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Population Analysis
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.
http://theory.cm.utexas.edu/henkelman/research/bader/
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.
Mulliken population analysis
Coulson's charges
Natural charges
CM1, CM2, CM3 charge models
Partitioning of a physical observable
derived from the wave function,
such as electron density
Bader charges
Density fitted atomic charges
Hirshfeld charges
Maslen's corrected Bader charges
Politzer's charges
Density charges
Mulliken population analysis
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
A
B
Overlap term
Discussion
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.
General Approach
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
= ∑
Gross atom population
= ∑
m
m
= ∑∑
n m
N/2
Total Charge at
atom A
=2 ∑ ∑∑
n m
i
N/2
= 2 ∑ ∑ ∑ CniCmi Smn
i
n m
Total Charge gain/loss
by atom A
Atomic number
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.
In computation
Gaussian
Mullikan
  r  = Total  r   { F  atom  r  +  Phenalene  r }
Bader
VASP
Charge of F = 
   r  dV
r 2
CASTEP
Hirshfeld
Mullikan
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
Thank You