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Family of phase fitted 3-step second-order BDF methods for solving periodic and orbital quantum chemistry problems

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Abstract

In this paper, we introduce a novel series of second-order Backward Differentiation Formulas (BDFs) specifically designed to address phase-lag and its first derivative in the numerical resolution of Initial Value Problems (IVPs) with orbital solutions. Our methodology commences with an in-depth analysis of phase-lag phenomena associated with second-order BDFs. Following this, we construct a suite of equations by embedding algebraic functions into the operational framework of the 3-step second-order BDF (SOBDF) method. Additionally, we elaborate on equations that precisely describe the phase-lag and its derivatives, with a concentrated focus on the 3-step SOBDF method. The culmination of this work is the presentation of six distinct methods, each methodically crafted to negate both the real and imaginary elements of phase-lag and its derivatives in numerical computations. The study advances with a meticulous examination of the local truncation error and the stability regions pertinent to the six phase-fitted methods introduced. Furthermore, we scrutinize their computational performance by deploying these methods across a spectrum of initial value problems, offering valuable insights into their effectiveness in varying contexts.

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Acknowledgements

The authors wish to thank the anonymous referees for the comments that greatly improved the manuscript.

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Author notes

  1. Sanaz Hami Hassan Kiyadeh and Ali Safaie contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, The University of Tabriz, Tabriz, Iran

    Hosein Saadat

  2. Department of Mathematics, The University of Alabama, Alabama, USA

    Sanaz Hami Hassan Kiyadeh

  3. Department of Computational and Information Sciences, Stillman College, Alabama, USA

    Ramin Goudarzi Karim

  4. Department of Mathematics, Faculty of Basic science, The Khatam-ol-Anbia (PBU) University, Tehran, Iran

    Ali Safaie

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  1. Hosein Saadat
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Contributions

A.B. and C.D. wrote the main manuscript text and E.F. prepared figures 1–3. All authors reviewed the manuscript.

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Correspondence to Hosein Saadat.

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Saadat, H., Kiyadeh, S.H.H., Karim, R.G. et al. Family of phase fitted 3-step second-order BDF methods for solving periodic and orbital quantum chemistry problems. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01619-3

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