PME Equations

The PME method replaces the reciprocal term with another set of equations that are lower-order. The direct replacement for $\mathcal{U}_{reciprocal}$ is:

$$\mathcal{U}_{\rm {reciprocal}} \simeq \sum_{k \in C, k \neq... ...(p)} \exp{(2\pi i \vec{k}\cdot \vec{s}_{klm})}\right\vert^{2}$$ (7)

¹¹. This depends on the charge mesh $Q_{H}^{(p)} \exp{(2\pi i \vec{k}\cdot \vec{s}_{klm})}$, and the sum over $C$ indexes over all mesh points. The mesh function is calculated calculated using 8

$$\begin{eqnarray*} \vec{s}_{klm} = & & \sum_{i=1}^{N} \sum_{n_{1},n_{2},n_{3}}... ..._{2i}-k_{2}-n_{2}K_{2}) \\ & & M_{n}(u_{3i}-k_{3}-n_{3}K_{3}) \end{eqnarray*}$$

where $M_n$ is the Cardinal B-Spline of order n given by the recursive function:

$$F = M_{n}(u) = \frac{u}{n-1} M_{n-1}(u) + \frac{n-u}{n-1} M_{n-1}(u-1)$$ (8)

and, for any real number, $u$, $M_{2}(u) = 1-\vert u-1\vert$ for $0\leq u \leq 2$ and $M_{2}(u)=0$ for $u<0$ $u>2$¹² The PME reciprocal space term, see Appendix A.5: recpTerm.F for code, calculates the reciprocal contribution using the Particle Mesh Ewald Method as discussed earlier in Section 2.3.