General subroutines for the calculation of optical matrix elements
V.A. Novikov* and A.V. Novikau
This is an abstract
for a presentation given at the
Foresight Conference on Molecular Nanotechnology.
There will be a link from here to the full article when it is
available on the web.
It is said about a universal calculation methodic of a calculation any kind of two-center matrix elements. The method was supposed first in . It does not use analytic procedures for changing initial orbitals to a typesetting of "useful" basis functions. That allows to unify a procedure of calculations and to control an accuracy reliable. This methodic in an algorithm realization is quicker than quickest analytic methods. And in addition, the analytic methods can not control the accuracy of calculation so effective.
A question of the accuracy's and the calculation speed's correlation was discussed in  by a calculation of electronic properties for silicon-bases structures. We began to model optical properties after our  publication and found out what extend of a correlation between matrix elements create a great influence to an accuracy of farther calculations. Errors of the matrix elements of methods using basis functions are great correlation. And because it, though an appearance is a rite result, own values of the getting matrix and the own values of a real matrix are great different. The errors of the matrix elements of our calculation method are not correlation almost. And therefore the own values of matrix do not have a visual trapezium dependent. This influence of the error looks like an influence of casual fluctuations to stability's curve of a great collective system. In an accordance of a synergetic theory , the influence of casual fluctuations to quality characteristics of the great collective system has a non-linear characteristic which is described a three extend polinom as minimum. A like effect was observing and in  where an energy gap is calculated without an essential error, if an assign relative error was not more then 0.05. Then we had an abrupt increase of the error of an unsteady character. A computer can not calculate even the own values of the matrix, if the relative error is about 0.5. We did not use known regularization methods  for a stabilization of the solve deliberative.
An optical matrix contains more then in 2.5 no-zero elements then the usual matrix. All additional elements are not diagonal and therefore the question of the correlation between the calculation accuracy and the errors is very impotent and actual. It is better to do not have a result at all, then to have knowingly a not control in accuracy result. It was found out, a limit of the unsteadying of the solve for optical matrix elements without using the regularization methods is not smaller then the "saveful" error about 10E-3.
We maid also a numerical experiment to get the solve with the regularization method (one of they is a method of a Tikhonov weak regularization with a small parameter Alpha). An increase of the Alpha parameter rubs off a partition between an area of an exact and an area of a false result really. And this confirms our hypothesis about the strong correlation of the matrix elements (with using a not big count of the basis functions of a sleather type for example). This correlation of the matrix elements is defined a difference between a class of a smoothness of a real orbitals function and a class of the smoothness of a basis functions combination. The differences disappear with an increase of the basis function's count slowly, and a time of the calculation is increased very quicker.
A separate attention in a realization of the algorithm's calculation of the optical matrix elements was directed to an elimination of a casual calculation errors. It is used a more exact procedure of a numerical integration with a splain interpolation of orbitals in this direction.
A time of the calculation of the optical matrix elements is more in 4 then time of a calculation of  ( is one of the best methods of the calculation of a usual matrix elements). And it is not able to modify  for a calculation of the optical elements.
The all calculations was maid at Fortran with a using of a double- precision arithmetic.
- Novikov V.A., Kholod A.N., Novikau A.V., Filinov A.V. Physics, Chemistry and Application of Nanostructures: Review and Short Notes to Nanomiting '97 (International Conference Nanomiting '97, Belarus, Minsk, 1997), pp. 123-131.
- Haken. H., Synergetic, 1980.
- Tikhonov A.N., Reports Of The USSR Science Academy, 151, 3, 1963, p. 501-505.
- W. Hierse, P.M. Oppeneer, Int. J. Quant. Chem. 52, 1249 (1994).
dr Novikov V.A.
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Tel: (RB, Minsk) 285-88-19