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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:

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

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

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:

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:

(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

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 wrote: ↑Sat May 10, 2025 1:36 pm

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).

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

Hi VASP community,

I would like to clarify the meaning of entropy-related energy values in VASP output:

In my calculation, I get:

werner@x13dai-t:~/Desktop/std_vasp_slurm/b1/fe/cd/b1fecd84-2a91-41fa-b7e0-f63215ca6329_1$ ug -B2 22428357 OUTCAR | tail -3
  free  energy   TOTEN  =       -24.22450672 eV

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

From the relationship: TOTEN = Energy without entropy + (entropy T*S EENTRO), I can verify: -24.22361411 + (-0.00089262) ≈ -24.22450673 eV

My question is: Why is "entropy T*S EENTRO" a negative value?

Since:
- Temperature T is always positive
- Entropy S is always positive (as it measures disorder)
- Therefore T×S should be positive

However, VASP outputs a negative value. Is this because:
1. VASP actually outputs "-T×S" (the contribution to free energy) rather than "T×S" itself?
2. Or is there something about how VASP calculates electronic entropy that I'm missing?

For now, I'm planning to use the "Energy without entropy" value as U(V) in QHA (Quasi-Harmonic Approximation) calculations, so I need to understand the correct relationship between these energy terms.

Thank you for any clarification!

Regards,
Zhao

See here for the related discussion.

Dear hszhao,

Yes you are correct in the line entropy TS EENTRO VASP writes the negative entropy. In this way the TOTEN can be obtained as a sum over the energy table in the OUTCAR file. Example:

   alpha Z        PSCENC =      1363.31358291
   Ewald energy   TEWEN  =    247838.17329901
   -Hartree energ DENC   =   -282067.70155624
   -exchange      EXHF   =         0.00000000
   -V(xc)+E(xc)   XCENC  =      1348.61735250
   PAW double counting   =     29410.74263967   -28786.08102701
   entropy T*S    EENTRO =        -0.01527901
   eigenvalues    EBANDS =      1076.85460609
   atomic energy  EATOM  =     34751.67368736
   Solvation  Ediel_sol  =         0.00000000
   ext. energy  EPLUGINS =         0.00000000
   ---------------------------------------------------
free energy    TOTEN  =      4935.57730528 eV

energy without entropy =     4935.59258429  energy(sigma->0) =     4935.58494478

And TOTEN can be obtained by 1363.31358291+247838.17329901-282067.70155624+0.00000000+1348.61735250+29410.74263967-28786.08102701-0.01527901+1076.85460609+34751.67368736+0.00000000+0.00000000.
So to have negative entropy is just a convention in VASP to make the table in the OUTCAR file a little easier to use.

All the best Jonathan

Dear Jonathan,

Thank you for your reply. You confirmed that "entropy T*S EENTRO" in VASP output is indeed the negative entropy, mainly as a convention to allow the energy table in the OUTCAR file to sum directly to TOTEN. I understand this point.

However, I would like to further clarify two aspects:

1. This negative entropy has essentially lost its physical meaning, correct? It's merely a convention adopted by VASP internally to satisfy the energy representation convention.

2. More importantly, this entropy is introduced due to the SIGMA parameter setting, which primarily aims to accelerate the convergence of electronic structure calculations - essentially a numerical technique. Therefore, the so-called "entropy" introduced this way is actually an artificial quantity without real physical significance.

Based on this understanding, for subsequent physical property calculations (such as QHA), I should use energy(sigma->0) as the only reliable physical quantity, rather than TOTEN or Energy without entropy, correct?

Thank you for your further explanation!

See below for the related discussions:

https://www.vasp.at/forum/viewtopic.php?t=20192
https://www.vasp.at/forum/viewtopic.php?t=18970
https://ww.vasp.at/forum/viewtopic.php?t=1333
https://wwww.vasp.at/forum/viewtopic.php?t=8658

Best regards,
Zhao

Dear

The entropy term for a Fermi gas can be computed from the Fermi Dirac distributions given by fik​=1/(1+exp((E_{ik}​−μ​)/kbT). The Boltzmann constant times temperature is set by SIGMA in vasp. Since there is no real temperature only some parameter you set in the INCAR file which helps to improve SFC stability as was explained to you in this post already https://ww.vasp.at/forum/viewtopic.php?t=19506. The extrapolation of energy value when extrapolating SIGMA->0 is the same as computing the electronic ground state really at T=0. So it is completely fine and also the correct way to use this values for your QHA calculations.

All the best Jonathan

Dear Jonathan,

Based on all the above discussions, we all agree that the entropy T*S EENTRO entry in VASP output is a fictional, non physical quantity. So, I think your comment below in one of our previous discussions is inappropriate: 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.

Based on our discussions so far, I think any further analysis and calculations should be conducted using energy(sigma>0).

Regards,
Zhao

Dear hszhao,

Yes sorry for the confusion. When I wrote the first response I was thinking about binding energies obtained from molecular dynamics simulations. In this case you can adjust your sigma properly because you have real temperature in your system. And in this case I would recommend to use TOTEN. But as you figured out already in your case it is best to use energy(sigma>0).

All the best Jonathan

hszhao.cn@gmail.com wrote: ↑Sat May 10, 2025 4:00 pm

hszhao.cn@gmail.com wrote: ↑Sat May 10, 2025 1:36 pm

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).

Based on the discussion above, the description here needs further correction: In standard DFT calculations, temperature effects are not considered, and calculations are actually performed at T=0K. On the other hand, the so-called entropy introduced by the sigma parameter setting in the calculation does not correspond to the entropy of the electron system in the actual physical sense. Therefore, F_elec(V) = U_elec - T_elec*S_elec actually corresponds to energy(sigma→0), which is the converged total energy E0 obtained from VASP calculations.

Regards,
Zhao

jonathan_lahnsteiner2 wrote: ↑Fri May 16, 2025 9:27 am

Dear hszhao,

Yes sorry for the confusion. When I wrote the first response I was thinking about binding energies obtained from molecular dynamics simulations. In this case you can adjust your sigma properly because you have real temperature in your system. And in this case I would recommend to use TOTEN. But as you figured out already in your case it is best to use energy(sigma>0).

All the best Jonathan

Dear Jonathan,

Thank you for your clarification regarding the usage of TOTEN versus energy(sigma->0) in different scenarios. I'd like to further discuss the entropy contributions in VASP calculations to ensure I fully understand the implications for my work.

From our previous exchange, I understand that the entropy term (T*S EENTRO) in VASP relates exclusively to electronic entropy arising from the smearing of states near the Fermi level. Even in molecular dynamics simulations where we have "real temperature" in the system, this entropy term still only accounts for the electronic subsystem and not the lattice/vibrational entropy of the nuclei.

Could you confirm if my understanding is correct? Additionally, I have a few follow-up questions:

1. When you recommend using TOTEN for binding energies in MD simulations with properly adjusted sigma, is this because the electronic entropy becomes physically meaningful when matched to the ionic temperature, rather than being just a computational aid for convergence?

2. For systems where both electronic and vibrational entropy might be significant (e.g., soft materials or systems near phase transitions), what would be your recommended approach for calculating binding energies that properly account for all entropy contributions?

I appreciate your insights, as they're helping me develop a more nuanced understanding of how to properly interpret and utilize the energy terms in VASP for different applications.

Best regards,
Zhao