Correlation Energy Estimators based on M{\o}ller-Plesset Perturbation Theory

Some methods for the convergence acceleration of the M{\o}ller-Plesset perturbation series for the correlation energy are discussed. The order-by-order summation is less effective than the Feenberg series. The latter is obtained by renormalizing the unperturbed Hamilton operator by a constant factor that is optimized for the third order energy. In the fifth order case, the Feenberg series can be improved by order-dependent optimization of the parameter. Alternatively, one may use Pad{\'e} approximants or a further method based on effective characteristic polynomials to accelerate the convergence of the perturbation series. Numerical evidence is presented that, besides the Feenberg-type approaches, suitable Pad{\'e} approximants, and also the effective second order characteristic polynomial, are excellent tools for correlation energy estimation.Comment: 21 pages, 87 references, LaTeX2e, elsart.cls, uuencoded compressed tar file created by csh script uufiles. J. Mol. Struct. (THEOCHEM), in press. Postscript and dvi also available via http://www.chemie.uni-regensburg.de/preprint.html and ftp://rchs1.uni-regensburg.de/pub/preprint/TC-QM-96-

Series Prediction based on Algebraic Approximants

It is described how the Hermite-Pad\'e polynomials corresponding to an algebraic approximant for a power series may be used to predict coefficients of the power series that have not been used to compute the Hermite-Pad\'e polynomials. A recursive algorithm is derived and some numerical examples are given.Comment: 9 pages, ISRN Applied Mathematics, in pres

Spin-Lattice Relaxation in Metal-Organic Platinum(II) Complexes

The dynamics of spin-lattice relaxation (slr) of metal-organic Pt(II) compounds is studied. Often, such systems are characterized by pronounced zero-field splittings (zfs) of the lowest-lying triplets. Previous expressions for the Orbach slr process do not allow to treat such splitting patterns properly. We discuss the behavior of a modified Orbach expression for a model system and present results of a fit of the temperature dependence of the spin-lattice relaxation rate of Pt(2-thpy)$_2$ based on the modified expression.Comment: 9 pages, 3 figures (made from 4 .eps files), elsart.cls. Using dvips (dvipsk 5.58f), it may be necessary to manually edit the generated file letter.ps to change in the first line from PS-Adobe-2.0 to PS-Adobe-3.0. Chemical Physics Letters, in pres

An Asymptotically Hierarchy-consistent, Iterative Sequence Transformation for Convergence Acceleration of Fourier Series

Introduction Slow convergence is a ubiquitous problem in numerical mathematics. Therefore, methods for the acceleration of convergence, for extrapolation to the limit, and also for the summation of divergent series are a promising and rapidly developing field. For detailed information, we mention especially the books of Brezinski and Redivo Zaglia [13] and Wimp [52] and also the work of Weniger Technical Report TC-NA-97-2, Institut fur Physikalische und Theoretische Chemie, Universitat Regensburg, 1997, Numer. Algo., in press Homepage: http://www.chemie.uni-regensburg.de/hoh05008 [46, 48--50], but also the books of Baker [1], Baker and Graves-Morris [2], Brezinski [8--12], Graves-Morris [15, 16], Graves-Morris, Saff und Varga [17], Khovanskii [30], Lorentzen and Waadeland [34], Nikishin and Sorokin [35], Petrushev and Popov [37], Ross [38], Saff and Varga [39], Wall [45], Werner and Buenger [51] and Wuytack [53]. Trigonometric Fourier series with partial sums s n = a 0

On Properties and the Application of Levin-type Sequence Transformations for the Convergence Acceleration of Fourier Series

We discuss Levin-type sequence transformations fsng ! fs0 ng that depend linearly on the sequence elements sn, and nonlinearly on an auxiliary sequence of remainder estimates f!ng. If the remainder estimates also depend on the sequence elements, non-linear transformations are obtained. The application of such transformations very often yields new sequences that are more rapidly convergent in the case of linearly and logarithmically convergent sequences. Also, divergent power series can often be summed, i.e., transformed to convergent sequences, by such transformations. The case of slowly convergent Fourier series is more di�cult and many known sequence transformations are not able to accelerate the convergence of Fourier series due to the more complicated sign pattern of the terms of the series in comparison to power series. In the present work, the Levin-type H transformation [H.H.H. Homeier, A Levin{type algo-rithm for accelerating the convergence of Fourier series, Numer. Algo. 3 (1992) 245{254] is studied that involves a frequency parameter �. In particular, properties of the H transformation are derived, and its implementation is discussed. We also present some generalization of it to the case of several frequency parameters. Finally, it is shown how to use the H transformation properly in the vicinity of singularities of the Fourier series