Complete in phase method for problems in chemistry
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Complete in phase method for problems in chemistry Chia‑Liang Lin1 · T. E. Simos2,3,4,5,6 Received: 17 April 2020 / Accepted: 3 May 2020 © Springer Nature Switzerland AG 2020
Abstract A new complete in phase method for the approximate solution of differential equations in Quantum Chemistry is proposed in this paper. Keywords Phase-lag · Derivative of the phase-lag · Initial value problems · Oscillating solution · Symmetric · Hybrid · Multistep · Schrödinger equation Mathematics subject classification 65L05
1 Introduction 1.1 The problem We pay attention on the following Systems of Differential Equations:
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. * T. E. Simos [email protected] http://theodoresimos.org/ 1
School of Art, Ningbo Polytechnic, Ningbo 315800, China
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South Ural State University, 76, Lenin Aven., Chelyabinsk, Russian Federation 454080
3
Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Dongtong Road 705, Neijiang 641100, China
4
Group of Modern Computational Methods, Ural Federal University, 19 Mira Street, Ekaterinburg, Russian Federation 620002
5
Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece
6
10 Konitsis Street, Amphithea ‑ Palaion Phaleroin, GR‑17564 Athens, Greece
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Vol.:(0123456789)
Journal of Mathematical Chemistry
[
] N ∑ li (li + 1) d2 2 POTENTim 𝛿mj − 𝛿 = + k − POTENT ij ii i dx2 x2 m=1
(1)
where 1 ≤ i ≤ N and m ≠ i. (3) ⟶ BVP ( = Boundary value problem) ⟶ boundary conditions:
𝛿ij = 0 at x = 0
𝛿ij ∼ ki xjli (ki x)𝜃ij +
( )1∕2 ki Kij ki x nli (ki x) kj
(2) (3)
where jl (x) and nl (x) ⟶ spherical Bessel and Neumann functions respectively (see [1]). Research disciplines (see [1]): • Open Channels • Close Channels problem (see [62]) Asymptotic form (4) of (3) for the open channels : ⟶
𝛅 ∼ 𝚵 + 𝐓𝐖� . where:
Wij�
( )1∕2 ki Kij = kj
Ξij = ki xjli (ki x)𝜃ij Tij = ki xnli (ki x)𝜃ij More details see in [2–11] and references therein. Areas of application of the problems of the form (1) can also be found in [12–20] and references therein (Cybernetics, Neural Networks, Information Sciences, Neurocomputing, Social networks, Computational Intelligence, Simulation Modelling, etc.) [1] ⟶: [ 2 ] l� (l� + 1) Jjl d 2 + kj� j − 𝛿j� l� (x) dx2 x2 2𝜈 ∑ ∑ Jjl < j� l� ;J ∣ POTENT ∣ j�� l�� ;J > 𝛿j�� l�� (x) = 2 � j�� l�� where (j, l), (j� , l� ) , J = j + l = j� + l� ⟶ [1]. and ] [ ℏ2 2𝜈 � � kj� j = 2 E + {j(j + 1) − j (j + 1)} . 2I ℏ where for E, I and 𝜈 ⟶ [1] and references therein.
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Journal of Mathematical Chemistry
2 Development of the method We study the family of Finite Difference Methods for systems of Second-Order ODEs:
𝛼 ̃n = 𝛼n − c0 h2 𝜑n 𝛼 ̂n = 𝛼n − c1 h2 𝜑 ̃n ( 𝛼 n−1 = 𝛼n−1 − a0 h2 𝜑n+1 ) − 2𝜑 ̂ n + 𝜑n−1 − 2 a1 h2 𝜑 ̂n ( ) 𝛼 n = 𝛼n − a2 h2 𝜑n+1 − 2 𝜑n +
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