Global Optimization
CHAPTER 3. GLOBAL OPTIMIZATION 4
Deaven e t al. [38] . I n his approach each structure generated by the GA i s subjected t o a gradient driven local minimization.
Deaven's approach is very significant , as it represents L amarckian rat her t han D ar winian evolut ion. Characteristics that are acquired by an individual in t he course of its lifetime can be p assed on to its offspring in t he Lamarckian concept of evolu t ion [43] . I n t he Darwinian concept o f evolution t he characteristics that are passed on to a n offspring are t he ones that t he parent possessed when it was born. The former is preferable since t he modified genetic information is passed from parents to offspring.
A GA can be divided into three essential steps: selection, mating (or crossover) and mutation. Figure (3. 1 ) depicts a flow chart of the operation of B E L P H E G O R . The initial
population (typically ranging from 10 - 20 individual struct ures) in t he 'mating pool' is generated in part randomly by placing the atoms in a cubic box, with edg lengths a that
are given by
(3 . 1 )
where N represents t he number of atoms and dmin t he minimal distance between any t wo atoms. Further initi al structures were constructed from global minima Lennard Jones structures with adapted bond lengt hs according to t he cluster type in question. Moreover, great effort has been made to ext ract t he predicted lowest-lying minima of t he clusters in question from various publications. These empirical structures represent t he final part of the initial population. All structures in t he initial population are then relaxed into t heir nearest local minimum using DFT and t he Los Alamos minimum basis set and corresponding shape-consistent scalar-relativistic pseudopotential ( L ANL2MB ) . I n t he algorit hm t he fit ness of each minimum structure i s determined based on its energy Vi (total electronic energy plus nuclear-nuclear repulsion energy) . The structure with t he lowest energy in a populat ion is assigned a fit ness of 1 and t hat with t he highest energy a fi tness of O. The fitness fi of each local minimum of the evolving population is calculated using a dynamically scaled potential energy Pi,
(3.2)
where Vmax and Vmin are the highest and lowest energies in the current population, re spectively. Using t he roulette wheel select ion method [39] two members of t he current p opulat ion are chosen for m at ing (i.e. to be p arents) according to t heir fitness. B E L P H E G Ol{. uses an exponential type fitness function, thus favoring t he choice of parents towards
CHAPTER 3. GLOBAL
42
I nitial Population Local M i n i m ization
Assi g n Fitness
Sel ectio n for Mating
M ating
-
-
Local M i n i m ization OFT ( LAN L2 M B ) No Term i n ation Criteria M et?I
M utation - ---1 Fi nal Population Isomers M i n i m ization OFT ( LANL2 0Z) Freq uency Analysis, - - - _ . ' T Lowest-lying isomers Fi nal Min i mization O FT ( La rg e Basis) Frequency Analysis
Figure 3. 1 : Flow chart for t he genet ic algorithm program B E L P H E G O R used in t he search
CHA PTER 3. G L OBA L OP TTi\l fTZA TTOJ\ t hose clusters wit h low energy.
f. t -- -ap, - e-3p, -
43
(3.3) In t he process of mat ing, t he chosen parent struct ures are first aligned according to t heir main axis of inertia, then bot h are rotated randomly about t he same angle and finally cut horizont ally through t heir ma centers into halves as depicted in figure (3.2) . Complementary halves are t hen merged to form an offspring, ensuring t hat the number of at oms remains constant and that t he lengths of the newly formed bonds are similar to t he bond lengt hs in the parent struct ures. This operation is a variant of the so called 'cut and splice' crossover operat ion introd uced by Deaven
et al.
[38] . The offspring is then relaxed into its local minimum . Th struct ural rearrangements during the minimization step are greatest in the region of the splice between t he fragment 'inherited' by its parents and it is ensured t hat the minimal(dmin)
and maximal(dmax)
bond lengt hs are not exceeded .. . ' .' . '. '
Figure 3.2: Pictorial repre entat ion of the cut and plice method. Two struct ures, here two AUlO isomers, are selected for mating according to their fit ness, rot ated randomly about the same angle and cut horizontally through t heir mass cen ters. Complementary halves are t hen merged, employing a random dihedral angle, to form an offspring.
CHAPTER 3. GLOBAL OPTIMIZATION 44