El objetivo del problema de Thomson es determinar la configuración mínima de energía potencial electrostática de N electrones restringidos a la superficie de una esfera unitaria que se repelen entre sí con una fuerza dada por la ley de Coulomb . El físico JJ Thomson planteó el problema en 1904 [1] después de proponer un modelo atómico , más tarde llamado modelo del pudín de ciruela , basado en su conocimiento de la existencia de electrones con carga negativa dentro de átomos con carga neutra.
Los problemas relacionados incluyen el estudio de la geometría de la configuración de energía mínima y el estudio del comportamiento del N grande de la energía mínima.
Declaración matemática
El sistema físico encarnado por el problema de Thomson es un caso especial de uno de los dieciocho problemas matemáticos sin resolver propuestos por el matemático Steve Smale : "Distribución de puntos en la 2-esfera". [2] La solución de cada problema de N -electrones se obtiene cuando la configuración de N -electrones restringida a la superficie de una esfera de unidad de radio,, produce un mínimo de energía potencial electrostática global ,.
La energía de interacción electrostática que ocurre entre cada par de electrones de cargas iguales (, con la carga elemental de un electrón) viene dada por la Ley de Coulomb,
Aquí, es la constante de Coulomb y es la distancia entre cada par de electrones ubicados en puntos de la esfera definidos por vectores y , respectivamente.
Unidades simplificadas de y se utilizan sin pérdida de generalidad. Luego,
La energía potencial electrostática total de cada configuración de N electrones se puede expresar como la suma de todas las interacciones por pares
La minimización global de sobre todas las colecciones posibles de N puntos distintos se encuentra típicamente mediante algoritmos de minimización numérica.
Ejemplo
La solución del problema de Thomson para dos electrones se obtiene cuando ambos electrones están lo más separados posible en lados opuestos del origen, , o
Soluciones conocidas
Las configuraciones de energía mínima se han identificado rigurosamente en solo un puñado de casos.
- Para N = 1, la solución es trivial ya que el electrón puede residir en cualquier punto de la superficie de la esfera unitaria. La energía total de la configuración se define como cero, ya que el electrón no está sujeto al campo eléctrico debido a otras fuentes de carga.
- Para N = 2, la configuración óptima consiste en electrones en los puntos antípodas .
- Para N = 3, los electrones residen en los vértices de un triángulo equilátero alrededor de un gran círculo . [3]
- Para N = 4, los electrones residen en los vértices de un tetraedro regular .
- Para N = 5, en 2010 se informó una solución asistida por computadora matemáticamente rigurosa con electrones que residen en los vértices de una bipirámide triangular . [4]
- Para N = 6, los electrones residen en los vértices de un octaedro regular . [5]
- Para N = 12, los electrones residen en los vértices de un icosaedro regular . [6]
En particular, las soluciones geométricas del problema de Thomson para N = 4, 6 y 12 electrones se conocen como sólidos platónicos cuyas caras son todos triángulos equiláteros congruentes. Las soluciones numéricas para N = 8 y 20 no son las configuraciones poliédricas convexas regulares de los dos sólidos platónicos restantes, cuyas caras son cuadradas y pentagonales, respectivamente [ cita requerida ] .
Generalizaciones
También se pueden preguntar los estados fundamentales de las partículas que interactúan con potenciales arbitrarios. Para ser matemáticamente precisos, sea f una función de valor real decreciente y defina la energía funcional
Tradicionalmente, uno considera también conocido como Riesz -núcleos. Para los núcleos de Riesz integrables, consulte; [7] para los granos de Riesz no integrables, se cumple el teorema del bagel con semillas de amapola , ver. [8] Los casos notables incluyen α = ∞, el problema de Tammes (empaque); α = 1, el problema de Thomson; α = 0, problema de Whyte (para maximizar el producto de las distancias).
También se pueden considerar configuraciones de N puntos en una esfera de mayor dimensión . Ver diseño esférico .
Relaciones con otros problemas científicos
El problema de Thomson es una consecuencia natural del modelo de pudín de ciruela de Thomson en ausencia de su carga de fondo positiva uniforme. [9]
—Sir JJ Thomson [10]
Aunque la evidencia experimental llevó al abandono del modelo de pudín de ciruela de Thomson como modelo atómico completo, se ha encontrado que las irregularidades observadas en las soluciones de energía numérica del problema de Thomson se corresponden con el llenado de capas de electrones en átomos naturales a lo largo de la tabla periódica de elementos. [11]
El problema de Thomson también juega un papel en el estudio de otros modelos físicos, incluidas las burbujas de múltiples electrones y el orden de la superficie de las gotas de metal líquido confinadas en trampas de Paul .
El problema generalizado de Thomson surge, por ejemplo, al determinar las disposiciones de las subunidades de proteínas que comprenden las capas de los virus esféricos . Las "partículas" en esta aplicación son grupos de subunidades de proteínas dispuestas en una cáscara. Otras realizaciones incluyen arreglos regulares de partículas coloides en coloidosomas , propuestos para la encapsulación de ingredientes activos como medicamentos, nutrientes o células vivas, patrones de fullereno de átomos de carbono y teoría VSEPR . Un ejemplo con interacciones logarítmicas de largo alcance lo proporcionan los vórtices de Abrikosov, que se formarían a bajas temperaturas en una capa metálica superconductora con un gran monopolo en el centro.
Configuraciones de menor energía conocida
En la siguiente tabla es el número de puntos (cargos) en una configuración, es la energía, el tipo de simetría se da en notación Schönflies (ver Grupos de puntos en tres dimensiones ), yson las posiciones de los cargos. La mayoría de los tipos de simetría requieren que la suma vectorial de las posiciones (y por lo tanto el momento dipolar eléctrico ) sea cero.
Se acostumbra considerar también el poliedro formado por el casco convexo de las puntas. Por lo tanto, es el número de vértices donde se encuentran el número dado de aristas, ' es el número total de aristas, es el número de caras triangulares, es el número de caras cuadriláteras, y es el ángulo más pequeño subtendido por vectores asociados con el par de carga más cercano. Tenga en cuenta que las longitudes de los bordes generalmente no son iguales; así (excepto en los casos N = 2, 3, 4, 6, 12 y los poliedros geodésicos ) el casco convexo es sólo topológicamente equivalente a la figura listada en la última columna. [12]
norte | Simetría | Poliedro equivalente | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 0.500000000 | 0 | - | - | - | - | - | - | 2 | - | - | 180.000 ° | excavar | |
3 | 1,732050808 | 0 | - | - | - | - | - | - | 3 | 2 | - | 120.000 ° | triángulo | |
4 | 3.674234614 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 6 | 4 | 0 | 109,471 ° | tetraedro | |
5 | 6.474691495 | 0 | 2 | 3 | 0 | 0 | 0 | 0 | 9 | 6 | 0 | 90.000 ° | bipirámide triangular | |
6 | 9.985281374 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 12 | 8 | 0 | 90.000 ° | octaedro | |
7 | 14.452977414 | 0 | 0 | 5 | 2 | 0 | 0 | 0 | 15 | 10 | 0 | 72.000 ° | bipirámide pentagonal | |
8 | 19.675287861 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | dieciséis | 8 | 2 | 71,694 ° | antiprisma cuadrado | |
9 | 25,759986531 | 0 | 0 | 3 | 6 | 0 | 0 | 0 | 21 | 14 | 0 | 69.190 ° | prisma triangular triaumentado | |
10 | 32.716949460 | 0 | 0 | 2 | 8 | 0 | 0 | 0 | 24 | dieciséis | 0 | 64,996 ° | bipirámide cuadrada giroelongada | |
11 | 40.596450510 | 0.013219635 | 0 | 2 | 8 | 1 | 0 | 0 | 27 | 18 | 0 | 58.540 ° | icosaedro de borde contraído | |
12 | 49.165253058 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 30 | 20 | 0 | 63.435 ° | icosaedro ( esfera geodésica {3,5+} 1,0 ) | |
13 | 58.853230612 | 0,008820367 | 0 | 1 | 10 | 2 | 0 | 0 | 33 | 22 | 0 | 52,317 ° | ||
14 | 69.306363297 | 0 | 0 | 0 | 12 | 2 | 0 | 0 | 36 | 24 | 0 | 52,866 ° | bipirámide hexagonal giroelongada | |
15 | 80.670244114 | 0 | 0 | 0 | 12 | 3 | 0 | 0 | 39 | 26 | 0 | 49,225 ° | ||
dieciséis | 92.911655302 | 0 | 0 | 0 | 12 | 4 | 0 | 0 | 42 | 28 | 0 | 48,936 ° | ||
17 | 106.050404829 | 0 | 0 | 0 | 12 | 5 | 0 | 0 | 45 | 30 | 0 | 50,108 ° | bipirámide pentagonal doble giroelongada | |
18 | 120.084467447 | 0 | 0 | 2 | 8 | 8 | 0 | 0 | 48 | 32 | 0 | 47.534 ° | ||
19 | 135.089467557 | 0.000135163 | 0 | 0 | 14 | 5 | 0 | 0 | 50 | 32 | 1 | 44,910 ° | ||
20 | 150,881568334 | 0 | 0 | 0 | 12 | 8 | 0 | 0 | 54 | 36 | 0 | 46.093 ° | ||
21 | 167,641622399 | 0,001406124 | 0 | 1 | 10 | 10 | 0 | 0 | 57 | 38 | 0 | 44,321 ° | ||
22 | 185.287536149 | 0 | 0 | 0 | 12 | 10 | 0 | 0 | 60 | 40 | 0 | 43.302 ° | ||
23 | 203.930190663 | 0 | 0 | 0 | 12 | 11 | 0 | 0 | 63 | 42 | 0 | 41.481 ° | ||
24 | 223.347074052 | 0 | 0 | 0 | 24 | 0 | 0 | 0 | 60 | 32 | 6 | 42.065 ° | cubo de desaire | |
25 | 243.812760299 | 0,001021305 | 0 | 0 | 14 | 11 | 0 | 0 | 68 | 44 | 1 | 39.610 ° | ||
26 | 265.133326317 | 0,001919065 | 0 | 0 | 12 | 14 | 0 | 0 | 72 | 48 | 0 | 38.842 ° | ||
27 | 287.302615033 | 0 | 0 | 0 | 12 | 15 | 0 | 0 | 75 | 50 | 0 | 39,940 ° | ||
28 | 310.491542358 | 0 | 0 | 0 | 12 | dieciséis | 0 | 0 | 78 | 52 | 0 | 37,824 ° | ||
29 | 334.634439920 | 0 | 0 | 0 | 12 | 17 | 0 | 0 | 81 | 54 | 0 | 36,391 ° | ||
30 | 359.603945904 | 0 | 0 | 0 | 12 | 18 | 0 | 0 | 84 | 56 | 0 | 36,942 ° | ||
31 | 385.530838063 | 0,003204712 | 0 | 0 | 12 | 19 | 0 | 0 | 87 | 58 | 0 | 36,373 ° | ||
32 | 412.261274651 | 0 | 0 | 0 | 12 | 20 | 0 | 0 | 90 | 60 | 0 | 37,377 ° | pentakis dodecaedro ( esfera geodésica {3,5+} 1,1 ) | |
33 | 440.204057448 | 0,004356481 | 0 | 0 | 15 | 17 | 1 | 0 | 92 | 60 | 1 | 33.700 ° | ||
34 | 468.904853281 | 0 | 0 | 0 | 12 | 22 | 0 | 0 | 96 | 64 | 0 | 33.273 ° | ||
35 | 498.569872491 | 0.000419208 | 0 | 0 | 12 | 23 | 0 | 0 | 99 | 66 | 0 | 33.100 ° | ||
36 | 529.122408375 | 0 | 0 | 0 | 12 | 24 | 0 | 0 | 102 | 68 | 0 | 33.229 ° | ||
37 | 560.618887731 | 0 | 0 | 0 | 12 | 25 | 0 | 0 | 105 | 70 | 0 | 32,332 ° | ||
38 | 593.038503566 | 0 | 0 | 0 | 12 | 26 | 0 | 0 | 108 | 72 | 0 | 33.236 ° | ||
39 | 626.389009017 | 0 | 0 | 0 | 12 | 27 | 0 | 0 | 111 | 74 | 0 | 32.053 ° | ||
40 | 660.675278835 | 0 | 0 | 0 | 12 | 28 | 0 | 0 | 114 | 76 | 0 | 31,916 ° | ||
41 | 695.916744342 | 0 | 0 | 0 | 12 | 29 | 0 | 0 | 117 | 78 | 0 | 31.528 ° | ||
42 | 732.078107544 | 0 | 0 | 0 | 12 | 30 | 0 | 0 | 120 | 80 | 0 | 31.245 ° | ||
43 | 769.190846459 | 0.000399668 | 0 | 0 | 12 | 31 | 0 | 0 | 123 | 82 | 0 | 30.867 ° | ||
44 | 807.174263085 | 0 | 0 | 0 | 24 | 20 | 0 | 0 | 120 | 72 | 6 | 31.258 ° | ||
45 | 846.188401061 | 0 | 0 | 0 | 12 | 33 | 0 | 0 | 129 | 86 | 0 | 30,207 ° | ||
46 | 886.167113639 | 0 | 0 | 0 | 12 | 34 | 0 | 0 | 132 | 88 | 0 | 29,790 ° | ||
47 | 927.059270680 | 0,002482914 | 0 | 0 | 14 | 33 | 0 | 0 | 134 | 88 | 1 | 28,787 ° | ||
48 | 968.713455344 | 0 | 0 | 0 | 24 | 24 | 0 | 0 | 132 | 80 | 6 | 29.690 ° | ||
49 | 1011.557182654 | 0,001529341 | 0 | 0 | 12 | 37 | 0 | 0 | 141 | 94 | 0 | 28,387 ° | ||
50 | 1055.182314726 | 0 | 0 | 0 | 12 | 38 | 0 | 0 | 144 | 96 | 0 | 29.231 ° | ||
51 | 1099.819290319 | 0 | 0 | 0 | 12 | 39 | 0 | 0 | 147 | 98 | 0 | 28.165 ° | ||
52 | 1145.418964319 | 0.000457327 | 0 | 0 | 12 | 40 | 0 | 0 | 150 | 100 | 0 | 27.670 ° | ||
53 | 1191.922290416 | 0.000278469 | 0 | 0 | 18 | 35 | 0 | 0 | 150 | 96 | 3 | 27.137 ° | ||
54 | 1239.361474729 | 0.000137870 | 0 | 0 | 12 | 42 | 0 | 0 | 156 | 104 | 0 | 27.030 ° | ||
55 | 1287.772720783 | 0.000391696 | 0 | 0 | 12 | 43 | 0 | 0 | 159 | 106 | 0 | 26.615 ° | ||
56 | 1337.094945276 | 0 | 0 | 0 | 12 | 44 | 0 | 0 | 162 | 108 | 0 | 26.683 ° | ||
57 | 1387.383229253 | 0 | 0 | 0 | 12 | 45 | 0 | 0 | 165 | 110 | 0 | 26,702 ° | ||
58 | 1438.618250640 | 0 | 0 | 0 | 12 | 46 | 0 | 0 | 168 | 112 | 0 | 26.155 ° | ||
59 | 1490.773335279 | 0.000154286 | 0 | 0 | 14 | 43 | 2 | 0 | 171 | 114 | 0 | 26.170 ° | ||
60 | 1543.830400976 | 0 | 0 | 0 | 12 | 48 | 0 | 0 | 174 | 116 | 0 | 25,958 ° | ||
61 | 1597.941830199 | 0,001091717 | 0 | 0 | 12 | 49 | 0 | 0 | 177 | 118 | 0 | 25,392 ° | ||
62 | 1652.909409898 | 0 | 0 | 0 | 12 | 50 | 0 | 0 | 180 | 120 | 0 | 25.880 ° | ||
63 | 1708.879681503 | 0 | 0 | 0 | 12 | 51 | 0 | 0 | 183 | 122 | 0 | 25,257 ° | ||
64 | 1765.802577927 | 0 | 0 | 0 | 12 | 52 | 0 | 0 | 186 | 124 | 0 | 24,920 ° | ||
sesenta y cinco | 1823.667960264 | 0.000399515 | 0 | 0 | 12 | 53 | 0 | 0 | 189 | 126 | 0 | 24.527 ° | ||
66 | 1882.441525304 | 0.000776245 | 0 | 0 | 12 | 54 | 0 | 0 | 192 | 128 | 0 | 24,765 ° | ||
67 | 1942.122700406 | 0 | 0 | 0 | 12 | 55 | 0 | 0 | 195 | 130 | 0 | 24,727 ° | ||
68 | 2002.874701749 | 0 | 0 | 0 | 12 | 56 | 0 | 0 | 198 | 132 | 0 | 24.433 ° | ||
69 | 2064.533483235 | 0 | 0 | 0 | 12 | 57 | 0 | 0 | 201 | 134 | 0 | 24.137 ° | ||
70 | 2127.100901551 | 0 | 0 | 0 | 12 | 50 | 0 | 0 | 200 | 128 | 4 | 24.291 ° | ||
71 | 2190.649906425 | 0,001256769 | 0 | 0 | 14 | 55 | 2 | 0 | 207 | 138 | 0 | 23,803 ° | ||
72 | 2255.001190975 | 0 | 0 | 0 | 12 | 60 | 0 | 0 | 210 | 140 | 0 | 24.492 ° | esfera geodésica {3,5+} 2,1 | |
73 | 2320.633883745 | 0,001572959 | 0 | 0 | 12 | 61 | 0 | 0 | 213 | 142 | 0 | 22.810 ° | ||
74 | 2387.072981838 | 0.000641539 | 0 | 0 | 12 | 62 | 0 | 0 | 216 | 144 | 0 | 22,966 ° | ||
75 | 2454.369689040 | 0 | 0 | 0 | 12 | 63 | 0 | 0 | 219 | 146 | 0 | 22.736 ° | ||
76 | 2522.674871841 | 0.000943474 | 0 | 0 | 12 | 64 | 0 | 0 | 222 | 148 | 0 | 22.886 ° | ||
77 | 2591.850152354 | 0 | 0 | 0 | 12 | sesenta y cinco | 0 | 0 | 225 | 150 | 0 | 23.286 ° | ||
78 | 2662.046474566 | 0 | 0 | 0 | 12 | 66 | 0 | 0 | 228 | 152 | 0 | 23.426 ° | ||
79 | 2733.248357479 | 0.000702921 | 0 | 0 | 12 | 63 | 1 | 0 | 230 | 152 | 1 | 22.636 ° | ||
80 | 2805.355875981 | 0 | 0 | 0 | dieciséis | 64 | 0 | 0 | 232 | 152 | 2 | 22,778 ° | ||
81 | 2878.522829664 | 0.000194289 | 0 | 0 | 12 | 69 | 0 | 0 | 237 | 158 | 0 | 21.892 ° | ||
82 | 2952.569675286 | 0 | 0 | 0 | 12 | 70 | 0 | 0 | 240 | 160 | 0 | 22,206 ° | ||
83 | 3027.528488921 | 0.000339815 | 0 | 0 | 14 | 67 | 2 | 0 | 243 | 162 | 0 | 21,646 ° | ||
84 | 3103.465124431 | 0.000401973 | 0 | 0 | 12 | 72 | 0 | 0 | 246 | 164 | 0 | 21.513 ° | ||
85 | 3180.361442939 | 0.000416581 | 0 | 0 | 12 | 73 | 0 | 0 | 249 | 166 | 0 | 21.498 ° | ||
86 | 3258.211605713 | 0,001378932 | 0 | 0 | 12 | 74 | 0 | 0 | 252 | 168 | 0 | 21.522 ° | ||
87 | 3337.000750014 | 0.000754863 | 0 | 0 | 12 | 75 | 0 | 0 | 255 | 170 | 0 | 21,456 ° | ||
88 | 3416.720196758 | 0 | 0 | 0 | 12 | 76 | 0 | 0 | 258 | 172 | 0 | 21,486 ° | ||
89 | 3497.439018625 | 0,000070891 | 0 | 0 | 12 | 77 | 0 | 0 | 261 | 174 | 0 | 21.182 ° | ||
90 | 3579.091222723 | 0 | 0 | 0 | 12 | 78 | 0 | 0 | 264 | 176 | 0 | 21.230 ° | ||
91 | 3661.713699320 | 0,000033221 | 0 | 0 | 12 | 79 | 0 | 0 | 267 | 178 | 0 | 21,105 ° | ||
92 | 3745.291636241 | 0 | 0 | 0 | 12 | 80 | 0 | 0 | 270 | 180 | 0 | 21.026 ° | ||
93 | 3829.844338421 | 0.000213246 | 0 | 0 | 12 | 81 | 0 | 0 | 273 | 182 | 0 | 20,751 ° | ||
94 | 3915.309269620 | 0 | 0 | 0 | 12 | 82 | 0 | 0 | 276 | 184 | 0 | 20,952 ° | ||
95 | 4001.771675565 | 0.000116638 | 0 | 0 | 12 | 83 | 0 | 0 | 279 | 186 | 0 | 20,711 ° | ||
96 | 4089.154010060 | 0,000036310 | 0 | 0 | 12 | 84 | 0 | 0 | 282 | 188 | 0 | 20.687 ° | ||
97 | 4177.533599622 | 0,000096437 | 0 | 0 | 12 | 85 | 0 | 0 | 285 | 190 | 0 | 20.450 ° | ||
98 | 4266.822464156 | 0.000112916 | 0 | 0 | 12 | 86 | 0 | 0 | 288 | 192 | 0 | 20.422 ° | ||
99 | 4357.139163132 | 0.000156508 | 0 | 0 | 12 | 87 | 0 | 0 | 291 | 194 | 0 | 20,284 ° | ||
100 | 4448.350634331 | 0 | 0 | 0 | 12 | 88 | 0 | 0 | 294 | 196 | 0 | 20,297 ° | ||
101 | 4540.590051694 | 0 | 0 | 0 | 12 | 89 | 0 | 0 | 297 | 198 | 0 | 20.011 ° | ||
102 | 4633.736565899 | 0 | 0 | 0 | 12 | 90 | 0 | 0 | 300 | 200 | 0 | 20.040 ° | ||
103 | 4727.836616833 | 0.000201245 | 0 | 0 | 12 | 91 | 0 | 0 | 303 | 202 | 0 | 19,907 ° | ||
104 | 4822.876522746 | 0 | 0 | 0 | 12 | 92 | 0 | 0 | 306 | 204 | 0 | 19,957 ° | ||
105 | 4919.000637616 | 0 | 0 | 0 | 12 | 93 | 0 | 0 | 309 | 206 | 0 | 19.842 ° | ||
106 | 5015.984595705 | 0 | 0 | 0 | 12 | 94 | 0 | 0 | 312 | 208 | 0 | 19,658 ° | ||
107 | 5113.953547724 | 0,000064137 | 0 | 0 | 12 | 95 | 0 | 0 | 315 | 210 | 0 | 19,327 ° | ||
108 | 5212.813507831 | 0.000432525 | 0 | 0 | 12 | 96 | 0 | 0 | 318 | 212 | 0 | 19,327 ° | ||
109 | 5312.735079920 | 0.000647299 | 0 | 0 | 14 | 93 | 2 | 0 | 321 | 214 | 0 | 19,103 ° | ||
110 | 5413.549294192 | 0 | 0 | 0 | 12 | 98 | 0 | 0 | 324 | 216 | 0 | 19.476 ° | ||
111 | 5515.293214587 | 0 | 0 | 0 | 12 | 99 | 0 | 0 | 327 | 218 | 0 | 19,255 ° | ||
112 | 5618.044882327 | 0 | 0 | 0 | 12 | 100 | 0 | 0 | 330 | 220 | 0 | 19,351 ° | ||
113 | 5721.824978027 | 0 | 0 | 0 | 12 | 101 | 0 | 0 | 333 | 222 | 0 | 18,978 ° | ||
114 | 5826.521572163 | 0.000149772 | 0 | 0 | 12 | 102 | 0 | 0 | 336 | 224 | 0 | 18.836 ° | ||
115 | 5932.181285777 | 0,000049972 | 0 | 0 | 12 | 103 | 0 | 0 | 339 | 226 | 0 | 18.458 ° | ||
116 | 6038.815593579 | 0.000259726 | 0 | 0 | 12 | 104 | 0 | 0 | 342 | 228 | 0 | 18,386 ° | ||
117 | 6146.342446579 | 0.000127609 | 0 | 0 | 12 | 105 | 0 | 0 | 345 | 230 | 0 | 18.566 ° | ||
118 | 6254.877027790 | 0.000332475 | 0 | 0 | 12 | 106 | 0 | 0 | 348 | 232 | 0 | 18.455 ° | ||
119 | 6364.347317479 | 0.000685590 | 0 | 0 | 12 | 107 | 0 | 0 | 351 | 234 | 0 | 18,336 ° | ||
120 | 6474.756324980 | 0,001373062 | 0 | 0 | 12 | 108 | 0 | 0 | 354 | 236 | 0 | 18.418 ° | ||
121 | 6586.121949584 | 0.000838863 | 0 | 0 | 12 | 109 | 0 | 0 | 357 | 238 | 0 | 18.199 ° | ||
122 | 6698.374499261 | 0 | 0 | 0 | 12 | 110 | 0 | 0 | 360 | 240 | 0 | 18.612 ° | esfera geodésica {3,5+} 2,2 | |
123 | 6811.827228174 | 0,001939754 | 0 | 0 | 14 | 107 | 2 | 0 | 363 | 242 | 0 | 17.840 ° | ||
124 | 6926.169974193 | 0 | 0 | 0 | 12 | 112 | 0 | 0 | 366 | 244 | 0 | 18.111 ° | ||
125 | 7041.473264023 | 0.000088274 | 0 | 0 | 12 | 113 | 0 | 0 | 369 | 246 | 0 | 17.867° | ||
126 | 7157.669224867 | 0 | 0 | 2 | 16 | 100 | 8 | 0 | 372 | 248 | 0 | 17.920° | ||
127 | 7274.819504675 | 0 | 0 | 0 | 12 | 115 | 0 | 0 | 375 | 250 | 0 | 17.877° | ||
128 | 7393.007443068 | 0.000054132 | 0 | 0 | 12 | 116 | 0 | 0 | 378 | 252 | 0 | 17.814° | ||
129 | 7512.107319268 | 0.000030099 | 0 | 0 | 12 | 117 | 0 | 0 | 381 | 254 | 0 | 17.743° | ||
130 | 7632.167378912 | 0.000025622 | 0 | 0 | 12 | 118 | 0 | 0 | 384 | 256 | 0 | 17.683° | ||
131 | 7753.205166941 | 0.000305133 | 0 | 0 | 12 | 119 | 0 | 0 | 387 | 258 | 0 | 17.511° | ||
132 | 7875.045342797 | 0 | 0 | 0 | 12 | 120 | 0 | 0 | 390 | 260 | 0 | 17.958° | geodesic sphere {3,5+}3,1 | |
133 | 7998.179212898 | 0.000591438 | 0 | 0 | 12 | 121 | 0 | 0 | 393 | 262 | 0 | 17.133° | ||
134 | 8122.089721194 | 0.000470268 | 0 | 0 | 12 | 122 | 0 | 0 | 396 | 264 | 0 | 17.214° | ||
135 | 8246.909486992 | 0 | 0 | 0 | 12 | 123 | 0 | 0 | 399 | 266 | 0 | 17.431° | ||
136 | 8372.743302539 | 0 | 0 | 0 | 12 | 124 | 0 | 0 | 402 | 268 | 0 | 17.485° | ||
137 | 8499.534494782 | 0 | 0 | 0 | 12 | 125 | 0 | 0 | 405 | 270 | 0 | 17.560° | ||
138 | 8627.406389880 | 0.000473576 | 0 | 0 | 12 | 126 | 0 | 0 | 408 | 272 | 0 | 16.924° | ||
139 | 8756.227056057 | 0.000404228 | 0 | 0 | 12 | 127 | 0 | 0 | 411 | 274 | 0 | 16.673° | ||
140 | 8885.980609041 | 0.000630351 | 0 | 0 | 13 | 126 | 1 | 0 | 414 | 276 | 0 | 16.773° | ||
141 | 9016.615349190 | 0.000376365 | 0 | 0 | 14 | 126 | 0 | 1 | 417 | 278 | 0 | 16.962° | ||
142 | 9148.271579993 | 0.000550138 | 0 | 0 | 12 | 130 | 0 | 0 | 420 | 280 | 0 | 16.840° | ||
143 | 9280.839851192 | 0.000255449 | 0 | 0 | 12 | 131 | 0 | 0 | 423 | 282 | 0 | 16.782° | ||
144 | 9414.371794460 | 0 | 0 | 0 | 12 | 132 | 0 | 0 | 426 | 284 | 0 | 16.953° | ||
145 | 9548.928837232 | 0.000094938 | 0 | 0 | 12 | 133 | 0 | 0 | 429 | 286 | 0 | 16.841° | ||
146 | 9684.381825575 | 0 | 0 | 0 | 12 | 134 | 0 | 0 | 432 | 288 | 0 | 16.905° | ||
147 | 9820.932378373 | 0.000636651 | 0 | 0 | 12 | 135 | 0 | 0 | 435 | 290 | 0 | 16.458° | ||
148 | 9958.406004270 | 0.000203701 | 0 | 0 | 12 | 136 | 0 | 0 | 438 | 292 | 0 | 16.627° | ||
149 | 10096.859907397 | 0.000638186 | 0 | 0 | 14 | 133 | 2 | 0 | 441 | 294 | 0 | 16.344° | ||
150 | 10236.196436701 | 0 | 0 | 0 | 12 | 138 | 0 | 0 | 444 | 296 | 0 | 16.405° | ||
151 | 10376.571469275 | 0.000153836 | 0 | 0 | 12 | 139 | 0 | 0 | 447 | 298 | 0 | 16.163° | ||
152 | 10517.867592878 | 0 | 0 | 0 | 12 | 140 | 0 | 0 | 450 | 300 | 0 | 16.117° | ||
153 | 10660.082748237 | 0 | 0 | 0 | 12 | 141 | 0 | 0 | 453 | 302 | 0 | 16.390° | ||
154 | 10803.372421141 | 0.000735800 | 0 | 0 | 12 | 142 | 0 | 0 | 456 | 304 | 0 | 16.078° | ||
155 | 10947.574692279 | 0.000603670 | 0 | 0 | 12 | 143 | 0 | 0 | 459 | 306 | 0 | 15.990° | ||
156 | 11092.798311456 | 0.000508534 | 0 | 0 | 12 | 144 | 0 | 0 | 462 | 308 | 0 | 15.822° | ||
157 | 11238.903041156 | 0.000357679 | 0 | 0 | 12 | 145 | 0 | 0 | 465 | 310 | 0 | 15.948° | ||
158 | 11385.990186197 | 0.000921918 | 0 | 0 | 12 | 146 | 0 | 0 | 468 | 312 | 0 | 15.987° | ||
159 | 11534.023960956 | 0.000381457 | 0 | 0 | 12 | 147 | 0 | 0 | 471 | 314 | 0 | 15.960° | ||
160 | 11683.054805549 | 0 | 0 | 0 | 12 | 148 | 0 | 0 | 474 | 316 | 0 | 15.961° | ||
161 | 11833.084739465 | 0.000056447 | 0 | 0 | 12 | 149 | 0 | 0 | 477 | 318 | 0 | 15.810° | ||
162 | 11984.050335814 | 0 | 0 | 0 | 12 | 150 | 0 | 0 | 480 | 320 | 0 | 15.813° | ||
163 | 12136.013053220 | 0.000120798 | 0 | 0 | 12 | 151 | 0 | 0 | 483 | 322 | 0 | 15.675° | ||
164 | 12288.930105320 | 0 | 0 | 0 | 12 | 152 | 0 | 0 | 486 | 324 | 0 | 15.655° | ||
165 | 12442.804451373 | 0.000091119 | 0 | 0 | 12 | 153 | 0 | 0 | 489 | 326 | 0 | 15.651° | ||
166 | 12597.649071323 | 0 | 0 | 0 | 16 | 146 | 4 | 0 | 492 | 328 | 0 | 15.607° | ||
167 | 12753.469429750 | 0.000097382 | 0 | 0 | 12 | 155 | 0 | 0 | 495 | 330 | 0 | 15.600° | ||
168 | 12910.212672268 | 0 | 0 | 0 | 12 | 156 | 0 | 0 | 498 | 332 | 0 | 15.655° | ||
169 | 13068.006451127 | 0.000068102 | 0 | 0 | 13 | 155 | 1 | 0 | 501 | 334 | 0 | 15.537° | ||
170 | 13226.681078541 | 0 | 0 | 0 | 12 | 158 | 0 | 0 | 504 | 336 | 0 | 15.569° | ||
171 | 13386.355930717 | 0 | 0 | 0 | 12 | 159 | 0 | 0 | 507 | 338 | 0 | 15.497° | ||
172 | 13547.018108787 | 0.000547291 | 0 | 0 | 14 | 156 | 2 | 0 | 510 | 340 | 0 | 15.292° | ||
173 | 13708.635243034 | 0.000286544 | 0 | 0 | 12 | 161 | 0 | 0 | 513 | 342 | 0 | 15.225° | ||
174 | 13871.187092292 | 0 | 0 | 0 | 12 | 162 | 0 | 0 | 516 | 344 | 0 | 15.366° | ||
175 | 14034.781306929 | 0.000026686 | 0 | 0 | 12 | 163 | 0 | 0 | 519 | 346 | 0 | 15.252° | ||
176 | 14199.354775632 | 0.000283978 | 0 | 0 | 12 | 164 | 0 | 0 | 522 | 348 | 0 | 15.101° | ||
177 | 14364.837545298 | 0 | 0 | 0 | 12 | 165 | 0 | 0 | 525 | 350 | 0 | 15.269° | ||
178 | 14531.309552587 | 0 | 0 | 0 | 12 | 166 | 0 | 0 | 528 | 352 | 0 | 15.145° | ||
179 | 14698.754594220 | 0.000125113 | 0 | 0 | 13 | 165 | 1 | 0 | 531 | 354 | 0 | 14.968° | ||
180 | 14867.099927525 | 0 | 0 | 0 | 12 | 168 | 0 | 0 | 534 | 356 | 0 | 15.067° | ||
181 | 15036.467239769 | 0.000304193 | 0 | 0 | 12 | 169 | 0 | 0 | 537 | 358 | 0 | 15.002° | ||
182 | 15206.730610906 | 0 | 0 | 0 | 12 | 170 | 0 | 0 | 540 | 360 | 0 | 15.155° | ||
183 | 15378.166571028 | 0.000467899 | 0 | 0 | 12 | 171 | 0 | 0 | 543 | 362 | 0 | 14.747° | ||
184 | 15550.421450311 | 0 | 0 | 0 | 12 | 172 | 0 | 0 | 546 | 364 | 0 | 14.932° | ||
185 | 15723.720074072 | 0.000389762 | 0 | 0 | 12 | 173 | 0 | 0 | 549 | 366 | 0 | 14.775° | ||
186 | 15897.897437048 | 0.000389762 | 0 | 0 | 12 | 174 | 0 | 0 | 552 | 368 | 0 | 14.739° | ||
187 | 16072.975186320 | 0 | 0 | 0 | 12 | 175 | 0 | 0 | 555 | 370 | 0 | 14.848° | ||
188 | 16249.222678879 | 0 | 0 | 0 | 12 | 176 | 0 | 0 | 558 | 372 | 0 | 14.740° | ||
189 | 16426.371938862 | 0.000020732 | 0 | 0 | 12 | 177 | 0 | 0 | 561 | 374 | 0 | 14.671° | ||
190 | 16604.428338501 | 0.000586804 | 0 | 0 | 12 | 178 | 0 | 0 | 564 | 376 | 0 | 14.501° | ||
191 | 16783.452219362 | 0.001129202 | 0 | 0 | 13 | 177 | 1 | 0 | 567 | 378 | 0 | 14.195° | ||
192 | 16963.338386460 | 0 | 0 | 0 | 12 | 180 | 0 | 0 | 570 | 380 | 0 | 14.819° | geodesic sphere {3,5+}3,2 | |
193 | 17144.564740880 | 0.000985192 | 0 | 0 | 12 | 181 | 0 | 0 | 573 | 382 | 0 | 14.144° | ||
194 | 17326.616136471 | 0.000322358 | 0 | 0 | 12 | 182 | 0 | 0 | 576 | 384 | 0 | 14.350° | ||
195 | 17509.489303930 | 0 | 0 | 0 | 12 | 183 | 0 | 0 | 579 | 386 | 0 | 14.375° | ||
196 | 17693.460548082 | 0.000315907 | 0 | 0 | 12 | 184 | 0 | 0 | 582 | 388 | 0 | 14.251° | ||
197 | 17878.340162571 | 0 | 0 | 0 | 12 | 185 | 0 | 0 | 585 | 390 | 0 | 14.147° | ||
198 | 18064.262177195 | 0.000011149 | 0 | 0 | 12 | 186 | 0 | 0 | 588 | 392 | 0 | 14.237° | ||
199 | 18251.082495640 | 0.000534779 | 0 | 0 | 12 | 187 | 0 | 0 | 591 | 394 | 0 | 14.153° | ||
200 | 18438.842717530 | 0 | 0 | 0 | 12 | 188 | 0 | 0 | 594 | 396 | 0 | 14.222° | ||
201 | 18627.591226244 | 0.001048859 | 0 | 0 | 13 | 187 | 1 | 0 | 597 | 398 | 0 | 13.830° | ||
202 | 18817.204718262 | 0 | 0 | 0 | 12 | 190 | 0 | 0 | 600 | 400 | 0 | 14.189° | ||
203 | 19007.981204580 | 0.000600343 | 0 | 0 | 12 | 191 | 0 | 0 | 603 | 402 | 0 | 13.977° | ||
204 | 19199.540775603 | 0 | 0 | 0 | 12 | 192 | 0 | 0 | 606 | 404 | 0 | 14.291° | ||
212 | 20768.053085964 | 0 | 0 | 0 | 12 | 200 | 0 | 0 | 630 | 420 | 0 | 14.118° | geodesic sphere {3,5+}4,1 | |
214 | 21169.910410375 | 0 | 0 | 0 | 12 | 202 | 0 | 0 | 636 | 424 | 0 | 13.771° | ||
216 | 21575.596377869 | 0 | 0 | 0 | 12 | 204 | 0 | 0 | 642 | 428 | 0 | 13.735° | ||
217 | 21779.856080418 | 0 | 0 | 0 | 12 | 205 | 0 | 0 | 645 | 430 | 0 | 13.902° | ||
232 | 24961.252318934 | 0 | 0 | 0 | 12 | 220 | 0 | 0 | 690 | 460 | 0 | 13.260° | ||
255 | 30264.424251281 | 0 | 0 | 0 | 12 | 243 | 0 | 0 | 759 | 506 | 0 | 12.565° | ||
256 | 30506.687515847 | 0 | 0 | 0 | 12 | 244 | 0 | 0 | 762 | 508 | 0 | 12.572° | ||
257 | 30749.941417346 | 0 | 0 | 0 | 12 | 245 | 0 | 0 | 765 | 510 | 0 | 12.672° | ||
272 | 34515.193292681 | 0 | 0 | 0 | 12 | 260 | 0 | 0 | 810 | 540 | 0 | 12.335° | geodesic sphere {3,5+}3,3 | |
282 | 37147.294418462 | 0 | 0 | 0 | 12 | 270 | 0 | 0 | 840 | 560 | 0 | 12.166° | geodesic sphere {3,5+}4,2 | |
292 | 39877.008012909 | 0 | 0 | 0 | 12 | 280 | 0 | 0 | 870 | 580 | 0 | 11.857° | ||
306 | 43862.569780797 | 0 | 0 | 0 | 12 | 294 | 0 | 0 | 912 | 608 | 0 | 11.628° | ||
312 | 45629.313804002 | 0.000306163 | 0 | 0 | 12 | 300 | 0 | 0 | 930 | 620 | 0 | 11.299° | ||
315 | 46525.825643432 | 0 | 0 | 0 | 12 | 303 | 0 | 0 | 939 | 626 | 0 | 11.337° | ||
317 | 47128.310344520 | 0 | 0 | 0 | 12 | 305 | 0 | 0 | 945 | 630 | 0 | 11.423° | ||
318 | 47431.056020043 | 0 | 0 | 0 | 12 | 306 | 0 | 0 | 948 | 632 | 0 | 11.219° | ||
334 | 52407.728127822 | 0 | 0 | 0 | 12 | 322 | 0 | 0 | 996 | 664 | 0 | 11.058° | ||
348 | 56967.472454334 | 0 | 0 | 0 | 12 | 336 | 0 | 0 | 1038 | 692 | 0 | 10.721° | ||
357 | 59999.922939598 | 0 | 0 | 0 | 12 | 345 | 0 | 0 | 1065 | 710 | 0 | 10.728° | ||
358 | 60341.830924588 | 0 | 0 | 0 | 12 | 346 | 0 | 0 | 1068 | 712 | 0 | 10.647° | ||
372 | 65230.027122557 | 0 | 0 | 0 | 12 | 360 | 0 | 0 | 1110 | 740 | 0 | 10.531° | geodesic sphere {3,5+}4,3 | |
382 | 68839.426839215 | 0 | 0 | 0 | 12 | 370 | 0 | 0 | 1140 | 760 | 0 | 10.379° | ||
390 | 71797.035335953 | 0 | 0 | 0 | 12 | 378 | 0 | 0 | 1164 | 776 | 0 | 10.222° | ||
392 | 72546.258370889 | 0 | 0 | 0 | 12 | 380 | 0 | 0 | 1170 | 780 | 0 | 10.278° | ||
400 | 75582.448512213 | 0 | 0 | 0 | 12 | 388 | 0 | 0 | 1194 | 796 | 0 | 10.068° | ||
402 | 76351.192432673 | 0 | 0 | 0 | 12 | 390 | 0 | 0 | 1200 | 800 | 0 | 10.099° | ||
432 | 88353.709681956 | 0 | 0 | 0 | 24 | 396 | 12 | 0 | 1290 | 860 | 0 | 9.556° | ||
448 | 95115.546986209 | 0 | 0 | 0 | 24 | 412 | 12 | 0 | 1338 | 892 | 0 | 9.322° | ||
460 | 100351.763108673 | 0 | 0 | 0 | 24 | 424 | 12 | 0 | 1374 | 916 | 0 | 9.297° | ||
468 | 103920.871715127 | 0 | 0 | 0 | 24 | 432 | 12 | 0 | 1398 | 932 | 0 | 9.120° | ||
470 | 104822.886324279 | 0 | 0 | 0 | 24 | 434 | 12 | 0 | 1404 | 936 | 0 | 9.059° |
According to a conjecture, if , p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p, then the solution for m electrons is f(m): .[13]
Referencias
- ^ Thomson, Joseph John (March 1904). "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure" (PDF). Philosophical Magazine. Series 6. 7 (39): 237–265. doi:10.1080/14786440409463107. Archived from the original (PDF) on 13 December 2013.
- ^ Smale, S. (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX 10.1.1.35.4101. doi:10.1007/bf03025291.
- ^ Föppl, L. (1912). "Stabile Anordnungen von Elektronen im Atom". J. Reine Angew. Math. (141): 251–301..
- ^ Schwartz, Richard (2010). "The 5 electron case of Thomson's Problem". arXiv:1001.3702 [math.MG].
- ^ Yudin, V.A. (1992). "The minimum of potential energy of a system of point charges". Discretnaya Matematika. 4 (2): 115–121 (in Russian).; Yudin, V. A. (1993). "The minimum of potential energy of a system of point charges". Discrete Math. Appl. 3 (1): 75–81. doi:10.1515/dma.1993.3.1.75.
- ^ Andreev, N.N. (1996). "An extremal property of the icosahedron". East J. Approximation. 2 (4): 459–462. MR1426716, Zbl 0877.51021
- ^ Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
- ^ Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
- ^ Levin, Y.; Arenzon, J. J. (2003). "Why charges go to the Surface: A generalized Thomson Problem". Europhys. Lett. 63 (3): 415. arXiv:cond-mat/0302524. Bibcode:2003EL.....63..415L. doi:10.1209/epl/i2003-00546-1.
- ^ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
- ^ LaFave Jr, Tim (2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure". Journal of Electrostatics. 71 (6): 1029–1035. arXiv:1403.2591. doi:10.1016/j.elstat.2013.10.001.
- ^ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
- ^ "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.
Notas
- Whyte, L.L. (1952). "Unique arrangements of points on a sphere". Amer. Math. Monthly. 59 (9): 606–611. doi:10.2307/2306764. JSTOR 2306764.
- Cohn, Harvey (1956). "Stability configurations of electrons on a sphere". Math. Comput. 10 (55): 117–120. doi:10.1090/S0025-5718-1956-0081133-0.
- Goldberg, Michael (1969). "Stability configurations of electrons on a sphere". Math. Comp. 23 (108): 785–786. doi:10.1090/S0025-5718-69-99642-2.
- Erber, T.; Hockney, G. M. (1991). "equilibrium configurations of N equal charges on a sphere". J. Phys. A: Math. Gen. 24 (23): L1369. Bibcode:1991JPhA...24L1369E. doi:10.1088/0305-4470/24/23/008.
- Morris, J. R.; Deaven, D. M.; Ho, K. M. (1996). "Genetic-algorithm energy minimization for point charges on a sphere". Phys. Rev. B. 53 (4): R1740–R1743. Bibcode:1996PhRvB..53.1740M. CiteSeerX 10.1.1.28.93. doi:10.1103/PhysRevB.53.R1740.
- Erber, T.; Hockney, G. M. (1997). Complex Systems: Equilibrium Configurations of Equal Charges on a Sphere . Advances in Chemical Physics. 98. pp. 495–594. doi:10.1002/9780470141571.ch5. ISBN 9780470141571..
- Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Tipton, R.; Stong, R.; Dowla, F.; Wooten, F. (1997). "Possible global minimum lattice configurations for Thomson's problem of charges on a sphere". Phys. Rev. Lett. 78 (14): 2681–2685. Bibcode:1997PhRvL..78.2681A. doi:10.1103/PhysRevLett.78.2681.
- Bowick, M.; Cacciuto, A.; Nelson, D. R.; Travesset, A. (2002). "Crystalline order on a sphere and the generalized Thomson Problem". Phys. Rev. Lett. 89 (18): 249902. arXiv:cond-mat/0206144. Bibcode:2002PhRvL..89r5502B. doi:10.1103/PhysRevLett.89.185502. PMID 12398614.
- Dragnev, P. D.; Legg, D. A.; Townsend, D. W. (2002). "Discrete logarithmic energy on the sphere". Pacific J. Math. 207 (2): 345–358. doi:10.2140/pjm.2002.207.345..
- Katanforoush, A.; Shahshahani, M. (2003). "Distributing points on the sphere. I". Exper. Math. 12 (2): 199–209. doi:10.1080/10586458.2003.10504492.
- Wales, David J.; Ulker, Sidika (2006). "Structure and dynamics of spherical crystals characterized for the Thomson problem". Phys. Rev. B. 74 (21): 212101. Bibcode:2006PhRvB..74u2101W. doi:10.1103/PhysRevB.74.212101. Configurations reprinted in Wales, D. J.; Ulker, S. "The Cambridge cluster database".
- Slosar, A.; Podgornik, R. (2006). "On the connected-charges Thomson problem". Europhys. Lett. 75 (4): 631. arXiv:cond-mat/0606765. Bibcode:2006EL.....75..631S. doi:10.1209/epl/i2006-10146-1.
- Cohn, Henry; Kumar, Abhinav (2007). "Universally optimal distribution of points on spheres". J. Amer. Math. Soc. 20 (1): 99–148. arXiv:math/0607446. Bibcode:2007JAMS...20...99C. doi:10.1090/S0894-0347-06-00546-7.
- Wales, D. J.; McKay, H.; Altschuler, E. L. (2009). "Defect motifs for spherical topologies". Phys. Rev. B. 79 (22): 224115. Bibcode:2009PhRvB..79v4115W. doi:10.1103/PhysRevB.79.224115.. Configurations reproduced in Wales, D. J.; Ulker, S. "The Cambridge cluster database".
- Ridgway, W. J. M.; Cheviakov, A. F. (2018). "An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere". Comput. Phys. Commun. 233: 84–109. doi:10.1016/j.cpc.2018.03.029.
- Cecka, Cris; Bowick, Mark J.; Middleton, Alan A. "Thomson Problem @ S.U."
- This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us.