Genetic algorithms as global optimizers
1.
Automated assignment of rovibronic spectra using GA
The task of performing a fit of the molecular Hamiltonian parameters to
an experimental spectrum
like the one shown below (actually, this is a rovibronic spectrum of
seven different tryptamine isotopomers) using
line position assigned fits can
be tedious or in some cases even impossible.
Details can be found in [1]. The
non-assigned fit of such a spectrum can be viewed as a classical
optimization process on a multidimensional parameter
surface. Genetic algorithms (GA), invented by J. Holland, are
optimizers [2], which have the advantage to end mostly in the global
minimum.
The GA mimick thereby a scheme traced out by nature:
- Representation of the parameters: The molecular
parameters are encoded as binary or as real data type, each parameter
representing a gene. A vector of all genes, which contains all
molecular parameters is called a chromosome. In an initial step
the values for all parameters are set to random values between lower
and upper limits which have to be chosen by the user. No prior
knowledge of the parameters is necessary. In our calculations in
general a total of 300 - 500 chromosomes are randomly generated,
forming a population.
- The solutions (chromosomes) are evaluated by a fitness
function (or cost function), which is a measure for the quality of the
individual solution.
- One optimization cycle, including evaluation of the cost
of all chromosomes is called a generation. Generally convergence
of the fit in our case is reached after 300 - 500 generations.
- Pairs of chromosomes are selected for reproduction and
their information is combined via a crossover process. This crossover
might take place as a one-point, two-point or uniform crossover. As
crossover just combines information from the parent generations. It
basically explores the error
landscape.
The performance of the GA depends on internal parameters like
mutation rate, elitism, crossover probability and population size,
which therefore should also be optimized for a given problem.
Fortunately this meta-optimization results in similar parameters for
quite different problems of optimization. The meta-optimization for
some of the parameters is described in ref. [3].
[1] Schmitt,
M., Böhm, M.,
Ratzer, C., Vu, C., Kalkman, I. and Meerts, W. L.: Structural selection
by microsolvation: conformational locking of tryptamine. J. Am. Chem.
Soc. 127 (2005), 10356).
[2] J. Holland: Adaption in Natural and Artificial Systems, MIT
Press (1994)
[3] Meerts, W. L. and Schmitt, M.:
A new automated assign and analyzing method for high resolution
rotational resolved spectra using Genetic Algorithms. Phys. Scripta 73
(2005), C47
This project is performed in collaboration with Leo
Meerts (University of Nijmegen). A nice slide show, describing the
GA can be found here.
2. Determination of molecular stuctural
parameters from rotational constants using GA
The rotational constants of a molecule can be determined from
several spectroscopic techniques that provide rotational resolution.
They are inversely proportional to the moments of inertia, which are
defined as:
The determination of the structure of a molecule from the
rotational constants of several isotopomers is a straightforward
procedure, which has been routinely used in microwave spectroscopy for
decades. The basic equations have been worked out by Kraitchman [1] and
Costain [2] and are nowadays standard textbook knowledge [3].
Basically, each atom in the moelcule has succesively to be replaced by
a (stable) isotope. Application of the Kraitchman equations using the
moments of inertia of the parent molecule with the normal isotopes and
of the singly substituted molecule yields the cartesian coordinates of
this atom in the inertial system of the parent molecule.
