arXiv:1403.2592v1 [physics.class-ph] 22 Feb 2014 h udai emi srbdt nrystored energy to ascribed is term quadratic The rpitsbitdt Elsevier to submitted Preprint aN with iiaadyednwisgt oaoi structure. atomic Keywords: to insights new yield and minima hre nteui pee hsst relatively a sets This limit, energy sphere. upper unit distant the on charges itiuino ueia ouin a enfit been has including solutions functions empirical numerical with of distribution orsodne ihselfiln eairi tmcsses hswo This obtain to systems. algorithms atomic search numerical in innovative and behavior efficient shell-filling with correspondences lblmnmmfreach a for toward minimum progress global iterations numerous which from oihsi h admdsrbto of distribution random the is gorithms r samnmmfreach for minimum a is hmo rbe a mre sabenchmark a algorithms as optimization emerged global for has Problem Thomson many- numerical for precise solutions yielded have algorithms tational surfaces. viruses, spherical nmdln , modeling in by infiatylwruprlmtt iiu nrysltoso h electr each the of of origin the solutions to energy introduced minimum is charge to point limit A upper reported. lower significantly A Abstract eea ouinrmisunknown. remains solution general mid-1900s the since interest Introduction 1. hs o hc h oa olm euso energy, repulsion Coulomb total the which for those j h nta odto fsm iiiainal- minimization some of condition initial The h hmo rbe a rw considerable drawn has Problem Thomson The ueia ouin fteTosnPolmare Problem Thomson the of solutions Numerical osrie otesraeo ntsphere. unit a of surface the to constrained 3 N / U (7) 2 − sa pe ii oec ( each to limit upper an as ( 0 N . (6) 5518(3 = ) and nteps e eae eea compu- several decades few past the In T aaeJr. LaFave “T. X i

Figure 1: The extreme upper energy limit of the Thomson 2. Discrete Derivatives Problem is given by N 2/2 for a continuous charge shell fol- lowed by N(N 1)/2, the energy associated with a random − + Two upper energy bounds of some interest are distribution of N point charges. Significantly lower, U(N ), those associated with a continuous charge shell (8) the energy of a given N-charge solution of the Thomson Problem with one charge at its origin is readily obtained consisting of infinite-many N charges such that by U(N) + N, where U(N) are solutions of the Thomson Problem.

N 2 U∞(N)= (2) 2 imposing additional arguments. If these intermedi- ate values of f(N) should have physical usefulness, and the random distribution of N discrete point for instance if an effective fractional charge on the charges across the surface of the unit sphere unit sphere surface is admitted as a discrete charge approaches or leaves the surface, f(N) should have local minima at integer values of N. Though these N N 2 N Ur(N)= (N 1) = . (3) fit functions, Eqs. 4 and 5 have no local minima at 2 − 2 − 2 integer values of N, they are coarsely instructive. Plots of Eqs. 2 and 3 are shown in Fig. 1 together It is potentially more fruitful to design a fit with a few numerical solutions (open circles) of the function in accordance with the physical nature of Thomson Problem for illustration. electrostatic charge configurations. In particular, The minimized global potential energy solutions knowledge of f(N) at integer values of N, given for up to N=65 point charges were previously fit the discrete nature of point charges is paramount. using, (7) Consider the discrete derivative of f(N) at integer values of N,

2 N 3/2 f1(N)= + aN (4) ∆f (N) f(N + ∆N) f(N) 2 i = − (6) ∆N ∆N in which a = 0.5510, and for up to N 100 of Eq. 4, which yields, after binomial expansion of using (8) − ∼ the half-integer term,

2 N 3/2 1/2 f2(N)= + aN + bN . (5) 1 2 ∆f1(N) = N +  2

These smooth, continuous fit functions are gener- a 1 3 − 1 1 − 3 + 3N 2 + N 2 N 2 + (7) ally unrepresentative of the absolute minimum en- 2 4 − 8 ··· ergy configurations of discrete charges due to a vari- ety of issues. Among them, discrete charges cannot for ∆N = 1. For comparison, the derivative of be infinitesimally subdivided so all points, f(N), for Eq. 5, with binomial expansion of the half-integer non-integer N have no physical significance without terms, 2 [N+ ] [N+ ]

1 1 1 7 1 2 − 2 ∆f2(N) = N + + 3aN + (a + b)N  2 2  4 q 0 N 1 − 3 N + + (a b)N 2 + (8) U (N+1) 8 − ··· ∆

for ∆N = 1. The first set of parentheses in Eqs. 7 U’(N+1)

∆ [N+1] and 8 is the discrete derivative of N 2/2, the energy of a continuous charge shell, Eq. 2. Energy U(N+1) q ∆ 3. Discrete Transformation Energies 0 To understand the physical charge distribu- tion represented by the discrete energy differences, [N] Eqs. 7 and 8, consider an N-charge solution, such N N+1 as that of [N] shown in Fig. 2 for which the total Charge Number minimized electrostatic energy is U(N) as shown in Fig. 1. The discrete energy changes needed to Figure 2: A well-defined intermediate charge configuration, + obtain the solution U(N + 1) in Fig. 1 may be ob- [N ], with q0 at the origin of [N] linearly increases the + tained by introducing an (N + 1)th point charge, energy by N. Consequently, U(N ) is readily accessible. + Displacement of q0 to the unit sphere surface and global q0, to the origin denoted [N ] in Fig. 2. Its contri- minimization of energy yields the new [N + 1] solution with bution to the total energy, given equal interaction ∆U +(N + 1) √N. with all N charges on the unit sphere, is N. The ∝ total energy, U(N +), (Fig. 1) is U(N)+ N as illus- trated in Fig. 2. Thus, the first term of each dis- as shown in Fig. 2. Consequently, ∆U +(N + 1) crete derivative, ∆f (N), is due to the appearance i accounts for the half-integer terms in Eqs. 7 and of q at the origin. The remaining terms are due 0 8 and the +1/2 term associated with the discrete to the displacement of q to the unit sphere surface 0 derivative of N 2/2. and subsequent (or in situ) global minimization of To check the validity of the discrete transforma- energy, U(N + 1) in Fig. 1, to yield the (N + 1) tions from each N-charge solution of the Thomson configuration (Fig. 2). Problem to its adjacent (N + 1)-charge solution, In the strictest spirit of the Thomson Problem, ∆U +(N + 1) in Eq. 9 is fit with the remaining introduction of q to the origin of the unit sphere 0 terms of the discrete derivatives of Eqs. 7 and 8. (Fig. 2b) increases the total energy of charges on The distribution of energy differences for up to 500- the unit sphere by N/2. However, since the goal is charge numerical solutions of the Thomson Problem to incorporate q in the final (N +1)-charge config- 0 (Fig. 3) is well-fit with the first three terms of Eq. 7 uration on the unit sphere, inclusion of the energy − for which a =0.5512 1 10 4, in very good agree- associated with q , that is N/2, while at the origin 0 ment with previous work± × in which a = 0.5510 for is necessary. up to N = 65. (7) Truncating Eq. 7 to only the first The example [N +] solution shown in Fig. 2, con- term, including the +1/2 term from the discrete sists of four charges at the vertices of a regular derivative of the quadratic term in Eq. 4, centered about q0 at the origin. Sym- metrically, the [N +] configuration is identical to the 3 1 [N] solution of the Thomson problem. In general, f +(N +1)= a√N + (10) U(N +) > U(N + 1). Therefore, the energy differ- 2 2 ence ∆U(N + 1) associated with the addition of a provides a reasonably good fit with a = 0.5518 charge in the Thomson problem may be expressed, 1 10−4. Fitting with half-integer terms in Eq.± 8 yields× large errors associated with b and larger val-

′ ues of a = 0.5525. These half-integer terms are, ∆U(N +1)=∆U (N + 1) ∆U +(N + 1) − however, useful for better fits to smaller N-charge = N ∆U +(N + 1) (9) solutions. (8) − 3 h nryo each of energy the h +1 the inof tion .The 4. in charges 100 to up for shown 3. as Fig. charges 500 to nrygvnb q ,the 3, Eq. by given energy lmns oteoii assteeeg by energy the raises origin the to elements) ee h rnfrainyed oe nrythan energy lower U a yields transformation the Here, sacnein tfnto o ∆ for function 10 fit Eq. convenient Problem, a Thomson is the charac- of of means solutions insufficient terizing an is set data tematic” to oee,teeeg ieec of difference energy the However, ∆ difference hw ntefirst the on shown t ieec rmteeeg farno ( random a of energy the from difference Its hredsrbto szr since zero is distribution charge iue3 h ucin∆ function The 3: Figure

∞ + ple oacniuu hresel introduc- shell, charge continuous a to Applied ple oprl admcag itiuin of distributions charge random purely to Applied ic tigsot ucin oadsrt,“sys- discrete, a to functions smooth fitting Since ∆ N −7 −6 −5 −4 −3 −2 −1 −8 U (N+1) 0 ( 0 0 0 080 60 40 20 0 50nmrclsltoso h hmo rbe,is Problem, Thomson the of solutions numerical =500 N U U U r ∞ ∞ / 1), + q ( N emi h nt ieec of difference finite the in term 2 0 ( ( q N N oeo nifiienme feulcharge equal of number infinite an of (one 0 ) U ) + ) → + Transformation → ( N U N U N r 1). + Number ofPointCharges,N ∞ ( − 10nmrclsltoso h energy the of solutions numerical =100 N ( N U N + ) cag ofiuainby configuration -charge ∞ f + ) ( ( N N N = ) N = )= 1) + q 0 − = N rnfrainraises transformation 0 . N 2 5518(3 2 U + − 2 − + 2 + 1 2 N / 1 2 ( / N consfor accounts 2 . . 2) N √ )frup for 1) + . N N 2 +1 N / N 2. 1)- + / N , (11) (12) (13) ,fit 2, 100 , 4 admydsrbtdcagso ntshr and sphere unit a N on charges distributed randomly we otnosaddsrt hresystems. charge discrete and continuous tween rt hressesi ncnitwt h mathe- them. the study with to conflict used in matics is systems charge dis- crete of nature physical underlying the Consequently, energy potential electrostatic mum global a sents yFg .I oedti,uigE.4 nlso of inclusion 4, Eq. using detail, more evident q In made as 3. sphere Fig. unit by the on charges all the with to configuration charge vorable by lution hs w ofiuain r nreial indistin- energetically guishable. are configurations two these hrecngrto the configuration charge vr steei oeeg ieec between difference energy no is How- there distributions. as charge ever, random purely of ergies nqaiyo h orefi parameter, fit coarse the on inequality tfnto a edsge orflc oa minima of local values reflect integer to at designed be may function fit o ozr oiieitgrvle of values integer positive nonzero For ettruhdsrt nrydffrne osatisfy to the differences to energy minimum discrete energy through one next from “hop” may tion omto assteeeg feach of energy the raises formation e of val- integer ues between charges fractional of possibility ria ni n[ an until arrival osqety the Consequently, parameter than greater l,the old, .FtFunctions Fit 5. Problem. Thomson the of solutions energy all 0 U ple oteTosnPolm the Problem, Thomson the to Applied sanwpitcag saddt given a to added is charge point new a As The oreydfie tfunctions fit Coarsely-defined oeach to hre naui peewith sphere unit a on charges ( N U hti o ersne yteefi functions. fit these by represented not is that aN r + ) q N ( 0 N q ned each Indeed, . 3 0 rnfrainuiul itnuse be- distinguishes uniquely transformation U > N / N )= 1) + N > a a 2 rnfrainyed nuprlmtto limit upper an yields transformation = cag ouinipsstefollowing the imposes solution -charge n ersnsa nreial unfa- energetically an represents and − > 0 − . N 1 ( 914 − 0 1 2 . ( 1 ofiuaini band A obtained. is configuration +1] ( 58i rae hnti thresh- this than greater is 5518 q 2 1 + N N 0 N h a 1) + ( transformation N osqety h tfunc- fit the Consequently, . 1) + ( 2 N N ) ic h orefit coarse the Since 1). = 1) + N N 1) + N cag ouinrepre- solution -charge hre epn oits to respond charges = 3 3 / / 2 N f 2 − i 2 ( q N 2 N 0 + N rsm the presume ) tteorigin, the at 3 N generates N N / solution 1 + a 2 , cag so- -charge . i , a − q 1 utbe must 0 . trans- N mini- (15) (14) 1 + en- N - (a) changes from that of [N] to [N + 1]. These discrete (b) symmetry transformations are responsible for cor- relations made previously (20–22) to electron shell- (c) filling throughout the periodic table of elements as discussed in the next section. Though not treated here, continuously smooth (d) functions having periodic (∆N = 1) local minima, Fig. 4c, or sharply defined (infinite derivative) be- havior at integer values of N as shown in Fig. 4d U(N+1) Potential Energy may also be useful to obtain solutions of the Thom- U(N) son Problem. Perhaps such a fit function may in- U(N−1) clude a variety of polynomial series exhibiting os- cillatory behavior. N−1 N N+1 Number of point charges 6. Correspondence with Atomic Structure

Figure 4: A few possible continuous fit functions to min- The discrete derivative prohibits electrostatic en- imized energy solutions of the Thomson Problem. a) q0 is introduced abruptly to each N-charge solution, and the ergy “states” between integer values of N. In- energy is gradually minimized to U(N + 1). b) q0 forms stead of beginning with an N-charge configura- gradually at the origin, and the total energy is minimized to tion we may begin with an (N + 1)-charge con- U(N + 1). c) Energy minima are obtained with a smooth, figuration in Fig. 2. The energy ∆U +(N + 1) periodic function having minima at integer values of N. d) Energy solutions are obtained with a piecewise-continuous represents an increase for the (N + 1) system by function with minima at integer N values. an amount approximately proportional to √N. The resulting U(N +) may then be associated with the N-charge configuration in the Thomson prob- lem (using a strict “surface-only” interpretation). the condition of local minima at integer values of When plotted, Fig. 5 (open circles), the distribu- N. A few possible fit functions are shown schemat- tion yields a pattern of energy differences, ∆U +(N) ically in Fig. 4. The saw-tooth fit function sketched (note the explicit association of energy with the in Fig. 4a reflects abrupt introduction of q0 at the N-charge configuration), consistent with electron origin of each N-charge configuration followed by shell-filling throughout the periodic table of ele- a gradual relaxation of the function to the neigh- ments as previously reported. (20–22) In particular, boring [N +1] configuration. This function may be as one progresses through the periodic table (in- useful if the strict definition of charges on the sur- creasing atomic number) the outermost electron in face of the unit sphere is to be maintained. How- respective atomic structures is characterized by a ever, if the number of charges within the volume wavefunction (or “orbital”) having different spatial of the unit sphere is of interest, then this function symmetries. For instance, the spherical s-shell as- is inappropriate as the [N +] configuration contains sociated with beryllium (Z=4) is followed by boron N + 1 charges and should be plotted to reflect this (Z=5), a system with an outermost electron having number. a dumbbell-shaped p-shell. These spatial symme- The sawtooth fit function sketched in Fig. 4b rises try differences among wavefunctions of neighboring linearly from each N-charge energy solution to a atomic systems are repeatedly coincident with no- new upper limit of each subsequent N + 1 energy ticeable disparities found in ∆U +(N) intrinsic to solution. If this linear rise is considered as a grad- the Thomson Problem. Moreover, a relatively lower ual “charging up” of q0 at the origin of each [N], energy is obtained at the “close” of each electron then the slope of the function is N/∆N = N. The shell followed by a disparity to a relatively higher generally shorter drop from U(N +), (cf. Fig. 1) energy which “opens” each subsequent shell. For plotted appropriately at each N +1, to U(N + 1) example, a lower energy appears at N=4 followed maintains the total number of charges in the sys- by an abrupt disparity to a higher energy for N=5. tem (including the spherical volume) as q0 is dis- The Thomson Problem treated within a dielec- placed from the origin to the surface. During this tric sphere, (20–22) using an appropriate model for transformation, the spatial symmetry of the system discrete charges in the presence of dielectrics (24) 5 10 0.04 not obtained by the insufficient mathematical de- 1s2s 2p3s 3p 4s 3d4p 5s 4d 5p 8 scription of coarse fit functions previously used to 0.03 model solutions of the Thomson Problem.

6 ε

+ The existence of possibly better fit functions U (N) 0.02 + ∆

U (N, ), (eV) based on the underlying physical properties of the 4 ∆ charge system described may lead to better ap- 0.01 2 proximations and potentially exact solutions of the Thomson Problem which appears to be intrinsically 0 2 4 10 12 18 20 30 36 38 48 shared with atomic structure. Electron Number, N Discrete transformations between neighboring [N] and [N + 1] configurations through an interme- Figure 5: Discrete energy differences in the Thomson Prob- + lem, ∆U +(N) (open circles) exhibit disparities uniquely cor- diate [N ] configuration utilizing q0 at the origin respondent to shell-filling behavior in natural atomic struc- include a trivial linear increase in energy followed ture. These disparities are enunciated when the Thomson by a non-trivial minimization of energy. The latter Problem is treated within a dielectric sphere (filled circles). transformation from a well-defined configuration to These data are given for a dielectric sphere of 100nm radius and dielectric constant, ε = 20ε0. the unknown [N + 1] configuration is the embod- iment of the Thomson Problem. Further develop- ments toward exact solutions of this transformation are merited, substantiated by its many correspon- these disparities become more pronounced as shown dences with electron shell-filling in atomic struc- in Fig. 5 (solid circles). The shift, N +1 N, → ture, and its usefulness to a wide range of other dis- in numerical solutions of the Thomson Problem is crete charge modeling applications. Identification unnecessary here. Note, for example, a dielectric of these discrete transformations in the Thomson sphere containing a single electron stores nonzero Problem may be the first step toward the develop- energy referenced to the empty, or electrically neu- ment of new and more efficient energy minimization tral, dielectric sphere. Enunciation of these fea- algorithms. tures when using a dielectric sphere is partly due to the use of a “Thomson radius” as a variational parameter rather than a static unit sphere. Both References frameworks may be traced to the classical “plum- pudding” model of the (25) in which [1] L. L. Whyte, Am. Math. Month. 59, 606 (1952). 62 reside within a uniform positively-charged sphere. [2] A. P´erez-Garrido, Phys. Rev. B , 6979 (2000). [3] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, A neutral dielectric sphere is used in place of the and R. E. Smalley, Nature 318, 162 (1985). positively-charged sphere, and the Thomson Prob- [4] H. Aranda-Espinoza, Y. Chen, N. Dan, T. C. Luben- lem is a natural consequence of the removal of sky, P. Nelson, L. Ramos, and D. A. Weitz, Science 285 this positively-charged sphere in which electrons fly , 394 (1999). [5] D. L. D. Caspar and A. Klug, Cold Spring Harbor (26) apart to form a spherical distribution. Symp. on Quant. Biology 27, 1 (1962). Many additional correspondences with atomic [6] M. Bowick, A. Cacciuto, D. R. Nelson, and A. Trav- shell-filling have been reported. (20–22) esset, Phys. Rev. Lett. 89, 185502 (2002). [7] T. Erber and G. M. Hockney, J. Phys. A: Math. Gen. 24, L1369 (1991). 7. Concluding Remarks [8] L. Glasser and A. G. Every, J. Phys. A: Math. Gen 25 2473-2482 (1992). [9] J. R. Edmundson, Acta Cryst. A 48, 60 (1992). In place of coarsely-defined fit functions to nu- [10] J. R. Edmundson, Acta Cryst. A 49, 648 (1993). merical solutions of the Thomson Problem, the [11] E. L. Altschuler, T. J. Williams, E. R. Ratner, F. discrete derivative of these fit functions has been Dowla, and F. Wooten, Phys. Rev. Lett. 72, 2671 demonstrated to correspond with the well-defined (1994). [12] T. Erber and G. M. Hockney, Phys. Rev Lett. 74, 1482 configuration of N charges on a unit sphere sur- (1995). rounding an (N + 1)th charge, q0, at its origin [13] A. P´erez-Garrido, M. Ortu˜no, E. Cuevas, and J. Ruiz, followed by relaxation of this volumetric problem J. Phys. A: Math. Gen. 29, 1973 (1996). 98 to each subsequent (N + 1)-charge solution of the [14] T. Erber and G. M. Hockney, Adv. Chem. Phys. , 495 (1997). Thomson Problem. This approach provides useful [15] A. P´erez-Garrido and M. A. Moore, Phys. Rev. B 60, new information about the discrete charge system 15628 (1999). 6 [16] E. L. Altschuler and A. P´erez-Garrido, Phys. Rev. E 71, 047703 (2005). [17] E. L. Altschuler and A. P´erez-Garrido, Phys. Rev. E 73, 036108 (2006). [18] D. J. Wales and S. Ulker, Phys. Rev. B 74, 212101 (2006). [19] S. Smale, Math. Intell. 20, 7 (1998). [20] T. P. LaFave Jr, Ph.D. dissertation, University of North Carolina, Charlotte, 2006. [21] T. LaFave Jr. and R. Tsu, Microelectron. J. 39, 617 (2008). [22] T. LaFave Jr., J. Electrostatics, submitted for review. [23] Thomson Applet, http://thomson.phy.syr.edu [24] T. LaFave Jr., J. Electrostatics 69, 414 (2011). [25] J. J. Thomson, Philos. Mag. 7, 237 (1904). [26] Y. Levin and J. J. Arenzon, Europhys. Lett. 63, 415 (2003).

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