0 = 5 =

A Discrete-Continuous Algorithm for Molecular Energy Minimization

R.S. Maier J.B. Rosen G.L. Xue

AHPCRC CSci Dept AHPCRC

University of Minnesota University of Minnesota University of Minnesota Minneapolis, MN 55415 Minneapolis, MN 55455 Minneapolis, MN 55415 maier@sl .arc.umn.edu rosen@cs. umn .edu xue@sl .arc.umn.edu

Abstract

We present a parallel algorithm for minimizing molecular energy potential functions applied to the case of pure Lennard- Jones clusters. The algorithm demonstrates the combination of discrete, lattice- based optimization with continuous optimization (re- laxation) techniques. The suggested approach is not restricted to the Lennard-Jones potential and is aimed at problems in which the potential of interest may be significantly more costly than the Lennard- Jones. The intended audience includes researchers interested in practical computational problems involving minimum energy cluster conformation, such as may arise in catalysis, and those interested in algorithm develop- ment. The advantage ^of the algorithm is that the time required t o find the minimum-energy structure for a relatively large cluster reduces to that of an interactive session. Our parallel implementation is capable ^of de- termining the best-known, previously published binding energies for n

5

150 L J clusters in a matter of seconds and has provided new results on minimum energies ^for clusters of up to n

=

1000 atoms using a massively- parallel processor, the Thinking Machines CM-5.

1 Introduction

Computational methods for minimizing molecular potential energy functions have attracted substantial interest among optimization researchers. One of the fundamental problems in this area is that of minimiz- ing the potential energy of the pure Lennard-Jones

(LJ)

cluster [5],[3].

A

number of research papers have been published dealing with optimization methods for this problem [2], [7], [15],[19]. In this paper, we call attention to an algorithmic approach [lo] which is rel- atively unknown among optimization researchers and has proven extremely successful for LJ clusters with up to n

=

150 atoms. The purpose of our research is to investigate the properties of this approach for large clusters and draw conclusions regarding its usefulness in a more general molecular energy minimization con- text.

The difficulty of the LJ cluster problem arises from the fact that it is a global optimization problem with an exponential number of local minima [5]. It is axiomatic that the solution of a global optimization problem, or multiple-minima problem, will be im- proved by any physical understanding of the problem which serves to restrict the search space, or domain, which contains the global solution. In the case of the pure LJ cluster, the critical assumption is that a well- defined set of lattice structures contains at least one initial cluster configuration which relaxes to the ground state. The support for this assumption comes primar- ily from numerical investigations [5],[10]. For the LJ cluster, the lattice structures consist of an icosahe- dral core and particular combinations of surface lattice points. Given this physical insight into the problem, the approach outlined below begins with a discrete optimization algorithm applied to a relatively small sample of initial states. The resulting lattice minirna are then relaxed as a continuous minimization prob- lem. The method we have used can be described at a

high level of abstraction:

1.

Define

S

as a

set

of lattices.

2.

Define a potential function for the discrete problem.

3.

For each element in

S,

Repeat

Perform lattice

search

based discrete optimization to identify minimum energy lattice conformations.

Until stopping criteria satisfied.

4. Define a potential function for continuous problem.

5 . Perform relaxation (continuous minimization) for each minimum energy lattice conformation.

In sections

2

and

3,

we introduce the lattice types and potential functions used in the algorithm. In sec- tion 4, we define the lattice search algorithm and a limited-memory variant of the BFGS minimization al- gorithm [8] used to relax the lattice configurations, including discussion of parallel implementation on a massively parallel processor. In section 5 we present computational results, including binding energies for up to n

= 1000.

In section

6

we draw some conclu- sions about the algorithm and its application to more general problems. An appendix is provided to permit reproduction of the lattices.

2 Lattices

We will denote

a

pure, n-atom, molecular cluster configuration by the vector

z

E

Pxn

=

{PI, *..,pn}, pi E

(1)

where the pi denote three-dimensional atomic coordi- nates. The sampling space of initial molecular con- figurations includes the

IC

and

FC

lattices (see ap- pendix), based on an icosahedral core but differing in the arrangements of atoms on the outer shell. The construction of these lattices is described in detail in the appendix. A sample or sampling unit from these lattice structures is an n-atom cluster selected in a certain way, using randomization. This procedure is discussed in section 4.

A main assumption is that the configurations con- tained in the

IC

and

FC

lattice include an initial state

i

in the region of attraction of the global minimum.

The evidence for this assumption is empirical. It is generally agreed that the n

= 13

and n

=

55 MacKay icosahedra relax into minimum-energy states for the pure

LJ

cluster. Further, the lowest binding energies in the published literature can be obtained from initial

states

selected from these lattice structures

[lo].

3 Potential F’unctions

We consider only tw-body forces between compo- nent atoms, leading to the general tw-body, n-atom Lennard- Jones potential

where ^VLJ

=

(r;:’

-2riTj6)

and r i j

=

^{llpj -pill2} is the pair distance, with gradient

n

vjvn(p) ^{= -12}

( ~ < ~ * - r ~ ~ ) ( p j - p i ) ,

i =

1,

...,

n.

i=l,i#j

(3)

For reproducing results by other researchers, we have also used the “nearest-neighbor”

(NN)

potential

100

r

< 0.8 -1

0 r > 1.3 0.8 5

r

5 1.3

The

N N

potential is used in addition to the

LJ

during the lattice optimization phase to identify candidate lattice configurations for subsequent relaxation. For certain cluster sizes, lattice minima of the NN poten- tial relax more freely (lead to lower binding energies) than the lattice minima of the

LJ

potential.

4 Algorithm

4.1 Discrete Optimization

Given

a

lattice of N points, the n-atom lattice opti- mization problem (with n

5 N)

can be formulated as

the nonconvex, quadratic, integer programming prob- lem

where the zero-one variables ai indicate an atom in the ith lattice position, and the pair potentials v ( r i j ) are fixed quantities corresponding to pairs of atoms a t the ith and j t h lattice points. Because this formulation is an indefinite,

0 -

1 integer program with many local minima, it is difficult to solve directly. In the next sec- tion, a heuristic algorithm for evaluating lattice local minima is described and analyzed.

4.2 Lattice Optimization Algorithm

Given a set of

N

lattice points, the following algo- rithm generates a sequence of random n-atom subsets from the lattice. Treating each random configuration as a starting point, the algorithm performs a local lat- tice search by iteratively dropping and adding lattice points, until a lattice local minimizer is obtained. The lattice minimizers with the lowest function value ob- tained from this process are saved for subsequent re- laxation. Following [lo], we use two kinds of lattices, the I C and the

FC.

It is assumed that one type of the lattices has been chosen before the algorithm is applied.

Algorithm 4.1

{

Lattice Optimization for n-atom cluster}

Initialization. Find the largest I C lattice which contains a t most n points and call this the lattice core. Let

Zcore

be the index set of the core with NCO,, = lIcoreI. If NCO,, = n, put the n atoms on the core points and stop.

Initialization. Find the next I C or

FC

lattice shell and let ISurj be its index set with N,,,,

=

IZbUr,l.

Define

N = NCO,, + NSu,j

as the total number of points in the lattice. For each lattice point on the surface layer, compute the potential energy between this point and all lattice points

in the core. For each pair of lattice points on the outer layer, compute the pair potential energy between these two points.

Generate n-atom cluster with random surface configuration. The NCO,, positions in the core are always filled. The remaining n

- NCore

atoms are randomly assigned to the surface sites.

Lattice search. Drop the atom on the outer shell which has the least contribution t o the total p c ~ tential (the most loosely bound atom) and re- place an atom a t an unfilled surface lattice point with the greatest contribution t o the total pc- tential (the most tightly binding empty lattice point). If this improves the total potential, go to Step 4; otherwise this is a lattice local minimum.

Stopping criteria. Store all the local minima found so far which have the lowest observed func- tion value. If the minimum function value has been observed k m a z times or the search limit m a z i t has been reached, stop, otherwise go to Step

3.

In Step 2, we compute and store the potential en- ergy value between each lattice point on the outer shell and the core atoms to reduce the time complexity of the lattice search. This initialization phase takes O(Nc,,, x N,,,,) time. Then, in step 4, the most loosely bound molecule can be found in O ( ( N -

TI)^)

time

,

which is no more than O(N;,,,,) = ( O ( n 3 ) ) . The most tightly binding empty lattice point can also be found in this amount of time. The most loosely bound molecule is defined by

and the most tightly binding empty lattice point is given by

The target architecture for the algorithm

is

the Thinking Machines CM-5, configured with up to 512 SPARC-2 processors in

a

tree network characterized by

5 -

20 Mb/sec interprocessor communication band- width. Our parallel implementationof the lattice algo- rithm depends mainly on an efficient implementation of the function evaluation (used in finding the most loosely bound molecule and the most tightly binding empty lattice point). The function evaluation is dis- tributed among all the available processor elements by

first

broadcasting the surface configuration to each processor. Each processor then computes a partial sum of the potential function. The full summation

is

accomplished with a global data reduction routine.

4.3 Local Minimization

Relaxation of a molecular configuration may be for- mulated as the unconstrained minimization problem

where ^{t o} denotes the starting point for

a

local mini- mization procedure and

L

is the set of lattice minima identified by the lattice optimization algorithm.

Local minimization procedures are designed to lo- cate stationary points of the potential function V(z) such that VV(z)

= 0.

There exist a number of good methods for local minimization without second deriva- tives, and because we are interested in large-scale min- imization, we have selected the limited-memory BFGS (LM-BFGS) method

[SI.

This method is suitable for parallel implementation due to the relatively simple interprocessor communication requirements [9].

Parallel implementation of the local minimization procedure depends on the efficient parallel implemen- tation of the LJ function evaluation as well as the LM- BFGS algorithm. As indicated in the previous section, the target architecture is the Thinking Machines CM- 5, with 512 SPARC-2 processors configured in a tree network with 5

-

20 Mb/sec interprocessor communi- cation bandwidth. We begin with an initial distribu- tion o f t ) among m processors, or roughly n/m _atoms per processor. For the LJ function evaluation (2), the approach which we have found to be most efficient for

clusters of up

to

n

=

15,000 (45,000 variables), is to broadcast the entire configuration ^{t k} to each proces-

sor.

Then, each processor computes na/m pair poten- tial values, providing n/m elements of ^g)

=

V V ( t ) ) and

a

partial value of V(z)). The full summation of V ( z ) ) is accomplished with a single, scalar global data reduction among the processors. On completion of the LJ function evaluation, the elements of ^{t h} and 9 k are already distributed among the processors and no further communication is required to prepare for the LM-BFGS iteration. The LM-BFGS iteration con-

sists

primarily of ^a sequence of inner products and linked-triad (saxpy) operations. The saxpy is a lo- cal operation and requires no synchronization. The inner-product requires a scalar global data reduction operation. No other data movement is required in the iteration. In general, the cost of the LJ function evaluation is O(n2/m) floating point operations, com- pared to O(n/m) operations for the LM-BFGS itera- tion. Therefore, for large n, the cost of the LJ function evaluation is dominant.

5 Computational Results

We have reproduced Northby’s published results [lo] for n

<

150 to verify the operation of the algo- rithm. For each value of n we have obtained the same minimum binding energy. As an illustration of per- formance, we have obtained the best-known binding energy for a cluster of size n

= 99, VW =

-550.6665, in approximately

6

seconds of Cray-2 CPU time.

Computational results for larger clusters obtained on the CM-5 are given in Table

1.

The results in- clude cluster sizes in the range (100

<

ⁿ

<

lOOO), binding energy for the lattice configuration(f,) and relaxed configuration(f), total time (sec), number of lattice searches, the number of times the lattice mini- mizer was found ( n r e p ) , and a mnemonic key indicat- ing which lattice type led to the lattice minimizer.

The results in Table 1 show that the amount of work performed by the algorithm does not increase mono- tonically with problem size (n). The number of lattice searches depends on how close the cluster size n is to

an icosahedral “magic number” (i.e., how close n is to

N).

For example, the case of n

=

900 solved very quickly because ICs

=

923. In this case, there were only 23 unfilled lattice sites, and the lattice minimizer was observed quite frequently. The combinatorial dif- ficulty of the lattice search was far greater for n

= 800,

where 123 lattice sites were unfilled. The value of nrep also indicates which cases were particularly difficult;

nrep

=

1 means that the lattice minimizer was only observed once in 2000 searches. In such cases, there is relatively little confidence that the global lattice min- imizer has been found.

As described in

[lo],

the

FC

lattice is useful for n such that only a few surface sites are filled. In Table 1, n

=

600 is such a case since only 39 out of 362 possible

IC

surface sites are occupied and we note the solution for n

=

600 was obtained by an

FC

lattice minimizer.

It is also interesting to note that in [lo], an alterna- tive potential function (4), called the

N N

potential, was used for lattice optimization in addition to the

LJ

with the result that for n

5

150, some solutions were found only with the N N potential and not with the

LJ.

In the larger cases that we have tried, we have always obtained the best solution using the

LJ

_poten- tial, although in some cases the

N N

potential has also identified the best solution. The results reported in Table 1 are based only on the

LJ

potential.

Here it should be noted that the total time for the algorithm divides naturally into the two phases of lat- tice search and continuous optimization. The time for continuous optimization tends to dominate for clus- ters of up t o about n

=

150. For larger clusters, the lattice search phase becomes increasingly difficult and the time increasingly dominant.

It should also be noted that the lattice search phase has no rigorous termination criteria. The stopping cri- teria used in generating the results for Table

1

were kmaz

=

10, or 10 replications of the same lattice min- imum function value, and matit

=

2000, or a limit of 2000 lattice searches. The first criteria is quite eas- ily satisfied for n

5

400, but extremely hard to sat- isfy for certain larger cases. To address some of these difficulties, we have experimented with various tech- niques from stochastic global optimization, including

clustering [12]

of

initial lattice states, sample reduc- tion, and the application of Bayesian stopping criteria [l]. Sample reduction is the most fundamental of these techniques, involving the generation of a large sam- ple of N lattice configurations, and choosing the qN

(0 <

7

<

1) configurations with lowest function value

as

starting points for lattice optimization. This tech- nique takes advantage of any correlation between func- tion values of initial

states

and local minima within regions of attraction. Bayesian stopping criteria are based on a generalized multinomial model of the sam- ple space of local minimizers. In the relaxation phase of the algorithm, the sample space is restricted to set of lattice minimizers L, which typically relax into minima with only one or a few distinct function val- ues. In this setting, the application of Bayesian stop- ping criteria can justify termination of the relaxat.ion phase. The results of these experiments are positive for clusters of moderate size. Sample reduction in- creases the observed frequency of the best lattice min- imum. Similarly, the application of Bayesian stopping criteria in the relaxation phase allows earlier termina- tion than the criteria used previously [lo]. However, for n

>

400, we found the difficulty of the lattice op- timization phase begins t o overwhelm the beneficial effect of these techniques.

n

fl f

sec srch lat ^{n r e p}

100 200 300 400 500 600 700 800 900 1000

-522.295 -1147.501 -1809.510 -2465.669 -3144.337 -3825.765 -4505.675 -5200.207 -5916.914 -6601.417

-557.040 -1229.185 -1942.107 -2650.432 -3382.693 -41 19.244 -4854.400 -5602.720 -6377.491 -7117.899

15 363 2 3366 1170 2383 7743 7083 176 14598

161 IC 10 1171

IC

10

22 IC

10 4000 IC

2

1408 IC 10 2777

FC

10 4000 IC 1 4000

IC

1

4000 IC 1 227 IC: 10

Table 1: Results for Discrete-Continuous Algo- rithm, 100

5

n

5

1000. Table includes binding energy for lattice minima

(f,)

and corresponding re- laxed configuration

(f),

CPU time, number of lattice

searches, and type of lattice and potential function.

CPU time represents singleproceseor CPU

seconds

uk

ing Thinking Machines CM-5 with

256

proceseocs and

SUN/4

front-end

(no

vector processing units) calcu- lated at

a

synchronization point.

fort compared

to

the lattice optimization phase. The results also show that relatively few LM-BFGS iter-

ations

are required to reach a stable state from the icosahedral structure.

A

stable state satisfies the ter- mination criteria

13 -44.32680 55

147 309 56 1 923 1415 2057 2869 3871 5083 6525 8217 10179 12431 14993

-279.2485 -876.4612 -2007.219 -3842.394 -6552.723 - 10308.89 -15281.55 -21641.35 -29558.91 -39204.88 -50749.87 -64364.50 -80219.39 -98485.16 -119332.4

0.2 0.3 0.5 0.7 1.9 2.8 6.4 12.5 24.7 52.0 83.8 170.6 230.1 441.1 656.8 1004.5

5 8 0

8 11 0

14 17 0 21 24 0 27 30 0 33 39 1 41 47 2 45 51 5 54 62 12 61 68 23 67 71 42 81 89 87 72 79 123 98 104 248 99 103 366 112 117 605

0 3 19 78 114 272 34 1 399 475 45 1 504 511 533 562 558 602

Table

2: LM-BFGS Performance

on

LJ Clus- ter Relaxation Problem.

Relaxation of IC2

-

^IC17

icosahedral structures using Thinking Machines

CM-5

with

512

processors, and

SUN/4

front-end (no vector processing units). Total seconds is single processor

CPU time calculated at a synchronization point.

In Table

2

we give performance statistics for relax- ation of icoeahedral structures IC2

-

IC17 using the LM-BFGS method. These structures are the special cases where

N = n

defined in step

1

of the lattice opti- mization algorithm, and no lattice search is required.

The statistics include binding energy

(f),

total sec- onds required for relaxation to a stable state, num- ber of LJ function-gradient evaluations

(f-g),

billions of floating point operations

(GfZ),

and the megaflop rate or million floating point operations per second (MfZs). The results clearly show that for n w

1,000,

the relaxation phase requires only

a

few seconds and is therefore a minor part of the total computational ef-

2, - - , 17,

the number of iterations ranges from

8

to

117,

or roughly O(d). Using

a

random starting config- uration, we would typically expect

O( lo3)

iterations for

n

w

50.

Figure

1

shows the binding energy as a function of cluster size, using the values of ^fi and

f

from Tables

1

and

2.

The graph of binding energy appears linear but in fact has several inflection points. Least-squares

fits

of V,

= f

and V,

=

fi to the model

V,

= a + p n r

(9)

yield ( a , p , 7 )

= (36.448,-5.6538, 1.0274)

for the lat- tice solution V,

=

fi and

(359.79,-6.1570,1.0271)

for the relaxed solution V, =

f.

The estimates of the exponent 7 are identical to four significant digits, ^sug- gesting that the two curves differ by a constant factor.

6 Conclusions

The lattice search and optimization approach to en- ergy minimization is the most successful method avail- able for the pure

LJ

cluster problem. For cluster size up to n FZ:

500,

the method can be employed interac- tively on a supercomputer. Extensions of the method to more complicated potentials, such as may arise in catalysis, would seem feasible using a more general lat- tice construction (e.g., diamond or hcp) in conjunction with iterative lattice refinement for clusters of up to a few hundred atoms.

The combinatorial difficulty of the underlying opti- mization problem emerges for large

LJ

cluster sizes which are not near the icosahedral “magic” num- bers. For cluster size of n

= 800,

which lies ap- proximately midway between the icosahedral numbers

544 < 800 < 923,

the algorithm failed to reproduce the

I10

-2

.

4 -

1

I

:I

^{4 -}

d -

-10

-

Figure 1 : Binding energy versus cluster size.

Dotted line represents lattice minimum function val- ues

f, ,

solid line represents relaxed configuration func- tion values

f

in Tables 1 and 2 .

same lattice minimum function value twice, giving lit- tle confidence in the resulting statistics on minimum binding energy. The difficulty would be accentuated on a more general lattice in the absence of the kind of specialized knowledge available for the pure

LJ

clus- ter.

The local minimization phase of the algorithm is significantly enhanced by the lattice search phase. It was observed that the number of iterations of the continuous minimization algorithm is vastly reduced by using a lattice minima as starting point. In fact, for large clusters, the ratio of time required for local minimization becomes trivial in comparison to lattice search. This suggests the possibility of somehow re- combining the positive features of lattice search and continuous minimization to improve performance for large clusters.

7 Appendix

This appendix gives details on construction of the IC and FC lattices. The I c d lattice consists of an in-

tegral number d of I C shells. IC1 consists of a single atom located

at

the origin. For d

= 2 , ...,

k , I c d con- sists of

ICd-1

surrounded by twelve vertices (forming an icosahedron) at distance

T

=

b ( d

-

1 ) _{( 1 0 )} from the origin. To uniquely define the lattice, we fix the first two vertices of the shell

at

p1

=

(0, 0 , T ) and pz

= (0, 0,

^{- T )} and the remaining vertices at

I

Z k + 3

=

T C O S ( ( - ~ ) ~ ~ ) C O S ( ~ T / ~ )

pk+3

=

yk+s

=

TCOS((-i)ke)Sifl(kT/5)

,

% k + 3

=

T S i f l ( ( - i ) k 8 )

( 1 1 )

k = 0, { ..., 9

where 0

=

~ / 2

-

t a n - ’ ( 2 ) . The 20 equilateral tri- angular faces of each shell are filled with atoms in a hexagonal close-packed lattice. For each pair of adja- cent vertices p a , ^Pb defining an edge, the coordinates of the d

-

2 interior edge atoms are

Pa

+

^i(pb

^-

p a ) / ( d

-

I ) , i

= 1, ...,

d

-

2 . ( 1 2 ) For a given triple of vertices, { P a , P b , P c } defining a face, the coordinates of the ( d

-

2 ) ( d

-

3 ) / 2 interior face atoms are given by

( b +

jpb

+

( d

-

1

-

i

-

j ) p c ) / ( d

-

I ) , ( 1 3 ) j = 1 ,

...,

d - i - 2 , i = l I

...,

d - 3 .

The number of atoms in I c d is

l r c d l = (10d3 -

152

+

l l d

-

3 ) / 3 .

7.1 FC Lattice

The FCd lattice consists of ICd-1 enclosed by a single stacking fault shell. The stacking fault outer shell consists of 12 icosahedral vertices and 10d(d

-

1 ) atoms in stacking fault positions on the faces of the I c d - 1 outer shell (the stacking fault positions are the

“deep holes” formed by the hexagonal close packing of the

IC

face). The F C outer shell has no “edge”

atoms between vertices. For a given triple of vertices, { p a , Pb, p c } forming a face, the coordinates of the ( d

-

2 ) ( d

-

1 ) / 2 interior F C face atoms are given by

(ilia

+

jPb

+

^{( d}

^{- i -}

j ) P e ) / d , ( 1 4 ) j = 1 ,

...,

d - i - l , i = l ,

...,

d - 2 ,

where

and

=

(Pa +pa

+ ^p,)/3.

Acknowledgment

This research was supported in part by the Army Research Office contract number DAAL03-894-0038 with the University of Minnesota Army High Perfor- mance Computing

Research

Center, the Air Force Of- fice of Scientific Research grant

AFOSR91-0147,

and the Minnesota Supercomputer Institute.

References

C.G.E. Boender and A.H.G. Rinooy Kan, Bayesian stopping rules for multistart global o p timization methods, Mathematical Programming

,

Vol.

37 (1987),

pp.

59-80.

R.H. Byrd,

E.

Eskow, R.B. Schnabel, and S.L.

Smith, Parallel global optimization: numerical methods, dynamic scheduling methods, and ap- plication to molecular configuration, Technical Report

CU-CS-553-91,

October

1991,

University of Colorado at Boulder, Department of Computer Science, Boulder, CO.

J.

Farges, M.F.

De

Feraudy, B. Raoult and G.

Torchet, Cluster models made of double icmahe dron units, Surface Science, Vol.

156(1985),

pp.

370-378.

Wilfred F. van Gunstern and Herman J.C.

Berendsen, Computer simulation of molecular dynamics: methodology, applications, and per- spectives in chemistry, Angewandte Chemie In- ternational Edition, Vol.

29(1990),

pp.

992-1023.

M.R. Hoare, Structure and dynamics of sim- ple microclusters, Advances in Chemical Physics, Vol.

40(1979),

pp.

49-135.

[SI J.

Danna Honeycutt and Hans C. Andersen, Molecular dynamics study of melting and freezing of small Lennard-Jones clusters, Journal of Phys- ical

Chemistry,

Vol.

91(1987),

pp.

4950-4963.

[7] R.S.

Judson, M.E. Colvin, J.C. Meza, A. Huf- fer, and

D.

Gutierrez, Do intelligent configuration search techniques outperform random search for large molecules?, Sandia Report

SAND91-8740,

Sandia National Laboratories, Center for Com- putational Engineering, Livermore, CA.

[8]

D. C. Liu and

J.

Nocedal, On the limited mem- ory BFGS method for large scale optimization, Mathematical Programming, Vol.

45( 1989),

pp.

503-528.

[9]

R.S. Maier, Large-scale.minimization on the CM-

200,

Journal of Optimization Methods and

Soft-

ware (to appear), Army High Performance Com- puting Research Center Preprint

91-86,

Univer- sity of Minnesota, Minneapolis, MN

55415.

[lo]

J.A. Northby, Structure and binding of Lennard- Jones clusters:

13 5

n

5 147,

Joumal of Chemi- cal Physics,

87(10), 6166-6178.

[ l l ]

L. Piela, J . Kostrowicki, and H.A. Scheraga, The multipleminima problem in the conforma- tion analysis of molecules: deformation of the po- tential energy hypersurface by the diffusion equa- tion method, Journal ^of Physical Chemistry Vol.

93(1989),

pp.

3339-3346.

[12]

A.H.G. Rinooy Kan and G.T. Timmer, Stochas- tic global optimization methods, Part

I:

cluster- ing methods, Mathematical Programming

,

Vol.

39( 1987),

pp.

27-56.

[13]

A.H.G. Rinooy Kan and G.T. Timmer, Stochas- tic global optimization methods, Part

11:

multi- level methods, Mathematical Programming,

Vol.

39( 1987) 57-78.

[14]

Tamar Schlick and Michael Overton, A powerful truncated Newton method for potential energy minimization, Journal of Computational Chem- istry, Vol.

8( 1987),

pp.

1025-1039.

[15] David Shalloway, Packet Annealing: A deter- ministic method for global minimization. appli- cation to molecular conformation, In press: Re- cent Advances in Global Optimization, C. Floudae and P. Pardalos, e&. Princeton University Press:

Princeton, NJ, 1991.

[16] D.G. Vlachos, L.D. Schmidt, and

R.

Aris, Struc- tures of small metal clusters: phase transitions and isomerization, Army High Performance Com- puting Research Center Preprint 91-69, Univer- sity of Minnesota, Minneapolis, 1991.

[17] D.G. Vlachos, L.D. Schmidt, and

R.

Aris, Struc- tures of small metal clusters: low temperature be- havior, Army High Performance Computing

Re-

search Center Preprint 91-70, University of Min- nesota, Minneapolis, 1991.

[18]

L.T.

Wille, Minimum-energy configurations of atomic clusters: new results obtained by sim- ulated annealing, Chemical Physics Letters Vol.

133( 1987), pp. 405410.

[19] G.L. Xue,

R.S.

Maier,

J.B.

Rosen, Minimizing the Lennard-Jones potential function on a mas- sively parallel computer, in Proceedings of the 1992

ACM

International Conference on Super- computing, Washington DC. July 19-23,1992, pp.

409-416.