka, phonon, contributions:
U = U_elec + U_vib
S = S_elec + S_vib

Substituting:
G = U_elec + U_vib + pV - T(S_elec + S_vib)
G = (U_elec - T*S_elec) + (U_vib - T*S_vib) + pV

Since TOTEN = F_elec = U_elec - T*S_elec, and defining F_vib = U_vib - T*S_vib:
G(T,p) = TOTEN(V) + F_vib(T,V) + pV

Using this paper's notation where E₀(V) = TOTEN(V):
G(T,p) = E₀(V) + F_vib(T,V) + pV

This ensures we properly account for electronic entropy in the total Gibbs free energy calculation.

Best regards,
Zhao

In my above description, TOTEN should have been written as energy without entropy. In detail:

(a). Regarding TOTEN: If I understand correctly, energy without entropy corresponds to the electronic Helmholtz free energy, aka, F_elec = U_elec - T_elec*S_elec, calculated by VASP for the electronic subsystem.
(b). TOTEN(V) should have been written as F_elec(V).

[hszhao.cn@gmail.com]

My additional supplement:

1. In general, the following relationship holds: energy without entropy >= energy(sigma->0) >= TOTEN.
2. The Gibbs free energy defined in phonopy-qha: $$G(T, p) = \min_V \left[ U(V) + F_\mathrm{phonon}(T;\,V) + pV \right]$$

Here, U(V) is exactly the F_elec(V) mentioned above, which is the zero-temperature electronic energy, corresponding to "energy without entropy" in VASP since it contains no thermal contributions.

Regards,
Zhao

[hszhao.cn@gmail.com]

See here for the related discussion.