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Just to add to the discussion: Mulliken charges are flawed in many aspects, but we know how and why, and therefore we accept its use, since it is simple and easily computed. But very dependent on the size of the basis set. Mulliken charges do not reproduce the dipole (or higher) moment, but can be made to do so easily: Thole, van Duijnen, "A general ...


17

Bader charge analysis In Bader's theory of Atoms in Molecules, we partition a molecule into "atoms" which are separated from each other by surfaces of minimum charge density:                                           You can then calculate the partial charges of the "atoms" in the molecule, e.g. H$_2$O might yield: \begin{array}{cc} \rm{...


16

Hirshfeld / CM5 (Charge Model 5) My main interest is in explaining CM5 charges, but to do that it's necessary to briefly explain what Hirshfeld charges are. Hirshfeld charges are obtained as: $$q_X=Z_X-\int\frac{\rho^0_X(\mathbf{r})}{\sum_Y\rho^0_Y(\mathbf{r})}\rho(r)d\mathbf{r}$$ where $Z_X$ is the atomic number of element $\ce{X}$, $\rho$ is the molecular ...


16

To give a example of the "random number generator" behavior of Mulliken charge analysis pointed by @SusiLehtola, below I'm showing the results of some test runs I did before on the CO2 molecule using Psi4 version 1:1.1-5 (default version on Ubuntu 18.04 repository as of july 2020). For reproducibility purposes, first my input files: user@machine:~/...


16

Electrostatic potential (ESP) derived charges Note that ESP1 derived charges include ChelpG (CHarges from ELectrostatic Potentials using a Grid-based method), the Merz–Kollman (MK) 2, and the RESP (restrained electrostatic potential) [3] scheme. While there are differences between the approaches the general idea is similar between the different methods. The ...


14

Mulliken population analysis The Mulliken charge scheme is based on the Linear Combination of Atomic Orbitals (LCAO), so, it is based on the system wave function and was described in a series of papers by R. S. Mulliken1,2,3,4. The idea is that the normalized Molecular Orbital (MO), $\phi_i$, of a diatomic molecule is written as a linear combination of ...


13

Mulliken and Löwdin population analysis In the atomic orbital basis set (enumerated in Greek indices), one finds that the number of electrons $N$ is equal to the trace of the product $\mathbf{PS}$ $$ N = \sum_\mu \left(\mathbf{PS}\right)_{\mu\mu} = \mathrm{Tr} \ \mathbf{PS} $$ where $\mathbf{P}$ is the density matrix, $\mathbf{S}$ is the AO overlap matrix, ...


11

Critic2 critic2 is a very nice QTAIM code that supports interfaces to many codes, including also periodic systems: "WIEN2k, elk, PI, Quantum ESPRESSO, abinit, VASP, DFTB+, Gaussian, psi4, siesta, and to any other program capable of writing the scalar field of interest to a grid" From personal experience, critic2 is very straight forward to compile ...


9

Don't use Mulliken for charge analysis. It is basically a random number generator, since it lacks a basis set limit. By choosing a different basis set representation, you can basically freely move the electrons around; e.g. in a one-center expansion all the electrons are counted for the expansion center whereas all the other nuclei in the system become bare. ...


9

The proof that (2) is the density arising from a Slater determinant wave function can be found in basically any quantum chemistry textbook. (2) does NOT hold for multiconfigurational wave functions, since the one-particle density matrix becomes non-diagonal. You can make the density diagonal like (2) by switching to natural orbitals, but then you have ...


9

Voronoi charges Voronoi charges (here called VC) are based on the partition of the real space in a system into Voronoi polyhedra.$^1$ A given point in space belongs to the polyhedron of some atom if the point is closer to that atom than to any other atom. This allows to partition the space and thus assign the point charge expectation value of a given point ...


8

Multiwfn Multiwfn is a QTAIM package which supports most quantum chemistry codes, seemingly focusing on non-periodic systems. For example, from a formatted Gaussian checkpoint file it is capable of calculating cube files of properties such as the electrostatic potential or electron localization function. These properties can then be used for many advanced ...


8

On the specific subject of the "EDIT", this was exactly the concern addressed by Mel Levy in the 1970s, e.g. M. Levy, "Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem", PNAS 76 (12) 6062-6065 (1979); https://doi.org/10.1073/pnas....


7

Density-Derived Electrostatic and Chemical approaches Net atomic charges have two primary applications (dual use): (1) to quantify charge transfer between atoms in materials; this identifies cations and anions and (2) to provide an electrostatic model in classical force fields using atomistic simulations (e.g., classical molecular dynamics or Monte Carlo ...


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