The objective of the **Thomson problem** is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904^{ [1] } after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.

- Mathematical statement
- Example
- Known exact solutions
- Generalizations
- Solution algorithms
- Continuous spherical shell charge
- Randomly distributed point charges
- Charge-centered distribution
- Relations to other scientific problems
- Configurations of smallest known energy
- References
- Notes

Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy.

The electrostatic interaction energy occurring between each pair of electrons of equal charges (, with the elementary charge of an electron) is given by Coulomb's law,

where, is the electric constant and is the distance between each pair of electrons located at points on the sphere defined by vectors and , respectively.

Simplified units of and (the Coulomb constant) are used without loss of generality. Then,

The total electrostatic potential energy of each *N*-electron configuration may then be expressed as the sum of all pair-wise interaction energies

The global minimization of over all possible configurations of *N* distinct points is typically found by numerical minimization algorithms.

Thomson's problem is related to the 7th of the eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere".^{ [2] } The main difference is that in Smale's problem the function to minimise is not the electrostatic potential but a logarithmic potential given by A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number *N* of points goes to infinity, not for concrete values of *N*.

The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin, , or

Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases.

- For
*N*= 1, the solution is trivial. The single electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge.

- For
*N*= 2, the optimal configuration consists of electrons at antipodal points. This represents the first one-dimensional solution.

- For
*N*= 3, electrons reside at the vertices of an equilateral triangle about any great circle.^{ [3] }The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-*N*electron solutions. Also, this represents the first two-dimensional solution.

- For
*N*= 4, electrons reside at the vertices of a regular tetrahedron. Of interest, this represents the first three-dimensional solution.

- For
*N*= 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid.^{ [4] }Of interest, it is impossible for any*N*solution with five or more electrons to exhibit global equidistance among all pairs of electrons.

- For
*N*= 6, electrons reside at vertices of a regular octahedron.^{ [5] }The configuration may be imagined as four electrons residing at the corners of a square about the equator and the remaining two residing at the poles.

- For
*N*= 12, electrons reside at the vertices of a regular icosahedron.^{ [6] }

Geometric solutions of the Thomson problem for *N* = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles. Numerical solutions for *N* = 8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids whose faces are square and pentagonal, respectively.^{[ citation needed ]}

One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let *f* be a decreasing real-valued function, and define the energy functional

Traditionally, one considers also known as Riesz -kernels. For integrable Riesz kernels see the 1972 work of Landkof.^{ [7] } For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.^{ [8] } Notable cases include:^{ [9] }

*α*= ∞, the Tammes problem (packing);*α*= 1, the Thomson problem;*α*= 0, to maximize the product of distances, latterly known as Whyte's problem;*α*= −1 : maximum average distance problem.

One may also consider configurations of *N* points on a sphere of higher dimension. See spherical design.

Several algorithms have been applied to this problem. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance:^{ [9] }

- constrained global optimization (Altschuler et al. 1994),
- steepest descent (Claxton and Benson 1966, Erber and Hockney 1991),
- random walk (Weinrach et al. 1990),
- genetic algorithm (Morris et al. 1996)

While the objective is to minimize the global electrostatic potential energy of each *N*-electron case, several algorithmic starting cases are of interest.

The energy of a continuous spherical shell of charge distributed across its surface is given by

and is, in general, greater than the energy of every Thomson problem solution. Note: Here *N* is used as a continuous variable that represents the infinitely divisible charge, *Q*, distributed across the spherical shell. For example, a spherical shell of represents the uniform distribution of a single electron's charge, , across the entire shell.

The global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by

and is, in general, greater than the energy of every Thomson problem solution.

Here, *N* is a discrete variable that counts the number of electrons in the system. As well, .

For every *N*th solution of the Thomson problem there is an th configuration that includes an electron at the origin of the sphere whose energy is simply the addition of *N* to the energy of the *N*th solution. That is,^{ [10] }

Thus, if is known exactly, then is known exactly.

In general, is greater than , but is remarkably closer to each th Thomson solution than and . Therefore, the charge-centered distribution represents a smaller "energy gap" to cross to arrive at a solution of each Thomson problem than algorithms that begin with the other two charge configurations.

The Thomson problem is a natural consequence of J. J. Thomson's plum pudding model in the absence of its uniform positive background charge.^{ [11] }

"No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."

—Sir J. J. Thomson^{ [12] }

Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements.^{ [13] }

The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps.

The generalized Thomson problem arises, for example, in determining arrangements of protein subunits that comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in *colloidosomes*, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR theory. An example with long-range logarithmic interactions is provided by Abrikosov vortices that form at low temperatures in a superconducting metal shell with a large monopole at its center.

In the following table is the number of points (charges) in a configuration, is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and are the positions of the charges. Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero.

It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, is the number of vertices where the given number of edges meet, is the total number of edges, is the number of triangular faces, is the number of quadrilateral faces, and is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal. Thus, except in the cases *N* = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last column.^{ [14] }

N | Symmetry | Equivalent polyhedron | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 0.500000000 | 0 | – | – | – | – | – | – | 2 | – | – | 180.000° | digon | |

3 | 1.732050808 | 0 | – | – | – | – | – | – | 3 | 2 | – | 120.000° | triangle | |

4 | 3.674234614 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 6 | 4 | 0 | 109.471° | tetrahedron | |

5 | 6.474691495 | 0 | 2 | 3 | 0 | 0 | 0 | 0 | 9 | 6 | 0 | 90.000° | triangular dipyramid | |

6 | 9.985281374 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 12 | 8 | 0 | 90.000° | octahedron | |

7 | 14.452977414 | 0 | 0 | 5 | 2 | 0 | 0 | 0 | 15 | 10 | 0 | 72.000° | pentagonal dipyramid | |

8 | 19.675287861 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 16 | 8 | 2 | 71.694° | square antiprism | |

9 | 25.759986531 | 0 | 0 | 3 | 6 | 0 | 0 | 0 | 21 | 14 | 0 | 69.190° | triaugmented triangular prism | |

10 | 32.716949460 | 0 | 0 | 2 | 8 | 0 | 0 | 0 | 24 | 16 | 0 | 64.996° | gyroelongated square dipyramid | |

11 | 40.596450510 | 0.013219635 | 0 | 2 | 8 | 1 | 0 | 0 | 27 | 18 | 0 | 58.540° | edge-contracted icosahedron | |

12 | 49.165253058 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 30 | 20 | 0 | 63.435° | icosahedron (geodesic sphere {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° | gyroelongated hexagonal dipyramid | |

15 | 80.670244114 | 0 | 0 | 0 | 12 | 3 | 0 | 0 | 39 | 26 | 0 | 49.225° | ||

16 | 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° | double-gyroelongated pentagonal dipyramid | |

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° | snub cube | |

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 | 16 | 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 dodecahedron (geodesic sphere {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° | ||

65 | 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° | geodesic sphere {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 | 65 | 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 | 16 | 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° | geodesic sphere {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*): .^{ [15] }^{[ clarification needed ]}

In atomic theory and quantum mechanics, an **atomic orbital** is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term *atomic orbital* may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital.

In quantum mechanics, the **Pauli exclusion principle** states that two or more identical particles with half-integer spins cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.

A **Van de Graaff generator** is an electrostatic generator which uses a moving belt to accumulate electric charge on a hollow metal globe on the top of an insulated column, creating very high electric potentials. It produces very high voltage direct current (DC) electricity at low current levels. It was invented by American physicist Robert J. Van de Graaff in 1929. The potential difference achieved by modern Van de Graaff generators can be as much as 5 megavolts. A tabletop version can produce on the order of 100 kV and can store enough energy to produce visible electric sparks. Small Van de Graaff machines are produced for entertainment, and for physics education to teach electrostatics; larger ones are displayed in some science museums.

**Capacitance** is the capability of a material object or device to store electric charge. It is measured by the charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related notions of capacitance: *self capacitance* and *mutual capacitance*. An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component designed to add capacitance to an electric circuit.

**Density-functional theory** (**DFT**) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

**Packing problems** are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.

**Electrostatics** is a branch of physics that studies electric charges at rest.

In computational physics and chemistry, the **Hartree–Fock** (**HF**) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

In solid-state physics, the **electronic band structure** of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have.

The **classical electron radius** is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energy of a homogeneous charge distribution to the electron's relativistic mass-energy. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The classical electron radius is given as

**Jellium**, also known as the **uniform electron gas** (**UEG**) or **homogeneous electron gas** (**HEG**), is a quantum mechanical model of interacting electrons in a solid where the positive charges are assumed to be uniformly distributed in space; the electron density is a uniform quantity as well in space. This model allows one to focus on the effects in solids that occur due to the quantum nature of electrons and their mutual repulsive interactions without explicit introduction of the atomic lattice and structure making up a real material. Jellium is often used in solid-state physics as a simple model of delocalized electrons in a metal, where it can qualitatively reproduce features of real metals such as screening, plasmons, Wigner crystallization and Friedel oscillations.

The **Madelung constant** is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist.

In theoretical chemistry, **Marcus theory** is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron acceptor). It was originally formulated to address outer sphere electron transfer reactions, in which the two chemical species only change in their charge with an electron jumping (e.g. the oxidation of an ion like Fe^{2+}/Fe^{3+}), but do not undergo large structural changes. It was extended to include inner sphere electron transfer contributions, in which a change of distances or geometry in the solvation or coordination shells of the two chemical species is taken into account (the Fe-O distances in Fe(H_{2}O)^{2+} and Fe(H_{2}O)^{3+} are different).

The **Poisson–Boltzmann equation** is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.

**Implicit solvation** is a method to represent solvent as a continuous medium instead of individual “explicit” solvent molecules, most often used in molecular dynamics simulations and in other applications of molecular mechanics. The method is often applied to estimate free energy of solute-solvent interactions in structural and chemical processes, such as folding or conformational transitions of proteins, DNA, RNA, and polysaccharides, association of biological macromolecules with ligands, or transport of drugs across biological membranes.

In computational chemistry, a **water model** is used to simulate and thermodynamically calculate water clusters, liquid water, and aqueous solutions with explicit solvent. The models are determined from quantum mechanics, molecular mechanics, experimental results, and these combinations. To imitate a specific nature of molecules, many types of models have been developed. In general, these can be classified by the following three points; (i) the number of interaction points called *site*, (ii) whether the model is rigid or flexible, (iii) whether the model includes polarization effects.

The **atomic nucleus** is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. Protons and neutrons are bound together to form a nucleus by the nuclear force.

**Biology Monte Carlo methods (BioMOCA)** have been developed at the University of Illinois at Urbana-Champaign to simulate ion transport in an electrolyte environment through ion channels or nano-pores embedded in membranes. It is a 3-D particle-based Monte Carlo simulator for analyzing and studying the ion transport problem in ion channel systems or similar nanopores in wet/biological environments. The system simulated consists of a protein forming an ion channel (or an artificial nanopores like a Carbon Nano Tube, CNT), with a membrane (i.e. lipid bilayer) that separates two ion baths on either side. BioMOCA is based on two methodologies, namely the Boltzmann transport Monte Carlo (BTMC) and particle-particle-particle-mesh (P^{3}M). The first one uses Monte Carlo method to solve the Boltzmann equation, while the later splits the electrostatic forces into short-range and long-range components.

The **trigonometric Rosen–Morse potential**, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials.

The **linearized augmented-plane-wave method** (**LAPW**) is an implementation of Kohn-Sham density functional theory (DFT) adapted to periodic materials. It typically goes along with the treatment of both valence and core electrons on the same footing in the context of DFT and the treatment of the full potential and charge density without any shape approximation. This is often referred to as the all-electron **full-potential linearized augmented-plane-wave method** (**FLAPW**). It does not rely on the pseudopotential approximation and employs a systematically extendable basis set. These features make it one of the most precise implementations of DFT, applicable to all crystalline materials, regardless of their chemical composition. It can be used as a reference for evaluating other approaches.

- ↑ 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".
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- ↑ 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. S2CID 117117450. - ↑ Andreev, N.N. (1996). "An extremal property of the icosahedron".
*East J. Approximation*.**2**(4): 459–462. MR 1426716, 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
- 1 2 Batagelj, Vladimir; Plestenjak, Bor. "Optimal arrangements of n points on a sphere and in a circle" (PDF). IMFM/TCS. Archived from the original (PDF) on 25 June 2018.
- ↑ LaFave Jr, Tim (February 2014). "Discrete transformations in the Thomson Problem".
*Journal of Electrostatics*.**72**(1): 39–43. arXiv: 1403.2592 . doi:10.1016/j.elstat.2013.11.007. S2CID 119309183. - ↑ 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. S2CID 18929981. - ↑ 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. S2CID 118480104. - ↑ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
- ↑ "Sloane's A008486 (see the comment from Feb 03 2017)".
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*Complex Systems: Equilibrium Configurations of Equal Charges on a Sphere*.*Advances in Chemical Physics*. Vol. 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".
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*Comput. Phys. Commun*.**233**: 84–109. Bibcode:2018CoPhC.233...84R. doi:10.1016/j.cpc.2018.03.029. S2CID 52097788. - 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.

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