Ewald Equations

$$\Phi_{{\rm dir}}(\vec{r};\alpha) = \sum_{n} \frac{ {\rm e... ...lpha \vert\vec{r}+ \vec{n}\vert)}{\vert\vec{r}+ \vec{n}\vert}$$ (2)

erfc is the complementary error function:

$${\rm erfc}(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^{2}} dt$$ (3)

(Computer implementation of erfc given in A.4.1: derfc.F)

$$\Phi_{{\rm rec}}(\vec{r};\alpha) = \frac{1}{\pi V} \sum_{... ... / \alpha^2)}{\vec{m}^2} \exp(2 \pi i \vec{m}\cdot \vec{r})$$ (4)

$$\mathcal{U}_{{\rm self}} = \frac{\alpha}{\sqrt{\pi}}\sum_{i=1}^N q_{i}^2$$ (5)

In these equations, $\vec{r}$ represents a distance between two particles⁸, $\alpha$ is a constant which can be adjusted to tweak the output values, and $V$ is the volume of the unit cell we are looking at. As mentioned under our description of the PME method, we replicate a particular region many times to create a realistic biological system. This is the volume of one such region. $m$ and $n$ are just unit vectors⁹ normal faces of the unit cell. The self-interaction term, see Appendix A.3: selfTerm.F for code, is a relatively simple function that simply corrects for the fact that the other terms calculate a particle's potential relative to its own charge field, which must be accounted for. The real space term, see Appendix A.4: realTerm.F for code, calculates short range interactions. It makes use of a cutoff radius to limit computational cost. Because this term is short range, the contributions to it from distant particles are very small, and neglecting them does not significantly impair accuracy. Because computing the interactions from all particles would make this term $\mathcal{O}(n^2)$ which would make it the slowest portion of the code, the cutoff radius is very important. To implement the cutoff radius we make use of integer rounding and division to rapidly compute which particles are within a certain radius of a given particle, and the calculate their effects on the given particle. Because the number of particles that must be calculated for each particle is $\mathcal{O}(1)$, this allows the real space term to be $\mathcal O(n)$.

$$\mathcal{U}_{{\rm reciprocal}}=\frac{1} {2}\sum_{i=1}^N \sum_{j=1}^N q_{i}q_j\Phi_{rec} (\vec{r}_j-\vec{r}_i;\alpha)$$ (6)

¹⁰. In this, $q_{i}$ and $q_{j}$ are charges of particles in the system and $\vec{r}_{i}$ and $\vec{r}_{j}$ refer to the respective positions of these charges. $\alpha$ and $\vec{n}$ are the same as referred to in the previous equation.