Charge Mixing Methods: Ver. 1.0

1 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

${\rho }_{n+1}^{(\mathrm{in})}=\alpha {\rho }_n^{(\mathrm{in})}+(1-\alpha ){\rho }_n^{(\mathrm{out})},$ (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.

2 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 $R$:

$R_n\equiv {\rho }_n^{(\mathrm{out})}-{\rho }_n^{(\mathrm{in})},$ (2)

we assume that the residual ${\overline{R}}_{n+1}$ at the next $(n+1)$th SCF step can be estimated by a linear combination of $\{R_m\}$

${\overline{R}}_{n+1}={\displaystyle \sum _{m=n-(p-1)}^n}{\alpha }_m{R}_m,$ (3)

where ${\alpha }_m$ is found by minimizing $⟨{\overline{R}}_n|{\overline{R}}_n⟩$ with a constraint ${\sum }_{m=n-(p-1)}^n{\alpha }_m=1$. According to Lagrange’s multiplier method, $F$ is defined by

$F$ $=$ $⟨{\overline{R}}_{n+1}|{\overline{R}}_{n+1}⟩-\lambda \left(1-{\displaystyle \sum _m^n}{\alpha }_m\right),$ (4)

$=$ ${\displaystyle \sum _{m,m^{\prime }}}{\alpha }_m{\alpha }_{m^{\prime }}⟨R_m|R_{m^{\prime }}⟩-\lambda \left(1-{\displaystyle \sum _m^n}{\alpha }_m\right).$

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:

(5)

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

${\rho }_{n+1}^{(\mathrm{in})}={\displaystyle \sum _{m=n-(p-1)}^n}{\alpha }_m{\rho }_m^{(\mathrm{in})}.$ (6)

3 Kerker mixing in momentum space

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

$\delta \tilde{n}({𝐪}_{{𝐩}^{\prime }})={\displaystyle \frac{1}{N_1{N}_2{N}_3}}{\displaystyle \sum _{𝐩}}\delta n({𝐫}_{𝐩}){\mathrm{e}}^{-i{𝐪}_{{𝐩}^{\prime }}\cdot {𝐫}_{𝐩}},$ (7)

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

$\delta {\tilde{n}}_{n+1}^{(\mathrm{in})}(𝐪)=\alpha w(𝐪)\delta {\tilde{n}}_n^{(\mathrm{in})}(𝐪)+\left(1-\alpha w(𝐪)\right)\delta {\tilde{n}}_n^{(\mathrm{out})}(𝐪)$ (8)

with the Kerker factor $w(𝐪)$.

$w(𝐪)={\displaystyle \frac{{|𝐪|}^2}{{|𝐪|}^2+q_0^2}},$ (9)

where $q_0=\gamma |{𝐪}_{\min }|$, and ${𝐪}_{\min }$ is the $𝐪$ 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 $𝐪$ vector, it is found that $w(𝐪)$ 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

$\delta n({𝐫}_{𝐩})={\displaystyle \sum _{{𝐩}^{\prime }}}\delta \tilde{n}({𝐪}_{{𝐩}^{\prime }}){\mathrm{e}}^{i{𝐪}_{{𝐩}^{\prime }}\cdot {𝐫}_{𝐩}}.$ (10)

4 RMM-DIIS in momentum space

By defining the residual vector $R(𝐪)$ in momentum space,

$R_n(𝐪)\equiv {\rho }_n^{(\mathrm{out})}(𝐪)-{\rho }_n^{(\mathrm{in})}(𝐪),$ (11)

and the norm with the Kerker metric as:

$⟨R_m|R_{m^{\prime }}⟩\equiv {\displaystyle \sum _{𝐪}}{\displaystyle \frac{R_m^{*}(𝐪)R_{m^{\prime }}(𝐪)}{w(𝐪)}},$ (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.