From vibrational spectroscopy and quantum tunnelling to periodic band structures – a self-supervised, all-purpose neural network approach to general quantum problems

Gamper, Jakob; Kluibenschedl, Florian; Weiss, Alexander K. H.; Hofer, Thomas S.

doi:10.1039/d2cp03921d

Cited by 6 publications

(4 citation statements)

References 69 publications

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“…Frequently, numerical methods prove to be the most suitable means to tackle this challenge. Within the scientific literature, numerous methods for solving the Schrödinger equation have been documented (Killingbeck, 1987;Koch et al, 2006;Gamper et al, 2023), with the choice of method contingent upon the specific characteristics of the nanostructure under investigation, each method harboring its own set of advantages and disadvantages. Notably, among these methods, the Numerov method emerges as a versatile solution capable of addressing the Schrödinger equation in 1D, 2D, and 3D dimensions (Kolagiratou et al, 2005;Graen & Grubmüller, 2016), thus providing more stable and reliable results.…”

Section: Introductionmentioning

confidence: 99%

Efficient Energy Level Calculations in InP 2D-Quantum Box with Two Distinct Potentials Using the Sparse Numerov Method

KOÇ

2024

Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this study, energy level calculations for an InP 2D quantum box structure with two distinct (infinite potential power-exponential) potential potentials have been conducted using the sparse Numerov method. The 2D Schrödinger equation has been transformed in accordance with the sparse Numerov approach, followed by the creation of the solution matrix employing appropriate finite difference expressions. A comparative analysis of calculation results has been performed with respect to CPU time, memory usage, and ground state energy for both O(h^4) and O(h^6) accuracy. The suitability of the sparse Numerov method for 2D nanostructures has been thoroughly discussed. The results revealed that the sparse Numerov approach yields physically meaningful and rational outcomes in the InP 2D quantum box structure. Importantly, it demands significantly lower CPU time and memory resources compared to the classical Numerov method, emphasizing its practical applicability in this context.

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Section: Introductionmentioning

confidence: 99%

Efficient Energy Level Calculations in InP 2D-Quantum Box with Two Distinct Potentials Using the Sparse Numerov Method

KOÇ

2024

Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…The calculation of PT rate in HB requires the knowledge of the energy levels which are the eigenvalues of the corresponding SE with DWP. PT is a typical example of a quantum particle in DWP which is an omnipresent problem in physics and chemistry [23,26,[29][30][31][32][33][34][35][36][37][38][39][40][41]. Most DWPs used for the analysis of HB and composed of polynomials, exponentials (e.g., the double Morse potential) or their combinations are amenable only to numerical solutions (even in onedimensional case let alone its two-dimensional generalization) or approximate analytic approaches like the quasi-classical (WKB) method.…”

Section: Introductionmentioning

confidence: 99%

Model for vibrationally enhanced tunneling of proton transfer in hydrogen bond

Sitnitsky

2023

Preprint

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Theoretical analysis of the effect of an external vibration on proton transfer (PT) in a hydrogen bond (HB) is carried out. It is based on the two-dimensional Schrödinger equation with trigonometric double-well potential. Its solution obtained within the framework of the standard adiabatic approximation is available. An analytic formula is derived that provides the calculation of PT rate with the help of elements implemented in Mathematica. We exemplify the general theory by calculating PT rate constant for the intermolecular HB in the Zundel ion H 5 O + 2 (oxonium hydrate). This object enables one to explore a wide range of the HB lengths. Below some critical value of the frequency of the external vibration the calculated PT rate yields extremely rich resonant behavior (multiple manifestations of bell-shaped peaks). It takes place at symmetric coupling of the external vibration to the proton coordinate. This phenomenon is absent for anti-symmetric and squeezed mode couplings.

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“…In past decades, neural networks have also been used to solve vibrational SE, however mostly for synthetic systems such as one or multi-dimensional harmonic oscillator problems (15,16,17,18,19,20,21,22,23,24,25,26). The only application to a triatomic molecule was carried out by Manzhos et al, who solved the vibrational SE for H2O by combining non-linear optimization of NN parameters with a linear matrix method (27).…”

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confidence: 99%

Solving the vibrational Schrödinger equation with artificial neural networks

Zhao,

Zhang

2024

Preprint

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Artificial neural networks (NN) are universal function approximators and have shown great ability in computing the ground state energy of the electronic Schrödinger equation, yet NN has not established itself as a practical and accurate approach to solve the vibrational Schrödinger equation for realistic polyatomic molecules to obtain vibrational energies and wave functions for the excited states. Here we purpose an efficient approach to use NN to solve the vibrational Schrödinger equation and demonstrate the new method on CH4, a five atom molecule with 9 degree of freedom. By using a NN with < 3,000 parameters, we are able to achieve vibration energies for the ground and excited states with an accuracy of less than 1 cm-1 as compared to the reference values obtained by using more than 109 basis functions. It is anticipated the new method is capable of providing highly accurate vibrational energies and wave functions for molecules with more than 10 atoms, beyond the limit for all the existing computational approaches.

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From vibrational spectroscopy and quantum tunnelling to periodic band structures – a self-supervised, all-purpose neural network approach to general quantum problems

Abstract: In this work, a feed-forward artificial neural network (FF-ANN) design capable of locating eigensolutions to Schrödinger’s equation via self-supervised learning is outlined. Based on the input potential determining the nature...

Cited by 6 publications

References 69 publications

Efficient Energy Level Calculations in InP 2D-Quantum Box with Two Distinct Potentials Using the Sparse Numerov Method

Efficient Energy Level Calculations in InP 2D-Quantum Box with Two Distinct Potentials Using the Sparse Numerov Method

Model for vibrationally enhanced tunneling of proton transfer in hydrogen bond

Solving the vibrational Schrödinger equation with artificial neural networks

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