Madelung Energy

Madelung Energy

The madelung energy $E_{mad}$ can be calculated as following:

$$E_{mad} = e^2 \sum_{\mu, \nu} Z_{\mu} Z_{\nu} M_{\mu\nu}$$ (81)

where madelung matrix $M_{\mu \nu}$ is

$$M_{\mu\nu} = \sum_{R_{\mu T} \neq r_\nu} \frac{\erfc (\eta \vert ... ...t] - \frac{2 \eta}{\sqrt{\pi}} \delta_{\mu \nu} - \frac{\pi}{\eta^2 \Omega_c} .$$ (82)

If the Ewald parameter $\eta$ is chosen approprietely, the summations for translation vectors $\bm{T}$ and reciprocal vectors $\bm{G}$ are converged rapidly. The $\eta$ has a dimension of $\mathrm{length}^{-1}$. So, we also use a dimension less parameter $\alpha = \eta \Omega_c^{1/3}$.

How to choose cutoff distances

Let us discuss cut-off distances of the summations. For large $\bm{T}$, the summation in first term may be approximated by the following integration:

$$\sum_{\vert\bm{T}\vert > S} \frac{\erfc(\eta \vert R_{\mu T} - r_... ...a T)}{T} = \frac{4 \pi}{\eta^2 \Omega_c} F(\eta S) \equiv \mathrm{Residual}(S).$$ (83)

where

$$F(x) = (\frac{1}{4} - \frac{x^2}{2}) \mathrm{erfc}(x) + \frac{2}{\sqrt{\pi}} \frac{x}{4} \exp(-x^2).$$ (84)

On the other hand, the largest term in the summation may be evaluated as

$$\frac{\mathrm{erfc}(\eta \Omega_c^{1/3})}{\Omega_c^{1/3}}$$ (85)

Therefore, the cutoff S must be chosen so that the following condition is satisfied:

$$\mathrm{Residual}(S) < \mathrm{tol} \cdot \frac{\mathrm{erfc}(\eta \Omega_c^{1/3})}{\Omega_c^{1/3}}$$ (86)

The above equation is simplified as

$$F(\eta S) < \mathrm{tol} \cdot \frac{1}{4\pi} \alpha^2 \erfc (\alpha).$$ (87)

In the program, the cutoff length $S$ is calculated by the function madFindCutOff which returns dimension less cut-off length $\tilde{S} = S/\Omega_c^{1/3}$.

/*
 *  return cutoff length for direct space summations.
 *  alpha = eta * (cell volume)^(1/3).
 *  ( in unit of cell volume = 1).
 */
double madFindCutoff(double alpha, double tol);

The cutoff length for reciprocal summation is estiamted in the same way.

$$\sum_{\vert\bm{G}\vert > Q} \frac{1}{G^2} \exp \left[ -i \bm{G} \... ...{2\eta}{\sqrt{\pi}} \mathrm{erfc} (\frac{Q}{2\eta}) \equiv \mathrm{Residual}(Q)$$ (88)

The largest term in the summation may be evaluated as

$$\frac{4\pi}{\Omega_c}\frac{1}{g^2}\exp\left[-\frac{g^2}{4\eta} \right]$$ (89)

where

$$g = \frac{2\pi}{\Omega_c^{1/3}}.$$ (90)

Thefore cut-off Q must satisfy the following condition:

$$\mathrm{Residual}(Q) < \mathrm{tol} \cdot \frac{4\pi}{\Omega_c}\frac{1}{g^2}\exp\left[-\frac{g^2}{4\eta} \right] .$$ (91)

The above equation is simplified as

$$\erfc (\frac{Q}{2\eta}) < \mathrm{tol} \cdot \frac{1}{2\sqrt{\pi}} \frac{1}{\alpha} \exp \left[ -\frac{\pi^2}{\alpha^2} \right]$$ (92)

In the program, the cutoff length $Q$ is calculated by the function madReciprocalFindCutOff which returns dimension less cut-off length $\tilde{Q} = Q \frac{\Omega_c^{1/3}}{2\pi}$.

/*
 *  return cutoff length for reciprocal space summations.
 *  alpha = eta * (cell volume)^(1/3).
 *  ( in unit of cell volume = 1 and factor 2pi omitted).
 */
double madFindReciprocalCutoff(double alpha, double tol);

How to choose Ewald parameter $\eta$

For effeciency, the Ewald parameter $\eta$ (or $\alpha$) must be chosen approprietely. From the analysis of previous subsection, number of terms in the summations are $\tilde{S}^3$ for the first term and $\tilde{Q}^3$ for the second term. Therefore appropriate $\alpha$ is the one which minimize $N = \tilde{S}^3 + \tilde{Q}^3$. For $\mathrm{tol} = 1.0e-12$, we calculated the $N$ numerically. The result is shown in Fig. [#!fig:madalpha!#]

Figure 1: number of summation terms as a function of dimensionless Ewald parameter $\alpha$.

From this result, appropriate range of $\alpha$ is $1.5 \leq \alpha \leq 2.5$.