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254 Tamás Vinkó and Arnold NeumaierLemma 1. Suppose that we have k small d-dimensional balls with radius r/2. Let r i be the distance ofthe ball i from the origin and suppose that r 1 ≤ . . . ≤ r k . Then a lower for the maximal distance r k ofa small ball from the origin is d √k−12r ≤ r k .Theorem 2. Let E i (x ∗ ) := ∑ i≠j v(‖x∗ i − x∗ j ‖) (i = 1, . . . , n) and assume that in the configurationtaken into account the minimal distance between the particles equals to q.(i) E 1 < −|v(s)| holds.(ii) E1 ∗ = ∑ r j ≤s v(r j) + ∑ r j >s v(r j) ≥ F (q) + S(q).(iii) If the inequality F (y) + S(y) > −|v(s)| holds for all y ∈ Y then r ∗ /∈ Y , where( (2s ) + y dF (y) := v(y) + v(s)− 1)yand S(y) :=∞∑j=⌈(2s/y+1) d ⌉( √ d) j − 1v y .2If we have a size independent lower bound on the minimal inter-particle distance thenlower bound on the values Ei∗ for all i = 1, . . . , n can be established. Thus linear (in thenumber of atoms) lower bound on the optimal values can be given as well [6].3. Lennard-Jones clusters[ (In general form the Lennard-Jones pair potential function is v σ,ɛ (r) = 4ɛ σ) 12 (r − σ) ] 6r,where ɛ is the pair well depth and 2 1/6 σ is the pair separation at equilibrium. In the globaloptimization literature the function v σ,ɛ with reduced units, i.e. ɛ = σ = 1 and s = 2 1/6 , or theso-called scaled Lennard-Jones pair potential (ɛ = 1, σ = 2 −1/6 , s = 1) is investigated.In the literature one can find previous results about the minimal distance in optimal scaledLennard-Jones clusters. These results are compared in the following table including that oneobtained in the present work. Note that all these results are independent of the number ofparticles in the configuration.dimension Xue [6] Blanc [1] Vinkó [5] present work2 – 0.7286 (0.7284) 0.75333 0.5 0.6108 0.6187 0.6536Based on these results linear lower bounds on the optimal values can be calculated. Theyare −67.88673405n · ɛ in dimension three and −9.565562565n · ɛ in dimension two (n=2, 3, . . . ).Note that for the scaled Lennard-Jones cluster even better lower bound (q ≥ 0.72997) isreported by Huang. This result is not included in the table above since one can prove that hisargument leading to such a good result is incorrect.4. Morse clustersThe pair potential function in Morse cluster is v ρ (r) = e ρ(1−r) ( e ρ(1−r) − 2 ) , where ρ > 0 is aparameter. In the context of global optimization, the cases ρ > 6 are interesting, since theseare more difficult problems than finding the optimal Lennard-Jones structures [2].We must emphasize that Theorem 2 gives an exclusion interval for r ∗ . For the Morse clusters(because it is defined even in the case r = 0) the function F (y) + S(y) + |v ρ (s)| in Theorem2 becomes negative for small y values. The very tiny y values are handled using [4].In Table 1 the results from [4] and form [5] collected and compared with the results can beachieved with the usage of the general technique introduced in this paper. Note that the new