Solving the Schrödinger equation of the hydrogen molecule with the free-complement variational theory: essentially exact potential curves and vibrational levels of the ground and excited states of Π symmetry

2020 ◽  
Vol 22 (24) ◽  
pp. 13489-13497
Author(s):  
Yusaku I. Kurokawa ◽  
Hiroyuki Nakashima ◽  
Hiroshi Nakatsuji
Keyword(s):  
Schrödinger Equation ◽  
Variational Method ◽  
Excited States ◽  
Schrodinger Equation ◽  
Hydrogen Molecule ◽  
Chem Phys ◽  
Variational Theory ◽  
Vibrational Levels ◽  
Phys Chem Chem Phys ◽  
Potential Curves

Following a previous study of the Σ states (Phys. Chem. Chem. Phys., 2019, 21, 6327), we solved the Schrödinger equation (SE) of the hydrogen molecule in the ground and excited Π states using the free complement (FC) variational method.

Solving the Schrödinger equation of hydrogen molecules with the free-complement variational theory: essentially exact potential curves and vibrational levels of the ground and excited states of the Σ symmetry

2019 ◽  
Vol 21 (12) ◽  
pp. 6327-6340 ◽  
Author(s):  
Yusaku I. Kurokawa ◽  
Hiroyuki Nakashima ◽  
Hiroshi Nakatsuji
Keyword(s):  
Schrödinger Equation ◽  
Potential Energy ◽  
Excited States ◽  
Schrodinger Equation ◽  
Potential Energy Curves ◽  
Variational Theory ◽  
Hydrogen Molecules ◽  
Vibrational Levels ◽  
Electronic Ground ◽  
Potential Curves

The Schrödinger equation of hydrogen molecules was solved essentially exactly and systematically for calculating the potential energy curves of the electronic ground and excited states of the 1Σg, 1Σu, 3Σg, and 3Σu symmetries.

HYPERSPHERICAL APPROACH TO STUDY THE SCHRÖDINGER EQUATION FOR AN N-PARTICLE SYSTEM

2009 ◽  
Vol 18 (07) ◽  
pp. 1497-1502
Author(s):  
H. HASSANABADI ◽  
A. A. RAJABI ◽  
M. M. SHOJAEI
Keyword(s):  
Analytical Solution ◽  
Schrödinger Equation ◽  
Excited States ◽  
Ground State ◽  
Schrodinger Equation ◽  
Particle System ◽  
Exact Analytical Solution ◽  
Hyperspherical Approach

In the present work we give an exact analytical solution of the Schrödinger equation for an N-particle system by using the hyperspherical approach, in the presence of the hypercentral potential of form V(R) = a1R2+b1R-4+c1R-6 for both the ground state and the excited states.

scholarly journals Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

2015 ◽  
Vol 18 (2) ◽  
pp. 321-350 ◽  
Author(s):  
Siwei Duo ◽  
Yanzhi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Excited States ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Euler Method ◽  
Potential Well ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation ◽  
Infinite Potential Well

AbstractIn this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

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