Improved bounds for interatomic distance in Morse clusters

Transcription

1 Improved bounds for interatomic distance in Morse clusters Bernardetta Addis Werner Schachinger October 2, 2008 Abstract We improve the best known lower bounds on the distance between two points of a Morse cluster in R 3, with ρ [4.967, 15]. Our method is a generalization of the one applied to the Lennard-Jones potential in [2], and it also leads to improvements of lower bounds for the energy of a Morse cluster. Some of the numerical results have been announced in [1]. 1 Introduction In the paper [2], a new method has been proposed to find lower bounds for the minimal interparticle distance for minimum energy configurations, when energy is expressed in terms of a pair potential such as the Lennard-Jones potential or the Morse potential. As a byproduct, the method also gives lower bounds for the total energy of minimum energy configurations. The basic idea is to consider the smallest distance occurring between two particles of an optimal configuration and to find bounds for the gain in energy that can be obtained moving one of the particles attaining the smallest distance. For the Lennard-Jones potential there is an extension of the method that further improves the bounds (see Section 3 of [2]). The basic idea here is to consider not only the first-smallest distance, but also the second-smallest, thirdsmallest and so on up to the k-smallest. The aim of the present paper is to demonstrate that the same strategy, with minor modifications, can be applied to the Morse potential, leading to significant improvements of the lower bounds present in the literature. Furthermore, our strategy has the potential to work also for other pair potentials from a nicely behaved class. For some parts of the proofs we will refer to [2]. Apart from that we tried to keep the paper self contained, and therefore have to resume some of the notation from [2]. addis@elet.polimi.it, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34, Milano, Italia Werner.Schachinger@univie.ac.at, Dept. of Statistics and Decision Support Systems, University of Vienna, Brünnerstr. 72, 1210 Wien, Austria 1

2 2 Refinements for Morse Potential We consider Morse clusters, i.e. minimum energy N-particle configurations x = (x 1,...,x N ) R3N, i.e. solutions of the global optimization problem { V N = V (x ) = min v( x i x j ) : x R }, 3N 1 i<j N where denotes the Euclidean norm and where v(t) = v ρ (t) = ( e ρ(1 r) 1 ) 2 1 denotes the pairwise Morse potential energy function depending on a positive parameter ρ, which we often suppress. We will also need the two inverses of v, which we denote by v< (y) = 1 ln(1+ 1+y) ρ and v> (y) = 1 ln(1 1+y) ρ and which are defined on [, [ and [,0[ respectively. Let ζ = 1 ln 2 ρ be the unique positive zero of v(t). We next turn to k-smallest distances. For a set x of N points in R 3, and x x, let 0 = d 0 x d 1 x dx N be the multiset { y x : y x} in increasing order, and define for 1 k < N d k = d k (x) := min x x dk x. In particular we have d 1 = min x,y x,x y y x, and clearly d k d l for k < l. We call d k x the k-smallest distance in x w.r.t. x, and d k (x) the minimal k-smallest distance in x. One of our main interests is then in finding lower bounds for d 1 (x ), holding uniformly for N 2. For 1 k < N let r = (r 1,...,r k ) R k, with 0 < r 1 r 2 r k. For A R 3 we define { ν r (A) := max n : x A n, min x i r 1, 1 i n d 1 (x) r 1,...,d k (x) r k }. For r,s R k we write r s (or s r) iff r l s l for 1 l k. Clearly r s implies ν r (A) ν s (A). Note that ν r (A) is hard to compute, so a manageable upper bound µ r (A) ν r (A) will be used later on. Define the spherical shell S y := {x R 3 : v( x ) y}. Given a configuration x, a point x x and a subset w x\{x}, then, as shown in [2], the contribution of pairs with one member x and the other member in w to the total energy of x can be bounded as follows, w w v( w x ) ν d(x) (S y )dy. (1) If x is an atom of an optimal configuration x having for some l its l-nearest neighbor at distance d l (x ) ζ, then we can strengthen the bound for the 2

3 contribution of x to the total energy of x, N v( w x ( ) lv d l (x ) ) ν d(x )(S y )dy, (2) w x \x which, if greater than, contradicts optimality of x, since or less is contributed to the total energy of a configuration by an atom whose distance to the convex hull of the remaining atoms equals 1. This observation results in the following lemma (cf. also [2, Lemma 2]). Lemma 1. If for some configuration x and some l, 1 l k we have d l (x) ζ and lv(d l (x)) µ d(x) (S y )dy + 1 > 0, (3) where µ is such that µ d(x) (S y ) ν d(x) (S y ) holds for any x and y, then neither x, nor any z satisfying d(z) d(x) and d l (z) = d l (x), can be optimal. This suggests to look for fixed k 1 at the solution r = (r 1,...,r k ) of the system of equations So we first let lv(r l ) µ r (S y )dy + 1 = 0, 1 l k. (4) r l = r l (t) = v < ( ) v(t) ]0,ζ] l be the unique solution of v(t) = lv(r l ) for 0 < t ζ and 1 l k. The vector of solutions r(t) = (r 1 (t),r 2 (t),...,r k (t)) then satisfies r(ζ) = (ζ,...,ζ), and r(t) r(u) for 0 < t u ζ. Also note that r 1 (t) = t. Moreover any solution of the equation U ρ,k (t) := v(t) µ r(t) (S y )dy + 1 = 0, (5) gives rise to a vector r(t) of solutions of (4). In [2] an upper bound µ r (S y ) ν r (S y ) was derived that in the case that r k ζ holds, simplifies to ( ) 2v 3 ( ) > (y) + r k 2v 3 < (y) r k µ r (S y ) := ( rk 3 1 ), (6) k 4 i=1 (r k r i ) 2 (2r k + r i ) r1 3 where a b denotes the maximum of a and b. This allows to derive along the lines of [2] the following explicit expression for U ρ,k (t) where U ρ,k (t) = (e ρ(1 t) 1) 2 W ρ (r k (t)) + ( rk 3(t) 1 ) (7) k 4 i=1 (r k(t) r i (t)) 2 (2r k (t) + r i (t)) r1 3(t) W ρ (r) = 2r 3 12 ( 1+4ln(2) ρ r2 2ρ ) ρ +1 r ( 7 6ln 2+2ln ρ ln 2 ρ ρ ). 3

4 U 4.5,15 U 5,19 U 6,27 U 8,46 U 10, t Figure 1: Distorted scale plots of the functions U ρ,k for various pairs (ρ,k). See Figure 1 for some plots of functions U ρ,k. At his point, it is possible to state the following Theorem. Theorem 2. Let ˆr ]0,ζ] be a lower bound for the minimal interatomic distance valid for optimal Morse clusters of any number of atoms, and, for some k 1, let r := v < (kv(ˆr)). If r > 0 and U ρ,k ( r) > 0, then there is a smallest solution t k > r of U ρ,k(t) = 0, such that d(x ) r := r(t k ) holds for any optimal Morse cluster x of N > k atoms. Moreover V (x ) N v(t k )+1 2 holds for any optimal Morse cluster x of N 2 atoms. Proof. Note that U ρ,k (ζ) 25, since µ r(ζ) (S y ) µ r(ζ) (S ) = ζ 26 2 holds. Since U ρ,k is continuous, there has to be a smallest positive zero t k of U ρ,k in the interval ] r,ζ]. We denote D := {r(t) : r < t t k }. Now assume that for an optimal Morse configuration x of at least k + 1 points the vector d = d(x ) does not satisfy d r. Clearly it has to satisfy d 1 ˆr, which implies r( r) d, since r l ( r) r k ( r) = ˆr d 1 d l holds for any 1 l k. This furthermore implies r r( r). Then r := max{r D : r d} = r(t ), for some t ] r,t k [, and for some l we have dl = r l. Therefore lv(d l ) µ d (S y )dy + 1 lv(r l) µ r (S y )dy + 1 = U ρ,k (t ) > 0, and by Lemma 1 we conclude that x is not optimal, a contradiction. Finally, the last assertion of the theorem follows for N > k from the following lower 4

5 bound for the contribution of a single particle x x to the total energy, which uses (1) and ν d ν r µ r, N w x \{x } v( w x ) ν d(x )(S y )dy µ r (S y )dy = v(t k) 1. To show that the same bound v(t k )+1 2 also holds for N k, we can use the subadditivity property V n+m V n + V m derived in [2], which implies Vn n V2n 2n for n 2. Remark 3. If the hypotheses of the theorem are satisfied for k = 1, then an improvement (t k > ˆr) of the lower bound for interatomic distance is guaranteed. In cases k > 1 it could turn out that r < t k ˆr holds, which would be no improvement. Luckily that does not happen to be the case for the Morse potentials (and the corresponding best up to date lower bounds) we consider, as is revealed in the following corollary, which has also been stated without proof in [1]. Corollary 4. For N 2, and ρ {4.967,5,6,7,8,9,10,11, 12,13,14,15}, any optimal Morse cluster x of N points satisfies min 1 i<j N x i x j t, (8) and V (x ) N V. (9) The values of t and V, which depend on ρ, are reported in Table 1. Proof. Let us consider for each value of ρ lower bounds ˆr as in Table 1. Values for ρ = and ρ = 5 are taken from [3], all the other values (which are actually zeros of U ρ,1 ) are taken from [2]. The first idea would be to look for K = K ρ such that the hypotheses of the theorem are satisfied for 1 k K, and then to use max 1 k K t k as a lower bound. That works fine for ρ 6: We can always find some large enough K such that the sequence (t k ) 1 k K increases up to some k = k opt, then decreases again, so that t k opt gives us a good bound. Unfortunately that approach does not give us the best bounds for ρ < 6: The value of k, where U ρ,k ( r) ceases to be positive, is smaller than k opt. So we need an intermediary step to improve the initial lower bound ˆr. Taking k = 4, we compute U ρ,4 ( r) and verify that it is positive, then from Theorem 2 a lower bound t 4 for minimal distance is obtained, computing the smallest zero larger than r of U ρ,4. It is trivial to prove that the bounds found for k = 4 are valid also for N k, in fact in these cases the global optimum is unique with all distances equal to 1. We now can repeat the procedure using t 4 as a new known lower bound ˆr for minimal distance. We verify that using these values, U( r) results positive for sufficiently large k also for ρ < 6, yielding an optimal value k opt and corresponding lower bound t k opt. (See Figure 2 for what happens in the case ρ = 5.) 5

6 0.5 t 19 t ˆr k Figure 2: Demonstration of our method of obtaining lower bounds in the case ρ = 5. For each k the vertical interval is the set {t ]0,1 ln 2 5 ] : U ρ,k(t) 0 and U ρ,k (v < (kv(t))) 0}. If a lower bound is known which lies in such an interval, it can be improved to the upper end of that interval. Because ˆr = does not lie in the interval corresponding to k opt = 19, an intermediate step via (e.g.) k = 4 is needed. ( ) According to Theorem 2, we can establish (9), setting V := 1 2 v(t k opt ) + 1. For the proof of the uniform lower bound (8) we use a combination of bounds t k valid for N > k and size dependent (and strictly decreasing) bounds t k := min{t 0 : v(t) k 3} holding at least for N k (cf. [2, page 342]). The best thing to do is let k unif := max{k k opt : t k t k} and use t := t k unif as a uniform lower bound. This completes the proof. In Table 1 we report values for minimal distance as found in [3] (fifth column), compared with our results for fixed and best uniform value of k (third and sixth column). We can observe that even with k = 4 the refinement we propose improves significantly the minimal distance bounds. The improvement is larger for small values of ρ. A similar statement is true for our improvement of the bound for the energy contribution per one atom (eighth and ninth column). Remark 5. As far as we know, ρ = is the smallest value for which there is a proof (by Vinko and Neumaier [3]) of a positive lower bound for minimal interparticle distance in Morse clusters of atoms interacting according to the 6

7 Table 1: Lower bounds for interatomic distance and for energy contribution per one atom in globally optimal Morse clusters ρ ˆr t 4 k unif l.b. for distance k opt l.b. for V N/N from[3] t (Eq. 8) from[3] V (Eq. 9) potential function v ρ. Still it is interesting to see what our method would yield, should there one day be a proof of such a lower bound also for some ρ < For instance, in the case ρ = 4.5, one has k opt = 15. See Figure 1 for a plot of U 4.5, 15. The analogue of Figure 2 would have nonempty intervals only for the values 3 k 15. Supposing that 0.2 is a lower bound, two intermediary improvements (e.g. via k = 4 and k = 13) are necessary until we can use Theorem 2 with k = 15 to obtain the lower bound t = , the best improvement achievable with our method in that case. As ρ becomes smaller, the set of k with nonempty intervals becomes smaller, and those nonempty intervals themselves become smaller. As ρ approaches the value , only one nonempty interval, corresponding to k = 6, remains, which shrinks to a single point, t = Remark 6. That our method be applicable to other pair potentials v depends on the existence of functions U v,k being substitutes for U ρ,k in (5), and on an initial lower bound ˆr > 0 being available. In [2] we introduced the class V consisting of functions v C 2 (R + ) such that v(1) =, (t 1) v(t) > 0 for t 1, and t 2 v(t) dt <. For v V the two inverses v 1 < and v > can be properly defined, giving a meaning to (6), moreover the integral in (5) will converge, giving rise to a function U v,k. For v V satisfying lim t 0 t 3 v(t) = (and more generally for v V satisfying liminf t 0 U v,1 (t) ]0, ]) we can apply [2, Corollary 1] to compute a lower bound ˆr > 0 from scratch. For v growing more slowly at 0 we have to rely on lower bounds ˆr coming from elsewhere. In cases where we have lim t 0 v(t) =, Theorem (2) simplifies: the condition r > 0 need not be checked. In any case, if the hypotheses of Theorem (2) are satisfied, the computation of t k, the solution of U v,k(t) = 0, is of course much easier if a closed form expression for the substitute for W ρ appearing in (7) is available. We performed all computations using Maple V Release 5.0. At http: //home.dei.polimi.it/addis/download/boundsmorse.mpl you can find the Maple code we used. 7

8 References [1] B. Addis and W. Schachinger, Morse potential energy minimization: Improved bounds for optimal configurations, Comput. Optim. Appl., published online 17 September 2008 (doi: /s ) [2] W. Schachinger, B. Addis, I. M. Bomze and F. Schoen, New results for molecular formation under pairwise potential minimization, Comput. Optim. Appl. 38 (3), (2007). [3] T. Vinko and A. Neumaier, New bounds for Morse clusters, J. Global Optimization 39, (2007) 8