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 $n$th SCF step as
    $\displaystyle \rho^{\rm (in)}_{n+1}
= \alpha \rho^{\rm (in)}_{n} + (1-\alpha)\rho^{\rm (out)}_{n},$ (1)

where $\alpha$ 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 $R$:
    $\displaystyle R_{n}
\equiv \rho^{\rm (out)}_{n} - \rho^{\rm (in)}_{n},$ (2)

we assume that the residual $\bar{R}_{n+1}$ at the next $(n+1)$th SCF step can be estimated by a linear combination of $\{ R_m \}$
    $\displaystyle \bar{R}_{n+1}
= \sum_{m=n-(p-1)}^{n}\alpha_{m} R_{m},$ (3)

where $\alpha_m$ is found by minimizing $\langle \bar{R}_{n} \vert \bar{R}_{n} \rangle$ with a constraint $\sum_{m=n-(p-1)}^{n} a_m=1$. According to Lagrange's multiplier method, $F$ is defined by
$\displaystyle F$ $\textstyle =$ $\displaystyle \langle \bar{R}_{n+1} \vert \bar{R}_{n+1} \rangle
1 - \sum_m^n a_m
  $\textstyle =$ $\displaystyle \sum_{m,m'}\alpha_m \alpha_{m'}
\langle R_{m} \vert R_{m'} \rangle
1 - \sum_m^n a_m
\right).$ (4)

Considering $\frac{\partial F}{\partial \alpha_k}=0$ and $\frac{\partial F}{\partial \lambda}=0$, an optimum set of $\{\alpha\}$ can be found by solving the following linear equation:
    $\displaystyle \left(
\langle R_{n-(p-1)} \vert R_{n-(p-1)} ...
\end{array}\right).\quad$ (5)

An optimum choice of the input density matrix $\rho^{\rm (in)}_{n+1}$ may be obtained by the set of coefficients $\{\alpha\}$ as
    $\displaystyle \rho^{\rm (in)}_{n+1} =
\sum_{m=n-(p-1)}^{n} \alpha_{m}\rho^{\rm (in)}_{m}.$ (6)

Kerker mixing in momentum space

After Fourier-transforming the difference charge density $\delta n({\bf r})$ by
    $\displaystyle \delta \tilde{n}({\bf q_{p'}})
\frac{1}{N_1 N_2 N_3}
\sum_{\bf p}
\delta n({\bf r}_{\bf p})
{\rm e}^{-i{\bf q_{p'}}\cdot {\bf r}_{\bf p} },$ (7)

$\delta \tilde{n}({\bf q})$ can be mixed in a simple mixing [3]:
    $\displaystyle \delta \tilde{n}_{n+1}^{\rm (in)}({\bf q})
\alpha w({\bf q})~\d...
\left(1-\alpha w({\bf q})\right)~
\delta \tilde{n}^{\rm (out)}_{n}({\bf q})$ (8)

with the Kerker factor $w({\bf q})$.
    $\displaystyle w({\bf q}) =
\frac{\vert {\bf q} \vert^2}{\vert {\bf q} \vert^2 + q_0^2 },$ (9)

where $q_0=\gamma \vert {\bf q}_{\rm min} \vert$, and ${\bf q}_{\rm min}$ is the ${\bf q}$ 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 ${\bf q}$ vector, it is found that $w({\bf q})$ 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
    $\displaystyle \delta n({\bf r_{p}})
\sum_{\bf p'}
\delta \tilde{n}({\bf q}_{\bf p'})
{\rm e}^{i{\bf q}_{\bf p'}\cdot {\bf r_{p}}}.$ (10)

RMM-DIIS in momentum space

By defining the residual vector $R({\bf q})$ in momentum space,
    $\displaystyle R_{n}({\bf q})
\equiv \rho^{\rm (out)}_{n} ({\bf q}) - \rho^{\rm (in)}_{n}({\bf q}),$ (11)

and the norm with the Kerker metric as:
    $\displaystyle \langle R_{m} \vert R_{m'} \rangle
\sum_{\bf q}
\frac{R^*_{m}({\bf q}) R_{m'}({\bf q})}{w({\bf q})},$ (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.


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

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

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

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