Charge Mixing Methods: Ver. 1.0
Taisuke Ozaki, RCIS, JAIST
A simple scheme of mixing charge densities is to mix the input matrix
and the output density matrix at the last th SCF step as



(1) 
where is a mixing parameter, and the optimum choice strongly
depends on the system under study. After mixing the density matrix,
the corresponding charge density is easily evaluated.
A more efficient scheme beyond the simple mixing method is
the residual minimization method in the direct inversion of
iterative subspace (RMMDIIS) [1,2].
By defining a residual :



(2) 
we assume that the residual at the next th SCF step
can be estimated by a linear combination of



(3) 
where is found by minimizing
with
a constraint
. According to
Lagrange's multiplier method, is defined by
Considering
and
, an optimum set of
can be found by solving the following linear equation:



(5) 
An optimum choice of the input density matrix
may be obtained by the set of coefficients as



(6) 
After Fouriertransforming the difference charge density
by



(7) 
can be mixed in a simple mixing [3]:



(8) 
with the Kerker factor .



(9) 
where
, and
is the vector with the minimum
magnitude except 0vector in the FFT.
Since the charge sloshing tends to be
introduced by charge components with a small vector,
it is found that is effective for avoiding
the charge sloshing. The back transformation of the mixed charge
density in momentum space gives the charge density in real space as



(10) 
By defining the residual vector in momentum space,



(11) 
and the norm with the Kerker metric as:



(12) 
we can apply the RMMDIIS to the charge density mixing
in momentum space with a care for the charge sloshing [4].
The procedure of finding an optimum charge density is same as in
the RMMDIIS for the density matrix.

 1

P. Csaszar and P. Pulay,
J. Mol. Struc. 114, 31 (1984).
 2

F. Eckert, P. Pulay, and H.J. Werner,
J. Comp. Chem. 18, 1473 (1997).
 3

G. P. Kerker, Phys. Rev. B 23, 3082 (1981).
 4

G. Kresse and J. Furthmeuller, Phys. Rev. B. 54, 11169 (1996).
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