New Approach for the Electronic Energies of the Hydrogen Molecular Ion (original) (raw)
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THE ENERGY EIGENVALUES OF THE TWO DIMENSIONAL HYDROGEN ATOM IN A MAGNETIC FIELD
International Journal of Modern Physics E, 2006
In this paper, the energy eigenvalues of the two dimensional hydrogen atom are presented for the arbitrary Larmor frequencies by using the asymptotic iteration method. We first show the energy eigenvalues for the no magnetic field case analytically, and then we obtain the energy eigenvalues for the strong and weak magnetic field cases within an iterative approach for n = 2 − 10 and m = 0 − 1 states for several different arbitrary Larmor frequencies. The effect of the magnetic field on the energy eigenvalues is determined precisely. The results are in excellent agreement with the findings of the other methods and our method works for the cases where the others fail.
Eigenvalue equations from the field of theoretical chemistry and correlation calculations
arXiv (Cornell University), 2017
Introduction 2.1.2. Properties of the multi-electron densities "between" the two Hohenberg-Kohn theorems and variational principle 2.1.3. Approximations for the functionals 2.1.4. The exact density functional operator for N=1 electron systems (H-like atoms and general one-electron systems), and Slater determinant level density functional operator for N=2 electron systems 2.1. References 2.2. Conversion of the non-relativistic electronic Schrödinger equation to scaling correct moment functional of ground state one-electron density to estimate ground state electronic energy .
Jurnal Fisika Indonesia, 2020
Calculation of energy eigen value of hydrogen negative ion (H − ) in 2p^2 configuration using the method of variation functions has been done. A work on H − can lead to calculations of electric multipole moments of a hydrogen molecule. The trial function is a linear combination of 8 expansion terms each of which is related to the Chandrasekhar’s basis. This work produces a series of 7 energy eigen values which converges to a value of −0.2468 whereas the value of this convergence is expected to be −0.2523. This deviation from the expected value is mainly due to the elimination of interelectronic distance (u) coordinate. The values of the exponent parameters used in this work contribute also to this deviation. This variational method will be applied to the construction of some energy eigen functions of Hv2 .
Lecture Notes in Computer Science, 2007
The boundary problem in cylindrical coordinates for the Schrödinger equation describing a hydrogen-like atom in a strong homogeneous magnetic field is reduced to the problem for a set of longitudinal equations in the framework of the Kantorovich method. The effective potentials of these equations are given by integrals over transversal variable of a product of transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives with respect to the parameter. A symbolic-numerical algorithm for evaluating the transverse basis functions and corresponding eigenvalues which depend on the parameter, their derivatives with respect to the parameter and corresponded effective potentials is presented. The efficiency and accuracy of the algorithm and of the numerical scheme derived are confirmed by computations of eigenenergies and eigenfunctions for the low-excited states of a hydrogen atom in the strong homogeneous magnetic field.
Open Journal of Microphysics, 2022
The objective of this work is to calculate and compare the energy eigenvalue of Hulthen Potential using the NU method and AIM method. Using these two methods the energy eigenvalue calculated from the NU method is less than 2 δ AIM method. Moreover, the energy eigenvalue calculated from both methods is charge independent and only depends upon the quantum numbers and screening parameters, while the third term of energy eigenvalue calculated using the NU method is only dependent on screening parameters.
Journal of Molecular Structure: THEOCHEM, 2006
Starting with the Hamilton-Jacobi equation we apply Hylleraas' method in association with the series established by Wind-Jaffe, to H 2 þ , D 2 þ , T 2 þ , HD + and DT + molecular ions in order to calculate the electronic energies as well as vibrational levels. In the later it is necessary to have the exact value of the electronic energy as a function of the nuclear distance. The dissociation of these molecular ions is important to study proton injection in magnetic mirrors and in Tokomak leading to the controlled thermonuclear fusion.
The H $ _2^+ $ molecular ion: low-lying states
As a continuation of our previous work [Phys. Rev. A 68, 012504 (2003)] an accurate study of the lowest 1 g and the low-lying excited 1 u , 2 g , 1 u,g , 1␦ g,u electronic states of the molecular ion H 2 + is made. Since the parallel configuration where the molecular axis coincides with the magnetic field direction is optimal, this is the only configuration which is considered. The variational method is applied and the same trial function is used for different magnetic fields. The magnetic field ranges from 10 9 to 4.414ϫ 10 13 G, where nonrelativistic considerations are justified. Particular attention is paid to the 1 u state which was studied for an arbitrary inclination. For this state a one-parameter vector potential is used which is then variationally optimized.
The Hydrogen Atom: Applications of Ordinary Differential Equations
The main idea of this report is to explore the Hydrogen atom by exploring its mathematics. In particular, we want a mathematical description of how the electron orbits around the proton. In the introduction, we will look at the postulates of Quantum Mechanics to see the role of the Schrodinger Equation, and then we will follow this by introducing the basic formulation for the atom. From here we step into a separation of variables on the partial differential equation that arises, so that the rest of the report consists of solving ordinary differential equations. We will encounter many famous differential equations, delving into the Legendre Equation, Euler's Equation, and the Radial Equation. In solving them we will run into Legendre Polynomials, Spherical Harmonics, and Laguerre Polynomials. These ultimately show the practical roles of infinite series methods for solving ordinary differential equations, as well as various other methods that will be presented. In the end, we piece everything together and explain the significance of all the results. We will be able to present visualizations of hydrogen through density plots, as well as find energy levels and descriptions of all possible hydrogen states.