Extrapolation of Charge Density: Ver. 1.0

Let us consider an extrapolation ${\overline{𝐱}}_{n+1}$ of the coordinate at the $(n+1)$th molecular dynamic or geometry optimization step [1, 2] by a linear combination of the previous coordinates $\{{𝐱}_m\}$ as

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

where for $p$ three is an optimum choice in many cases. To fit well the coordinate ${\overline{𝐱}}_{n+1}$ to the real coordinate ${𝐱}_{n+1}$, we consider the minimization of a function $F$:

$F={|{\overline{𝐱}}_{n+1}-{𝐱}_{n+1}|}^2-\lambda \left(1-{\displaystyle \sum _{m=n-(p-1)}^n}{\alpha }_m\right)$ (2)

with respect to $\{{\alpha }_i\}$ and $\lambda$. The conditions $\frac{\partial F}{\partial {\alpha }_m}=0$ and $\frac{\partial F}{\partial \lambda }=0$ leads to

(3)

By solving the linear equation, we may have an optimum choice of a set of $\{{\alpha }_m\}$. Then, it is assumed that the difference charge density $\mathrm{\Delta }{\rho }_i^{(\mathrm{out})}$ can be extrapolated well by the same set of coefficients $\{{\alpha }_m\}$ as

${\rho }_{n+1}^{(\mathrm{in})}={\rho }_{n+1}^{(\mathrm{atom})}+{\displaystyle \sum _{m=n-(p-1)}^n}{\alpha }_m\mathrm{\Delta }{\rho }_m^{(\mathrm{out})},$ (4)

where ${\rho }_{n+1}^{(\mathrm{atom})}$ is given by the superposition of atomic charge densities at ${𝐱}_{n+1}$. Using Eq. (4) it can be possible to estimate a good input charge density at the $(n+1)$th step in molecular dynamic simulations or geometry optimizations.