Coverage for gpaw/spherical_harmonics.py: 99%

1r""" 

2Real-valued spherical harmonics 

3 

4 

5=== === === ======= 

6 L l m 

7=== === === ======= 

8 0 0 0 1 

9 1 1 -1 y 

10 2 1 0 z 

11 3 1 1 x 

12 4 2 -2 xy 

13 5 2 -1 yz 

14 6 2 0 3z2-r2 

15 7 2 1 zx 

16 8 2 2 x2-y2 

17=== === === ======= 

18 

19For a more complete list, see c/bmgs/sharmonic.py 

20""" 

21 

22import numpy as np 

23 

24from math import pi 

25from collections import defaultdict 

26from gpaw import debug, GPAW_NO_C_EXTENSION 

27 

28__all__ = ['Y', 'YL', 'nablarlYL', 'Yl'] 

29 

30if GPAW_NO_C_EXTENSION: 

31 Yl = None 

32else: 

33 from gpaw.cgpaw import spherical_harmonics 

34 if debug: 

35 def Yl(l: int, R_v: np.ndarray, rlY_m: np.ndarray) -> None: 

36 assert R_v.dtype == float and R_v.shape == (3,) 

37 assert rlY_m.dtype == float and rlY_m.shape == (2 * l + 1,) 

38 spherical_harmonics(l, R_v, rlY_m) 

39 else: 

40 Yl = spherical_harmonics 

41 

42names = [['1'], 

43 ['y', 'z', 'x'], 

44 ['xy', 'yz', '3z2-r2', 'zx', 'x2-y2'], 

45 ['3x2y-y3', 'xyz', '4yz2-y3-x2y', '2z3-3x2z-3y2z', '4xz2-x3-xy2', 

46 'x2z-y2z', 'x3-3xy2']] 

47 

48 

49def Y(L, x, y, z): 

50 result = 0.0 

51 for c, n in YL[L]: 

52 result += c * x**n[0] * y**n[1] * z**n[2] 

53 return result 

54 

55 

56def Yarr(L_M, R_Av): 

57 """ 

58 Calculate spherical harmonics L_M at positions R_Av, where 

59 A is some array like index. 

60 """ 

61 Y_MA = np.zeros((len(L_M), *R_Av.shape[:-1])) 

62 for M, L in enumerate(L_M): 

63 for c, n in YL[L]: # could be vectorized further 

64 Y_MA[M] += c * np.prod(np.power(R_Av, n), axis=-1) 

65 return Y_MA 

66 

67 

68def nablarlYL(L, R): 

69 """Calculate the gradient of a real solid spherical harmonic.""" 

70 x, y, z = R 

71 dYdx = dYdy = dYdz = 0.0 

72 terms = YL[L] 

73 # The 'abs' avoids error in case powx == 0 

74 for N, (powx, powy, powz) in terms: 

75 dYdx += N * powx * x**abs(powx - 1) * y**powy * z**powz 

76 dYdy += N * powy * x**powx * y**abs(powy - 1) * z**powz 

77 dYdz += N * powz * x**powx * y**powy * z**abs(powz - 1) 

78 return dYdx, dYdy, dYdz 

79 

80 

81g = [1.0] 

82for l in range(11): 

83 g.append(g[-1] * (l + 0.5)) 

84 

85 

86def gam(n0, n1, n2): 

87 h0 = n0 // 2 

88 h1 = n1 // 2 

89 h2 = n2 // 2 

90 if 2 * h0 != n0 or 2 * h1 != n1 or 2 * h2 != n2: 

91 return 0.0 

92 return 2.0 * pi * g[h0] * g[h1] * g[h2] / g[1 + h0 + h1 + h2] 

93 

94 

95def Y0(l, m): 

96 """Sympy version of spherical harmonics.""" 

97 from fractions import Fraction 

98 from sympy import assoc_legendre, sqrt, simplify, factorial as fac, I, pi 

99 from sympy.abc import x, y, z 

100 c = sqrt((2 * l + 1) * fac(l - m) / fac(l + m) / 4 / pi) 

101 if m > 0: 

102 return simplify(c * (x + I * y)**m / (1 - z**2)**Fraction(m, 2) * 

103 assoc_legendre(l, m, z)) 

104 return simplify(c * (x - I * y)**(-m) / (1 - z**2)**Fraction(-m, 2) * 

105 assoc_legendre(l, m, z)) 

106 

107 

108def S(l, m): 

109 """Sympy version of real valued spherical harmonics.""" 

110 from sympy import I, Number, sqrt 

111 if m > 0: 

112 return (Y0(l, m) + (-1)**m * Y0(l, -m)) / sqrt(2) * (-1)**m 

113 if m < 0: 

114 return -(Y0(l, m) - Number(-1)**m * Y0(l, -m)) / (sqrt(2) * I) 

115 return Y0(l, m) 

116 

117 

118def poly_coeffs(l, m): 

119 """Sympy coefficients for polynomiunm in x, y and z.""" 

120 from sympy import Poly 

121 from sympy.abc import x, y, z 

122 Y = S(l, m) 

123 coeffs = {} 

124 for nx, coef in enumerate(reversed(Poly(Y, x).all_coeffs())): 

125 for ny, coef in enumerate(reversed(Poly(coef, y).all_coeffs())): 

126 for nz, coef in enumerate(reversed(Poly(coef, z).all_coeffs())): 

127 if coef: 

128 coeffs[(nx, ny, nz)] = coef 

129 return coeffs 

130 

131 

132def fix_exponents(coeffs, l): 

133 """Make sure exponents add up to l.""" 

134 from sympy import Number 

135 new = defaultdict(lambda: Number(0)) 

136 for (nx, ny, nz), coef in coeffs.items(): 

137 if nx + ny + nz == l: 

138 new[(nx, ny, nz)] += coef 

139 else: 

140 new[(nx + 2, ny, nz)] += coef 

141 new[(nx, ny + 2, nz)] += coef 

142 new[(nx, ny, nz + 2)] += coef 

143 

144 new = {nxyz: coef for nxyz, coef in new.items() if coef} 

145 

146 if not all(sum(nxyz) == l for nxyz in new): 

147 new = fix_exponents(new, l) 

148 

149 return new 

150 

151 

152def print_YL_table_code(): 

153 """Generate YL table using sympy. 

154 

155 This will generate slightly more accurate numbers, but we will not update 

156 right now because then we would also have to update 

157 c/bmgs/spherical_harminics.c. 

158 """ 

159 print('# Computer generated table - do not touch!') 

160 print('YL = [') 

161 print(' # s, l=0:') 

162 print(f' [({(4 * pi)**-0.5}, (0, 0, 0))],') 

163 for l in range(1, 9): 

164 s = 'spdfghijk'[l] 

165 print(f' # {s}, l={l}:') 

166 for m in range(-l, l + 1): 

167 e = poly_coeffs(l, m) 

168 e = fix_exponents(e, l) 

169 if l**2 + m + l < len(YL): 

170 assert len(e) == len(YL[l**2 + m + l]) 

171 for c0, n in YL[l**2 + m + l]: 

172 c = e[n].evalf() 

173 assert abs(c0 - c) < 1e-10 

174 terms = [] 

175 for n, en in e.items(): 

176 c = float(en) 

177 terms.append(f'({c!r}, {n})') 

178 print(' [' + ',\n '.join(terms) + '],') 

179 print(']') 

180 

181 

182def write_c_code(l: int) -> None: 

183 print(f' else if (l == {l})') 

184 print(' {') 

185 for m in range(2 * l + 1): 

186 terms = [] 

187 for c, n in YL[l**2 + m]: 

188 terms.append(f'{c!r} * ' + '*'.join('x' * n[0] + 

189 'y' * n[1] + 

190 'z' * n[2])) 

191 print(f' Y_m[{m}] = ' + ' + '.join(terms) + ';') 

192 print(' }') 

193 

194 

195# Computer generated table - do not touch! 

196# The numbers match those in c/bmgs/spherical_harmonics.c and were 

197# originally generated with c/bmgs/sharmonic.py (old Python 2 code). 

198YL = [ 

199 # s, l=0: 

200 [(0.28209479177387814, (0, 0, 0))], 

201 # p, l=1: 

202 [(0.4886025119029199, (0, 1, 0))], 

203 [(0.4886025119029199, (0, 0, 1))], 

204 [(0.4886025119029199, (1, 0, 0))], 

205 # d, l=2: 

206 [(1.0925484305920792, (1, 1, 0))], 

207 [(1.0925484305920792, (0, 1, 1))], 

208 [(0.6307831305050401, (0, 0, 2)), 

209 (-0.31539156525252005, (0, 2, 0)), 

210 (-0.31539156525252005, (2, 0, 0))], 

211 [(1.0925484305920792, (1, 0, 1))], 

212 [(0.5462742152960396, (2, 0, 0)), 

213 (-0.5462742152960396, (0, 2, 0))], 

214 # f, l=3: 

215 [(-0.5900435899266435, (0, 3, 0)), 

216 (1.7701307697799304, (2, 1, 0))], 

217 [(2.890611442640554, (1, 1, 1))], 

218 [(-0.4570457994644658, (0, 3, 0)), 

219 (1.828183197857863, (0, 1, 2)), 

220 (-0.4570457994644658, (2, 1, 0))], 

221 [(0.7463526651802308, (0, 0, 3)), 

222 (-1.1195289977703462, (2, 0, 1)), 

223 (-1.1195289977703462, (0, 2, 1))], 

224 [(1.828183197857863, (1, 0, 2)), 

225 (-0.4570457994644658, (3, 0, 0)), 

226 (-0.4570457994644658, (1, 2, 0))], 

227 [(1.445305721320277, (2, 0, 1)), 

228 (-1.445305721320277, (0, 2, 1))], 

229 [(0.5900435899266435, (3, 0, 0)), 

230 (-1.7701307697799304, (1, 2, 0))], 

231 # g, l=4: 

232 [(2.5033429417967046, (3, 1, 0)), 

233 (-2.5033429417967046, (1, 3, 0))], 

234 [(-1.7701307697799307, (0, 3, 1)), 

235 (5.310392309339792, (2, 1, 1))], 

236 [(-0.9461746957575601, (3, 1, 0)), 

237 (-0.9461746957575601, (1, 3, 0)), 

238 (5.6770481745453605, (1, 1, 2))], 

239 [(-2.0071396306718676, (2, 1, 1)), 

240 (2.676186174229157, (0, 1, 3)), 

241 (-2.0071396306718676, (0, 3, 1))], 

242 [(0.6347132814912259, (2, 2, 0)), 

243 (-2.5388531259649034, (2, 0, 2)), 

244 (0.31735664074561293, (0, 4, 0)), 

245 (-2.5388531259649034, (0, 2, 2)), 

246 (0.31735664074561293, (4, 0, 0)), 

247 (0.8462843753216345, (0, 0, 4))], 

248 [(2.676186174229157, (1, 0, 3)), 

249 (-2.0071396306718676, (3, 0, 1)), 

250 (-2.0071396306718676, (1, 2, 1))], 

251 [(2.8385240872726802, (2, 0, 2)), 

252 (0.47308734787878004, (0, 4, 0)), 

253 (-0.47308734787878004, (4, 0, 0)), 

254 (-2.8385240872726802, (0, 2, 2))], 

255 [(1.7701307697799307, (3, 0, 1)), 

256 (-5.310392309339792, (1, 2, 1))], 

257 [(-3.755014412695057, (2, 2, 0)), 

258 (0.6258357354491761, (0, 4, 0)), 

259 (0.6258357354491761, (4, 0, 0))], 

260 # h, l=5: 

261 [(-6.5638205684017015, (2, 3, 0)), 

262 (3.2819102842008507, (4, 1, 0)), 

263 (0.6563820568401701, (0, 5, 0))], 

264 [(8.302649259524165, (3, 1, 1)), 

265 (-8.302649259524165, (1, 3, 1))], 

266 [(-3.913906395482003, (0, 3, 2)), 

267 (0.4892382994352504, (0, 5, 0)), 

268 (-1.467714898305751, (4, 1, 0)), 

269 (-0.9784765988705008, (2, 3, 0)), 

270 (11.741719186446009, (2, 1, 2))], 

271 [(-4.793536784973324, (3, 1, 1)), 

272 (-4.793536784973324, (1, 3, 1)), 

273 (9.587073569946648, (1, 1, 3))], 

274 [(-5.435359814348363, (0, 3, 2)), 

275 (0.9058933023913939, (2, 3, 0)), 

276 (-5.435359814348363, (2, 1, 2)), 

277 (3.6235732095655755, (0, 1, 4)), 

278 (0.45294665119569694, (4, 1, 0)), 

279 (0.45294665119569694, (0, 5, 0))], 

280 [(3.508509673602708, (2, 2, 1)), 

281 (-4.678012898136944, (0, 2, 3)), 

282 (1.754254836801354, (0, 4, 1)), 

283 (-4.678012898136944, (2, 0, 3)), 

284 (1.754254836801354, (4, 0, 1)), 

285 (0.9356025796273888, (0, 0, 5))], 

286 [(-5.435359814348363, (3, 0, 2)), 

287 (3.6235732095655755, (1, 0, 4)), 

288 (0.45294665119569694, (5, 0, 0)), 

289 (0.9058933023913939, (3, 2, 0)), 

290 (-5.435359814348363, (1, 2, 2)), 

291 (0.45294665119569694, (1, 4, 0))], 

292 [(-2.396768392486662, (4, 0, 1)), 

293 (2.396768392486662, (0, 4, 1)), 

294 (4.793536784973324, (2, 0, 3)), 

295 (-4.793536784973324, (0, 2, 3))], 

296 [(3.913906395482003, (3, 0, 2)), 

297 (-0.4892382994352504, (5, 0, 0)), 

298 (0.9784765988705008, (3, 2, 0)), 

299 (-11.741719186446009, (1, 2, 2)), 

300 (1.467714898305751, (1, 4, 0))], 

301 [(2.075662314881041, (4, 0, 1)), 

302 (-12.453973889286246, (2, 2, 1)), 

303 (2.075662314881041, (0, 4, 1))], 

304 [(-6.5638205684017015, (3, 2, 0)), 

305 (0.6563820568401701, (5, 0, 0)), 

306 (3.2819102842008507, (1, 4, 0))], 

307 # i, l=6: 

308 [(4.099104631151485, (5, 1, 0)), 

309 (-13.663682103838287, (3, 3, 0)), 

310 (4.099104631151485, (1, 5, 0))], 

311 [(11.83309581115876, (4, 1, 1)), 

312 (-23.66619162231752, (2, 3, 1)), 

313 (2.366619162231752, (0, 5, 1))], 

314 [(20.182596029148968, (3, 1, 2)), 

315 (-2.0182596029148967, (5, 1, 0)), 

316 (2.0182596029148967, (1, 5, 0)), 

317 (-20.182596029148968, (1, 3, 2))], 

318 [(-7.369642076119388, (0, 3, 3)), 

319 (-5.527231557089541, (2, 3, 1)), 

320 (2.7636157785447706, (0, 5, 1)), 

321 (22.108926228358165, (2, 1, 3)), 

322 (-8.29084733563431, (4, 1, 1))], 

323 [(-14.739284152238776, (3, 1, 2)), 

324 (14.739284152238776, (1, 1, 4)), 

325 (1.842410519029847, (3, 3, 0)), 

326 (0.9212052595149235, (5, 1, 0)), 

327 (-14.739284152238776, (1, 3, 2)), 

328 (0.9212052595149235, (1, 5, 0))], 

329 [(2.9131068125936572, (0, 5, 1)), 

330 (-11.652427250374629, (0, 3, 3)), 

331 (5.8262136251873144, (2, 3, 1)), 

332 (-11.652427250374629, (2, 1, 3)), 

333 (2.9131068125936572, (4, 1, 1)), 

334 (4.660970900149851, (0, 1, 5))], 

335 [(5.721228204086558, (4, 0, 2)), 

336 (-7.628304272115411, (0, 2, 4)), 

337 (-0.9535380340144264, (2, 4, 0)), 

338 (1.0171072362820548, (0, 0, 6)), 

339 (-0.9535380340144264, (4, 2, 0)), 

340 (5.721228204086558, (0, 4, 2)), 

341 (-0.3178460113381421, (0, 6, 0)), 

342 (-7.628304272115411, (2, 0, 4)), 

343 (-0.3178460113381421, (6, 0, 0)), 

344 (11.442456408173117, (2, 2, 2))], 

345 [(-11.652427250374629, (3, 0, 3)), 

346 (4.660970900149851, (1, 0, 5)), 

347 (2.9131068125936572, (5, 0, 1)), 

348 (5.8262136251873144, (3, 2, 1)), 

349 (-11.652427250374629, (1, 2, 3)), 

350 (2.9131068125936572, (1, 4, 1))], 

351 [(7.369642076119388, (2, 0, 4)), 

352 (-7.369642076119388, (0, 2, 4)), 

353 (-0.46060262975746175, (2, 4, 0)), 

354 (-7.369642076119388, (4, 0, 2)), 

355 (0.46060262975746175, (4, 2, 0)), 

356 (-0.46060262975746175, (0, 6, 0)), 

357 (0.46060262975746175, (6, 0, 0)), 

358 (7.369642076119388, (0, 4, 2))], 

359 [(7.369642076119388, (3, 0, 3)), 

360 (-2.7636157785447706, (5, 0, 1)), 

361 (5.527231557089541, (3, 2, 1)), 

362 (-22.108926228358165, (1, 2, 3)), 

363 (8.29084733563431, (1, 4, 1))], 

364 [(2.522824503643621, (4, 2, 0)), 

365 (5.045649007287242, (0, 4, 2)), 

366 (-30.273894043723452, (2, 2, 2)), 

367 (-0.5045649007287242, (0, 6, 0)), 

368 (-0.5045649007287242, (6, 0, 0)), 

369 (5.045649007287242, (4, 0, 2)), 

370 (2.522824503643621, (2, 4, 0))], 

371 [(2.366619162231752, (5, 0, 1)), 

372 (-23.66619162231752, (3, 2, 1)), 

373 (11.83309581115876, (1, 4, 1))], 

374 [(-10.247761577878714, (4, 2, 0)), 

375 (-0.6831841051919143, (0, 6, 0)), 

376 (0.6831841051919143, (6, 0, 0)), 

377 (10.247761577878714, (2, 4, 0))], 

378 # j, l=7: 

379 [(14.850417383016522, (2, 5, 0)), 

380 (4.950139127672174, (6, 1, 0)), 

381 (-24.75069563836087, (4, 3, 0)), 

382 (-0.7071627325245963, (0, 7, 0))], 

383 [(-52.91921323603801, (3, 3, 1)), 

384 (15.875763970811402, (1, 5, 1)), 

385 (15.875763970811402, (5, 1, 1))], 

386 [(-2.5945778936013015, (6, 1, 0)), 

387 (2.5945778936013015, (4, 3, 0)), 

388 (-62.26986944643124, (2, 3, 2)), 

389 (4.670240208482342, (2, 5, 0)), 

390 (6.226986944643123, (0, 5, 2)), 

391 (31.13493472321562, (4, 1, 2)), 

392 (-0.5189155787202603, (0, 7, 0))], 

393 [(41.513246297620825, (3, 1, 3)), 

394 (12.453973889286246, (1, 5, 1)), 

395 (-41.513246297620825, (1, 3, 3)), 

396 (-12.453973889286246, (5, 1, 1))], 

397 [(-18.775072063475285, (2, 3, 2)), 

398 (-0.4693768015868821, (0, 7, 0)), 

399 (0.4693768015868821, (2, 5, 0)), 

400 (2.3468840079344107, (4, 3, 0)), 

401 (-12.516714708983523, (0, 3, 4)), 

402 (37.55014412695057, (2, 1, 4)), 

403 (1.4081304047606462, (6, 1, 0)), 

404 (9.387536031737643, (0, 5, 2)), 

405 (-28.162608095212928, (4, 1, 2))], 

406 [(13.27598077334948, (3, 3, 1)), 

407 (6.63799038667474, (1, 5, 1)), 

408 (-35.402615395598616, (3, 1, 3)), 

409 (21.24156923735917, (1, 1, 5)), 

410 (-35.402615395598616, (1, 3, 3)), 

411 (6.63799038667474, (5, 1, 1))], 

412 [(-0.4516580379125865, (0, 7, 0)), 

413 (10.839792909902076, (0, 5, 2)), 

414 (-1.3549741137377596, (2, 5, 0)), 

415 (-1.3549741137377596, (4, 3, 0)), 

416 (-21.679585819804153, (0, 3, 4)), 

417 (-21.679585819804153, (2, 1, 4)), 

418 (5.781222885281108, (0, 1, 6)), 

419 (-0.4516580379125865, (6, 1, 0)), 

420 (21.679585819804153, (2, 3, 2)), 

421 (10.839792909902076, (4, 1, 2))], 

422 [(-11.471758521216831, (2, 0, 5)), 

423 (1.0925484305920792, (0, 0, 7)), 

424 (-11.471758521216831, (0, 2, 5)), 

425 (28.67939630304208, (2, 2, 3)), 

426 (-2.3899496919201733, (6, 0, 1)), 

427 (-7.16984907576052, (4, 2, 1)), 

428 (14.33969815152104, (4, 0, 3)), 

429 (-2.3899496919201733, (0, 6, 1)), 

430 (-7.16984907576052, (2, 4, 1)), 

431 (14.33969815152104, (0, 4, 3))], 

432 [(10.839792909902076, (1, 4, 2)), 

433 (-0.4516580379125865, (7, 0, 0)), 

434 (21.679585819804153, (3, 2, 2)), 

435 (-1.3549741137377596, (5, 2, 0)), 

436 (-0.4516580379125865, (1, 6, 0)), 

437 (-21.679585819804153, (3, 0, 4)), 

438 (-1.3549741137377596, (3, 4, 0)), 

439 (5.781222885281108, (1, 0, 6)), 

440 (-21.679585819804153, (1, 2, 4)), 

441 (10.839792909902076, (5, 0, 2))], 

442 [(10.620784618679584, (2, 0, 5)), 

443 (-10.620784618679584, (0, 2, 5)), 

444 (3.31899519333737, (6, 0, 1)), 

445 (3.31899519333737, (4, 2, 1)), 

446 (-17.701307697799308, (4, 0, 3)), 

447 (-3.31899519333737, (0, 6, 1)), 

448 (-3.31899519333737, (2, 4, 1)), 

449 (17.701307697799308, (0, 4, 3))], 

450 [(-1.4081304047606462, (1, 6, 0)), 

451 (0.4693768015868821, (7, 0, 0)), 

452 (18.775072063475285, (3, 2, 2)), 

453 (-0.4693768015868821, (5, 2, 0)), 

454 (12.516714708983523, (3, 0, 4)), 

455 (-2.3468840079344107, (3, 4, 0)), 

456 (28.162608095212928, (1, 4, 2)), 

457 (-37.55014412695057, (1, 2, 4)), 

458 (-9.387536031737643, (5, 0, 2))], 

459 [(10.378311574405206, (4, 0, 3)), 

460 (-3.1134934723215615, (0, 6, 1)), 

461 (15.56746736160781, (2, 4, 1)), 

462 (-62.26986944643124, (2, 2, 3)), 

463 (10.378311574405206, (0, 4, 3)), 

464 (-3.1134934723215615, (6, 0, 1)), 

465 (15.56746736160781, (4, 2, 1))], 

466 [(-2.5945778936013015, (1, 6, 0)), 

467 (-62.26986944643124, (3, 2, 2)), 

468 (-0.5189155787202603, (7, 0, 0)), 

469 (31.13493472321562, (1, 4, 2)), 

470 (2.5945778936013015, (3, 4, 0)), 

471 (6.226986944643123, (5, 0, 2)), 

472 (4.670240208482342, (5, 2, 0))], 

473 [(2.6459606618019005, (6, 0, 1)), 

474 (-2.6459606618019005, (0, 6, 1)), 

475 (-39.68940992702851, (4, 2, 1)), 

476 (39.68940992702851, (2, 4, 1))], 

477 [(0.7071627325245963, (7, 0, 0)), 

478 (-14.850417383016522, (5, 2, 0)), 

479 (24.75069563836087, (3, 4, 0)), 

480 (-4.950139127672174, (1, 6, 0))], 

481 # k, l=8: 

482 [(-5.831413281398639, (1, 7, 0)), 

483 (40.81989296979047, (3, 5, 0)), 

484 (-40.81989296979047, (5, 3, 0)), 

485 (5.831413281398639, (7, 1, 0))], 

486 [(-2.9157066406993195, (0, 7, 1)), 

487 (61.22983945468571, (2, 5, 1)), 

488 (-102.04973242447618, (4, 3, 1)), 

489 (20.409946484895237, (6, 1, 1))], 

490 [(7.452658724833596, (3, 5, 0)), 

491 (-3.1939965963572554, (1, 7, 0)), 

492 (44.715952349001576, (1, 5, 2)), 

493 (7.452658724833596, (5, 3, 0)), 

494 (-149.05317449667191, (3, 3, 2)), 

495 (-3.1939965963572554, (7, 1, 0)), 

496 (44.715952349001576, (5, 1, 2))], 

497 [(31.04919559888297, (2, 5, 1)), 

498 (-3.449910622098108, (0, 7, 1)), 

499 (13.799642488392433, (0, 5, 3)), 

500 (17.24955311049054, (4, 3, 1)), 

501 (-137.99642488392433, (2, 3, 3)), 

502 (-17.24955311049054, (6, 1, 1)), 

503 (68.99821244196217, (4, 1, 3))], 

504 [(-1.9136660990373229, (3, 5, 0)), 

505 (-1.9136660990373229, (1, 7, 0)), 

506 (45.927986376895745, (1, 5, 2)), 

507 (-76.54664396149292, (1, 3, 4)), 

508 (1.9136660990373229, (7, 1, 0)), 

509 (1.9136660990373229, (5, 3, 0)), 

510 (-45.927986376895745, (5, 1, 2)), 

511 (76.54664396149292, (3, 1, 4))], 

512 [(18.528992329433162, (4, 3, 1)), 

513 (3.705798465886632, (2, 5, 1)), 

514 (-49.41064621182176, (2, 3, 3)), 

515 (-3.705798465886632, (0, 7, 1)), 

516 (24.70532310591088, (0, 5, 3)), 

517 (-19.764258484728707, (0, 3, 5)), 

518 (11.117395397659896, (6, 1, 1)), 

519 (-74.11596931773265, (4, 1, 3)), 

520 (59.29277545418611, (2, 1, 5))],