WIP Ewald 2D in 3D general z != 0 constant factors.
Former-commit-id: 787689f357bd8670948ba8ce7d8dc1205ca77d0f
This commit is contained in:
Marek Nečada
parent
975d23b557
commit
97977dbb46
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\begin{eqnarray}
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\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\times\nonumber \\
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& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\nonumber \\
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& = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\sqrt{\pi}2^{n+1}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
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& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\nonumber \\
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& = & -\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
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& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}\label{eq:2D Ewald in 3D long-range part}
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\end{eqnarray}
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For
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\begin_inset Formula $z\ne0$
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\begin{align*}
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& =-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\\
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& \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{n-\left|m\right|}\frac{\Delta_{npq}}{j!}\left(-1\right)^{j}\left(\gamma_{pq}\right)^{2j-1}\sum_{s\overset{*}{=}j}^{\min(2j,n-\left|m\right|)}\binom{j}{2j-s}\frac{\left(-\kappa z\right)^{2j-s}\left(\beta_{pq}/k\right)^{n-s}}{\left(\frac{1}{2}\left(n-m-s\right)\right)!\left(\frac{1}{2}\left(n+m-s\right)\right)!}
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@ -0,0 +1,375 @@
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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Here and in Kambe's papers,
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\begin_inset Formula $\kappa$
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\end_inset
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is the wavenumber (
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\begin_inset Formula $k$
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\end_inset
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in Linton).
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Here
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is a point of the reciprocal lattice (
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)
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Gammas
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For
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positive,
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\begin_inset Formula
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\[
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\Kambe{\Gamma_{p}}\equiv\begin{cases}
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\sqrt{\kappa^{2}-\left|\vect K_{p}\right|^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}>0\\
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i\sqrt{\left|\vect K_{p}\right|^{2}-\kappa^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}<0
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\]
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\[
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\Linton{\gamma_{\mu}}\equiv\begin{cases}
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\sqrt{\left(\frac{\vect K_{p}}{\kappa}\right)^{2}-1} & \kappa-\left|\vect K_{p}\right|\le0\\
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-i\sqrt{1-\left(\frac{\vect K_{p}}{\kappa}\right)^{2}} & \kappa-\left|\vect K_{p}\right|>0
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\end{cases}
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\]
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\end_inset
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hence
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\begin_inset Formula
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\[
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\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}},
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\]
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\begin_inset Formula
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\[
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\Linton{\gamma_{\mu}}=i\frac{\Kambe{\Gamma_{p}}}{\kappa}.
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\]
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\end_inset
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\begin_layout Section
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D vs sigma
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\begin_layout Standard
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In-plane sums [Linton 2009, (4.5)], replacing
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\begin_inset Formula $n,m\rightarrow L,M$
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\end_inset
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,
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\begin_inset Formula $k\rightarrow\kappa$
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\lang english
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\begin_inset Formula
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\begin{eqnarray*}
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\sigma_{L}^{M(1)} & = & -\frac{i^{L+1}}{2\kappa^{2}\mathscr{A}}\left(-1\right)^{\left(L+M\right)/2}\sqrt{\left(2L+1\right)\left(L-M\right)!\left(L+M\right)!}\times\\
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& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(L-\left|M\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2\kappa\right)^{L-2j}e^{iM\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(L-M\right)-j\right)!\left(\frac{1}{2}\left(L+M\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}
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\end{eqnarray*}
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\end_inset
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[Kambe II, (3.17)], replacing
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\lang finnish
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\begin_inset Formula $n\rightarrow j$
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\end_inset
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\lang english
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,
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\lang finnish
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\begin_inset Formula $A\rightarrow\mathscr{A}$
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\end_inset
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,
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\begin_inset Formula $\vect K_{pt}\to\vect K_{p}$
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\end_inset
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,
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\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,e^{-i\pi}\Gamma_{p}^{2}\omega/2\right)\to\Gamma_{j,p}$
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\end_inset
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and performing little typographic modifications
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\lang english
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\begin_inset Formula
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\begin{align*}
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D_{LM} & =-\frac{1}{\mathscr{A}\kappa}i^{\left|M\right|+1}2^{-L}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
|
||||
& \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ijt}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(\Gamma_{p}/\kappa\right)^{2j-1}\left(K_{p}/\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
Using the relations between
|
||||
\begin_inset Formula $\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}}$
|
||||
\end_inset
|
||||
|
||||
, we have (also, we replace the
|
||||
\begin_inset Formula $\mu$
|
||||
\end_inset
|
||||
|
||||
index with
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
)
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
D_{LM} & =-\frac{1}{\mathscr{A}\kappa}i^{\left|M\right|+1}2^{-L}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
|
||||
& \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ijt}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(-i\gamma_{p}\right)^{2j-1}\left(K_{p}/\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
and now, trying to make the exponents look the same as in Linton,
|
||||
\begin_inset Formula $2^{-1}2^{2j-L}2^{1-2j}=2^{-L}$
|
||||
\end_inset
|
||||
|
||||
(OK),
|
||||
\begin_inset Formula $K_{p}^{L-2j}=K_{p}^{L-2j}$
|
||||
\end_inset
|
||||
|
||||
(OK),
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
D_{LM} & =-\frac{1}{2\kappa\mathscr{A}}i^{\left|M\right|+1}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
|
||||
& \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ij}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(-i\right)^{2j-1}\left(K_{p}/2\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}\left(\frac{\gamma_{p}}{2}\right)^{2j-1}
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
There are now these differences left:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
|
||||
\lang english
|
||||
Additional
|
||||
\begin_inset Formula $\kappa$
|
||||
\end_inset
|
||||
|
||||
factor in
|
||||
\begin_inset Formula $D_{LM}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
|
||||
\lang english
|
||||
\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}\left(-1\right)^{j}$
|
||||
\end_inset
|
||||
|
||||
vs.
|
||||
|
||||
\begin_inset Formula $i^{\left|M\right|+1}\left(-i\right)^{2j-1}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
|
||||
\lang english
|
||||
Opposite phase in the angular part.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
|
||||
\lang english
|
||||
Plane wave factor in
|
||||
\begin_inset Formula $D_{LM}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
Let's look at the
|
||||
\begin_inset Formula $i,-1$
|
||||
\end_inset
|
||||
|
||||
factors (note that
|
||||
\begin_inset Formula $L+M$
|
||||
\end_inset
|
||||
|
||||
is odd):
|
||||
\begin_inset Formula $\left(-i\right)^{2j}=\left(-1\right)^{j},$
|
||||
\end_inset
|
||||
|
||||
leaving
|
||||
\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}$
|
||||
\end_inset
|
||||
|
||||
vs.
|
||||
|
||||
\begin_inset Formula $i^{\left|M\right|+1}i$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
So there is might be a phase difference due to different conventions, but
|
||||
it does not depend on
|
||||
\begin_inset Formula $j$
|
||||
\end_inset
|
||||
|
||||
, so one should be able to transplant the
|
||||
\begin_inset Formula $z\ne0$
|
||||
\end_inset
|
||||
|
||||
sum from Kambe without major problems.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Ewald parameter (integration limits)
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
\end_document
|
||||
41
qpms/ewald.c
41
qpms/ewald.c
|
|
@ -26,11 +26,12 @@
|
|||
#endif
|
||||
|
||||
#ifndef M_SQRTPI
|
||||
#define M_SQRTPI 1.7724538509055160272981674833411452
|
||||
#define M_SQRTPI 1.7724538509055160272981674833411452L
|
||||
#endif
|
||||
|
||||
|
||||
// sloppy implementation of factorial
|
||||
// We prefer to avoid tgamma/lgamma, as their errors are about 4+ bits
|
||||
static inline double factorial(const int n) {
|
||||
assert(n >= 0);
|
||||
if (n < 0)
|
||||
|
|
@ -45,6 +46,44 @@ static inline double factorial(const int n) {
|
|||
return tgamma(n + 1); // hope it's precise and that overflow does not happen
|
||||
}
|
||||
|
||||
// sloppy implementation of double factorial n!!
|
||||
static inline double double_factorial(int n) {
|
||||
assert(n >= 0);
|
||||
if (n <= 25) {
|
||||
double fac = 1;
|
||||
while (n > 0) {
|
||||
fac *= n;
|
||||
n -= 2;
|
||||
}
|
||||
return fac;
|
||||
} else {
|
||||
if (n % 2) { // odd, (2*k - 1)!! = 2**k * Γ(k + 0.5) / sqrt(п)
|
||||
const int k = n / 2 + 1;
|
||||
return pow(2, k) * tgamma(k + 0.5) / M_SQRTPI;
|
||||
} else { // even, n!! = 2**(n/2) * (n/2)!
|
||||
const int k = n/2;
|
||||
return pow(2, k) * factorial(k);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// sloppy implementation of (n/2)! = Γ(n/2 + 1)
|
||||
// It is _slightly_ more precise than direct call of tgamma for small odd n
|
||||
static inline double factorial_of_half(const int n2) {
|
||||
assert(n2 >= 0);
|
||||
if (n2 % 2 == 0) return factorial(n2/2);
|
||||
else {
|
||||
if (n2 <= 50) { // odd, use (k - 0.5)! = Γ(k + 0.5) = 2**(-k) (2*k - 1)!! sqrt(п) for small n2
|
||||
const int k = n2 / 2 + 1;
|
||||
double fac2 = 1;
|
||||
for(int j = 2*k - 1; j > 0; j -= 2)
|
||||
fac2 *= j;
|
||||
return fac2 * pow(2, -k) * M_SQRTPI;
|
||||
}
|
||||
else return tgamma(1. + 0.5*n2);
|
||||
}
|
||||
}
|
||||
|
||||
static inline complex double csq(complex double x) { return x * x; }
|
||||
static inline double sq(double x) { return x * x; }
|
||||
|
||||
|
|
|
|||
28
qpms/ewald.h
28
qpms/ewald.h
|
|
@ -80,6 +80,34 @@ typedef struct qpms_ewald3_constants_t {
|
|||
* 2D sum with additional factor of
|
||||
* \f$ \sqrt{pi} \kappa \gamma(\abs{\vect{k}+\vect{K}}/\kappa) \f$.
|
||||
*/
|
||||
|
||||
///=============== NEW GENERATION GENERAL 2D-IN-3D, including z != 0 =========================
|
||||
|
||||
// TODO indexing mechanisms
|
||||
|
||||
/// The constant factors for the long range part of a 2D Ewald sum.
|
||||
complex double **S1_constfacs; // indices [y][j] where j is same as in [1, (4.5)]
|
||||
/* These are the actual numbers now: (in the EWALD32_CONSTANTS_AGNOSTIC version)
|
||||
* for m + n EVEN:
|
||||
*
|
||||
* s1_constfacs[y(m,n)][x(j,s)] =
|
||||
*
|
||||
* -2 * I**(n+1) * sqrt(π) * ((n-m)/2)! * ((n+m)/2)! * (-1)**j / j \
|
||||
* ----------------------------------------------------------- | |
|
||||
* j! * ((n - m - s)/2)! * ((n + m - s)/2)! \ 2j - s /
|
||||
*
|
||||
* for m + n ODD:
|
||||
*
|
||||
* s1_constfacs[y(m,n)][j] = 0
|
||||
*/
|
||||
complex double *S1_constfacs_base; ///< Internal pointer holding memory for the 2D Ewald sum constant factors.
|
||||
/// The constant factors for the long range part of a 1D Ewald sum along the \a z axis.
|
||||
/** If the summation points lie along a different direction, use the formula for
|
||||
* 2D sum with additional factor of
|
||||
* \f$ \sqrt{pi} \kappa \gamma(\abs{\vect{k}+\vect{K}}/\kappa) \f$.
|
||||
*/
|
||||
|
||||
|
||||
complex double **s1_constfacs_1Dz;
|
||||
/* These are the actual numbers now:
|
||||
* s1_constfacs_1Dz[n][j] =
|
||||
|
|
|
|||
Loading…
Reference in New Issue