Thomson problem

Last updated

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.

Contents

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.

Mathematical statement

The physical system embodied by the Thomson problem is a special case of one of eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere". [2] The solution of each N-electron problem is obtained when the N-electron configuration constrained to the surface of a sphere of unit radius, , yields a global electrostatic potential energy minimum, .

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,

Here, is Coulomb's 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 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 interactions

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

Example

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

Known solutions

Schematic geometric solutions of the mathematical Thomson Problem for up to N = 5 electrons. N 2 to 5 ThomsonSolutions.png
Schematic geometric solutions of the mathematical Thomson Problem for up to N = 5 electrons.

Minimum energy configurations have been rigorously identified in only a handful of cases.

The geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are 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 ]

Generalizations

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]

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

Solution algorithms

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]

Relations to other scientific problems

The Thomson problem is a natural consequence of Thomson's plum pudding model in the absence of its uniform positive background charge. [10]

"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 [11]

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. [12]

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 the arrangements of the protein subunits which 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 the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.

Configurations of smallest known energy

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. [13]

N Symmetry Equivalent polyhedron
20.50000000002180.000° digon
31.732050808032120.000° triangle
43.6742346140400000640109.471° tetrahedron
56.474691495023000096090.000° triangular dipyramid
69.9852813740060000128090.000° octahedron
714.45297741400520001510072.000° pentagonal dipyramid
819.6752878610080000168271.694° square antiprism
925.75998653100360002114069.190° triaugmented triangular prism
1032.71694946000280002416064.996° gyroelongated square dipyramid
1140.5964505100.0132196350281002718058.540° edge-contracted icosahedron
1249.165253058000120003020063.435° icosahedron
(geodesic sphere {3,5+}1,0)
1358.8532306120.00882036701102003322052.317°
1469.306363297000122003624052.866°gyroelongated hexagonal dipyramid
1580.670244114000123003926049.225°
1692.911655302000124004228048.936°
17106.050404829000125004530050.108°double-gyroelongated pentagonal dipyramid
18120.08446744700288004832047.534°
19135.0894675570.00013516300145005032144.910°
20150.881568334000128005436046.093°
21167.6416223990.001406124011010005738044.321°
22185.2875361490001210006040043.302°
23203.9301906630001211006342041.481°
24223.347074052000240006032642.065° snub cube
25243.8127602990.001021305001411006844139.610°
26265.1333263170.001919065001214007248038.842°
27287.3026150330001215007550039.940°
28310.4915423580001216007852037.824°
29334.6344399200001217008154036.391°
30359.6039459040001218008456036.942°
31385.5308380630.003204712001219008758036.373°
32412.2612746510001220009060037.377° pentakis dodecahedron
(geodesic sphere {3,5+}1,1)
33440.2040574480.004356481001517109260133.700°
34468.9048532810001222009664033.273°
35498.5698724910.000419208001223009966033.100°
36529.12240837500012240010268033.229°
37560.61888773100012250010570032.332°
38593.03850356600012260010872033.236°
39626.38900901700012270011174032.053°
40660.67527883500012280011476031.916°
41695.91674434200012290011778031.528°
42732.07810754400012300012080031.245°
43769.1908464590.0003996680012310012382030.867°
44807.17426308500024200012072631.258°
45846.18840106100012330012986030.207°
46886.16711363900012340013288029.790°
47927.0592706800.0024829140014330013488128.787°
48968.71345534400024240013280629.690°
491011.5571826540.0015293410012370014194028.387°
501055.18231472600012380014496029.231°
511099.81929031900012390014798028.165°
521145.4189643190.00045732700124000150100027.670°
531191.9222904160.0002784690018350015096327.137°
541239.3614747290.00013787000124200156104027.030°
551287.7727207830.00039169600124300159106026.615°
561337.094945276000124400162108026.683°
571387.383229253000124500165110026.702°
581438.618250640000124600168112026.155°
591490.7733352790.00015428600144320171114026.170°
601543.830400976000124800174116025.958°
611597.9418301990.00109171700124900177118025.392°
621652.909409898000125000180120025.880°
631708.879681503000125100183122025.257°
641765.802577927000125200186124024.920°
651823.6679602640.00039951500125300189126024.527°
661882.4415253040.00077624500125400192128024.765°
671942.122700406000125500195130024.727°
682002.874701749000125600198132024.433°
692064.533483235000125700201134024.137°
702127.100901551000125000200128424.291°
712190.6499064250.00125676900145520207138023.803°
722255.001190975000126000210140024.492° geodesic sphere {3,5+}2,1
732320.6338837450.00157295900126100213142022.810°
742387.0729818380.00064153900126200216144022.966°
752454.369689040000126300219146022.736°
762522.6748718410.00094347400126400222148022.886°
772591.850152354000126500225150023.286°
782662.046474566000126600228152023.426°
792733.2483574790.00070292100126310230152122.636°
802805.355875981000166400232152222.778°
812878.5228296640.00019428900126900237158021.892°
822952.569675286000127000240160022.206°
833027.5284889210.00033981500146720243162021.646°
843103.4651244310.00040197300127200246164021.513°
853180.3614429390.00041658100127300249166021.498°
863258.2116057130.00137893200127400252168021.522°
873337.0007500140.00075486300127500255170021.456°
883416.720196758000127600258172021.486°
893497.4390186250.00007089100127700261174021.182°
903579.091222723000127800264176021.230°
913661.7136993200.00003322100127900267178021.105°
923745.291636241000128000270180021.026°
933829.8443384210.00021324600128100273182020.751°
943915.309269620000128200276184020.952°
954001.7716755650.00011663800128300279186020.711°
964089.1540100600.00003631000128400282188020.687°
974177.5335996220.00009643700128500285190020.450°
984266.8224641560.00011291600128600288192020.422°
994357.1391631320.00015650800128700291194020.284°
1004448.350634331000128800294196020.297°
1014540.590051694000128900297198020.011°
1024633.736565899000129000300200020.040°
1034727.8366168330.00020124500129100303202019.907°
1044822.876522746000129200306204019.957°
1054919.000637616000129300309206019.842°
1065015.984595705000129400312208019.658°
1075113.9535477240.00006413700129500315210019.327°
1085212.8135078310.00043252500129600318212019.327°
1095312.7350799200.00064729900149320321214019.103°
1105413.549294192000129800324216019.476°
1115515.293214587000129900327218019.255°
1125618.0448823270001210000330220019.351°
1135721.8249780270001210100333222018.978°
1145826.5215721630.000149772001210200336224018.836°
1155932.1812857770.000049972001210300339226018.458°
1166038.8155935790.000259726001210400342228018.386°
1176146.3424465790.000127609001210500345230018.566°
1186254.8770277900.000332475001210600348232018.455°
1196364.3473174790.000685590001210700351234018.336°
1206474.7563249800.001373062001210800354236018.418°
1216586.1219495840.000838863001210900357238018.199°
1226698.3744992610001211000360240018.612° geodesic sphere {3,5+}2,2
1236811.8272281740.001939754001410720363242017.840°
1246926.1699741930001211200366244018.111°
1257041.4732640230.000088274001211300369246017.867°
1267157.6692248670021610080372248017.920°
1277274.8195046750001211500375250017.877°
1287393.0074430680.000054132001211600378252017.814°
1297512.1073192680.000030099001211700381254017.743°
1307632.1673789120.000025622001211800384256017.683°
1317753.2051669410.000305133001211900387258017.511°
1327875.0453427970001212000390260017.958° geodesic sphere {3,5+}3,1
1337998.1792128980.000591438001212100393262017.133°
1348122.0897211940.000470268001212200396264017.214°
1358246.9094869920001212300399266017.431°
1368372.7433025390001212400402268017.485°
1378499.5344947820001212500405270017.560°
1388627.4063898800.000473576001212600408272016.924°
1398756.2270560570.000404228001212700411274016.673°
1408885.9806090410.000630351001312610414276016.773°
1419016.6153491900.000376365001412601417278016.962°
1429148.2715799930.000550138001213000420280016.840°
1439280.8398511920.000255449001213100423282016.782°
1449414.3717944600001213200426284016.953°
1459548.9288372320.000094938001213300429286016.841°
1469684.3818255750001213400432288016.905°
1479820.9323783730.000636651001213500435290016.458°
1489958.4060042700.000203701001213600438292016.627°
14910096.8599073970.000638186001413320441294016.344°
15010236.1964367010001213800444296016.405°
15110376.5714692750.000153836001213900447298016.163°
15210517.8675928780001214000450300016.117°
15310660.0827482370001214100453302016.390°
15410803.3724211410.000735800001214200456304016.078°
15510947.5746922790.000603670001214300459306015.990°
15611092.7983114560.000508534001214400462308015.822°
15711238.9030411560.000357679001214500465310015.948°
15811385.9901861970.000921918001214600468312015.987°
15911534.0239609560.000381457001214700471314015.960°
16011683.0548055490001214800474316015.961°
16111833.0847394650.000056447001214900477318015.810°
16211984.0503358140001215000480320015.813°
16312136.0130532200.000120798001215100483322015.675°
16412288.9301053200001215200486324015.655°
16512442.8044513730.000091119001215300489326015.651°
16612597.6490713230001614640492328015.607°
16712753.4694297500.000097382001215500495330015.600°
16812910.2126722680001215600498332015.655°
16913068.0064511270.000068102001315510501334015.537°
17013226.6810785410001215800504336015.569°
17113386.3559307170001215900507338015.497°
17213547.0181087870.000547291001415620510340015.292°
17313708.6352430340.000286544001216100513342015.225°
17413871.1870922920001216200516344015.366°
17514034.7813069290.000026686001216300519346015.252°
17614199.3547756320.000283978001216400522348015.101°
17714364.8375452980001216500525350015.269°
17814531.3095525870001216600528352015.145°
17914698.7545942200.000125113001316510531354014.968°
18014867.0999275250001216800534356015.067°
18115036.4672397690.000304193001216900537358015.002°
18215206.7306109060001217000540360015.155°
18315378.1665710280.000467899001217100543362014.747°
18415550.4214503110001217200546364014.932°
18515723.7200740720.000389762001217300549366014.775°
18615897.8974370480.000389762001217400552368014.739°
18716072.9751863200001217500555370014.848°
18816249.2226788790001217600558372014.740°
18916426.3719388620.000020732001217700561374014.671°
19016604.4283385010.000586804001217800564376014.501°
19116783.4522193620.001129202001317710567378014.195°
19216963.3383864600001218000570380014.819° geodesic sphere {3,5+}3,2
19317144.5647408800.000985192001218100573382014.144°
19417326.6161364710.000322358001218200576384014.350°
19517509.4893039300001218300579386014.375°
19617693.4605480820.000315907001218400582388014.251°
19717878.3401625710001218500585390014.147°
19818064.2621771950.000011149001218600588392014.237°
19918251.0824956400.000534779001218700591394014.153°
20018438.8427175300001218800594396014.222°
20118627.5912262440.001048859001318710597398013.830°
20218817.2047182620001219000600400014.189°
20319007.9812045800.000600343001219100603402013.977°
20419199.5407756030001219200606404014.291°
21220768.0530859640001220000630420014.118° geodesic sphere {3,5+}4,1
21421169.9104103750001220200636424013.771°
21621575.5963778690001220400642428013.735°
21721779.8560804180001220500645430013.902°
23224961.2523189340001222000690460013.260°
25530264.4242512810001224300759506012.565°
25630506.6875158470001224400762508012.572°
25730749.9414173460001224500765510012.672°
27234515.1932926810001226000810540012.335° geodesic sphere {3,5+}3,3
28237147.2944184620001227000840560012.166° geodesic sphere {3,5+}4,2
29239877.0080129090001228000870580011.857°
30643862.5697807970001229400912608011.628°
31245629.3138040020.000306163001230000930620011.299°
31546525.8256434320001230300939626011.337°
31747128.3103445200001230500945630011.423°
31847431.0560200430001230600948632011.219°
33452407.7281278220001232200996664011.058°
34856967.47245433400012336001038692010.721°
35759999.92293959800012345001065710010.728°
35860341.83092458800012346001068712010.647°
37265230.02712255700012360001110740010.531° geodesic sphere {3,5+}4,3
38268839.42683921500012370001140760010.379°
39071797.03533595300012378001164776010.222°
39272546.25837088900012380001170780010.278°
40075582.44851221300012388001194796010.068°
40276351.19243267300012390001200800010.099°
43288353.70968195600024396120129086009.556°
44895115.54698620900024412120133889209.322°
460100351.76310867300024424120137491609.297°
468103920.87171512700024432120139893209.120°
470104822.88632427900024434120140493609.059°

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): . [14] [ clarification needed ]

Related Research Articles

Plum pudding model Obsolete model of the atom

The plum pudding model is one of several historical scientific models of the atom. First proposed by J. J. Thomson in 1904 soon after the discovery of the electron, but before the discovery of the atomic nucleus, the model tried to explain two properties of atoms then known: that electrons are negatively charged particles and that atoms have no net electric charge. The plum pudding model has electrons surrounded by a volume of positive charge, like negatively charged "plums" embedded in a positively charged "pudding".

In solid-state physics, the work function is the minimum thermodynamic work needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electric fields in the vacuum. The work function is not a characteristic of a bulk material, but rather a property of the surface of the material.

Ionization Process by which atoms or molecules acquire charge by gaining or losing electrons

Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule is called an ion. Ionization can result from the loss of an electron after collisions with subatomic particles, collisions with other atoms, molecules and ions, or through the interaction with electromagnetic radiation. Heterolytic bond cleavage and heterolytic substitution reactions can result in the formation of ion pairs. Ionization can occur through radioactive decay by the internal conversion process, in which an excited nucleus transfers its energy to one of the inner-shell electrons causing it to be ejected.

In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.

Capacitance Ability of a body to store an electrical charge

Capacitance is the ratio of the amount of electric charge stored on a conductor to a difference in electric potential. There are two closely related notions of capacitance: self capacitance and mutual capacitance. Any object that can be electrically charged exhibits self capacitance. In this case the electric potential difference is measured between the object and ground. A material with a large self capacitance holds more electric charge at a given potential difference than one with low capacitance. The notion of mutual capacitance is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components. In a typical capacitor, two conductors are used to separate electric charge, with one conductor being positively charged and the other negatively charged, but the system having a total charge of zero. The ratio in this case is the magnitude of the electric charge on either conductor and the potential difference is that measured between the two conductors.

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 Problems which attempt to find the most efficient way to pack objects into containers

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.

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 atomic, molecular, and solid-state physics, the electric field gradient (EFG) measures the rate of change of the electric field at an atomic nucleus generated by the electronic charge distribution and the other nuclei. The EFG couples with the nuclear electric quadrupole moment of quadrupolar nuclei to generate an effect which can be measured using several spectroscopic methods, such as nuclear magnetic resonance (NMR), microwave spectroscopy, electron paramagnetic resonance, nuclear quadrupole resonance (NQR), Mössbauer spectroscopy or perturbed angular correlation (PAC). The EFG is non-zero only if the charges surrounding the nucleus violate cubic symmetry and therefore generate an inhomogeneous electric field at the position of the nucleus.

Polaron Quasiparticle in condensed matter physics

A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was proposed by Lev Landau in 1933 and Solomon Pekar in 1946 to describe an electron moving in a dielectric crystal where the atoms displace from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. For comparison of the models proposed in these papers see M. I. Dykman and E. I. Rashba, The roots of polaron theory, Physics Today 68, 10 (2015). This lowers the electron mobility and increases the electron's effective mass.

In condensed matter physics, the term geometrical frustration refers to a phenomenon where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces lead to quite complex structures. As a consequence of the frustration in the geometry or in the forces, a plenitude of distinct ground states may result at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much studied examples are amorphous materials, glasses, or dilute magnets.

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 Jahn–Teller effect is an important mechanism of spontaneous symmetry breaking in molecular and solid-state systems which has far-reaching consequences in different fields, and is responsible for a variety of phenomena in spectroscopy, stereochemistry, crystal chemistry, molecular and solid-state physics, and materials science. The effect is named for Hermann Arthur Jahn and Edward Teller, who first reported studies about it in 1937. The Jahn–Teller effect, and the related Renner–Teller effect, are discussed in Section 13.4 of the spectroscopy textbook by Bunker and Jensen.

Madelung constant

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.

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 Fe2+/Fe3+), 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(H2O)2+ and Fe(H2O)3+ are different).

Wigner crystal

A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934. A gas of electrons moving in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value. This is because the potential energy dominates the kinetic energy at low densities, so the detailed spatial arrangement of the electrons becomes important. To minimize the potential energy, the electrons form a bcc lattice in 3D, a triangular lattice in 2D and an evenly spaced lattice in 1D. Most experimentally observed Wigner clusters exist due to the presence of the external confinement, i.e. external potential trap. As a consequence, deviations from the b.c.c or triangular lattice are observed. A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field. However, it is still not clear whether it is the Wigner crystallization that has led to observation of insulating behaviour in magnetotransport measurements on 2D electron systems, since other candidates are present, such as Anderson localization.

Car–Parrinello molecular dynamics or CPMD refers to either a method used in molecular dynamics or the computational chemistry software package used to implement this method.

First introduced by M. Pollak, the Coulomb gap is a soft gap in the single-particle density of states (DOS) of a system of interacting localized electrons. Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap.

Interatomic potential Functions for calculating potential energy

Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.

Stochastic block model

The stochastic block model is a generative model for random graphs. This model tends to produce graphs containing communities, subsets of nodes characterized by being connected with one another with particular edge densities. For example, edges may be more common within communities than between communities. Its mathematical formulation has been firstly introduced in 1983 in the field of social network by Holland et al. The stochastic block model is important in statistics, machine learning, and network science, where it serves as a useful benchmark for the task of recovering community structure in graph data.

References

  1. 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.
  2. 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. S2CID   1331144.
  3. Föppl, L. (1912). "Stabile Anordnungen von Elektronen im Atom". J. Reine Angew. Math. (141): 251–301..
  4. Schwartz, Richard (2010). "The 5 electron case of Thomson's Problem". arXiv: 1001.3702 [math.MG].
  5. 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.
  6. Andreev, N.N. (1996). "An extremal property of the icosahedron". East J. Approximation. 2 (4): 459–462. MR 1426716, Zbl   0877.51021
  7. 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.
  8. Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
  9. 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.
  10. 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.
  11. Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
  12. 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.
  13. Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
  14. "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.

Notes