A. G. Yagola

Joint treatment of ab initio and experimental data in molecular force field calculations with Tikhonov’s method of regularization

We describe a novel method for employing calculated ab initio potential data together with Tikhonov’s variational procedure to extract fundamental molecular force field parameters from experimental spectral data, the formal ‘‘inverse problem’’ of vibrational spectroscopy. In this approach, the ab initio quantities serve to ‘‘regularize’’ the initially ill‐posed problem (in the sense of Tikhonov), leading to variationally stable and unique force field parameters that optimally mimic overall patterns of the (approximate) ab initio quantities, but exactly reproduce the available experimental data within specified experimental precision. In this manner, ab initio and experimental data can be jointly combined to produce more stable and reliable force fields (improvable to any degree through higher level ab initio treatment, additional experimental data, etc.) than could be attained by theoretical or experimental methods alone. The proposed procedure allows use of any system of generalized coordinates, includin...

The equilibrium structure of thiophene by the combined use of electron diffraction, vibrational spectroscopy and microwave spectroscopy guided by theoretical calculations

Abstract The equilibrium molecular geometry of thiophene has been determined from a combination of gas-phase electron diffraction, vibrational and microwave data and ab initio and DFT calculations. The quadratic and cubic force constants of thiophene calculated theoretically and empirically improved by harmonic scale factors were incorporated in the analysis in which equilibrium distances and harmonic scale factors were refined simultaneously. The diffraction intensities were calculated by the use of first-order perturbation theory. The commonly used r a distances and amplitudes of vibration were also estimated and found to agree reasonably well with those from an earlier investigation. Anharmonic phase shift parameters for all atom pairs and the various distance correction terms are presented.

Regularizing algorithm for determination of equilibrium geometry and harmonic force field of free molecules from joint use of electron diffraction, vibrational spectroscopy and ab initio data with application to benzene

Abstract A novel integrated algorithm is suggested for joint treatment of gas-phase electron diffraction and spectroscopic data. This algorithm develops the idea of the regularized quantum mechanical force field approach based on the theory of nonlinear illposed problems. The main advantage of the algorithm is that it provides a unique and stable solution for the equilibrium geometry and intramolecular harmonic force field of quasi-rigid systems. The check calculations were carried out with Oslo intensity data on benzene collected with improved precision. Infrared frequencies of benzene and its isotopomer were taken from the literature.

Data Errors and an Error Estimation for Ill-Posed Problems

In this paper we shall discuss the problem how to use a priori information for constructing regularizing algorithms and error estimation while solving ill-posed problems. We shall consider the following types of a priori information: (1) a compactness of a set of solutions; (2) a sourcewise representation of a solution with a compact operator.

An analytical formula for ring artefact suppression in X-ray tomography

A method of ring artefact suppression in X-ray computerised tomography (CT) reconstructions is proposed. The method is based on the assumption that a sinogram is a smooth function along the horizontal spatial coordinate. Methods based on the theory of ill-posed problems are applied to find a regularised solution. An analytical formula for the solution is proposed allowing fast ring artefact suppression. Its performance is demonstrated for parallel beam synchrotron X-ray tomography.

Image reconstruction technique and optical monitoring of the QSO2237+0305 from Maidanak Observatory in 2002-2003

We have observed the gravitational lens system Q2237+0305 from the Maidanak Observatory over the period from 2002 August to 2003 November. Here we report the results of our observations. We implemented a two-stage technique that has been developed specifically for the purpose of gravitational lens image reconstruction. The technique is based on the Tikhonov regularization approach and allows one to obtain astrometric and photometric characteristics of the gravitational lens system. Light curves with 78 data points for the four quasar components are obtained. Slow brightness variations over the observational period are found in all components. Images A, C and D have a tendency to decrease in brightness. Image B does not vary more than 0.05 mag. The observations did not reveal evidence for large variations in brightness of the components due to microlensing effects. To provide an overall picture of the photometry behaviour, our data are combined with the Maidanak observations published for 1995-2000.

Extension of a regularizing algorithm for the determination of equilibrium geometry and force field of free molecules from joint use of electron diffraction, molecular spectroscopy and ab initio data on systems with large-amplitude oscillatory motion

Abstract The previously developed integrated algorithm for the joint treatment of gas-phase electron diffraction and vibrational spectroscopic data is extended to include systems with large-amplitude oscillatory motion. In addition, the treatment is augmented by the inclusion of microwave rotational constants. As in the previous work, the analysis of data from experimental sources is guided by quantum mechanical molecular geometry and force field optimization results. The computed force field matrix can be corrected empirically with the aid of suitable scale factors. Centrifugal distortion corrections to interatomic distances are included. The standard deviations of the parameters determined and the correlation coefficients can now be estimated. The principal design of the developed computer program is outlined, and some methodological problems associated with diffraction analysis of molecules with large-amplitude motion are discussed. To provide an example of a problem susceptible to attack by the present method an account is made of the re-analysis of diffraction data for 4-fluorobenzaldehyde collected earlier on the Balzers apparatus in Oslo.

The use of ab initio anharmonic force fields in experimental studies of equilibrium molecular geometry

Abstract The use of ab initio methods has been investigated for obtaining physically meaningful anharmonic force fields applicable in structure analysis of molecules by electron diffraction. The quadratic and cubic force constants for the sample molecule SF 6 chosen as a suitable test case were theoretically estimated and improved by an empirical scaling based on a quadratic force constant scale factors. It was confirmed that if theoretical calculations are made with well selected basis sets the accuracy of the individual values of the computed cubic constants established by reference to precise spectroscopic data is practically sufficient to experimentally determine the accurate equilibrium S–F distance and to theoretically estimate the amplitudes of vibration and the phase shift parameters for all internuclear distances. Calculations based on a Morse-like anharmonic model function were also performed for comparison. The present calculations show that determination of an accurate equilibrium molecular structure by electron diffraction is possible through the appropriate combination of experimental and theoretical data. The best equilibrium geometry results, if empirically scaled ab initio quadratic and cubic force constants are used in a regularizing algorithm developed earlier for the effective interaction of electron diffraction with vibrational and microwave spectroscopy techniques.

New approaches to error estimation to ill-posed problems with applications to inverse problems of heat conductivity

Abstract - In this paper several new approaches for error estimation of an approximate solution of linear ill-posed problems are offered in the presence of the additional a priori information about the exact solution. This information is used when the exact solution belongs to a compact set or is sourcewise represented. All considered problems are reduced to convex programming problems in finite dimensional Euclidean spaces. The method to cut convex polyhedrons is applied to solve these problems and to construct areas, which the exact solutions belong to. Several model inverse problems for the heat conduction equation are considered.

Stable numerical methods of solving certain inverse problems of vibrational spectroscopy

Abstract Stable numerical methods of constructing the force fields of multi-atomic molecules from experimental data based on Tikhonovs method of regularization of ill-posed problems are proposed.