One of the main output data of an atomistic simulation based on density functional theory is the total energy. This energy includes contributions the kinetic energy of the electrons, as the potential energy for nucleus-nucleus, nucleus-electron, and electron-electron interactions. In simulations of optimization of geometries, the kinetic energy of the nuclei is not taken into account. Considering the system in which nuclei and electrons are totally dissociated as the reference of energy, those of the total energies found would be very high negative values. However, due to the use of the pseudopotential technique to smooth the wave functions in the core region, keeping the valence region unchanged, the total energies can vary widely to lower negative values. This is not a problem, since we can do data analysis of various simulations using total energy differences calculated with the same pseudopotential, exchange-correlation functional and parameters. One of the most immediate results that can be calculated via the total energy provided by the density functional theory comes from the Janak’s theorem [Phys. Rev. B 18, 7165 (1978)]:
Based on this relation, we can calculate the ionization potentials (IP) and electron affinity (EA) as:
In theses equations, the variable δ is a small value so that this finite difference approximates the eigenvalues of Janak’s theorem.
Bandgap and Mulliken Electronegativity
Two electronic properties that can be derived from electron affinity and ionization potential are the bandgap and the Mulliken electronegativity. These properties are given by:
for the bandgap, and
for Mulliken electronegativity.
Bandgap is a property that is directly associated with electromagnetic wave frequencies that can be absorbed by the atom, molecule or solid. It is interesting to note that in general the bandgap calculated via DFT is underestimated when compared to those obtained experimentally.
Example 04 (Phenol)
Example 05 (Ionic liquid pair)
Basis Set Superposition Error
Counterpoise correction (CC)