Can the the relaxed energy of a material system (eV/atom) computed by DFT using VASP be positive?

Question

If it is positive, why is it so? Is it due to specific convergence criteria or choice of pseudopotentials? I cannot check the convergence of the DFT calculations as they have been downloaded from a dataset.

Answer 1

When DFT is being used for electronic structure energy calculations, a negative energy would be indicative of bound electrons. There's nothing physically, chemically or mathematically illegal about having a "positive electronic energy" in theory, but it's not something that we see or study often. I'll also note that it doesn't matter whether you're looking at the total energy or the energy/atom, because the "number of atoms" is a positive number so if the energy/atom is positive, it also means that the total energy is positive, and vice versa. If you're obtaining positive electronic energies in VASP's DFT calculations, then sure, this could have been due to the convergence criteria or the use of pseudopotentials, or other possible things (it's hard to know without more details about what exactly was calculated).

So what about a positive energy after relaxation (as in the title of your question)? Relaxation in this context usually means geometry optimization, meaning that calculations are done to find the geometry at which the electronic total energy (or energy/atom if you wish) is lowest or in a local minimum. This energy can still in theory be positive for the same reason as given in the first paragraph of this answer. The point about metastability in the answer by Jaafar can be related to local minima, but there's nothing theoretically illegal about having a positive energy at a global minimum either.

The answer by Jaafar mentioned "reference energies" too. If a single-point energy calculation is done using DFT, it's fair for us to assume that this is an electronic energy calculation, in which the program would usually give you a "total energy" (or in this case a "total energy per atom") compared to a reference energy of 0 which would only indicate the existence of 0 electrons. However, the point about reference energies can be useful to illustrate the phenomenon of unbound states, or even just the idea of positive energies relative to a different state, the latter being shown in the diagram below from this paper of mine:

In the above figure, the "relaxed" energy for the gerade state is at around +8000cm-1 , or +1 eV, or +0.5 eV/atom, which is a positive number, but any energy can be made positive like this by shifting the "reference" or "zero" energy to make that happen.

Answer 2

Firstly, by expressing the energy per atom (eV/atom), it allows for a more convenient comparison between different materials or systems, as it provides a measure of the energy on a per-atom basis. This is particularly useful when comparing the stability or energetics of different crystal structures, polymorphs, or chemical compositions.

For example, if you have a system with 100 atoms and the total energy of the system is -1000 eV, then the energy per atom would be -10 eV/atom. This means that, on average, each atom contributes -10 eV to the total energy of the system.

It's important to note that the energy per atom is a relative quantity and can be positive or negative, depending on the system's stability and reference energy chosen.

If the energy per atom is positive, it generally indicates that the system is not in its lowest energy state and may not be thermodynamically stable in its current form. A positive energy per atom suggests that the system has excess energy compared to some reference state or configuration. This excess energy could arise from various factors such as:

• Metastability: The system might be in a metastable state, meaning it is kinetically trapped in a higher energy configuration, even though a lower energy state exists. In this case, the positive energy per atom indicates that the system has not reached its most stable state.
• High-energy configuration: The system might have a crystal structure or arrangement of atoms that is energetically unfavorable or less stable compared to other structures. The positive energy per atom reflects the fact that the current configuration is not the lowest energy configuration.
• Unfavorable interactions: The system could involve interactions or chemical bonding that are energetically unfavorable. This can lead to a positive energy per atom, indicating that the system is not in an energetically favorable state.

It's important to note that a positive energy per atom does not necessarily imply that the system is completely unstable or unusable. It could still exist in a metastable state or have other desirable properties. However, from an energetic standpoint, a positive energy per atom suggests that there may be more stable configurations or compositions that the system could adopt.