1. Population Analysis

3. Bader Charge & Bader volume
Richard Bader from McMaster University, developed an intuitive way of dividing molecules into atoms called the Quantum Theory of Atoms in Molecules (QTAIM). His definition of an atom is based purely on the electronic charge density. Bader uses what are called zero flux surfaces to divide atoms. A zero flux surface is a 2-D surface on which the charge density is a minimum perpendicular to the surface. Typically in molecular systems, the charge density reaches a minimum between atoms and this is a natural place to separate atoms from each other.

4. A partial charge is a charge with an absolute value of less than one elementary charge unit (that is, smaller than the charge of the electron).
 Partial charges are a property only of zones within the distribution, and not the assemblage as a whole
The concept of a partial atomic charge is somewhat arbitrary, because it depends on the method used to delimit between one atom and the next (in reality, atoms have no clear boundaries).
Partitioning the molecular wave function using some arbitrary, orbital based scheme.
Partitioning of a physical observable
derived from the wave function,
such as electron density
Mulliken population analysis
Coulson's charges
Natural charges
CM1, CM2, CM3 charge models
Bader charges
Density fitted atomic charges
Hirshfeld charges
Maslen's corrected Bader charges
Politzer's charges
Density charges

5. m n Mulliken population analysis A B The molecular Orbital {j}
For Simplicity
Two atom each have one atomic orbital A B n m
The molecular Orbital {j}
j = Cn Yn+ CmYm
Molecular Orbital is occupied by N (2) electrons, suppose this population may be
Considered as divided into three sub-populations. In space the detail distribution is
Nj2 = NCn 2Yn2+ 2N CnCm Smn (YnYm /Smn) + NCm2 Ym2
The three wavefunction are normalized So,
N = NCn 2+ 2N CnCm Smn + NCm2
Overlap term

6. In Previous slide we assume there is only two atom with single atomic orbital
each (example- D2), but for more then single orbital and more then two atom, we have
to generalized this.
Discussion
If the coefficients of the basis functions in the molecular orbital are Cμi for the μ'th
basis function in the i'th molecular orbital, the density matrix terms are:
m,n atomic orbital of atoms
Population Matrix
Overlap Matrix of the basis
function
Gross Orbital Product for
Orbital n = ∑ m n
Gross atom population
General Approach

7. ∑ CniCmi Smn = 2 Total Charge gain/loss by atom A Atomic number m n i
Total Charge at
atom A
Total Charge gain/loss
by atom A
Atomic number

9. Bader charges
The chemical bonding of a system based on the topology of the quantum charge density.
In addition to bonding, atoms in molecules (AIM) allows the calculation of certain physical properties on a per-atom basis, by dividing space up into atomic volumes containing exactly one nucleus.
In quantum theory of atoms in molecules (QTAIM) an atom is defined as a proper open system, i.e. a system that can share energy and electron density, which is localized in the 3D space. Each atom acts as a local attractor of the electron density, and therefore it can be defined in terms of the local curvatures of the electron density. 

10. In computation Mullikan Bader Mullikan Hirshfeld Gaussian VASP
Population analysis in CASTEP is performed using a projection of the PW states onto a localized basis using a technique described by Sanchez-Portal et al. (1995). Population analysis of the resulting projected states is then performed using the Mulliken formalism (Mulliken, 1955). This technique is widely used in the analysis of electronic structure calculations performed with LCAO basis sets
CASTEP
Mullikan
Hirshfeld

11. Thank You