MGAC (Modified Genetic Algorithm for Crystals and Clusters); blind test in molecular crystals

VÍCTOR BAZTERRA; MARTA FERRARO; JULIO C. FACELLI

Isla MArgarita, Venezuela

Congreso; Congreso internacional de Químicos teóricos de expresión latina; 2005

IVIC, Venezuela

Genetic Algorithms (GAs) are a family of search techniques rooted on the ideas of Darwinian evolution.1 Operators analogue to crossover, mutation and natural selection are employed to perform a search able to explore and learn the multidimensional parameter space and determine which regions of the space provide good solutions to a problem. To improve their convergence GAs are commonly coupled with local optimisations at each generation. This combined approach has been implemented in our program MGAC (Modified Genetic Algorithm for Crystal and Clusters).2 Crystal structure coding             When using GAs for the prediction of crystal structures, these structures have to be encoded in a “genome” that can be manipulated by the genetic operators as well as used to calculate the energy of the crystal. For organic crystals the molecular geometries are highly constrained by its strong covalent bonds, this allows a reduction of the number of parameters for which a global search is necessary. It is possible to consider that all the internal bond lengths and bond angles are fixed during the GA search, performing only local optimisations of these parameters. Because the rotational barriers around single covalent bonds are comparable to the intermolecular energies, these dihedral angles should be included both in the GA driven global and local optimisations. Therefore, for the n-molecules in the asymmetric cell the MGAC uses their molecule center of mass , its orientation  and the values of the relevant dihedral angles  as the parameters, describing the molecular coordinates, included in the global optimisations. In addition it is necessary to specify the space group and lattice parameters to define a crystal. In the MGAC program the space group is fixed and only the lattice angles are considered as independent parameters in the GA optimisation. Given the molecular coordinates and the lattice angles  that define the crystal system, the molecules are translated using the space group operations and the values of  chosen to define the smaller asymmetric cell that encloses the set n-molecules. To reduce the possibility of producing crystals with very short inter-molecular distances, the asymmetric cell is 8% larger than smallest possible cell. Once a crystal has been built using this procedure, its structure undergoes a full local minimization employing all the inter- and intra- molecular parameters. This last procedure allows that the effects of molecular interactions be included in the local refinement of the entire crystal structure.             The MGA scheme for molecular cluster optimisation previously proposed by Niesse, Mayne, and White3 for clusters was applied in this work.  This scheme uses one-point-crossover, two-point-crossover, N-point-crossover, uniform-crossover, arithmetic-crossover, inversion-crossover, geometric-crossover and Gaussian mutation. These operators were implemented in MGAC under the condition that they only act on the molecular coordinates. For  only the arithmetic-crossover, geometric-crossover and Gaussian mutation operators are used, because they are the only ones that preserve the crystal system, i.e. they transform a triclinic crystal into another triclinic crystal, monoclinic into monoclinic, and so on. During the evolution, when these operators are used to construct new candidates for crystal structures, a new set of  are chosen to fit the new structures in their corresponding asymmetric cells. This is followed by the verification that  effectively define a set of linear independent lattice vectors and that the resulting crystal energies are under a certain energy cut-off limit to avoid structures with unphysical intermolecular distances. Those structures that do not fulfil these conditions are eliminated and replaced by entire new randomly chosen ones. After each GA evolution all the new structures are relaxed by a local optimisation within the desired space group, creating a new set of candidate solutions that compete with the existing solutions in the population. This competition is implemented by combining the new solutions with the old solutions in a larger population from which the worst ones are eliminated until the number of candidates equals the desired population size. This procedure is repeated and ends when the diversity of the population reaches the desired threshold or a predetermined number of generations have been completed. The MGAC calculations were done using 30 individuals, evolving for 60 generations with a mutation probability of 0.1 and a replacement factor of 0.5 for a set of I, II and III, which crystal structure is considered unknown. Calculations were performed twice for each of the most common fourteen space groups: P1, P-1, P21, C2, Pc, Cc, P21/c, C2/c, P212121, Pca21, Pna21, Pbcn, Pbca and Pnma with one and two molecules per asymmetric unit cell, respectively.             MGAC uses the CHARMM program 9, 10 for the energy evaluation and local optimisation of the crystal structures. Within the CHARMM program the GAFF (Generic Amber Force Field)11 was used in all the calculations with atomic charges determined with the RESP (Restrained Electrostatic Potential)12 method using the Gaussian program13 with the B3LYP exchange correlation functional and 6-31G* basis sets. References 1-      D. E. Goldberg, Genetic Algorithms in Search, Optimisation and Machine Learning (Addison-Wesley, New York, 1989). . L. Johnson and C. Robers, in Soft Computing Approaches in Chemistry, edited by H. M. Cartwright and L. M. Sztandera (Springer-Verlag, Heidelberg, 2003), Vol. 120, p. 161. 2-      V. E. Bazterra, M. B. Ferraro, and J. C. Facelli, J. Chem. Phys. 116, 5984 (2002); 116, 5992 (2002); Int. J.  Quantum Chem. 96, 312 (2004). 3-      J. A. Niesse and H. R. Mayne, Journal of Computational Chemistry 18, 1233 (1996).               R. P. White, J. A. Niesse, and H. R. Mayne, Journal of Chemical Physics 108, 2208 (1998). A. D. MacKerell, J. Brooks, B. , C. L. Brooks III, et al., in The Encyclopedia of Computational Chemistry, edited by P. v. R. S. e. al. (John Wiley & Sons, Chichester, 1998), p. 271.;B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, et al., J. Comp. Chem. 4, 187 (1983).