3066:
2662:
3962:. To visualize this most clearly, think of a unit cube; the upper face is effectively in contact with the lower face, the right with the left face, and the front with the back face. As a result, the unit cell size must be carefully chosen to be large enough to avoid improper motion correlations between two faces "in contact", but still small enough to be computationally feasible. The definition of the cutoff between short- and long-range interactions can also introduce artifacts.
3061:{\displaystyle E_{\ell r}=\int {\frac {d\mathbf {k} }{\left(2\pi \right)^{3}}}\ {\tilde {\rho }}_{\text{TOT}}^{*}(\mathbf {k} ){\tilde {V}}(\mathbf {k} )=\int {\frac {d\mathbf {k} }{\left(2\pi \right)^{3}}}{\tilde {L}}^{*}(\mathbf {k} )\left|{\tilde {\rho }}_{uc}(\mathbf {k} )\right|^{2}{\tilde {\Phi }}(\mathbf {k} )={\frac {1}{\Omega }}\sum _{m_{1},m_{2},m_{3}}\left|{\tilde {\rho }}_{uc}(\mathbf {k} )\right|^{2}{\tilde {\Phi }}(\mathbf {k} )}
4014:, i.e. depends on the order of the summation. For example, if the dipole-dipole interactions of a central unit cell with unit cells located on an ever-increasing cube, the energy converges to a different value than if the interaction energies had been summed spherically. Roughly speaking, this conditional convergence arises because (1) the number of interacting dipoles on a shell of radius
1025:
4825:
1898:
452:
769:
2332:
1539:
4646:
4388:
3957:
simulations this is normally accomplished by deliberately constructing a charge-neutral unit cell that can be infinitely "tiled" to form images; however, to properly account for the effects of this approximation, these images are reincorporated back into the original simulation cell. The overall
66:
of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge
695:
2460:
1698:
302:
3801:
188:
3436:
2101:
1999:
2184:
3965:
The restriction of the density field to a mesh makes the PME method more efficient for systems with "smooth" variations in density, or continuous potential functions. Localized systems or those with large fluctuations in density may be treated more efficiently with the
4149:
This somewhat surprising result can be reconciled with the finite energy of real crystals because such crystals are not infinite, i.e. have a particular boundary. More specifically, the boundary of a polar crystal has an effective surface charge density on its surface
1693:
1351:
2560:
3952:
Due to the periodicity assumption implicit in Ewald summation, applications of the PME method to physical systems require the imposition of periodic symmetry. Thus, the method is best suited to systems that can be simulated as infinite in spatial extent. In
4253:
3550:
1020:{\displaystyle \rho _{\text{TOT}}(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n_{1},n_{2},n_{3}}\sum _{\mathrm {charges} \ k}q_{k}\delta (\mathbf {r} -\mathbf {r} _{k}-n_{1}\mathbf {a} _{1}-n_{2}\mathbf {a} _{2}-n_{3}\mathbf {a} _{3})}
67:
neutrality of the molecular system to accurately calculate the total
Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.
280:. The method assumes that the short-range part can be summed easily; hence, the problem becomes the summation of the long-range term. Due to the use of the Fourier sum, the method implicitly assumes that the system under study is infinitely
2651:
3165:
3900:
part). Since both summations converge quickly in their respective spaces (real and
Fourier), they may be truncated with little loss of accuracy and great improvement in required computational time. To evaluate the Fourier transform
554:
2339:
3675:
4820:{\displaystyle d\mathbf {E} \ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {-1}{4\pi \epsilon }}\right){\frac {dq\ \mathbf {r} }{r^{3}}}=\left({\frac {-1}{4\pi \epsilon }}\right){\frac {\sigma \,dS\ \mathbf {r} }{r^{3}}}}
3688:
78:
1592:
4637:
3329:
2465:
2004:
1903:
4550:
4184:
1893:{\displaystyle {\tilde {L}}(\mathbf {k} )={\frac {\left(2\pi \right)^{3}}{\Omega }}\sum _{m_{1},m_{2},m_{3}}\delta (\mathbf {k} -m_{1}\mathbf {b} _{1}-m_{2}\mathbf {b} _{2}-m_{3}\mathbf {b} _{3})}
447:{\displaystyle E_{\ell r}=\iint d\mathbf {r} \,d\mathbf {r} ^{\prime }\,\rho _{\text{TOT}}(\mathbf {r} )\rho _{uc}(\mathbf {r} ^{\prime })\ \varphi _{\ell r}(\mathbf {r} -\mathbf {r} ^{\prime })}
3218:
4144:
1587:
1268:
737:
270:
3939:
3882:
3441:
2327:{\displaystyle v(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} ^{\prime }\,\rho _{uc}(\mathbf {r} ^{\prime })\ \varphi _{\ell r}(\mathbf {r} -\mathbf {r} ^{\prime })}
229:
2176:
3842:
295:
The long-range interaction energy is the sum of interaction energies between the charges of a central unit cell and all the charges of the lattice. Hence, it can be represented as a
1313:
493:
2571:
1058:
3242:
is straightforward and should converge quickly. The most common reason for lack of convergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums.
1534:{\displaystyle L(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n_{1},n_{2},n_{3}}\delta (\mathbf {r} -n_{1}\mathbf {a} _{1}-n_{2}\mathbf {a} _{2}-n_{3}\mathbf {a} _{3})}
4420:
4008:
1149:
1120:
1091:
522:
2136:
1347:
4575:
4970:
4847:
4495:
4228:
4206:
3240:
3585:
4911:
4450:
4096:
3580:
3305:
3073:
1230:
1203:
1176:
764:
549:
5048:
Di Pierro, M.; Elber, R.; Leimkuhler, B. (2015), "A Stochastic
Algorithm for the Isobaric-Isothermal Ensemble with Ewald Summations for all Long Range Forces.",
4473:
4059:
272:
represents the long-range term whose sum quickly converges in
Fourier (reciprocal) space. The long-ranged part should be finite for all arguments (most notably
4931:
4383:{\displaystyle U={\frac {1}{2V_{uc}}}\int {\frac {\left(\mathbf {p} _{uc}\cdot \mathbf {r} \right)\left(\mathbf {p} _{uc}\cdot \mathbf {n} \right)}{r^{3}}}\,dS}
4248:
4032:
5050:
54:. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a
690:{\displaystyle \rho _{uc}(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{\mathrm {charges} \ k}q_{k}\delta (\mathbf {r} -\mathbf {r} _{k})}
5302:
2455:{\displaystyle {\tilde {V}}(\mathbf {k} )\ {\stackrel {\mathrm {def} }{=}}\ {\tilde {\rho }}_{uc}(\mathbf {k} ){\tilde {\Phi }}(\mathbf {k} )}
5307:
5138:
5219:"New tricks for modelers from the crystallography toolkit: the particle mesh Ewald algorithm and its use in nucleic acid simulations"
3796:{\displaystyle E_{\ell r}=\sum _{\mathbf {k} }{\tilde {\Phi }}_{\ell r}(\mathbf {k} )\left|{\tilde {\rho }}(\mathbf {k} )\right|^{2}}
3438:. The basic idea of particle mesh Ewald summation is to replace the direct summation of interaction energies between point particles
183:{\displaystyle \varphi (\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \varphi _{sr}(\mathbf {r} )+\varphi _{\ell r}(\mathbf {r} ),}
4580:
3431:{\displaystyle \varphi (\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \varphi _{sr}(\mathbf {r} )+\varphi _{\ell r}(\mathbf {r} )}
2096:{\displaystyle \Omega \ {\stackrel {\mathrm {def} }{=}}\ \mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)}
1994:{\displaystyle \mathbf {b} _{1}\ {\stackrel {\mathrm {def} }{=}}\ 2\pi {\frac {\mathbf {a} _{2}\times \mathbf {a} _{3}}{\Omega }}}
5292:
3326:
In the particle mesh method, just as in standard Ewald summation, the generic interaction potential is separated into two terms
4500:
5297:
4153:
284:(a sensible assumption for the interiors of crystals). One repeating unit of this hypothetical periodic system is called a
5021:
Kolafa, Jiri; Perram, John W. (September 1992). "Cutoff Errors in the Ewald
Summation Formulae for Point Charge Systems".
3173:
1688:{\displaystyle {\tilde {\rho }}_{\text{TOT}}(\mathbf {k} )={\tilde {L}}(\mathbf {k} ){\tilde {\rho }}_{uc}(\mathbf {k} )}
3959:
281:
1554:
1235:
704:
58:. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a
4497:
is the vector from the central unit cell to the infinitesimal area. This formula results from integrating the energy
4101:
234:
3904:
3847:
2555:{\displaystyle {\tilde {V}}(\mathbf {k} )=\int d\mathbf {r} \ v(\mathbf {r} )\ e^{-i\mathbf {k} \cdot \mathbf {r} }}
5287:
5282:
5277:
4991:
193:
63:
55:
47:
35:
2141:
4011:
3808:
1277:
457:
3308:
43:
1032:
5094:
Herce, HD; Garcia, AE; Darden, T (28 March 2007). "The electrostatic surface term: (I) periodic systems".
3942:
1548:
38:) in periodic systems. It was first developed as the method for calculating the electrostatic energies of
299:
integral over two charge density fields representing the fields of the unit cell and the crystal lattice
3967:
277:
4393:
3981:
3545:{\displaystyle E_{\text{TOT}}=\sum _{i,j}\varphi (\mathbf {r} _{j}-\mathbf {r} _{i})=E_{sr}+E_{\ell r}}
1125:
1096:
1067:
498:
5197:
5150:
5103:
2110:
1321:
1061:
4555:
3259:
3251:
4936:
4830:
4478:
4211:
4189:
3223:
5248:
4996:
3954:
3945:, which requires that the density field be evaluated on a discrete lattice in space (this is the
3263:
2656:
288:. One such cell is chosen as the "central cell" for reference and the remaining cells are called
2646:{\displaystyle E_{\ell r}=\int d\mathbf {r} \ \rho _{\text{TOT}}(\mathbf {r} )\ v(\mathbf {r} )}
50:, replacing the summation of interaction energies in real space with an equivalent summation in
2336:
Since this is also a convolution, the
Fourier transformation of the same equation is a product
5240:
5166:
5119:
5077:
4986:
4880:
4862:
3885:
3160:{\displaystyle \mathbf {k} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
59:
5230:
5205:
5158:
5111:
5067:
5059:
5030:
4981:
4858:
4640:
4425:
3555:
3312:
3280:
31:
4064:
1208:
1181:
1154:
742:
527:
4874:
4577:
represents the infinitesimal electric field generated by an infinitesimal surface charge
5201:
5154:
5107:
4455:
1695:
where the
Fourier transform of the lattice function is another sum over delta functions
5072:
5001:
4916:
4233:
4037:
4017:
3893:
3311:
in order to eliminate artifacts due to truncation. Applications include simulations of
3274:
2104:
5235:
5218:
4861:
in 1921 (see
References below) to determine the electrostatic energy (and, hence, the
5271:
4250:
of the dipole in a central unit cell with that surface charge density can be written
51:
39:
5252:
17:
3670:{\displaystyle E_{sr}=\sum _{i,j}\varphi _{sr}(\mathbf {r} _{j}-\mathbf {r} _{i})}
1544:
1271:
276: = 0) but may have any convenient mathematical form, most typically a
5034:
3978:
The electrostatic energy of a polar crystal (i.e. a crystal with a net dipole
231:
represents the short-range term whose sum quickly converges in real space and
5209:
5170:
5063:
5139:"Particle mesh Ewald: An N ⋅log( N ) method for Ewald sums in large systems"
3889:
3320:
75:
Ewald summation rewrites the interaction potential as the sum of two terms,
5244:
5123:
5081:
3258:. However, the Ewald method has enjoyed widespread use since the 1970s in
3255:
3270:
5115:
3262:
of particle systems, especially those whose particles interact via an
42:, and is now commonly used for calculating long-range interactions in
5162:
3316:
5185:
4230:
represents the net dipole moment per volume. The interaction energy
4061:; (2) the strength of a single dipole-dipole interaction falls like
5261:
Understanding molecular simulation: from algorithms to applications
3266:
5186:"Die Berechnung optischer und elektrostatischer Gitterpotentiale"
2103:
is the volume of the central unit cell (if it is geometrically a
2107:, which is often but not necessarily the case). Note that both
4632:{\displaystyle dq\ {\stackrel {\mathrm {def} }{=}}\ \sigma dS}
4933:
is the number of atoms in the system. The PME method gives
4873:
Generally, different Ewald summation methods give different
2316:
2274:
2245:
436:
394:
344:
3685:) and a summation in Fourier space of the long-ranged part
2181:
For brevity, define an effective single-particle potential
34:, is a method for computing long-range interactions (e.g.
4545:{\displaystyle dU=-\mathbf {p} _{uc}\cdot d\mathbf {E} }
5137:
Darden, Tom; York, Darrin; Pedersen, Lee (1993-06-15).
4452:
are the net dipole moment and volume of the unit cell,
4179:{\displaystyle \sigma =\mathbf {P} \cdot \mathbf {n} }
4104:
4067:
4040:
3941:
of the charge density field efficiently, one uses the
4939:
4919:
4883:
4849:, which points towards the charge, not away from it.
4833:
4649:
4583:
4558:
4503:
4481:
4458:
4428:
4396:
4256:
4236:
4214:
4192:
4156:
4020:
3984:
3907:
3850:
3811:
3691:
3588:
3558:
3444:
3332:
3283:
3226:
3176:
3076:
2665:
2574:
2468:
2342:
2187:
2144:
2113:
2007:
1906:
1701:
1595:
1557:
1354:
1324:
1280:
1238:
1211:
1184:
1157:
1128:
1099:
1070:
1035:
772:
745:
707:
557:
530:
501:
460:
305:
237:
196:
81:
4475:
is an infinitesimal area on the crystal surface and
3277:. Recently, PME has also been used to calculate the
3213:{\displaystyle {\tilde {\rho }}_{uc}(\mathbf {k} )}
4964:
4925:
4905:
4841:
4819:
4631:
4569:
4544:
4489:
4467:
4444:
4414:
4382:
4242:
4222:
4200:
4178:
4138:
4090:
4053:
4026:
4002:
3933:
3876:
3836:
3795:
3669:
3574:
3544:
3430:
3299:
3234:
3212:
3159:
3060:
2645:
2554:
2454:
2326:
2170:
2130:
2095:
1993:
1892:
1687:
1581:
1533:
1341:
1307:
1262:
1224:
1197:
1170:
1143:
1114:
1085:
1052:
1019:
758:
731:
689:
543:
516:
487:
446:
264:
223:
182:
5217:Darden, T; Perera, L; Li, L; Pedersen, L (1999).
4827:The negative sign derives from the definition of
2659:, the energy can also be summed in Fourier space
4139:{\textstyle \sum _{n=1}^{\infty }{\frac {1}{n}}}
1582:{\displaystyle \rho _{\text{TOT}}(\mathbf {r} )}
1263:{\displaystyle \rho _{\text{TOT}}(\mathbf {r} )}
732:{\displaystyle \rho _{\text{TOT}}(\mathbf {r} )}
1900:where the reciprocal space vectors are defined
265:{\displaystyle \varphi _{\ell r}(\mathbf {r} )}
3934:{\displaystyle {\tilde {\rho }}(\mathbf {k} )}
3877:{\displaystyle {\tilde {\rho }}(\mathbf {k} )}
3220:is calculated, the summation/integration over
3250:Ewald summation was developed as a method in
8:
3582:of the short-ranged potential in real space
62:. The advantage of this method is the rapid
739:is the same sum over the unit-cell charges
224:{\displaystyle \varphi _{sr}(\mathbf {r} )}
46:. Ewald summation is a special case of the
5051:Journal of Chemical Theory and Computation
2171:{\displaystyle {\tilde {L}}(\mathbf {k} )}
1232:range over all integers. The total field
5234:
5071:
4949:
4938:
4918:
4894:
4882:
4834:
4832:
4809:
4799:
4789:
4783:
4756:
4741:
4731:
4719:
4692:
4670:
4669:
4664:
4662:
4661:
4653:
4648:
4602:
4601:
4596:
4594:
4593:
4582:
4562:
4557:
4537:
4522:
4517:
4502:
4482:
4480:
4457:
4433:
4427:
4403:
4398:
4395:
4373:
4365:
4350:
4338:
4333:
4317:
4305:
4300:
4291:
4276:
4263:
4255:
4235:
4215:
4213:
4193:
4191:
4171:
4163:
4155:
4126:
4120:
4109:
4103:
4081:
4076:
4071:
4066:
4045:
4039:
4019:
3991:
3986:
3983:
3923:
3909:
3908:
3906:
3866:
3852:
3851:
3849:
3837:{\displaystyle {\tilde {\Phi }}_{\ell r}}
3825:
3814:
3813:
3810:
3787:
3774:
3760:
3759:
3745:
3733:
3722:
3721:
3713:
3712:
3696:
3690:
3658:
3653:
3643:
3638:
3625:
3609:
3593:
3587:
3563:
3557:
3533:
3517:
3501:
3496:
3486:
3481:
3462:
3449:
3443:
3420:
3408:
3393:
3381:
3359:
3358:
3353:
3351:
3350:
3339:
3331:
3288:
3282:
3227:
3225:
3202:
3190:
3179:
3178:
3175:
3151:
3146:
3139:
3126:
3121:
3114:
3101:
3096:
3089:
3077:
3075:
3050:
3036:
3035:
3029:
3016:
3004:
2993:
2992:
2977:
2964:
2951:
2946:
2932:
2921:
2907:
2906:
2900:
2887:
2875:
2864:
2863:
2848:
2839:
2828:
2827:
2818:
2795:
2789:
2775:
2761:
2760:
2752:
2743:
2738:
2727:
2726:
2714:
2691:
2685:
2670:
2664:
2635:
2618:
2609:
2597:
2579:
2573:
2545:
2537:
2530:
2515:
2501:
2484:
2470:
2469:
2467:
2444:
2430:
2429:
2421:
2409:
2398:
2397:
2378:
2377:
2372:
2370:
2369:
2358:
2344:
2343:
2341:
2315:
2310:
2301:
2289:
2273:
2268:
2255:
2250:
2244:
2239:
2214:
2213:
2208:
2206:
2205:
2194:
2186:
2160:
2146:
2145:
2143:
2120:
2112:
2082:
2077:
2067:
2062:
2047:
2042:
2023:
2022:
2017:
2015:
2014:
2006:
1979:
1974:
1964:
1959:
1955:
1931:
1930:
1925:
1923:
1922:
1913:
1908:
1905:
1881:
1876:
1869:
1856:
1851:
1844:
1831:
1826:
1819:
1807:
1793:
1780:
1767:
1762:
1747:
1728:
1717:
1703:
1702:
1700:
1677:
1665:
1654:
1653:
1644:
1630:
1629:
1618:
1609:
1598:
1597:
1594:
1571:
1562:
1556:
1522:
1517:
1510:
1497:
1492:
1485:
1472:
1467:
1460:
1448:
1434:
1421:
1408:
1403:
1381:
1380:
1375:
1373:
1372:
1361:
1353:
1331:
1323:
1297:
1285:
1279:
1252:
1243:
1237:
1216:
1210:
1189:
1183:
1162:
1156:
1135:
1130:
1127:
1106:
1101:
1098:
1077:
1072:
1069:
1042:
1034:
1008:
1003:
996:
983:
978:
971:
958:
953:
946:
933:
928:
919:
907:
872:
871:
859:
846:
833:
828:
806:
805:
800:
798:
797:
786:
777:
771:
750:
744:
721:
712:
706:
678:
673:
664:
652:
617:
616:
594:
593:
588:
586:
585:
574:
562:
556:
535:
529:
508:
503:
500:
477:
465:
459:
454:where the unit-cell charge density field
435:
430:
421:
409:
393:
388:
375:
363:
354:
349:
343:
338:
333:
328:
310:
304:
254:
242:
236:
213:
201:
195:
169:
157:
142:
130:
108:
107:
102:
100:
99:
88:
80:
1308:{\displaystyle \rho _{uc}(\mathbf {r} )}
488:{\displaystyle \rho _{uc}(\mathbf {r} )}
5013:
2462:where the Fourier transform is defined
4857:The Ewald summation was developed by
4098:; and (3) the mathematical summation
1053:{\displaystyle \delta (\mathbf {x} )}
7:
5259:Frenkel, D., & Smit, B. (2001).
3170:This is the essential result. Once
2564:The energy can now be written as a
4677:
4674:
4671:
4609:
4606:
4603:
4121:
3816:
3724:
3552:with two summations, a direct sum
3366:
3363:
3360:
3038:
2937:
2909:
2432:
2385:
2382:
2379:
2221:
2218:
2215:
2030:
2027:
2024:
2008:
1986:
1938:
1935:
1932:
1753:
1388:
1385:
1382:
891:
888:
885:
882:
879:
876:
873:
813:
810:
807:
636:
633:
630:
627:
624:
621:
618:
601:
598:
595:
115:
112:
109:
25:
4415:{\displaystyle \mathbf {p} _{uc}}
4208:is the surface normal vector and
4003:{\displaystyle \mathbf {p} _{uc}}
4835:
4800:
4732:
4654:
4563:
4538:
4518:
4483:
4399:
4351:
4334:
4318:
4301:
4216:
4194:
4172:
4164:
3987:
3924:
3867:
3775:
3746:
3714:
3654:
3639:
3497:
3482:
3421:
3394:
3340:
3246:Particle mesh Ewald (PME) method
3228:
3203:
3147:
3122:
3097:
3078:
3051:
3017:
2922:
2888:
2849:
2796:
2776:
2753:
2692:
2636:
2619:
2598:
2546:
2538:
2516:
2502:
2485:
2445:
2422:
2359:
2311:
2302:
2269:
2240:
2195:
2161:
2121:
2078:
2063:
2043:
2001:(and cyclic permutations) where
1975:
1960:
1909:
1877:
1852:
1827:
1808:
1718:
1678:
1645:
1619:
1572:
1518:
1493:
1468:
1449:
1362:
1332:
1298:
1253:
1144:{\displaystyle \mathbf {a} _{3}}
1131:
1115:{\displaystyle \mathbf {a} _{2}}
1102:
1086:{\displaystyle \mathbf {a} _{1}}
1073:
1043:
1004:
979:
954:
929:
920:
787:
722:
674:
665:
575:
517:{\displaystyle \mathbf {r} _{k}}
504:
478:
431:
422:
389:
364:
339:
329:
255:
214:
170:
143:
89:
5143:The Journal of Chemical Physics
5096:The Journal of Chemical Physics
2131:{\displaystyle L(\mathbf {r} )}
1342:{\displaystyle L(\mathbf {r} )}
5303:Computational electromagnetics
4959:
4943:
4900:
4887:
3928:
3920:
3914:
3871:
3863:
3857:
3819:
3779:
3771:
3765:
3750:
3742:
3727:
3664:
3634:
3507:
3477:
3425:
3417:
3398:
3390:
3344:
3336:
3207:
3199:
3184:
3055:
3047:
3041:
3021:
3013:
2998:
2926:
2918:
2912:
2892:
2884:
2869:
2853:
2845:
2833:
2780:
2772:
2766:
2757:
2749:
2732:
2640:
2632:
2623:
2615:
2520:
2512:
2489:
2481:
2475:
2449:
2441:
2435:
2426:
2418:
2403:
2363:
2355:
2349:
2321:
2298:
2279:
2264:
2199:
2191:
2165:
2157:
2151:
2125:
2117:
1887:
1804:
1722:
1714:
1708:
1682:
1674:
1659:
1649:
1641:
1635:
1623:
1615:
1603:
1576:
1568:
1528:
1445:
1366:
1358:
1336:
1328:
1302:
1294:
1257:
1249:
1047:
1039:
1014:
916:
791:
783:
726:
718:
684:
661:
579:
571:
482:
474:
441:
418:
399:
384:
368:
360:
259:
251:
218:
210:
174:
166:
147:
139:
93:
85:
1:
5236:10.1016/S0969-2126(99)80033-1
4570:{\displaystyle d\mathbf {E} }
4965:{\displaystyle O(N\,\log N)}
4842:{\displaystyle \mathbf {r} }
4490:{\displaystyle \mathbf {r} }
4223:{\displaystyle \mathbf {P} }
4201:{\displaystyle \mathbf {n} }
3254:, long before the advent of
3235:{\displaystyle \mathbf {k} }
1151:are the lattice vectors and
495:is a sum over the positions
5308:Series acceleration methods
4877:. Direct calculation gives
3960:periodic boundary condition
5324:
3970:of Greengard and Rokhlin.
2178:are real, even functions.
766:and their periodic images
36:electrostatic interactions
5035:10.1080/08927029208049126
4992:Poisson summation formula
551:in the central unit cell
48:Poisson summation formula
5210:10.1002/andp.19213690304
5064:10.1021/acs.jctc.5b00648
4906:{\displaystyle O(N^{2})}
4012:conditionally convergent
3167:in the final summation.
1270:can be represented as a
5293:Computational chemistry
3309:Lennard-Jones potential
44:computational chemistry
4966:
4927:
4907:
4843:
4821:
4633:
4571:
4546:
4491:
4469:
4446:
4445:{\displaystyle V_{uc}}
4416:
4384:
4244:
4224:
4202:
4180:
4140:
4125:
4092:
4091:{\textstyle 1/{R^{3}}}
4055:
4028:
4004:
3943:fast Fourier transform
3935:
3878:
3838:
3797:
3671:
3576:
3575:{\displaystyle E_{sr}}
3546:
3432:
3301:
3300:{\displaystyle r^{-6}}
3236:
3214:
3161:
3062:
2647:
2556:
2456:
2328:
2172:
2132:
2097:
1995:
1894:
1689:
1583:
1549:Fourier transformation
1535:
1343:
1309:
1264:
1226:
1199:
1172:
1145:
1116:
1087:
1054:
1021:
760:
733:
691:
545:
518:
489:
448:
266:
225:
184:
5298:Theoretical chemistry
4967:
4928:
4908:
4865:) of ionic crystals.
4844:
4822:
4634:
4572:
4547:
4492:
4470:
4447:
4417:
4385:
4245:
4225:
4203:
4181:
4141:
4105:
4093:
4056:
4029:
4010:in the unit cell) is
4005:
3968:fast multipole method
3936:
3879:
3839:
3798:
3672:
3577:
3547:
3433:
3302:
3237:
3215:
3162:
3063:
2648:
2557:
2457:
2329:
2173:
2133:
2098:
1996:
1895:
1690:
1584:
1536:
1344:
1310:
1265:
1227:
1225:{\displaystyle n_{3}}
1200:
1198:{\displaystyle n_{2}}
1173:
1171:{\displaystyle n_{1}}
1146:
1117:
1088:
1055:
1022:
761:
759:{\displaystyle q_{k}}
734:
701:charge density field
692:
546:
544:{\displaystyle q_{k}}
519:
490:
449:
278:Gaussian distribution
267:
226:
185:
5023:Molecular Simulation
4937:
4917:
4881:
4831:
4647:
4581:
4556:
4501:
4479:
4456:
4426:
4394:
4254:
4234:
4212:
4190:
4154:
4102:
4065:
4038:
4018:
3982:
3905:
3848:
3809:
3689:
3586:
3556:
3442:
3330:
3281:
3260:computer simulations
3224:
3174:
3074:
2663:
2572:
2466:
2340:
2185:
2142:
2111:
2005:
1904:
1699:
1593:
1555:
1352:
1322:
1278:
1236:
1209:
1182:
1155:
1126:
1097:
1068:
1062:Dirac delta function
1033:
770:
743:
705:
555:
528:
499:
458:
303:
235:
194:
79:
5202:1921AnP...369..253E
5155:1993JChPh..9810089D
5149:(12): 10089–10092.
5108:2007JChPh.126l4106H
3958:effect is called a
3683:particle mesh Ewald
3252:theoretical physics
2748:
18:Particle mesh Ewald
4997:Molecular modeling
4962:
4923:
4903:
4839:
4817:
4629:
4567:
4542:
4487:
4468:{\displaystyle dS}
4465:
4442:
4412:
4380:
4240:
4220:
4198:
4176:
4136:
4088:
4054:{\textstyle R^{2}}
4051:
4024:
4000:
3955:molecular dynamics
3931:
3886:Fourier transforms
3874:
3834:
3793:
3719:
3667:
3620:
3572:
3542:
3473:
3428:
3297:
3232:
3210:
3157:
3058:
2984:
2725:
2657:Plancherel theorem
2643:
2552:
2452:
2324:
2168:
2128:
2093:
1991:
1890:
1800:
1685:
1579:
1531:
1441:
1339:
1305:
1260:
1222:
1195:
1168:
1141:
1112:
1083:
1050:
1017:
902:
866:
756:
729:
687:
647:
541:
514:
485:
444:
262:
221:
180:
5263:, Academic press.
5184:Ewald, P (1921).
5116:10.1063/1.2714527
5058:(12): 5624–5637,
4987:Madelung constant
4926:{\displaystyle N}
4875:time complexities
4863:Madelung constant
4815:
4798:
4777:
4747:
4730:
4713:
4687:
4682:
4660:
4619:
4614:
4592:
4371:
4286:
4243:{\displaystyle U}
4134:
4027:{\displaystyle R}
3917:
3860:
3822:
3768:
3730:
3708:
3605:
3458:
3452:
3376:
3371:
3349:
3187:
3044:
3001:
2942:
2940:
2915:
2872:
2836:
2824:
2769:
2741:
2735:
2724:
2720:
2628:
2612:
2604:
2525:
2508:
2478:
2438:
2406:
2395:
2390:
2368:
2352:
2284:
2231:
2226:
2204:
2154:
2040:
2035:
2013:
1989:
1948:
1943:
1921:
1758:
1756:
1711:
1662:
1638:
1612:
1606:
1565:
1399:
1398:
1393:
1371:
1246:
897:
867:
824:
823:
818:
796:
780:
715:
642:
612:
611:
606:
584:
404:
357:
125:
120:
98:
60:Fourier transform
16:(Redirected from
5315:
5288:Fourier analysis
5283:Potential theory
5278:Electromagnetism
5256:
5238:
5213:
5175:
5174:
5163:10.1063/1.464397
5134:
5128:
5127:
5091:
5085:
5084:
5075:
5045:
5039:
5038:
5018:
4982:Paul Peter Ewald
4971:
4969:
4968:
4963:
4932:
4930:
4929:
4924:
4912:
4910:
4909:
4904:
4899:
4898:
4859:Paul Peter Ewald
4848:
4846:
4845:
4840:
4838:
4826:
4824:
4823:
4818:
4816:
4814:
4813:
4804:
4803:
4796:
4784:
4782:
4778:
4776:
4765:
4757:
4748:
4746:
4745:
4736:
4735:
4728:
4720:
4718:
4714:
4712:
4701:
4693:
4685:
4684:
4683:
4681:
4680:
4668:
4663:
4658:
4657:
4638:
4636:
4635:
4630:
4617:
4616:
4615:
4613:
4612:
4600:
4595:
4590:
4576:
4574:
4573:
4568:
4566:
4551:
4549:
4548:
4543:
4541:
4530:
4529:
4521:
4496:
4494:
4493:
4488:
4486:
4474:
4472:
4471:
4466:
4451:
4449:
4448:
4443:
4441:
4440:
4421:
4419:
4418:
4413:
4411:
4410:
4402:
4389:
4387:
4386:
4381:
4372:
4370:
4369:
4360:
4359:
4355:
4354:
4346:
4345:
4337:
4326:
4322:
4321:
4313:
4312:
4304:
4292:
4287:
4285:
4284:
4283:
4264:
4249:
4247:
4246:
4241:
4229:
4227:
4226:
4221:
4219:
4207:
4205:
4204:
4199:
4197:
4185:
4183:
4182:
4177:
4175:
4167:
4145:
4143:
4142:
4137:
4135:
4127:
4124:
4119:
4097:
4095:
4094:
4089:
4087:
4086:
4085:
4075:
4060:
4058:
4057:
4052:
4050:
4049:
4033:
4031:
4030:
4025:
4009:
4007:
4006:
4001:
3999:
3998:
3990:
3940:
3938:
3937:
3932:
3927:
3919:
3918:
3910:
3883:
3881:
3880:
3875:
3870:
3862:
3861:
3853:
3843:
3841:
3840:
3835:
3833:
3832:
3824:
3823:
3815:
3802:
3800:
3799:
3794:
3792:
3791:
3786:
3782:
3778:
3770:
3769:
3761:
3749:
3741:
3740:
3732:
3731:
3723:
3718:
3717:
3704:
3703:
3676:
3674:
3673:
3668:
3663:
3662:
3657:
3648:
3647:
3642:
3633:
3632:
3619:
3601:
3600:
3581:
3579:
3578:
3573:
3571:
3570:
3551:
3549:
3548:
3543:
3541:
3540:
3525:
3524:
3506:
3505:
3500:
3491:
3490:
3485:
3472:
3454:
3453:
3450:
3437:
3435:
3434:
3429:
3424:
3416:
3415:
3397:
3389:
3388:
3374:
3373:
3372:
3370:
3369:
3357:
3352:
3347:
3343:
3306:
3304:
3303:
3298:
3296:
3295:
3241:
3239:
3238:
3233:
3231:
3219:
3217:
3216:
3211:
3206:
3198:
3197:
3189:
3188:
3180:
3166:
3164:
3163:
3158:
3156:
3155:
3150:
3144:
3143:
3131:
3130:
3125:
3119:
3118:
3106:
3105:
3100:
3094:
3093:
3081:
3067:
3065:
3064:
3059:
3054:
3046:
3045:
3037:
3034:
3033:
3028:
3024:
3020:
3012:
3011:
3003:
3002:
2994:
2983:
2982:
2981:
2969:
2968:
2956:
2955:
2941:
2933:
2925:
2917:
2916:
2908:
2905:
2904:
2899:
2895:
2891:
2883:
2882:
2874:
2873:
2865:
2852:
2844:
2843:
2838:
2837:
2829:
2825:
2823:
2822:
2817:
2813:
2800:
2799:
2790:
2779:
2771:
2770:
2762:
2756:
2747:
2742:
2739:
2737:
2736:
2728:
2722:
2721:
2719:
2718:
2713:
2709:
2696:
2695:
2686:
2678:
2677:
2652:
2650:
2649:
2644:
2639:
2626:
2622:
2614:
2613:
2610:
2602:
2601:
2587:
2586:
2561:
2559:
2558:
2553:
2551:
2550:
2549:
2541:
2523:
2519:
2506:
2505:
2488:
2480:
2479:
2471:
2461:
2459:
2458:
2453:
2448:
2440:
2439:
2431:
2425:
2417:
2416:
2408:
2407:
2399:
2393:
2392:
2391:
2389:
2388:
2376:
2371:
2366:
2362:
2354:
2353:
2345:
2333:
2331:
2330:
2325:
2320:
2319:
2314:
2305:
2297:
2296:
2282:
2278:
2277:
2272:
2263:
2262:
2249:
2248:
2243:
2229:
2228:
2227:
2225:
2224:
2212:
2207:
2202:
2198:
2177:
2175:
2174:
2169:
2164:
2156:
2155:
2147:
2137:
2135:
2134:
2129:
2124:
2102:
2100:
2099:
2094:
2092:
2088:
2087:
2086:
2081:
2072:
2071:
2066:
2052:
2051:
2046:
2038:
2037:
2036:
2034:
2033:
2021:
2016:
2011:
2000:
1998:
1997:
1992:
1990:
1985:
1984:
1983:
1978:
1969:
1968:
1963:
1956:
1946:
1945:
1944:
1942:
1941:
1929:
1924:
1919:
1918:
1917:
1912:
1899:
1897:
1896:
1891:
1886:
1885:
1880:
1874:
1873:
1861:
1860:
1855:
1849:
1848:
1836:
1835:
1830:
1824:
1823:
1811:
1799:
1798:
1797:
1785:
1784:
1772:
1771:
1757:
1752:
1751:
1746:
1742:
1729:
1721:
1713:
1712:
1704:
1694:
1692:
1691:
1686:
1681:
1673:
1672:
1664:
1663:
1655:
1648:
1640:
1639:
1631:
1622:
1614:
1613:
1610:
1608:
1607:
1599:
1588:
1586:
1585:
1580:
1575:
1567:
1566:
1563:
1543:Since this is a
1540:
1538:
1537:
1532:
1527:
1526:
1521:
1515:
1514:
1502:
1501:
1496:
1490:
1489:
1477:
1476:
1471:
1465:
1464:
1452:
1440:
1439:
1438:
1426:
1425:
1413:
1412:
1396:
1395:
1394:
1392:
1391:
1379:
1374:
1369:
1365:
1348:
1346:
1345:
1340:
1335:
1317:lattice function
1314:
1312:
1311:
1306:
1301:
1293:
1292:
1269:
1267:
1266:
1261:
1256:
1248:
1247:
1244:
1231:
1229:
1228:
1223:
1221:
1220:
1204:
1202:
1201:
1196:
1194:
1193:
1177:
1175:
1174:
1169:
1167:
1166:
1150:
1148:
1147:
1142:
1140:
1139:
1134:
1121:
1119:
1118:
1113:
1111:
1110:
1105:
1092:
1090:
1089:
1084:
1082:
1081:
1076:
1059:
1057:
1056:
1051:
1046:
1026:
1024:
1023:
1018:
1013:
1012:
1007:
1001:
1000:
988:
987:
982:
976:
975:
963:
962:
957:
951:
950:
938:
937:
932:
923:
912:
911:
901:
895:
894:
865:
864:
863:
851:
850:
838:
837:
821:
820:
819:
817:
816:
804:
799:
794:
790:
782:
781:
778:
765:
763:
762:
757:
755:
754:
738:
736:
735:
730:
725:
717:
716:
713:
696:
694:
693:
688:
683:
682:
677:
668:
657:
656:
646:
640:
639:
609:
608:
607:
605:
604:
592:
587:
582:
578:
570:
569:
550:
548:
547:
542:
540:
539:
523:
521:
520:
515:
513:
512:
507:
494:
492:
491:
486:
481:
473:
472:
453:
451:
450:
445:
440:
439:
434:
425:
417:
416:
402:
398:
397:
392:
383:
382:
367:
359:
358:
355:
348:
347:
342:
332:
318:
317:
271:
269:
268:
263:
258:
250:
249:
230:
228:
227:
222:
217:
209:
208:
189:
187:
186:
181:
173:
165:
164:
146:
138:
137:
123:
122:
121:
119:
118:
106:
101:
96:
92:
32:Paul Peter Ewald
21:
5323:
5322:
5318:
5317:
5316:
5314:
5313:
5312:
5268:
5267:
5266:
5216:
5183:
5179:
5178:
5136:
5135:
5131:
5093:
5092:
5088:
5047:
5046:
5042:
5020:
5019:
5015:
5010:
4978:
4935:
4934:
4915:
4914:
4890:
4879:
4878:
4871:
4855:
4829:
4828:
4805:
4785:
4766:
4758:
4752:
4737:
4721:
4702:
4694:
4688:
4645:
4644:
4579:
4578:
4554:
4553:
4516:
4499:
4498:
4477:
4476:
4454:
4453:
4429:
4424:
4423:
4397:
4392:
4391:
4361:
4332:
4331:
4327:
4299:
4298:
4294:
4293:
4272:
4268:
4252:
4251:
4232:
4231:
4210:
4209:
4188:
4187:
4152:
4151:
4100:
4099:
4077:
4063:
4062:
4041:
4036:
4035:
4016:
4015:
3985:
3980:
3979:
3976:
3903:
3902:
3846:
3845:
3812:
3807:
3806:
3758:
3754:
3753:
3720:
3692:
3687:
3686:
3652:
3637:
3621:
3589:
3584:
3583:
3559:
3554:
3553:
3529:
3513:
3495:
3480:
3445:
3440:
3439:
3404:
3377:
3328:
3327:
3284:
3279:
3278:
3248:
3222:
3221:
3177:
3172:
3171:
3145:
3135:
3120:
3110:
3095:
3085:
3072:
3071:
2991:
2990:
2986:
2985:
2973:
2960:
2947:
2862:
2861:
2857:
2856:
2826:
2806:
2802:
2801:
2791:
2702:
2698:
2697:
2687:
2666:
2661:
2660:
2605:
2575:
2570:
2569:
2568:field integral
2526:
2464:
2463:
2396:
2338:
2337:
2309:
2285:
2267:
2251:
2238:
2183:
2182:
2140:
2139:
2109:
2108:
2076:
2061:
2060:
2056:
2041:
2003:
2002:
1973:
1958:
1957:
1907:
1902:
1901:
1875:
1865:
1850:
1840:
1825:
1815:
1789:
1776:
1763:
1735:
1731:
1730:
1697:
1696:
1652:
1596:
1591:
1590:
1558:
1553:
1552:
1516:
1506:
1491:
1481:
1466:
1456:
1430:
1417:
1404:
1350:
1349:
1320:
1319:
1281:
1276:
1275:
1239:
1234:
1233:
1212:
1207:
1206:
1185:
1180:
1179:
1158:
1153:
1152:
1129:
1124:
1123:
1100:
1095:
1094:
1071:
1066:
1065:
1031:
1030:
1002:
992:
977:
967:
952:
942:
927:
903:
855:
842:
829:
773:
768:
767:
746:
741:
740:
708:
703:
702:
672:
648:
558:
553:
552:
531:
526:
525:
524:of the charges
502:
497:
496:
461:
456:
455:
429:
405:
387:
371:
350:
337:
306:
301:
300:
238:
233:
232:
197:
192:
191:
153:
126:
77:
76:
73:
28:Ewald summation
23:
22:
15:
12:
11:
5:
5321:
5319:
5311:
5310:
5305:
5300:
5295:
5290:
5285:
5280:
5270:
5269:
5265:
5264:
5257:
5229:(3): R55–R60.
5214:
5196:(3): 253–287.
5180:
5177:
5176:
5129:
5102:(12): 124106.
5086:
5040:
5029:(5): 351–368.
5012:
5011:
5009:
5006:
5005:
5004:
5002:Wolf summation
4999:
4994:
4989:
4984:
4977:
4974:
4961:
4958:
4955:
4952:
4948:
4945:
4942:
4922:
4902:
4897:
4893:
4889:
4886:
4870:
4867:
4854:
4851:
4837:
4812:
4808:
4802:
4795:
4792:
4788:
4781:
4775:
4772:
4769:
4764:
4761:
4755:
4751:
4744:
4740:
4734:
4727:
4724:
4717:
4711:
4708:
4705:
4700:
4697:
4691:
4679:
4676:
4673:
4667:
4656:
4652:
4628:
4625:
4622:
4611:
4608:
4605:
4599:
4589:
4586:
4565:
4561:
4540:
4536:
4533:
4528:
4525:
4520:
4515:
4512:
4509:
4506:
4485:
4464:
4461:
4439:
4436:
4432:
4409:
4406:
4401:
4379:
4376:
4368:
4364:
4358:
4353:
4349:
4344:
4341:
4336:
4330:
4325:
4320:
4316:
4311:
4308:
4303:
4297:
4290:
4282:
4279:
4275:
4271:
4267:
4262:
4259:
4239:
4218:
4196:
4174:
4170:
4166:
4162:
4159:
4133:
4130:
4123:
4118:
4115:
4112:
4108:
4084:
4080:
4074:
4070:
4048:
4044:
4023:
3997:
3994:
3989:
3975:
3972:
3930:
3926:
3922:
3916:
3913:
3894:charge density
3884:represent the
3873:
3869:
3865:
3859:
3856:
3831:
3828:
3821:
3818:
3790:
3785:
3781:
3777:
3773:
3767:
3764:
3757:
3752:
3748:
3744:
3739:
3736:
3729:
3726:
3716:
3711:
3707:
3702:
3699:
3695:
3666:
3661:
3656:
3651:
3646:
3641:
3636:
3631:
3628:
3624:
3618:
3615:
3612:
3608:
3604:
3599:
3596:
3592:
3569:
3566:
3562:
3539:
3536:
3532:
3528:
3523:
3520:
3516:
3512:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3476:
3471:
3468:
3465:
3461:
3457:
3448:
3427:
3423:
3419:
3414:
3411:
3407:
3403:
3400:
3396:
3392:
3387:
3384:
3380:
3368:
3365:
3362:
3356:
3346:
3342:
3338:
3335:
3294:
3291:
3287:
3275:electrostatics
3264:inverse square
3247:
3244:
3230:
3209:
3205:
3201:
3196:
3193:
3186:
3183:
3154:
3149:
3142:
3138:
3134:
3129:
3124:
3117:
3113:
3109:
3104:
3099:
3092:
3088:
3084:
3080:
3057:
3053:
3049:
3043:
3040:
3032:
3027:
3023:
3019:
3015:
3010:
3007:
3000:
2997:
2989:
2980:
2976:
2972:
2967:
2963:
2959:
2954:
2950:
2945:
2939:
2936:
2931:
2928:
2924:
2920:
2914:
2911:
2903:
2898:
2894:
2890:
2886:
2881:
2878:
2871:
2868:
2860:
2855:
2851:
2847:
2842:
2835:
2832:
2821:
2816:
2812:
2809:
2805:
2798:
2794:
2788:
2785:
2782:
2778:
2774:
2768:
2765:
2759:
2755:
2751:
2746:
2734:
2731:
2717:
2712:
2708:
2705:
2701:
2694:
2690:
2684:
2681:
2676:
2673:
2669:
2642:
2638:
2634:
2631:
2625:
2621:
2617:
2608:
2600:
2596:
2593:
2590:
2585:
2582:
2578:
2548:
2544:
2540:
2536:
2533:
2529:
2522:
2518:
2514:
2511:
2504:
2500:
2497:
2494:
2491:
2487:
2483:
2477:
2474:
2451:
2447:
2443:
2437:
2434:
2428:
2424:
2420:
2415:
2412:
2405:
2402:
2387:
2384:
2381:
2375:
2365:
2361:
2357:
2351:
2348:
2323:
2318:
2313:
2308:
2304:
2300:
2295:
2292:
2288:
2281:
2276:
2271:
2266:
2261:
2258:
2254:
2247:
2242:
2237:
2234:
2223:
2220:
2217:
2211:
2201:
2197:
2193:
2190:
2167:
2163:
2159:
2153:
2150:
2127:
2123:
2119:
2116:
2105:parallelepiped
2091:
2085:
2080:
2075:
2070:
2065:
2059:
2055:
2050:
2045:
2032:
2029:
2026:
2020:
2010:
1988:
1982:
1977:
1972:
1967:
1962:
1954:
1951:
1940:
1937:
1934:
1928:
1916:
1911:
1889:
1884:
1879:
1872:
1868:
1864:
1859:
1854:
1847:
1843:
1839:
1834:
1829:
1822:
1818:
1814:
1810:
1806:
1803:
1796:
1792:
1788:
1783:
1779:
1775:
1770:
1766:
1761:
1755:
1750:
1745:
1741:
1738:
1734:
1727:
1724:
1720:
1716:
1710:
1707:
1684:
1680:
1676:
1671:
1668:
1661:
1658:
1651:
1647:
1643:
1637:
1634:
1628:
1625:
1621:
1617:
1605:
1602:
1578:
1574:
1570:
1561:
1530:
1525:
1520:
1513:
1509:
1505:
1500:
1495:
1488:
1484:
1480:
1475:
1470:
1463:
1459:
1455:
1451:
1447:
1444:
1437:
1433:
1429:
1424:
1420:
1416:
1411:
1407:
1402:
1390:
1387:
1384:
1378:
1368:
1364:
1360:
1357:
1338:
1334:
1330:
1327:
1304:
1300:
1296:
1291:
1288:
1284:
1259:
1255:
1251:
1242:
1219:
1215:
1192:
1188:
1165:
1161:
1138:
1133:
1109:
1104:
1080:
1075:
1049:
1045:
1041:
1038:
1016:
1011:
1006:
999:
995:
991:
986:
981:
974:
970:
966:
961:
956:
949:
945:
941:
936:
931:
926:
922:
918:
915:
910:
906:
900:
893:
890:
887:
884:
881:
878:
875:
870:
862:
858:
854:
849:
845:
841:
836:
832:
827:
815:
812:
809:
803:
793:
789:
785:
776:
753:
749:
728:
724:
720:
711:
686:
681:
676:
671:
667:
663:
660:
655:
651:
645:
638:
635:
632:
629:
626:
623:
620:
615:
603:
600:
597:
591:
581:
577:
573:
568:
565:
561:
538:
534:
511:
506:
484:
480:
476:
471:
468:
464:
443:
438:
433:
428:
424:
420:
415:
412:
408:
401:
396:
391:
386:
381:
378:
374:
370:
366:
362:
353:
346:
341:
336:
331:
327:
324:
321:
316:
313:
309:
261:
257:
253:
248:
245:
241:
220:
216:
212:
207:
204:
200:
179:
176:
172:
168:
163:
160:
156:
152:
149:
145:
141:
136:
133:
129:
117:
114:
111:
105:
95:
91:
87:
84:
72:
69:
40:ionic crystals
30:, named after
24:
14:
13:
10:
9:
6:
4:
3:
2:
5320:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5291:
5289:
5286:
5284:
5281:
5279:
5276:
5275:
5273:
5262:
5258:
5254:
5250:
5246:
5242:
5237:
5232:
5228:
5224:
5220:
5215:
5211:
5207:
5203:
5199:
5195:
5191:
5187:
5182:
5181:
5172:
5168:
5164:
5160:
5156:
5152:
5148:
5144:
5140:
5133:
5130:
5125:
5121:
5117:
5113:
5109:
5105:
5101:
5097:
5090:
5087:
5083:
5079:
5074:
5069:
5065:
5061:
5057:
5053:
5052:
5044:
5041:
5036:
5032:
5028:
5024:
5017:
5014:
5007:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4979:
4975:
4973:
4956:
4953:
4950:
4946:
4940:
4920:
4895:
4891:
4884:
4876:
4868:
4866:
4864:
4860:
4852:
4850:
4810:
4806:
4793:
4790:
4786:
4779:
4773:
4770:
4767:
4762:
4759:
4753:
4749:
4742:
4738:
4725:
4722:
4715:
4709:
4706:
4703:
4698:
4695:
4689:
4665:
4650:
4642:
4641:Coulomb's law
4626:
4623:
4620:
4597:
4587:
4584:
4559:
4534:
4531:
4526:
4523:
4513:
4510:
4507:
4504:
4462:
4459:
4437:
4434:
4430:
4407:
4404:
4377:
4374:
4366:
4362:
4356:
4347:
4342:
4339:
4328:
4323:
4314:
4309:
4306:
4295:
4288:
4280:
4277:
4273:
4269:
4265:
4260:
4257:
4237:
4168:
4160:
4157:
4147:
4131:
4128:
4116:
4113:
4110:
4106:
4082:
4078:
4072:
4068:
4046:
4042:
4021:
4013:
3995:
3992:
3973:
3971:
3969:
3963:
3961:
3956:
3950:
3948:
3944:
3911:
3899:
3896:(this is the
3895:
3891:
3887:
3854:
3829:
3826:
3803:
3788:
3783:
3762:
3755:
3737:
3734:
3709:
3705:
3700:
3697:
3693:
3684:
3680:
3677:(this is the
3659:
3649:
3644:
3629:
3626:
3622:
3616:
3613:
3610:
3606:
3602:
3597:
3594:
3590:
3567:
3564:
3560:
3537:
3534:
3530:
3526:
3521:
3518:
3514:
3510:
3502:
3492:
3487:
3474:
3469:
3466:
3463:
3459:
3455:
3446:
3412:
3409:
3405:
3401:
3385:
3382:
3378:
3354:
3333:
3324:
3322:
3318:
3314:
3310:
3292:
3289:
3285:
3276:
3272:
3268:
3265:
3261:
3257:
3253:
3245:
3243:
3194:
3191:
3181:
3168:
3152:
3140:
3136:
3132:
3127:
3115:
3111:
3107:
3102:
3090:
3086:
3082:
3068:
3030:
3025:
3008:
3005:
2995:
2987:
2978:
2974:
2970:
2965:
2961:
2957:
2952:
2948:
2943:
2934:
2929:
2901:
2896:
2879:
2876:
2866:
2858:
2840:
2830:
2819:
2814:
2810:
2807:
2803:
2792:
2786:
2783:
2763:
2744:
2729:
2715:
2710:
2706:
2703:
2699:
2688:
2682:
2679:
2674:
2671:
2667:
2658:
2653:
2629:
2606:
2594:
2591:
2588:
2583:
2580:
2576:
2567:
2562:
2542:
2534:
2531:
2527:
2509:
2498:
2495:
2492:
2472:
2413:
2410:
2400:
2373:
2346:
2334:
2306:
2293:
2290:
2286:
2259:
2256:
2252:
2235:
2232:
2209:
2188:
2179:
2148:
2114:
2106:
2089:
2083:
2073:
2068:
2057:
2053:
2048:
2018:
1980:
1970:
1965:
1952:
1949:
1926:
1914:
1882:
1870:
1866:
1862:
1857:
1845:
1841:
1837:
1832:
1820:
1816:
1812:
1801:
1794:
1790:
1786:
1781:
1777:
1773:
1768:
1764:
1759:
1748:
1743:
1739:
1736:
1732:
1725:
1705:
1669:
1666:
1656:
1632:
1626:
1600:
1589:is a product
1559:
1550:
1546:
1541:
1523:
1511:
1507:
1503:
1498:
1486:
1482:
1478:
1473:
1461:
1457:
1453:
1442:
1435:
1431:
1427:
1422:
1418:
1414:
1409:
1405:
1400:
1376:
1355:
1325:
1318:
1289:
1286:
1282:
1273:
1240:
1217:
1213:
1190:
1186:
1163:
1159:
1136:
1107:
1078:
1063:
1036:
1027:
1009:
997:
993:
989:
984:
972:
968:
964:
959:
947:
943:
939:
934:
924:
913:
908:
904:
898:
868:
860:
856:
852:
847:
843:
839:
834:
830:
825:
801:
774:
751:
747:
709:
700:
679:
669:
658:
653:
649:
643:
613:
589:
566:
563:
559:
536:
532:
509:
469:
466:
462:
426:
413:
410:
406:
379:
376:
372:
351:
334:
325:
322:
319:
314:
311:
307:
298:
293:
291:
287:
283:
279:
275:
246:
243:
239:
205:
202:
198:
177:
161:
158:
154:
150:
134:
131:
127:
103:
82:
70:
68:
65:
61:
57:
53:
52:Fourier space
49:
45:
41:
37:
33:
29:
19:
5260:
5226:
5222:
5193:
5189:
5146:
5142:
5132:
5099:
5095:
5089:
5055:
5049:
5043:
5026:
5022:
5016:
4872:
4856:
4148:
3977:
3964:
3951:
3946:
3897:
3804:
3682:
3678:
3325:
3307:part of the
3269:law such as
3249:
3169:
3069:
2654:
2565:
2563:
2335:
2180:
1542:
1316:
1028:
698:
296:
294:
289:
285:
273:
74:
27:
26:
4034:grows like
3974:Dipole term
1545:convolution
1272:convolution
64:convergence
56:singularity
5272:Categories
5008:References
4146:diverges.
71:Derivation
5223:Structure
5190:Ann. Phys
5171:0021-9606
4954:
4787:σ
4774:ϵ
4771:π
4760:−
4710:ϵ
4707:π
4696:−
4621:σ
4532:⋅
4514:−
4348:⋅
4315:⋅
4289:∫
4169:⋅
4158:σ
4122:∞
4107:∑
3915:~
3912:ρ
3890:potential
3858:~
3855:ρ
3827:ℓ
3820:~
3817:Φ
3766:~
3763:ρ
3735:ℓ
3728:~
3725:Φ
3710:∑
3698:ℓ
3650:−
3623:φ
3607:∑
3535:ℓ
3493:−
3475:φ
3460:∑
3410:ℓ
3406:φ
3379:φ
3334:φ
3321:molecules
3290:−
3256:computers
3185:~
3182:ρ
3042:~
3039:Φ
2999:~
2996:ρ
2944:∑
2938:Ω
2913:~
2910:Φ
2870:~
2867:ρ
2841:∗
2834:~
2811:π
2787:∫
2767:~
2745:∗
2733:~
2730:ρ
2707:π
2683:∫
2672:ℓ
2607:ρ
2592:∫
2581:ℓ
2543:⋅
2532:−
2496:∫
2476:~
2436:~
2433:Φ
2404:~
2401:ρ
2350:~
2317:′
2307:−
2291:ℓ
2287:φ
2275:′
2253:ρ
2246:′
2233:∫
2152:~
2074:×
2054:⋅
2009:Ω
1987:Ω
1971:×
1953:π
1863:−
1838:−
1813:−
1802:δ
1760:∑
1754:Ω
1740:π
1709:~
1660:~
1657:ρ
1636:~
1604:~
1601:ρ
1560:ρ
1504:−
1479:−
1454:−
1443:δ
1401:∑
1283:ρ
1241:ρ
1037:δ
990:−
965:−
940:−
925:−
914:δ
869:∑
826:∑
775:ρ
710:ρ
670:−
659:δ
614:∑
560:ρ
463:ρ
437:′
427:−
411:ℓ
407:φ
395:′
373:ρ
352:ρ
345:′
323:∬
312:ℓ
286:unit cell
244:ℓ
240:φ
199:φ
159:ℓ
155:φ
128:φ
83:φ
5253:40964921
5245:10368306
5124:17411107
5082:26616351
4976:See also
4913:, where
3892:and the
3681:part of
3679:particle
3317:galaxies
697:and the
282:periodic
5198:Bibcode
5151:Bibcode
5104:Bibcode
5073:4890727
4869:Scaling
4853:History
3949:part).
3888:of the
3313:plasmas
3271:gravity
1315:with a
1060:is the
5251:
5243:
5169:
5122:
5080:
5070:
4797:
4729:
4686:
4659:
4618:
4591:
4552:where
4390:where
4186:where
3805:where
3375:
3348:
3070:where
2723:
2655:Using
2627:
2603:
2566:single
2524:
2507:
2394:
2367:
2283:
2230:
2203:
2039:
2012:
1947:
1920:
1547:, the
1397:
1370:
1029:Here,
896:
822:
795:
641:
610:
583:
403:
297:double
290:images
190:where
124:
97:
5249:S2CID
3898:Ewald
3267:force
699:total
5241:PMID
5167:ISSN
5120:PMID
5078:PMID
4422:and
3947:mesh
3844:and
3319:and
2138:and
1205:and
1122:and
5231:doi
5206:doi
5194:369
5159:doi
5112:doi
5100:126
5068:PMC
5060:doi
5031:doi
4951:log
3451:TOT
3273:or
2740:TOT
2611:TOT
1611:TOT
1564:TOT
1551:of
1274:of
1245:TOT
779:TOT
714:TOT
356:TOT
5274::
5247:.
5239:.
5225:.
5221:.
5204:.
5192:.
5188:.
5165:.
5157:.
5147:98
5145:.
5141:.
5118:.
5110:.
5098:.
5076:,
5066:,
5056:11
5054:,
5025:.
4972:.
4643:)
3323:.
3315:,
1178:,
1093:,
1064:,
292:.
5255:.
5233::
5227:7
5212:.
5208::
5200::
5173:.
5161::
5153::
5126:.
5114::
5106::
5062::
5037:.
5033::
5027:9
4960:)
4957:N
4947:N
4944:(
4941:O
4921:N
4901:)
4896:2
4892:N
4888:(
4885:O
4836:r
4811:3
4807:r
4801:r
4794:S
4791:d
4780:)
4768:4
4763:1
4754:(
4750:=
4743:3
4739:r
4733:r
4726:q
4723:d
4716:)
4704:4
4699:1
4690:(
4678:f
4675:e
4672:d
4666:=
4655:E
4651:d
4639:(
4627:S
4624:d
4610:f
4607:e
4604:d
4598:=
4588:q
4585:d
4564:E
4560:d
4539:E
4535:d
4527:c
4524:u
4519:p
4511:=
4508:U
4505:d
4484:r
4463:S
4460:d
4438:c
4435:u
4431:V
4408:c
4405:u
4400:p
4378:S
4375:d
4367:3
4363:r
4357:)
4352:n
4343:c
4340:u
4335:p
4329:(
4324:)
4319:r
4310:c
4307:u
4302:p
4296:(
4281:c
4278:u
4274:V
4270:2
4266:1
4261:=
4258:U
4238:U
4217:P
4195:n
4173:n
4165:P
4161:=
4132:n
4129:1
4117:1
4114:=
4111:n
4083:3
4079:R
4073:/
4069:1
4047:2
4043:R
4022:R
3996:c
3993:u
3988:p
3929:)
3925:k
3921:(
3872:)
3868:k
3864:(
3830:r
3789:2
3784:|
3780:)
3776:k
3772:(
3756:|
3751:)
3747:k
3743:(
3738:r
3715:k
3706:=
3701:r
3694:E
3665:)
3660:i
3655:r
3645:j
3640:r
3635:(
3630:r
3627:s
3617:j
3614:,
3611:i
3603:=
3598:r
3595:s
3591:E
3568:r
3565:s
3561:E
3538:r
3531:E
3527:+
3522:r
3519:s
3515:E
3511:=
3508:)
3503:i
3498:r
3488:j
3483:r
3478:(
3470:j
3467:,
3464:i
3456:=
3447:E
3426:)
3422:r
3418:(
3413:r
3402:+
3399:)
3395:r
3391:(
3386:r
3383:s
3367:f
3364:e
3361:d
3355:=
3345:)
3341:r
3337:(
3293:6
3286:r
3229:k
3208:)
3204:k
3200:(
3195:c
3192:u
3153:3
3148:b
3141:3
3137:m
3133:+
3128:2
3123:b
3116:2
3112:m
3108:+
3103:1
3098:b
3091:1
3087:m
3083:=
3079:k
3056:)
3052:k
3048:(
3031:2
3026:|
3022:)
3018:k
3014:(
3009:c
3006:u
2988:|
2979:3
2975:m
2971:,
2966:2
2962:m
2958:,
2953:1
2949:m
2935:1
2930:=
2927:)
2923:k
2919:(
2902:2
2897:|
2893:)
2889:k
2885:(
2880:c
2877:u
2859:|
2854:)
2850:k
2846:(
2831:L
2820:3
2815:)
2808:2
2804:(
2797:k
2793:d
2784:=
2781:)
2777:k
2773:(
2764:V
2758:)
2754:k
2750:(
2716:3
2711:)
2704:2
2700:(
2693:k
2689:d
2680:=
2675:r
2668:E
2641:)
2637:r
2633:(
2630:v
2624:)
2620:r
2616:(
2599:r
2595:d
2589:=
2584:r
2577:E
2547:r
2539:k
2535:i
2528:e
2521:)
2517:r
2513:(
2510:v
2503:r
2499:d
2493:=
2490:)
2486:k
2482:(
2473:V
2450:)
2446:k
2442:(
2427:)
2423:k
2419:(
2414:c
2411:u
2386:f
2383:e
2380:d
2374:=
2364:)
2360:k
2356:(
2347:V
2322:)
2312:r
2303:r
2299:(
2294:r
2280:)
2270:r
2265:(
2260:c
2257:u
2241:r
2236:d
2222:f
2219:e
2216:d
2210:=
2200:)
2196:r
2192:(
2189:v
2166:)
2162:k
2158:(
2149:L
2126:)
2122:r
2118:(
2115:L
2090:)
2084:3
2079:a
2069:2
2064:a
2058:(
2049:1
2044:a
2031:f
2028:e
2025:d
2019:=
1981:3
1976:a
1966:2
1961:a
1950:2
1939:f
1936:e
1933:d
1927:=
1915:1
1910:b
1888:)
1883:3
1878:b
1871:3
1867:m
1858:2
1853:b
1846:2
1842:m
1833:1
1828:b
1821:1
1817:m
1809:k
1805:(
1795:3
1791:m
1787:,
1782:2
1778:m
1774:,
1769:1
1765:m
1749:3
1744:)
1737:2
1733:(
1726:=
1723:)
1719:k
1715:(
1706:L
1683:)
1679:k
1675:(
1670:c
1667:u
1650:)
1646:k
1642:(
1633:L
1627:=
1624:)
1620:k
1616:(
1577:)
1573:r
1569:(
1529:)
1524:3
1519:a
1512:3
1508:n
1499:2
1494:a
1487:2
1483:n
1474:1
1469:a
1462:1
1458:n
1450:r
1446:(
1436:3
1432:n
1428:,
1423:2
1419:n
1415:,
1410:1
1406:n
1389:f
1386:e
1383:d
1377:=
1367:)
1363:r
1359:(
1356:L
1337:)
1333:r
1329:(
1326:L
1303:)
1299:r
1295:(
1290:c
1287:u
1258:)
1254:r
1250:(
1218:3
1214:n
1191:2
1187:n
1164:1
1160:n
1137:3
1132:a
1108:2
1103:a
1079:1
1074:a
1048:)
1044:x
1040:(
1015:)
1010:3
1005:a
998:3
994:n
985:2
980:a
973:2
969:n
960:1
955:a
948:1
944:n
935:k
930:r
921:r
917:(
909:k
905:q
899:k
892:s
889:e
886:g
883:r
880:a
877:h
874:c
861:3
857:n
853:,
848:2
844:n
840:,
835:1
831:n
814:f
811:e
808:d
802:=
792:)
788:r
784:(
752:k
748:q
727:)
723:r
719:(
685:)
680:k
675:r
666:r
662:(
654:k
650:q
644:k
637:s
634:e
631:g
628:r
625:a
622:h
619:c
602:f
599:e
596:d
590:=
580:)
576:r
572:(
567:c
564:u
537:k
533:q
510:k
505:r
483:)
479:r
475:(
470:c
467:u
442:)
432:r
423:r
419:(
414:r
400:)
390:r
385:(
380:c
377:u
369:)
365:r
361:(
340:r
335:d
330:r
326:d
320:=
315:r
308:E
274:r
260:)
256:r
252:(
247:r
219:)
215:r
211:(
206:r
203:s
178:,
175:)
171:r
167:(
162:r
151:+
148:)
144:r
140:(
135:r
132:s
116:f
113:e
110:d
104:=
94:)
90:r
86:(
20:)
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