Convergence Theory for Preconditioned Eigenvalue Solvers in a Nutshell

Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using multilevel preconditioning for elliptic differential operators or graph Laplacian eigenvalue problems, exhibit almost optimal complexity in practice; i.e., their computational costs to calculate a fixed number of eigenvalues and eigenvectors grow linearly with the matrix problem size. Theoretical justification of their optimality requires convergence rate bounds that do not deteriorate with the increase of the problem size. Such bounds were pioneered by E. D’yakonov over three decades ago, but to date only a handful have been derived, mostly for symmetric eigenvalue problems. Just a few of known bounds are sharp. One of them is proved in doi: 10.1016/S0024-3795(01)00461-X for the simplest preconditioned eigensolver with a fixed step size. The original proof has been greatly simplified and shortened in doi: 10.1137/080727567 by using a gradient flow integration approach. In the present work, we give an even more succinct proof, using novel ideas based on Karush–Kuhn–Tucker theory and nonlinear programming.     

Estimation of the contributions of absorption bands for overtones and composite frequencies in infrared spectra of brominated and oxygen-containing organic compounds

Intensities of fundamental, overtone, and composite absorption bands for 27 brominated hydrocarbons and 20 oxygen-containing organic compounds are calculated in an anharmonic approach. The first and second derivatives of the electric dipole moment of the molecule with respect to normal coordinates are determined using ab initio quantum-chemical MP2/6-31G(1d) calculations. For the studied compounds, the average contributions of overtones and composite frequencies to absorption in the region 100–4000 cm⁻¹ is 4.8% for brominated hydrocarbons and 3.2% for oxygen-containing compounds. The major part of the contribution of overtones and composite frequencies falls into the regions (mainly from 1600 to 2800 cm⁻¹) where fundamental transitions are observed rarely. The calculation performed well describe the positions of maxima and the intensities of fundamental, overtone, and composite absorption bands and can be used for the standardless spectrochemical analysis of compounds by their overtone spectra.     

Simulation of photochemical reactions of butadiene and calculation of the quantum yields

The photochemical reactions of butadiene-1,3 have been simulated. The calculated quantum yields of the products quantitatively agree with experimental data (deviation from the experimental data in on average less than 30%). The efficacy of the method in predicting photochemical processes of various types (isomerization, decomposition) occurring simultaneously via different pathways has been shown. It has been confirmed that the quantum beat frequency of resonating states can be used as a parameter of the method and, as such, allows obtainable quantitative estimates to be refined.     

Conference on the theory of the structure and spectra of complex molecules

Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, p. 203, March–April, 1995.     

This virtual molecular world

The existing rules and computing techniques are of little use for directed research dedicated to the real molecular world whose objects are enormous in number and diversity. Any progress is possible only through introducing forward-looking numerical computer experiments into the research practice. This requires changing from analysis of single objects and calculation of individual characteristics and properties to designing a virtual molecular world. Its objects, unlimited in number, can, similarly to real objects, respond to external actions and differ from one another. At the same time, by contrast to real objects, they can grow in complexity through interactions, accumulate energy, transform the initial information, etc. In this context, the prospects of designing such a world were discussed, as well as the methodological requirements on problem statement and solving as regards the theoretical and program construction aspects.