# Charge Mixing Methods: Ver. 1.0

Taisuke Ozaki, RCIS, JAIST

# Simple mixing

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.

# RMM-DIIS for density matrix

A more efficient scheme beyond the simple mixing method is the residual minimization method in the direct inversion of iterative subspace (RMM-DIIS) [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
 (4)

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)

# Kerker mixing in momentum space

After Fourier-transforming 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 0-vector 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)

# RMM-DIIS in momentum space

By defining the residual vector in momentum space,
 (11)

and the norm with the Kerker metric as:
 (12)

we can apply the RMM-DIIS 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 RMM-DIIS for the density matrix.

## Bibliography

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).

2007-08-20