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Exploring chemical space with discrete, gradient, and hybrid optimization methods
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10.1063/1.2987711
/content/aip/journal/jcp/129/17/10.1063/1.2987711
http://aip.metastore.ingenta.com/content/aip/journal/jcp/129/17/10.1063/1.2987711

Figures

Image of FIG. 1.
FIG. 1.

Flow chart for BFS method. The example assumes a system with site-specific libraries. The libraries (A1, A2, A3), (B1, B2, B3), and (C1, C2, C3) are specific to the sites 1, 2, and, 3, respectively. In the example, the variable sites of a molecule are represented by circles and the searched molecular configuration is indicated in the brackets. The selected molecule is represented with an asterisk. The dotted arrow on a site indicates that the configuration is not changed during the search.

Image of FIG. 2.
FIG. 2.

The computational costs for optimizing the binary substitution problem as a function of size. Results are shown for enumeration, BB, LCAP with analytical derivatives, and LCAP with the numerical derivatives (see the text for details).

Image of FIG. 3.
FIG. 3.

Molecular frameworks used for the optimization. The variable sites are numbered for each framework.

Image of FIG. 4.
FIG. 4.

BFS optimization of the polyene framework hyperpolarizability as a function of iteration number with two random initial configurations using the 13 atom library.

Image of FIG. 5.
FIG. 5.

Performance of discrete, gradient, and hybrid methods in optimizing the static first hyperpolarizability of polyene, tetracene, and coronene frameworks using the five atom library. From top to bottom, the size of the molecule increases. The hyperpolarizability predicted by a method is scaled to the highest value and given as a percentage. The symbols G1, G2, and G3 on the -axis represent the gradient methods GDMC, LCAP-J, and LCAP-SJ, respectively. H represents a hybrid method . The symbols B1, B2, B3, B4, and B5 represent the BB methods DEE, BFS, BFS-NR, BFS-HR, and BFS-C, respectively. GA represents the ECGA method.

Image of FIG. 6.
FIG. 6.

The gradients and the property values as a function of LCAP coefficient on a polyene site. A LCAP atom is defined by X with or O or P or S.

Image of FIG. 7.
FIG. 7.

Performance of discrete methods in optimizing the static first hyperpolarizability of polyene, tetracene, and coronene frameworks using a 13 atom library. The framework, number of variable sites, and the library size are given on each plot. From top to bottom, the size of the molecule increases. The hyperpolarizability predicted by a method is scaled to the highest value and given as a percentage. The symbols B1, B2, B3, B4, and B5 represent the BB methods DEE, BFS, BFS-NR, BFS-HR, and BFS-C, respectively. GA represents the ECGA method.

Image of FIG. 8.
FIG. 8.

Performance of the methods with reduced computational cost in optimizing for the polyene framework with 13 atoms. The optimized values for each method were scaled to the maximum value. The symbols B2, B3, B4, and B5 on the -axis represent the BFS, BFS-NR, BFS-HR, and BFS-C methods, respectively.

Image of FIG. 9.
FIG. 9.

Flow chart for the BFS-NR (top) and BFS-HR (bottom) methods with an example system with site-specific libraries. The libraries (A1, A2, A3), (B1, B2, B3), and (C1, C2, C3) are specific to the sites. The variable sites are represented by circles, the searched molecular configuration is indicated in the brackets, and the selected molecule is represented with an asterisk. The symbol R on a site represents the random assignment and the dotted arrow on a site indicates that the configuration is not changed during the search.

Image of FIG. 10.
FIG. 10.

Flow chart and example of BFS-C method for a nine site (S1-S9) structure. Three sites are formed into a cluster indicated by the ellipse. Each cluster is optimized using BFS on site basis as shown by arrows. Once a cluster is optimized, the search continues to the next nearest cluster. The steps are iterated to convergence.

Image of FIG. 11.
FIG. 11.

Flow chart for the DEE method (top) and library reduction (bottom). For a random seed configuration, BFS arranges the library (L1-Lm) on all sites (S1-Sn) by descending property value. Defining a “cut” value, the library size is reduced on each site. With the reduced libraries, an extensive search is then performed.

Image of FIG. 12.
FIG. 12.

Flow chart for the ECGA method. In addition to the genetic operations, the selection and crossover moves, ECGA uses the marginal product model (MPM) (Refs. 42 and 43). In the first step of the ECGA, process an initial random population is generated and finesses are evaluated. Some of the individuals are selected from the population based on their fitness. The selected individuals may have blocks of variables that are relevant for the global solution. Therefore, the essential blocks need to be identified and preserved in the population evolution. The essential blocks are identified based on the dependencies among the variables in the selected individuals. A probabilistic model is built from evaluation of the dependencies among the variables. The MPM is a probabilistic model that is built by evaluating higher-order correlations among the variables in the selected individuals.

Image of FIG. 13.
FIG. 13.

Flow chart for the hybrid method combining the LCAP-J and BFS. A random seed molecule begins the optimization based on a first phase of LCAP-J optimization followed by BFS optimization.

Tables

Generic image for table
Table I.

Library of five atom types with the same number of electrons.

Generic image for table
Table II.

Library of 13 atom types.

Generic image for table
Table III.

ECGA optimization details for the frameworks using two libraries. Each framework is optimized twice using the two different libraries. The population size (Refs. 28 and 29) is 4000 for all of the frameworks.

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/content/aip/journal/jcp/129/17/10.1063/1.2987711
2008-11-06
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Exploring chemical space with discrete, gradient, and hybrid optimization methods
http://aip.metastore.ingenta.com/content/aip/journal/jcp/129/17/10.1063/1.2987711
10.1063/1.2987711
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