4. Population analysis¶
4.1. Methods¶
At the heart of of any point charge embedding scheme is the assumption that the electrostatic potential generated by an atom can be approximated at a certain distance by that of a point charge:
With \(\Psi_i(\mathbf{r})\) being the wavefunction of atom \(i\), \(q_i\) its point charge value and \(\mathbf{r_i}\) its position. This approximation naturally breaks down at small \(|\mathbf{r_i} - \mathbf{r}|\) but by including ground state exchange via the rl term, this situation is avoided.
The choice of \(q_i\) is a non trivial matter. Considerable effort is spent on parametrising forcefields for various applications, however in this case we intend to use the point charges as one-electron terms in the mh and ml Hamiltonians.
A first approximation can be the use of Mulliken charges. This has the advantage of being trivial to compute in both the molecular and crystal calculations. Unfortunately the well-known basis-set dependence of Mulliken charge values can become a real problem.
Charge-density based methods represent the most direct way of fulfilling the equation above. Hirshfeld and AIM charges[10] fall in this category. RESP are practically defined by the equation, albeit in a radial range around the atoms.
4.2. Crystal or molecular¶
The population analysis methods described above can be applied to periodic crystal calculations or single molecule calculations. Using the crystal calculation has the advantage of including an interacting charge distribution at equilibrium.
On the other hand, charges from molecular calculations could be more judicious since they can be matched to the level of theory of the molecular calculation that they are embedding. In this way if the high level of theory is TD-DFT B3LYP and the low level of theory is DFT PBE, the charges assigned to mh can come from B3LYP and the ones to ml from PBE. This is particularly important in order to most totally cancel out intersystem electrostatic interactions from rl.