Searching for distinct classes of atomic and molecular states using convergence and separability criteria
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Searching for distinct classes of atomic and molecular states using convergence and separability criteria Alejandro López‑Castillo1 Received: 16 March 2020 / Accepted: 5 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We demonstrated that the solution ψ to the Schrodinger equation (SE) Ĥψ = εψ converges logarithmically in the classically forbidden region, i.e., as ε′ approaches to the correct eigenvalue ε, the approximate solution ψ′ logarithmically converges to ψ. Knowing that this approximate eigenvalue procedure generates a straightforward but inefficient method to solve a general problem of n-bodies, we thereby discuss the main characteristic of the usual methods to obtain a better convergence for atoms and molecules. Such usual methods consider that the solution of SE can be approached through a linear combination of solutions of systems with analytical solutions. Hydrogen-like atom solutions are used to describe atoms and molecules. This solution avoids the convergence problem of ψ′ even for approximate eigenvalues ε′ since all the terms of the linear combination decay asymptotically to zero. We argue that this type of solution works very well for a large class of almost separable atomic and molecular states in which the separation of electronic (and nucleus) movements occurs. We also establish a comparison of this separability and other systems, like gravitational, in which separability is only possible in particular classes of restricted systems. Finally, we consider the existence of distinct atomic and molecular states that may not be described using usual methods that apply this type of solution, which implies the separability of restricted problems. Therefore, usual methods to describe atoms and molecules may be insufficient to reach solutions with different or more general electronic correlations, as discussed in the text. Strategies to achieve general or distinctive solutions, although approximated, should be further studied and developed. Keywords Schrödinger equation · Usual methods · Convergent and divergent solutions · Separable and non-separable states · Non-systematic methods
1 Introduction Quantum calculation of atoms and molecules is not a trivial task. Sophisticated numerical techniques need to be applied to obtain the electronic description of atoms and molecules [1, 2], considering the impossibility of an exact analytical “Festschrift in honor of Prof. Fernando R. Ornellas” Guest Edited by Adélia Justino Aguiar Aquino, Antonio Gustavo Sampaio de Oliveira Filho & Francisco Bolivar Correto Machado. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00214-020-02661-5) contains supplementary material, which is available to authorized users. * Alejandro López‑Castillo [email protected] 1
Chemistry Department, UFSCar, São Carlos, SP 13565‑905, Brazil
calculation of many-electron systems. Many computational packages are available to incorporate these numerical techniques to solve the Schrödin
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