https://www.cup.uni-muenchen.de/oc/zipse/teaching/computational-chemistry-2/topics/determining-atomic-charges-using-the-merz-singh-kollman-mk-scheme/

https://www.cup.uni-muenchen.de/oc/zipse/teaching/computational-chemistry-2/topics/mulliken-population-analysis/

Interesting question.

Regards

Mulliken charges are partial atomic charges based on the linear combination of atomic orbitals molecular orbital method. Mulliken charges are basis set dependent. In order to calculate the exactly Mulliken charges, one has to use a complete basis set for a molecule by placing a large set of functions on a single atom. These problems can be addressed by modern methods for computing net atomic charges, such as density derived electrostatic and chemical analysis, electrostatic potential analysis, and natural population analysis.

MK provides atomic charges that can be fitted to reproduce the molecular electrostatic potential (MEP) at a number of points around the molecule using pop=MK keyword in Gaussian software. MK charge is comparable to charges derived from electrostatic potential (CHELP) or charges derived from electrostatic potential gradient (CHELPG).

Hope this helps,

Regards,

Karthick

Mulliken charge is calculated directly based on density matrix (a special representation of electronic wavefunction), it is the oldest method for calculating atomic charge. Although it is very fast, there are many well drawbacks of this method, such as poor reproducibility of dipole moment and electrostatic potential(ESP), heavily underestimate ionicity of highly ionic bonds, very high sensitivity to basis set and nonconvergence with respect to increase of basis set size.

MK is a popular practical implementation of the idea "electrostatic potential fitting". Calculation of MK is more expensive than Mulliken. The most prominent point of MK (or other ESP fitting charges, such as CHELP and CHELPG) is that it can well reproduce ESP around molecular van der Waals surface, this great feature makes MK very suitable for approximately representing interatomic electrostatic interaction. One of major shortcomings of MK (including other ESP fitting methods) is that the charge of heavily embeded atoms cannot be well determined.

If you want to find more detailed description, especially the formula and algorithms of these two kinds of atomic charges, you are suggested to consult Section 3.9.3 and 3.9.10 of Multiwfn manual (available at http://sobereva.com/multiwfn)