New bound for More cluter Tamá Vinkó Advanced Concept Team, European Space Agency, ESTEC Keplerlaan 1, 2201 AZ Noordwijk, The Netherland Tama.Vinko@ea.int and Arnold Neumaier Fakultät für Mathematik, Univerität Wien Nordbergtrae 15, A-1090 Wien, Autria Arnold.Neumaier@univie.ac.at http://www.mat.univie.ac.at/~neum January 10, 2007 Abtract Thi paper preent improved lower bound for the total energy and the minimal inter-particle ditance in minimal energy atom cluter problem with interaction given by a More potential. The method ue imple argument and can be employed directly for More cluter, where the atom eparation problem i difficult due to the finite energy at zero atom eparation. The theoretical reult are applied numerically achieving harper reult than thoe previouly known for variou More potential, even for ρ 4.967 parameter. Mot reult hold for more general pair potential. Keyword: atom cluter, lower bound, minimal ditance, More cluter. 1 Introduction Given a cluter of n atom in d-dimenional pace (d > 1, define the coordinate vector x i R d (i = 1,..., n a the center of the ith atom. The potential energy of the cluter x = Thi work ha been upported by the grant OTKA T 048377 and AÖU 60oü6. 1
(x 1,..., x n R dn i then defined a the um of the two-body inter-particle pair potential over all of the pair, i.e., E(x = v(r ij, (1 i<j where r ij = x i x j 2 i the Euclidean ditance of x i and x j and v(r i the value of the pair potential for two particle at ditance r. The aim of thi paper i to obtain lower bound for the total energy and for the minimal interatomic ditance in the tructure with minimal total energy (1 in cae that the pair potential i a More potential, v ρ (r = e (e ρ(1 r ρ(1 r 2, (2 where ρ > ln 2 i a parameter. Mot reult hold more generally for pair potential v(r which are a continuou, piecewie differentiable function trictly decreaing for r and increaing for r, with global minimum value v( < 0, poitive for mall r, and approaching zero from below for r. v ha a unique zero, which i denoted by t. Clearly, t < and v(t = 0, v(r < 0 for r > t. For the More potential, = 1, v( = 1, and t = 1 ln 2 ρ. The exitence of a poitive zero require ρ > ln 2. Previou reult. The book by Ruelle [9] contain in Section 2.3 (ee alo the reference given there invetigation for general pair potential. Ruelle call a pair potential function table if the aociated total energy of a cluter i bounded from below by a linear function of the cluter ize, and prove ufficient condition for tability (ee Propoition 3.2.7 and 3.2.8 in [9] but without giving explicit formula for the reulting bound. Ruelle reult apply to the More potential for ρ > ln 16( 2.7726, giving a linear lower bound on the energy but nothing about atom eparation. Mot other rigorou work wa done for the Lennard-Jone interaction; ee, e.g., [1, 6, 10, 13, 14]. For More cluter, the atom eparation problem i ignificantly more difficult due to the finite energy at zero atom eparation. Indeed, the firt ize-independent lower bound for the interatomic ditance in the optimal tructure were obtained only in 2002 by Locatelli & Schoen [5], uing complicated geometric argument etablihing uch a bound for d = 3 and ρ 6. Then Vinkó [11], obtained for general potential function atifying natural aymptotic propertie ize-independent lower bound on the minimal ditance and linear lower bound on the total energy which improved the reult of [5], and Schachinger et al. [10] improved thee further. All thee reult work only for ρ 6, ince they are baed upon crucial etimate of [5]. The method preented in thi paper improve thee lower bound further, uing argument much impler than thoe of [5]. Moreover, we find ize-independent lower bound on the 2
minimal ditance already for ρ 4.967. Mot argument apply to more general pair potential. All numerical computation were done both with MuPAD [7] and Mathematica [12], to be ure of the correctne of the numerical reult we obtained. Note that Mathematica provided fater evaluation time than MuPad. Notation. The following notation will be ued. A global minimizer of the function E i any configuration x R dn with E := E(x = min E(x, (3 x R dn where d > 1 i the dimenion of the pace containing the cluter. (Of interet are mainly d = 2 and d = 3. Let r ij be the Euclidean ditance of the point x i and x j (i, j = 1,..., n. The potential energy of particle i in an arbitrary configuration x i defined a E i (x = i j v( x i x j (i = 1,..., n and we put Clearly, the total energy i E i = E i (x. E(x = 1 2 E i (x. (4 If the number of atom i to be emphaized, the notation E (n and E i (n i ued for the optimal total energy and for the optimal potential energy of particle i, repectively. We write R k for the minimum over i of the kth mallet ditance of ome atom from x i. Then R 1 = 0, and R 2 = r min := min r ij (i, j = 1,..., n (5 i,j i the minimal ditance in the optimal configuration. The R k form a not necearily increaing equence. We give ome atom (to be determined later the label 1 and label the remaining atom uch that r i := r 1i atifie Then (5 implie r i R i for i = 2,..., n. i=1 0 = r 1 r 2... r n. 2 Energy bound In thi ection we prove bound on the optimal total energy. We firt generalize conideration by Marana & Flouda [6] for the Lennard-Jone potential. Lemma 1. An optimal n-atom cluter ha total energy bounded by n(n 1 v( E (n d(n d + 1 v(. (6 2 3
Proof. Since v(r ij v( 0, we have E (n = i<j = i<j ( v(r ij v( + v( ( v(r ij v( + v( i<j giving the lower bound. n(n 1 v(, 2 If we contruct a cluter containing n atom where n d particle are in a poition that each of them touche (i.e., ha minimal ditance to d other, tarting with d particle in uch a way that the ditance between thee point are (i.e., a line egment in dimenion 2, an equilateral triangle in dimenion 3, and o on, we get a cluter of total energy d v( d(n d v( + M d(n d + 1 v( with nonpoitive M, which i the um of the pair potential value v(r in cae of r i greater than. Thu thi i an upper bound for the total energy of the optimal tructure. Since M < 0, the upper bound follow. In the following, we hall aume that, in the optimal configuration, the potential energy of particle i i bounded by (n 1 v( E i (n < e d v( (7 for ome contant e d > 0 independent of the dimenion and the ize of the given optimal cluter. It i likely that (7 hold for n > d = 3 with e d = d ince in the optimal tructure, every atom ha mot likely at leat d contact. But howing thi rigorouly eem to be nontrivial, and we only etablih Lemma 2. (7 hold with e d = 1. Proof. To prove the upper bound, let k = n if i n and k = n 1 if i = n, and define the configuration z = (z 1,..., z n in uch a way that z j = x j for all j i, z i z k = and z i z l for all l i. Then place the atom z i on the line determined by the origin and the coordinate of z k in uch a way that z i ha the maximal r j value. Thu E i (z < v(. By contruction of z, Since E i (z < v( and E E i = E(z E i (z. E E i = E(z E i (z > E(z + v( E + v(, we find the upper bound Ei < v(. The lower bound come from the fact that v(r i monotone decreaing in the interval [0, ] and from the definition of Ei (n. Indeed, the formula for Ei (n contain n 1 term and all of them have the lower bound v(. 4
To get ize-independent lower bound on Ei and linear lower bound on the total energy, we proceed to find upper and lower bound on um of the form Σ m := m v(r k. (8 k=2 Let N d (r be the maximal number of dijoint open unit ball fitting into a ball of radiu r. By a imple volume comparion one can eaily find the upper bound N d (r r d, (9 which we hall ue in the following. Any improvement in thi geometric packing bound would reult in correponding improvement of our etimate depending on it. Propoition 1. Let K(r := Then K i an increaing function of r, and ( 2r min (m 1N d + 1. m N,R m >0 R m k K(r k for all k = 1, 2,.... (10 In particular, ( 2r d K(r (m 1 + 1 R m for all m = 2, 3,.... (11 Proof. Fix k 1 and m 2. We conider the et S coniting of the k atom cloet to atom 1. We recurively pick an atom from S, tarting with atom 1, and remove it and the m 2 atom nearet to it from S, until S i empty. Thi pick a et of κ = k/(m 1 atom at mutual ditance at leat R m. Thu the open ball of radiu R m /2 around thee atom are dijoint and inide the open ball of radiu r k + R m /2 = (2r k + R m /2 around the atom labelled 1. A caling argument give ( 2rk κ N d + 1, R m hence ( 2rk ( 2rk d k (m 1κ (m 1N d + 1 (m 1 + 1. R m R m Propoition 2. If r m then Σ m m v( + E1 + Moreover, if m 2 and R m then (m 1v(R m + (m + e d v( K(rv (rdr. (12 K(rv (rdr. (13 5
Proof. Let firt m be the larget integer with r m. Then K(r K(r m m for r by Propoition 1, and r m+1 >, hence v(r k+1 v(r k 0 for k m + 1. Therefore, with r n+1 =, v( = 0, we have k(v(r k+1 v(r k k=m+1 The left hand ide equal k=m+1 rk+1 k=m+1 rk+1 K(r k r k v (rdr r k K(rv (rdr = r m+1 K(rv (rdr. mv(r m+1 k=m+1 and ince v (rdr = v(r, we find r Σ m E 1 + E 1 + mv( + v(r k = mv(r m+1 E 1 + Σ m, r m+1 (K(r mv (rdr E 1 + K(rv (rdr. Thi prove (12 for the maximal allowed value of m. Since (K(r mv (rdr Σ m mv( = m (v(r k v( v( k=2 i a um of nonnegative number, the left hand ide i monotone increaing in m; thu (12 alo hold for all maller value of m. By definition of R m, one can label ome atom a 1 uch that r m = R m. In thi cae, we have for k < m the trivial lower bound Σ m (m 1v(R m. Combining thi inequality with (12 and with E 1 < e d v( give (13. The above argument can be improved lightly with the following conideration. For integer m 1 and real number r, r, let K m (r, r be the number of k > m uch that max(t, r r k r. Clearly, K m (r, r i a decreaing function of m and K m (r, r K(r m for all r r m. (14 A bound on the value K m (r, r can be found a follow: Conider each r k a a center of an open ball with radiu r min /2. The number of uch ball that can be packed into the big ball with radiu r + r min /2 cannot exceed (2r/r min + 1 d. On the other hand, ince r r k, we 6
can drop out the mall ball from the ball with radiu r r min /2. Thi etimation baed on the volume of pherical hell give K m (r, r ( 2r d + 1 max (m, 2r d 1. (15 r min r min In the following, we hall alway define r in term of r by v(r = v(r, r r. (16 (Cloed formula for r exit for the pecific pair potential in the application part. Propoition 3. The um (8 i bounded by Σ m E1 + Proof. Indeed, K m (r, rv (rdr. Σ m E 1 = k>m = = v(rk 0 v(r k k>m,r k t 0 v(rk dρ {k > m r k t; v(r k ρ} dρ K m (r, rv (rdr, (17 where we made the ubtitution ρ = v(r with r. Note that thi etimate combined with (14 for r m implie Propoition 2. However, combining it with (15 give tronger bound. Theorem 1. If then B := Moreover, for any contant B atifying (19, K 1 (r, rv (rdr < (18 E i B for all i = 1,..., n. (19 B 2 n E. (20 Proof. The pecial cae m = 1 of (17 give 0 = Σ 1 E 1 + K 1 (r, rv (rdr = E 1 + B which lead to (19 for i = 1. Since the choice of the label 1 i arbitrary, (19 hold for all i. Finally, (20 follow from (4. 7
Corollary 1. If q i a lower bound on the minimal inter-particle ditance r min then (19 hold with ( 2r d ( B := q + 1 max 2, 2r d q 1 v (rdr. (21 Proof. Uing (15 for m = 2, we can bound B a defined in (18 by B ( 2r d + 1 max (2, 2r d 1 v (rdr r min r min ( 2r d ( q + 1 max 2, 2r d q 1 v (rdr. A mentioned in the introduction, Ruelle [9] call a potential function table if the energy of the optimal cluter i bounded below by a multiple of the cluter ize. We ummarize hi ufficient condition for tability in [9, Section 3.2.6]. For More cluter, it give a lower bound for the energy proportional to n under weaker condition than other known argument. Propoition 4. (Ruelle [9] If the pair potential v i of poitive type and v(0 i finite then it i table, and n 2 v(0 E. (22 Here a continuou function f i of poitive type if, for arbitrary x 1,..., x n, f(x i x j 0. (23 i=1 j=1 Proof. For the optimal configuration x, 0 v( x i x j = nv(0 + 2 i<j i=1 j=1 v( x i x j, hence v(0 2 n v( x i x j. i<j In general it i not trivial to how that a pair potential function i of poitive type, but Ruelle [9] quote the known reult (Bochner [2] that f i of poitive type if and only if the Fourier tranform of f i of poitive type. More generally, it clearly uffice for the deired concluion that f i bounded below by a function whoe Fourier tranform of f i of poitive type. 8
3 Bound on the minimal ditance Corollary 1 depend on a lower bound for the minimal inter-particle ditance. Thi ection i devoted to obtain uch lower bound. Note that by Lemma 2 the following certainly hold with e d = 1. Lemma 3. If n > 2 + e d then ( q(n = w (n 2 e d v( i a lower bound for the minimal inter-particle ditance in the optimal configuration. Here w, defined by { r iff x = v(r and r, w(x = (25 0 otherwie. i the unique olution of v(w(x = min(x, v(0. Proof. Let E1(n be the term which contain the minimal ditance in the optimal tructure. Uing (7, we find e d v( > v(r j = j=2 v(r j + v(r 2 j=3 (n 2 v( + v(r 2. Rearranging the inequalitie one obtain v(r 2 < (n 2 e d v(, which implie the bound. (24 Lemma 4. In the optimal configuration the minimal interatomic ditance i alway le than or equal to the minimizer point of the pair potential function, i.e., r min hold. Proof. Suppoe that in the optimal configuration r min >. We know that function v i increaing for r. Hence, recaling all of the ditance uch that r min = decreae the total energy. Thu r min. Theorem 2. Let [R, R] [0, ] be uch that (2r d R + 1 (2r d R + 1 Then the function defined by v (rdr v(r + v( for all R [R, R], (26 { v (rdr < min v(r + v(, 1 2 v(r + (1 + e } d 2 v(. (27 f(q := v(q + (2 + e d v( ha a mallet zero q in ]R, [, and we have r min q. 9 ( 2r q + 1 d v (rdr (28
Proof. For any integer m 2 we find from (11 and (13 that R = R m atifie (2r d (m 1v(R + (m + e d v( (m 1 R + 1 v (rdr, hence v(r + v( < v(r + m + e d m 1 v( Thi contradict (26 unle R m < R or R m > R. (2r R + 1 d v (rdr. If the firt cae can happen for ome m 2, let m be the larget integer uch that R m < R. Then R m+1 > R, hence ( 2r d (2r d K(r m + 1 m R m+1 R + 1, and ince v(r v(r m, we find from (13 that 1 ( (m 1v(R + (m + e d v( m (2r R + 1 d v (rdr. The left hand ide i monotone in m, hence extremal at the boundary, and ince m 2, thi contradict (27. Thu the firt cae cannot happen. In particular, we find for m = 2 that r min = R 2 > R. Since (13 implie for m = 2 that f(r min 0 and (26 implie f(r > 2 v( > 0, the intermediate value theorem implie that f ha a zero in ]R, [, and that r min cannot be maller than the mallet uch zero. Corollary 2. If there i ome R [0, ] uch that (2r d R + 1 v (rdr < min {v(r + v(, 1 2 v(r + 3 } 2 v( then the aumption of the theorem are atifiable, and there i a poitive n-independent lower bound on r min. Proof. Take R = R = R. Note that the aumption i automatically atified with R = 0 if the potential V (r diverge for r 0, but i a nontrivial retriction for the More potential. By Theorem 2 we can compute lower bound on the minimal inter-particle ditance. If we take m = 2 in formula (15 and in Propoition 3, it lead to better reult. Namely, the function defined by ( 2r d ( f(q := v(q + e d v( q + 1 max 2, 2r d q 1 v (rdr (29 alo ha a mallet zero in ]R, [ and then we have r min q. 10
4 Numerical reult In thi ection the numerical reult are hown for different ρ value. For ρ = 6, the More and the caled Lennard-Jone pair potential are related; they have the ame curvature at the minimum point r = 1. In the context of global optimization, the cae ρ > 6 are mot intereting, ince thee are more difficult problem than finding the optimal Lennard-Jone tructure (ee Doye et al. [3]. On the other hand, finding minimal interatomic ditance in the optimal More cluter become more difficult a ρ become maller and the pair potential become le repulive at mall ditance. Size dependent bound for the minimal ditance. Lemma 3 give (e ρ(1 r 1 2 1 (n 2 e d v ρ (. Since = 1, we have v ρ ( = 1, and we conclude that { q(n = max 0, 1 ρ 1 ln (1 + } n 2 e d (30 i a lower bound for the minimal inter-particle ditance of an optimal More cluter with n > 2 + e d particle. Thi formula yield a trictly poitive bound if n (2 + e d + eρ (e ρ 2. v ρ ( q from q from q from new bound ρ t R R formula (29 L&S [5] Sch. [10] for Eρ 15 0.95379 0.00001 0.86424 0.865691 0.715166 0.865230 20.6118n 14 0.95049 0.00197 0.85320 0.854724 0.694918 0.854303 21.5684n 13 0.94668 0.00039 0.84018 0.841826 0.671606 0.841431 22.7337n 12 0.94224 0.00077 0.82460 0.826440 0.644492 0.826034 24.1842n 11 0.93699 0.00152 0.80559 0.807692 0.612565 0.807275 26.0923n 10 0.93068 0.00302 0.78187 0.784318 0.574381 0.783890 28.7126n 9 0.92298 0.00608 0.75135 0.754293 0.527627 0.753859 32.5332n 8 0.91336 0.01250 0.71045 0.714143 0.467709 0.713711 38.6074n 7 0.90097 0.02663 0.65212 0.657166 0.375988 0.656752 49.7140n 6 0.88448 0.06167 0.55928 0.567463 0.113522 0.567097 76.3745n 5 0.86137 0.20982 0.33235 0.371212 245.8110n 4.967 0.86045 0.23439 0.30471 0.356997 272.8310n Table 1: Lower bound on minimum inter-particle ditance and total energy of optimal More cluter. Size independent bound and linear lower bound for the energy. Ruelle [8] proved that if ρ > ln 16 2.7726, then the Fourier tranform of the pair potential v ρ i of poitive type, hence it i table by Bochner theorem [2] and Propoition 4. The reulting linear lower bound, v ρ(0 2 n M ρ (ρ > ln 16 (31 11
i quite poor: For ρ = 4.967 (the mallet value for which the condition in Corollary 2 hold and for ρ = 15 formula (31 give 1.0166 10 4 n and 5.3432 10 12 n, repectively. No bound on the minimal ditance i available from Ruelle argument. However, our theory applie. Table 1 contain the reult of the application of formula (29 for More cluter, together with the previou reult from Locatelli & Schoen [5] and Schachinger et al. [10]. Thoe of Vinkó [11] are intermediate in quality, and are not reported for pace reaon. The lat line (ρ = 4.967 how the mallet ρ where formula (29 could be applied according to Theorem 2. The linear lower bound for E ρ obtained from Theorem 1 are alo preented. 5 Concluion The method were introduced in thi paper are able to make lower bound on the minimal interatomic ditance and on the total energy in optimal tructure of More cluter. With thee method ize dependent bound (for mall configuration and ize independent bound (for arbitrary large cluter can be obtained. Numerical computation how that thee bound are better than the known one for More cluter. Moreover, the ize independent method ha the advantage that i able to handle More cluter directly, even for mall ρ parameter. Reference [1] X. Blanc. Lower bound for the interatomic ditance in Lennard-Jone cluter. Comput. Optimization Appl. 29:5 12, 2004. [2] S. Bochner. Lecture on Fourier Integral. Princeton Univerity Pre, 1959. [3] J.P.K. Doye, R.H. Leary, M. Locatelli and F. Schoen. The global optimization of More cluter by potential energy tranformation. INFORMS J. Computing, 16:371 379, 2004. [4] H.X. Huang, P. Pardalo and Z.J. Shen. Equivalent formulation and neceary optimality condition for the Lennard-Jone problem. J. Global Optimization 22:97 118, 2002. [5] M. Locatelli and F. Schoen. Minimal interatomic ditance in More-cluter. J. Global Optimization 22:175 190, 2002. [6] C. Marana and C. Flouda. A global optimization approach for Lennard-Jone microcluter. J. Chemical Phyic. 97:7667 7678, 1992. [7] MuPAD Reearch Group. http://www.mupad.de [8] D. Ruelle. Claical tatitical mechanic of a ytem of particle. Helvetica Phyica Acta 36:183 197, 1963. 12
[9] D. Ruelle. Statitical Mechanic Rigorou Reult. Benjamin, New York. 1969. [10] W. Schachinger, B. Addi, I. M. Bomze and F. Schoen. New reult for molecular formation under pairwie potential minimization. Comput. Optimization Appl., to appear, 2006. [11] T. Vinkó. Minimal inter-particle ditance in atom cluter. Acta Cybernetica 17:105-119, 2005. http://www.inf.u-zeged.hu/~tvinko/mindit vinko.pdf [12] Wolfram Reearch. Mathematica. http://www.wolfram.com/product/mathematica/index.html [13] G. L. Xue, R. S. Maier, and J. B. Roen. Minimizing the Lennard-Jone potential function on a maively parallel computer. Proc. 6th Int. Conf. Supercomputing. 409 416, 1992. [14] G.L. Xue. Minimum inter-particle ditance at global minimizer of Lennard-Jone cluter. J. Global Optimization, 11:83 90, 1997. 13