Charge Analysis: Ver. 1.0
Taisuke Ozaki, RCIS, JAIST
The KohnSham (KS) Bloch functions are expanded
in a form of linear combination of pseudoatomic basis functions (LCPAO)
centered on site by



(1) 
where and are an expansion coefficient and pseudoatomic
function,
a lattice vector, a site index,
( or )
spin index,
an organized orbital
index with a multiplicity index , an angular momentum quantum number ,
a magnetic quantum number , and the number of repeated cells.
The charge density operator
for
the spin index is given by



(2) 
where means the integration over the first
Brillouin zone of which volume is , and
means the summation over occupied states.
The charge density
with the spin index
is found as
with a density matrix defined by



(4) 
Then, Mulliken populations
are given by



(5) 
where
is an overlap integral.
Since the Mulliken population can be obtained by integrating Eq. (3) over
real space, and by decomposing it into each contribution
specified with and , it can be confirmed that
the sum of
gives the number of electron
per unit cell as follows:



(6) 
The fuzzy cell method decomposes real space into smeared
Voronoi cells, called the fuzzy cell [2].
The fuzzy cell at the site is determined by
a weighting function
:



(7) 
with defined by
where is chosen in OpenMX. As increases the fuzzy cells
defined by approach to Voronoi cells (WignerSeitz cells).
From the definition Eq. (7) it is clear that



(13) 
Thus, the integration of the charge density Eq. (3) over real space
can be decomposed by employing the weighting functions as follows:
Thus, the Voronoi charge at the site
can be defined by



(15) 
From Eq. (14), it is confirmed that



(16) 
Let us consider to express the Hartree potential in a system by
the sum of Coulomb potentials with an effective point charge
located on each atomic site as follows:



(17) 
where is the number of atoms in the system.
The can be found by a least square fitting with a constraint
[3,4,5],
where is the total charge
in the system. The Lagrange multiplier method casts this to
a minimization problem of the following function :



(18) 
where
and
are the Hartree potential
calculated by the DFT calculation and Eq. (17), respectively,
is a set of sampling points, and is the number
of the sampling points.
The sampling points are given by the grids in the real space between
two shells of the first and second scale factors times
van der Waals radii [6].
The conditions
and
lead to



(19) 
with



(20) 
and



(21) 
By solving the linear equation Eq. (19), we can find
the electrostatic potential fitting (ESP) charges.
It is noted that the ESP charge is an effective charge on each atom
including the contribution of the core charge compared to
the Mulliken and Voronoi charges.

 1

R. S. Mulliken, J. Chem. Phys. 23, 1833 (1955).
 2

A. D. Becke and R. M. Dickson, J. Chem. Phys. 89, 2993 (1988).
 3

U. C. Singh and P. A. Kollman, J. Comp. Chem. 5, 129(1984).
 4

L. E. Chirlian and M. M. Francl, J. Com. Chem. 8, 894(1987).
 5

B. H. Besler, K. M. Merz Jr. and P. A. Kollman,
J. Comp. Chem. 11, 431 (1990).
 6

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