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

$$\rho^{\rm (in)}_{n+1} = \alpha \rho^{\rm (in)}_{n} + (1-\alpha)\rho^{\rm (out)}_{n},$$ (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 $R$:

$$R_{n} \equiv \rho^{\rm (out)}_{n} - \rho^{\rm (in)}_{n},$$ (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 $\{ R_m \}$

$$\bar{R}_{n+1} = \sum_{m=n-(p-1)}^{n}\alpha_{m} R_{m},$$ (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

$$F = \langle \bar{R}_{n+1} \vert \bar{R}_{n+1} \rangle -\lambda \left( 1 - \sum_m^n a_m \right),$$

$$= \sum_{m,m'}\alpha_m \alpha_{m'} \langle R_{m} \vert R_{m'} \rangle -\lambda \left( 1 - \sum_m^n a_m \right).$$ (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:

$$\left( \begin{array}{cccc} \langle R_{n-(p-1)} \vert R_{n-(p-1)} ... ...\right) = \left( \begin{array}{c} 0\\ 0\\ \cdot\\ 1 \end{array}\right).\quad$$ (5)

An optimum choice of the input density matrix $\rho^{\rm (in)}_{n+1}$ may be obtained by the set of coefficients $\{\alpha\}$ as

$$\rho^{\rm (in)}_{n+1} = \sum_{m=n-(p-1)}^{n} \alpha_{m}\rho^{\rm (in)}_{m}.$$ (6)

Kerker mixing in momentum space

After Fourier-transforming the difference charge density $\delta n({\bf r})$ by

$$\delta \tilde{n}({\bf q_{p'}}) = \frac{1}{N_1 N_2 N_3} \sum_{\bf p} \delta n({\bf r}_{\bf p}) {\rm e}^{-i{\bf q_{p'}}\cdot {\bf r}_{\bf p} },$$ (7)

$\delta \tilde{n}({\bf q})$ can be mixed in a simple mixing [3]:

$$\delta \tilde{n}_{n+1}^{\rm (in)}({\bf q}) = \alpha w({\bf q})~\d... ...) + \left(1-\alpha w({\bf q})\right)~ \delta \tilde{n}^{\rm (out)}_{n}({\bf q})$$ (8)

with the Kerker factor $w({\bf q})$.

$$w({\bf q}) = \frac{\vert {\bf q} \vert^2}{\vert {\bf q} \vert^2 + q_0^2 },$$ (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

$$\delta n({\bf r_{p}}) = \sum_{\bf p'} \delta \tilde{n}({\bf q}_{\bf p'}) {\rm e}^{i{\bf q}_{\bf p'}\cdot {\bf r_{p}}}.$$ (10)

RMM-DIIS in momentum space

By defining the residual vector $R({\bf q})$ in momentum space,

$$R_{n}({\bf q}) \equiv \rho^{\rm (out)}_{n} ({\bf q}) - \rho^{\rm (in)}_{n}({\bf q}),$$ (11)

and the norm with the Kerker metric as:

$$\langle R_{m} \vert R_{m'} \rangle \equiv \sum_{\bf q} \frac{R^*_{m}({\bf q}) R_{m'}({\bf q})}{w({\bf q})},$$ (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.

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).