This is my notes when refreshing up on pCOHP theory. Emphasis is on intuition rather than rigor.
DFT in periodic boundary conditions
In periodic DFT, we calculate the non-interacting one-particle states that give the same electron density as the fully interacting states. This yields the one-electron bands and the band energies:
\(\hat{H}\psi_j(\vec{k}, \vec{r}) = \epsilon_j(\vec{k})\psi_j(\vec{k}, \vec{r})\)
The density of states is the number of allowed states between energies \(E\) and \(E+\delta E\):
\(\mathrm{DOS}(E) = \sum_{j=1}^{N_\mathrm{band}}\int_{\mathcal{V}}\delta(E-\epsilon_j(\vec{k}))\)
The integration is over reciprocal space volume. If the set of \(\psi_j\)’s are self-consistently obtained, the system’s energy is calculated by filling them up to \(N_\mathrm{elec}\). When this is done, the highest band’s energy is the Fermi level \(E_F\) (in the convention for metals). DOS has the dimensions of \(\mathrm{E^{-1}\mathrm{V^{-1}}}\).
In the solid state, the natural basis is bloch functions, aka plane waves modulated by the reciprocal lattice:
\(\psi_j(\vec{k}, \vec{r})=\sum_{\vec{G}} C_{j \vec{G}}(\vec{k}) \exp \{\mathrm{i}(\vec{k}+\vec{G}) \cdot \vec{r}\}\)
However, the plane waves are not eigenstates of the Hamiltonian, and they do not have a chemically relevant notion of energy.
Bonding in molecules
In molecules, the DFT part is very similar, except we no longer need to worry about the reciprocal space:
\(\hat{H}\Phi_(\vec{r}) = \epsilon_j\Phi_j(\vec{r})\)
Density of states becomes discretized into a spectrum of orbital energies, where you have the HOMO and LUMO as the frontier orbitals, instead of \(E_F\). Here, the natural basis set are atom-centered functions:
\(\Phi_j(\vec{r}) = \sum_\mu C_\mu \phi_\mu(\vec{r})\)
The crucial difference is that the \(\phi_\mu(\vec{r})\) now have a chemical meaning, if we define them to be ‘atomic orbitals’ (which can then be constructed out of a contracted gaussian basis set, but that is for numerics), then there is an associated orbital energy \(\epsilon’_\mu\). \(\epsilon’_\mu\) can be compared to \(\epsilon_j\), the molecular orbital energy, to distinguish between ‘bonding’ and ‘antibonding’ orbitals. This sort of chemical reasoning is not possible from the DOS.
projected DOS
Projected DOS is the overlap of the periodic bands with atom-centered orbitals
COOP/COHP
the energy integral yields the overlap population between 2 atoms/orbitals. In COOP, positive means bonding, negative means antibonding. Bonding and antibonding means occupying these states would raise/lower system energy, respectively. How do you obtain that just from the occupations? The more each band contributes to the overlap between two atoms, the more it contributes to the bonding. Some orbitals can contribute negatively and thus they are antibonding.
Clearly, COHP is better because some contributions to overlap will actually raise the energies and thus be antibonding.
COOP is dimensionless, COHP has energy dimensions since Hamiltonian elements is used. In both cases, the integrated IpCOOP and IpCOHP, when compared to the number of electrons / total energy, respectively, shows the contribution of this ‘bond’ to the whole system. It is thus an indication of whether there is a bond.
So the xlim for COHP and pCOHP in the plotting script are rather arbitrary. Depending on bond strength they should be changed.