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Two-step method with vanished phase-lag and its derivatives for problems in quantum chemistry: an economical case

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Abstract

A new ECON2STEP (Economical Two-Step Method) method with vanished phase-lag and its derivatives up to order five is introduced in this paper, for initial or boundary value problems with solutions of oscillating and/or periodical behavior, with an application on problems in Chemistry.

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Author information

Authors and Affiliations

  1. Ural Federal University, Mira 19, Yekaterinburg, Russia

    Maxim A. Medvedev

  2. Institute of Industrial Ecology UB RAS, Kovalevskaya st. 20, Yekaterinburg, Russia

    Maxim A. Medvedev

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    T. E. Simos

  4. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Dongtong Road 705, Neijiang, 641100, China

    T. E. Simos

  5. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 10 Konitsis Street, Amphithea - Paleon Phaleron, GR-175 64, Xanthi, Athens, Greece

    T. E. Simos

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  1. Maxim A. Medvedev
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T.E. Simos: Highly Cited Researcher (2001-2013 List, 2017 List and 2018 List), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of European Academy of Arts, Sciences and Humanities.

Appendix

Appendix

$$ \begin{aligned} LTE_{{CL}} & = LTE_{{NM142S4SPD1}} = LTE_{{NM142S4SPD2}} = LTE_{{NM142S4SPD3}} = LTE_{{NM142S4SPD4}} = LTE_{{NM142S4SPD5}} \approx \approx h^{{16}} {\mkern 1mu} \ell _{0} \\ & = h^{{16}} [\frac{{14893{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} }}{{52484889600}} + \frac{{33443{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{288666892800}} \\ & + \frac{{371{\mkern 1mu} \left( {g\left( x \right)} \right)^{4} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{28866689280}} + \frac{{38213{\mkern 1mu} \left( {g\left( x \right)} \right)^{4} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{1154667571200}} \\ & + \frac{{65243{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{91848556800}} + \frac{{23797{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{8} }}{{{\text{d}}x^{8} }}g\left( x \right)} \right)\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)}}{{673556083200}} \\ & + \frac{{371{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{{10}} }}{{{\text{d}}x^{{10}} }}g\left( x \right)}}{{104969779200}} + \frac{{1007{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{9} }}{{{\text{d}}x^{9} }}g\left( x \right)}}{{88820582400}} + \frac{{53{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)y\left( x \right)\frac{{{\text{d}}^{{11}} }}{{{\text{d}}x^{{11}} }}g\left( x \right)}}{{85530931200}} \\ & + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)y\left( x \right)\frac{{{\text{d}}^{{10}} }}{{{\text{d}}x^{{10}} }}g\left( x \right)}}{{29606860800}} + \frac{{522739{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)^{2} }}{{1154667571200}} + \frac{{39167{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{4} y\left( x \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{288666892800}} \\ & + \frac{{4823{\mkern 1mu} \left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)^{2} y\left( x \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{22205145600}} + \frac{{8533{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)y\left( x \right)\left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)^{2} }}{{44410291200}} + \frac{{371{\mkern 1mu} \left( {g\left( x \right)} \right)^{5} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{288666892800}} \\ & + \frac{{53{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)^{2} }}{{159045120}} + \frac{{1641569{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{3} }}{{8082672998400}} + \frac{{23479{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} y\left( x \right)\frac{{{\text{d}}^{8} }}{{{\text{d}}x^{8} }}g\left( x \right)}}{{4041336499200}} \\ & + \frac{{16589{\mkern 1mu} \left( {g\left( x \right)} \right)^{4} y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} }}{{769778380800}} + \frac{{53{\mkern 1mu} \left( {g\left( x \right)} \right)^{5} y\left( x \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{18041680800}} + \frac{{1643{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{9} }}{{{\text{d}}x^{9} }}g\left( x \right)}}{{449037388800}} \\ & + \frac{{2491{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{{11}} }}{{{\text{d}}x^{{11}} }}g\left( x \right)}}{{4041336499200}} + \frac{{624181{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)^{2} }}{{4041336499200}} + \frac{{54007{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)^{2} }}{{734788454400}} \\ & + \frac{{53{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{198806400}} + \frac{{35351{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{192444595200}} \\ \end{aligned} $$
$$ \begin{aligned} & + \frac{{2173{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{4} }}{{57733378560}} + \frac{{1007{\mkern 1mu} \left( {g\left( x \right)} \right)^{5} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} }}{{577333785600}} + \frac{{53{\mkern 1mu} \left( {g\left( x \right)} \right)^{6} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)}}{{577333785600}} \\ & + \frac{{371{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{3} }}{{28866689280}} + \frac{{168169{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{2020668249600}} + \frac{{2491{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} y\left( x \right)\frac{{{\text{d}}^{8} }}{{{\text{d}}x^{8} }}g\left( x \right)}}{{104969779200}} \\ & + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)}}{{1009324800}} + \frac{{689{\mkern 1mu} \left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)y\left( x \right)\frac{{{\text{d}}^{9} }}{{{\text{d}}x^{9} }}g\left( x \right)}}{{177641164800}} + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)y\left( x \right)\frac{{{\text{d}}^{8} }}{{{\text{d}}x^{8} }}g\left( x \right)}}{{8074598400}} \\ & + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)}}{{1284595200}} + \frac{{371{\mkern 1mu} \left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{8} }}{{{\text{d}}x^{8} }}g\left( x \right)}}{{14803430400}} + \frac{{1007{\mkern 1mu} \left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)y\left( x \right)\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)}}{{113044377600}} \\ & + \frac{{2491{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{9} }}{{{\text{d}}x^{9} }}g\left( x \right)} \right)\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)}}{{168389020800}} + \frac{{265{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} y\left( x \right)\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)}}{{2368548864}} + \frac{{3551{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{9868953600}} \\ & + \frac{{7897{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{3} y\left( x \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{52484889600}} + \frac{{583{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{2385676800}} + \frac{{97997{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{1010334124800}} \\ & + \frac{{26129{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{144333446400}} + \frac{{148771{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{144333446400}} \\ & + \frac{{26977{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)}}{{367394227200}} + \frac{{53{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{3} }}{{106913664}} + \frac{{265{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{3} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{1049697792}} \\ & + \frac{{48707{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} y\left( x \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)^{2} }}{{505167062400}} + \frac{{31747{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{202066824960}} + \frac{{53{\mkern 1mu} \left( {g\left( x \right)} \right)^{6} y\left( x \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{128296396800}} \\ & + \frac{{1219{\mkern 1mu} g\left( x \right)y\left( x \right)\frac{{{\text{d}}^{{12}} }}{{{\text{d}}x^{{12}} }}g\left( x \right)}}{{8082672998400}} + \frac{{4717{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\frac{{{\text{d}}^{{10}} }}{{{\text{d}}x^{{10}} }}g\left( x \right)}}{{2694224332800}} + \frac{{901{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)}}{{126291765600}} \\ & + \frac{{110399{\mkern 1mu} \left( {g\left( x \right)} \right)^{4} y\left( x \right)\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)}}{{16165345996800}} + \frac{{1961{\mkern 1mu} \left( {g\left( x \right)} \right)^{4} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{384889190400}} + \frac{{169441{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{144333446400}} \\ \end{aligned} $$
$$ \begin{aligned} & + \frac{{16271{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{15395567616}} + \frac{{2491{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)y\left( x \right)\left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{9542707200}} \\ & + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{4} y\left( x \right)}}{{526344192}} + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)} \right)^{2} y\left( x \right)}}{{10766131200}} + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{{14}} }}{{{\text{d}}x^{{14}} }}g\left( x \right)} \right)y\left( x \right)}}{{32330691993600}} + \frac{{53{\mkern 1mu} \left( {\frac{{{\text{d}}^{{13}} }}{{{\text{d}}x^{{13}} }}g\left( x \right)} \right)\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)}}{{2309335142400}} \\ & + \frac{{53{\mkern 1mu} \left( {g\left( x \right)} \right)^{8} y\left( x \right)}}{{32330691993600}} + \frac{{53{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{5} \frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)}}{{2624244480}} + \frac{{371{\mkern 1mu} \left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)^{3} \frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)}}{{1850428800}} + \frac{{170713{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{8} }}{{{\text{d}}x^{8} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{4041336499200}} \\ & + \frac{{346037{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{4041336499200}} + \frac{{65773{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{80826729984}} \\ & + \frac{{25387{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)y\left( x \right)\left( {\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{128296396800}} + \frac{{12137{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)y\left( x \right)\left( {\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{107768973312}} \\ \end{aligned} $$
$$ \begin{aligned} & + \frac{{298867{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{310872038400}} + \frac{{5671{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)^{2} \frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)}}{{9020840400}} \\ & + \frac{{8851{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{32074099200}} + \frac{{61427{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)}}{{310872038400}} \\ & + \frac{{2173{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{36083361600}} + \frac{{2809{\mkern 1mu} \left( {g\left( x \right)} \right)^{3} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{28866689280}} \\ & + \frac{{63017{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{288666892800}} + \frac{{622697{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{2020668249600}} \\ & + \frac{{164353{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)} \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{288666892800}} + \frac{{20087{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{3} \frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{72166723200}} \\ & + \frac{{793781{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{3} \frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{2020668249600}} + \frac{{11819{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{91848556800}} \\ & + \frac{{29839{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)y\left( x \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{88820582400}} + \frac{{67363{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{192444595200}} \\ & + \frac{{11183{\mkern 1mu} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)}}{{11102572800}} + \frac{{438787{\mkern 1mu} \left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)}}{{577333785600}} \\ & + \frac{{5353{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} \left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)}}{{64148198400}} + \frac{{176543{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\frac{{{\text{d}}^{5} }}{{{\text{d}}x^{5} }}g\left( x \right)}}{{673556083200}} \\ & + \frac{{1277989{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\frac{{{\text{d}}^{6} }}{{{\text{d}}x^{6} }}g\left( x \right)}}{{8082672998400}} + \frac{{87821{\mkern 1mu} \left( {g\left( x \right)} \right)^{2} y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)\frac{{{\text{d}}^{7} }}{{{\text{d}}x^{7} }}g\left( x \right)}}{{1347112166400}} \\ & + \frac{{232511{\mkern 1mu} g\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}y\left( x \right)} \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} \frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)}}{{288666892800}} + \frac{{6307{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)^{2} }}{{7401715200}} \\ & + \frac{{270883{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)} \right)^{2} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}g\left( x \right)} \right)^{2} }}{{367394227200}} + \frac{{2334809{\mkern 1mu} g\left( x \right)y\left( x \right)\left( {\frac{{{\text{d}}^{3} }}{{{\text{d}}x^{3} }}g\left( x \right)} \right)\left( {\frac{{{\text{d}}^{4} }}{{{\text{d}}x^{4} }}g\left( x \right)} \right)\frac{{\text{d}}}{{{\text{d}}x}}g\left( x \right)}}{{2020668249600}}] \\ \end{aligned} $$

where \(\varepsilon \left( x \right) = \varepsilon _{n} = y \left( x \right) \,and \,g \left( x \right) = PTN (x)\).

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Medvedev, M.A., Simos, T.E. Two-step method with vanished phase-lag and its derivatives for problems in quantum chemistry: an economical case. J Math Chem 59, 1880–1916 (2021). https://doi.org/10.1007/s10910-021-01263-1

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