arXiv subjects
Publications and source records attributed to Herbert H. H. Homeier.
It is described how the Hermite-Padé 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-Padé polynomials. A recursive algorithm is derived and some numerical examples are given.
We discuss certain special cases of algebraic approximants that are given as zeroes of so-called "effective characteristic polynomials" and their generalization to a multiseries setting. These approximants are useful for the convergence acceleration or summation of quantum mechanical perturbation series. Examples will be given and some properties will be discussed.
Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $ω_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $ω_n$. Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.
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.
Estimators $Πn$ for the correlation energy can be computed as roots of effective characteristic polynomials of degree $n$. The coefficients of these polynomials are derived from the terms of the perturbation series of the energy. From a fourth-order Møller-Plesset (MP4) calculation one can calculate with negligible effort a size-extensive estimator $Π2$ that is in many cases much closer to the full CI correlation energy of the ground state than the MP4 value. [H.H.H. Homeier, J. Mol. Struct. (Theochem) 366, 161 (1996)] Here, we prove that the estimators $Πn$ for $n>2$ are size-extensive if they are calculated from the MP series.
Some methods for the convergence acceleration of the Møller-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{é} 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{é} approximants, and also the effective second order characteristic polynomial, are excellent tools for correlation energy estimation.
The solution of the Ornstein-Zernike equation with various closure approximations is studied. This problem is rewritten as an integral equation that can be solved iteratively on a grid. The convergence of the fixed point iterations is relatively slow. We consider transformations of the sequence of solution vectors using non-linear sequence transformations, so-called vector extrapolation processes. An example is the vector J transformation. The transformed vector sequences turn out to converge considerably faster than the original sequences.