Extrapolation Methods for Improving the Convergence of Oligomer Calculations to the Infinite Chain Limit of Quasi-Onedimensional Stereoregular Polymers

TL;DR

This work addresses the challenge of modeling infinite chain properties of quasi-onedimensional stereoregular polymers by extrapolating finite oligomer calculations. It advocates using sequence transformations, grounded in the Euler-Maclaurin framework, to accelerate convergence of properties such as energies to their infinite-chain limits. Through a polyacetylene HF example, the authors demonstrate that energy differences converge exponentially and that Wynn's epsilon algorithm effectively extrapolates to the infinite limit, while the average energy contains a dominant 1/N term that can be removed by forming differences. The results provide a practical path to accurate infinite-chain predictions with modest computational effort, enabling broader application to nonlinear optical properties and electronic transitions in these systems.

Abstract

Quasi-onedimensional stereoregular polymers as for example polyacetylene are currently of considerable interest. There are basically two different approaches for doing electronic structure calculations: One method is essentially based on concepts of solid state theory. The other method is essentially a quantum chemical method since it approximates the polymer by oligomers consisting of a finite number of monomer units. In this way, the highly developed technology of quantum chemical molecular programs can be used. Unfortunately, oligomers of finite size are not necessarily able to model those features of a polymer which crucially depend of its in principle infinite extension. In such a case extrapolation techniques can be extremely helpful. For example, one can perform electronic structure calculations for a sequence of oligomers with an increasing number of monomer units. In the next step, one then can try to determine the limit of this sequence for an oligomer of infinite length with the help of suitable extrapolation methods. Several different extrapolation methods are discussed which are able to accomplish an extrapolation of energies and properties of oligomers to the infinite chain limit. Calculations for the ground state energy of polyacetylene are presented which demonstrate the practical usefulness of extrapolation methods.