Hydrogenic Wave Functions Research Articles

Analytical estimate of effective charge and ground-state energies of two to five electron sequences up to atomic number 20 utilizing the variational method

BackgroundThe variational method, a quantum mechanical approach, estimates effective charge distributions and ground-state energy by minimizing the Hamiltonian's expectation value using trial wave functions with adjustable parameters. This method provides valuable insights into system behavior and is widely used in theoretical chemistry and physics. This paper aims to investigate ground-state energies and isoelectronic sequences using the variational method, introducing a novel approach for analyzing multi-electron systems. This technique allows for determining effective charge values and ground-state energies for 2–5 electrons sequence up to Z ≤ 20. Hydrogenic wave functions are used as a trial wave function to calculate effective charge in 1 s, 2 s, and 2p states. Two varying parameters were used to calculate an approximate wave function for the system. These values are then used in non-relativistic Hamiltonian with electron–electron interaction terms to calculate the ground-state energy of an atom.ResultThe results align with the reported experimental values, showing a marginal 1% error.ConclusionA Python algorithm is established based on the variational principle. It was found that, based on a few selected parameters in scripting the program, a very promising result was obtained. Furthermore, adding more variational parameters can minimize the difference between experimental and theoretical values, and this technique can be extended to elements with higher atomic numbers.

Time evolution as an optimization problem: The hydrogen atom in strong laser fields in a basis of time-dependent Gaussian wave packets.

Recent advances in attosecond science have made it increasingly important to develop stable, reliable, and accurate algorithms and methods to model the time evolution of atoms and molecules in intense laser fields. A key process in attosecond science is high-harmonic generation, which is challenging to model with fixed Gaussian basis sets, as it produces high-energy electrons, with a resulting rapidly varying and highly oscillatory wave function that extends over dozens of ångström. Recently, Rothe's method, where time evolution is rephrased as an optimization problem, has been applied to the one-dimensional Schrödinger equation. Here, we apply Rothe's method to the hydrogen wave function and demonstrate that thawed, complex-valued Gaussian wave packets with time-dependent width, center, and momentum parameters are able to reproduce spectra obtained from essentially exact grid calculations for high-harmonic generation with only 50-181 Gaussians for field strengths up to 5 × 1014 W/cm2. This paves the way for the inclusion of continuum contributions into real-time, time-dependent electronic-structure theory with Gaussian basis sets for strong fields and eventually accurate simulations of the time evolution of molecules without the Born-Oppenheimer approximation.

On the wavefunction cutoff factors of atomic hydrogen confined by an impenetrable spherical cavity

AbstractThe Schrödinger equation for the hydrogen atom enclosed by an impenetrable spherical cavity is solved through a Finite‐Differences approach to gain an insight on the actual nature and structure of the ansatz wavefunction cutoff factor widely used in an ad hoc manner in corresponding variational calculations to comply with the Dirichlet boundary conditions. The results of this work provide a theoretical foundation for the choice of the appropriate analytical cutoff functions that fulfill the boundary conditions. We find three different regions for the behavior of the cutoff functions. Small cavity radius where the cutoff function has a parabolic behavior, an intermediate region where the cutoff function is quasi‐linear, and a large cavity region where the cutoff function is a step‐like function. We deduce the traditional linear and quadratic cutoff functions used in the literature as well as its validity region for the confining radius. Finally, we provide a mathematical deduction of the exact cutoff function in terms of the nodal structure of the free hydrogenic wavefunctions and a relation to the Laguerre polynomials for some cavity radii where the free atomic energy level coincides with a confined energy level. We find that the cutoff function transit over several unconfined solutions in terms of its nodal structure as the system is compressed.

Hydrogenic spin-valley states of the bromine donor in 2H-MoTe2

In semiconductors, the identification of doping atomic elements allowing to encode a qubit within spin states is of intense interest for quantum technologies. In transition metal dichalcogenides semiconductors, the strong spin-orbit coupling produces locked spin-valley states with expected long coherence time. Here we study the substitutional Bromine BrTe dopant in 2H-MoTe2. Electron spin resonance measurements show that this dopant carries a spin with long-lived nanoseconds coherence time. Using scanning tunneling spectroscopy, we find that the hydrogenic wavefunctions associated with the dopant levels have characteristics spatial modulations that result from their hybridization to the Q-valleys of the conduction band. From a Fourier analysis of the conductance maps, we find that the amplitude and phase of the Fourier components change with energy according to the different irreducible representations of the impurity-site point-group symmetry. These results demonstrate that a dopant can inherit the locked spin-valley properties of the semiconductor and so exhibit long spin-coherence time.

Hydrogen vibration excitations of ZrH1.8 and TiH1.84 up to 21 GPa by incoherent inelastic neutron scattering

Hydrogen vibration excitations of fluorite-type $\mathrm{Zr}{\mathrm{H}}_{1.8}$ and $\mathrm{Ti}{\mathrm{H}}_{1.84}$ were investigated at pressures up to 21 and 4 GPa, respectively, by incoherent inelastic neutron scattering experiments. The excitations of both the samples were well described by quantum harmonic oscillator (QHO) over the entire pressure region of this study. The first excitation energies increased with increasing pressure, as described by the equations ${E}_{1}(\mathrm{meV})=141.4(2)+1.02(2)P$ (GPa) and ${E}_{1}\phantom{\rule{0.16em}{0ex}}(\mathrm{meV})=149.4(1)+1.21(8)P$ (GPa) for $\mathrm{Zr}{\mathrm{H}}_{1.8}$ and $\mathrm{Ti}{\mathrm{H}}_{1.84}$, respectively. Coupling with pressure dependence of lattice parameters determined by diffraction experiments, the relations between metal-hydrogen distance $({d}_{\mathrm{M}\text{\ensuremath{-}}\mathrm{H}})$ and ${E}_{1}$ of $\mathrm{Zr}{\mathrm{H}}_{1.8}$ and $\mathrm{Ti}{\mathrm{H}}_{1.84}$ at high pressures are found to be well described by the equations ${E}_{1}(\mathrm{meV})=1.62(9)\ifmmode\times\else\texttimes\fi{}{10}^{3}\phantom{\rule{0.16em}{0ex}}{d}_{\mathrm{M}\text{\ensuremath{-}}\mathrm{H}}^{\ensuremath{-}3.32\phantom{\rule{0.16em}{0ex}}(7)}(\AA{})$ and ${E}_{1} (\mathrm{meV})=1.47(21)\ifmmode\times\else\texttimes\fi{}{10}^{3}\phantom{\rule{0.16em}{0ex}}{d}_{\mathrm{M}\text{\ensuremath{-}}\mathrm{H}}^{\ensuremath{-}3.5(2)}(\AA{})$, respectively. The slopes of these curves are very steep compared to the previously reported trend in various fluorite-type metal hydrides at ambient pressure, suggesting that pressure and chemical substitution affect the hydrogen vibration excitations differently. The hydrogen wave function spreading estimated from the ${E}_{1}$ value assuming the QHO model showed the preferential shrinkage of the wave function to the tetrahedral sites, suggesting that the local potential field for a hydrogen atom shrinks more intensively than the tetrahedral site. The preferential shrinkage of the hydrogen wave function and the steep rise in the ${E}_{1}$ at a small ${d}_{\mathrm{M}\text{\ensuremath{-}}\mathrm{H}}$ under pressure are likely caused by the rigid metal ion core compared to hydrogen atoms and the resulting confinement of the hydrogen atom in the narrower potential field at high pressures.

Electron shake probability for arbitrary shell

AbstractGeneral expressions for electron shakeup and shakeoff probabilities of arbitrary atomic shells under the sudden change in central potential have been obtained in the sudden approximation by the use of non‐relativistic hydrogenic wave functions. The matrix elements for the shakeup and shakeoff processes are expressed analytically in the similar form in terms of the Gauss‐type hypergeometric functions. Many‐electron effect is taken into account through the screened hydrogenic model and the effect of the screening constants is examined. It is found that the shakeup and total shakeoff probabilities can be written in universal form by the use of a single parameter. The dependence of the shake probabilities on this parameter is discussed.

Quantum states of a confined hydrogen atom calculated in a basis of explicitly correlated Gaussians

A quantum-mechanical model for describing a hydrogen atom confined to a soft-wall cuboidal potential energy trap is implemented. Explicitly correlated Gaussian functions are used to expand the hydrogen wave functions that are symmetry-adapted with respect to the symmetry elements of the trapping potential. The calculations are performed without assuming the Born-Oppenheimer approximation. The electronic and nuclear densities of the calculated states are visualized using one-particle density plots. The approach enables to describe the behavior of a trapped hydrogen atom and, when extended to multiple hydrogen molecules, has potential for application in the theoretical modeling of the hydrogen storage phenomenon.

COMPLEX-ROTATION CALCULATIONS FOR DOUBLY EXCITED STATES (Nℓ N’ℓ’) 2S+1L ᴨ [(2≤N≤ 3 ) ; ( 3 ≤N’≤4); (0 ≤ ℓ ≤ 2 ) AND ( 0 ≤ ℓ’ ≤ 2)] OF HELIUM ISOELECTRONIC SERIES

In this paper, we report the energies and resonant widths of the [(2s3s 1Se and 2s3s 3Se) (2s4s 1Se and 2s4s 3Se) (2s3p 1P0 and 2s3p 3P0) (2s4p 1P0 and 2s4p 3P0) (2p3p 1De and 2p3p 3De) (2p4p 1De and 2p4p 3De) (3s4s 1Se and 3s4s 3Se) (3s4p 1P0 and 3s4p 3P0) (3p4p 1De and 3p4p 3De) (3d4d 1Ge and 3d4d 1Ge)] Doubly Excited States of Helium isoelectronic series with nuclear charge Z (2 ≤ Z ≤ 10).Calculations are performedusing the Complex Rotation Method (CRM) in the framework of a variational procedure. The purpose of this study required a new correlated hydrogenic radial wave function combined with a Hylleraas wave function. The study leads to analytical expressions which are carried out under special MAXIMA computational program. This proposed variational procedure, leads to accurate results in good agreement with available other theoretical results.The present accurate data may be a useful guideline for future experimental and theoretical studies in the (Nℓ Nℓ) 2S+1Lᴨsystems.

A strong-field approach with realistic wave functions to the above-threshold ionization of Ba+

We study the above-threshold ionization of atoms in intense circularly polarized laser pulses. In order to compute photoelectron energy spectra, we apply the strong-field approximation with different models for the initial state wave function. Specifically, we compare the spectra for singly ionized Barium (Ba+) using hydrogenic wave functions and realistic one-particle wave functions obtained by multiconfiguration Dirac–Hartree–Fock computations, respectively. As a particular example, we discuss the dependence of the photoelectron spectra on the magnetic quantum number m of the initial state and we reproduce the well known m-selectivity in strong-field ionization. Here, we show that the photoelectron spectra exhibit noticeable differences for the two models of the initial state and that the m-selectivity is enhanced when realistic wave functions are used. We conclude that the description of strong-field processes within the strong-field approximation will benefit from a realistic description of the initial atomic state.

Analytical estimate of effective charge and ground-state energy of beryllium atom utilizing variational method

In this paper, we present an analytical procedure to determine the effective charge and ground-state energy of a four-electron beryllium atom. The procedure followed is based on a variant of variational technique using modified hydrogenic one-electron wave function to calculate effective charge in 1s and 2s states of beryllium atom. The effective charge in 1s state is 3.68609 and in 2s state is 1.82362. Further, we have used these values in non-relativistic Hamiltonian with interaction terms to calculate the ground-state energy of beryllium atom. The estimated value of ground-state energy (− 14.3423 au) is in good agreement with the experimental results and with other theoretically obtained values. The calculated results without using any computationally intensive technique are the essence of this research work.

Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

Numerical variational solution of hydrogen molecule and ions using one-dimensional hydrogen as basis functions

The ground state solution of hydrogen molecule and ions are numerically obtained as an application of our scheme to solve many-electron multi-center potential Schrödinger equation by using one-dimensional hydrogen wavefunctions as basis functions. The all-electron sparse Hamiltonian matrix for the given system is generated with the standard order finite-difference method, then the electronic trial wavefunction to describe the ground state is constructed based on the molecular orbital treatment, and finally an effective and accurate iteration process is implemented to systematically improve the result. Many problems associated with the evaluation of the matrix elements of the Hamiltonian in more general basis and potential are circumvented. Compared with the standard results, the variationally obtained energy of H2+ is within 0.1 mhartree accuracy, while that of H2 and H3+ include the electron correlation effect. The equilibrium bond length is highly consistent with the accurate results and the virial theorem is satisfied to an accuracy of −V/T = 2.0.

The completeness properties of Gaussian‐type orbitals in quantum chemistry

AbstractIn this work, I extend results on the convergence of Gaussian basis sets in quantum chemistry, previously shown for ground‐state hydrogenic wavefunctions, to orbitals of arbitrary angular momentum. I give rigorous proofs of their asymptotic behavior, and demonstrate for methods with regular potential operators—in particular, Hartree–Fock and Kohn–Sham density functional theory—that the assumption of completeness is correct under fairly lenient conditions. The final result under the correct norm is that the convergence in energy follows , where M is the number of Gaussians and k is a positive constant, generalizing previous results due to Kutzelnigg. This then yields prescriptions for accelerated convergence using even‐tempered Gaussians, which could be used as initial guesses in future basis set optimizations.

Long-range interactions of the ground state muonium with atoms.

The scaling relations for the dispersion coefficients of long-range interactions between the Mu(1s)-Mu(1s, 2s, or 2p) systems and the H(1s)-H(1s, 2s, or 2p) systems are obtained using analytical properties of hydrogenic wavefunctions, which allows us to obtain the dispersion coefficients for Mu(1s)-Mu(1s, 2s, or 2p) systems from the corresponding H(1s)-H(1s, 2s, or 2p) systems. Additionally, the dispersion coefficients of long-range interactions of Mu(1s) with the ground-state H, noble gas atoms He, Ne, Ar, Kr, and Xe, alkali-metal atoms Li, Na, K, and Rb, alkaline-earth atoms Be, Mg, Ca, and Sr, and Cu, Ag, F, and Cl atoms are calculated.

Resonant and bound states of charged defects in two-dimensional semiconductors

A detailed understanding of charged defects in two-dimensional semiconductors is needed for the development of ultrathin electronic devices. Here, we study negatively charged acceptor impurities in monolayer WS$_2$ using a combination of scanning tunnelling spectroscopy and large-scale atomistic electronic structure calculations. We observe several localized defect states of hydrogenic wave function character in the vicinity of the valence band edge. Some of these defect states are bound, while others are resonant. The resonant states result from the multi-valley valence band structure of WS$_2$, whereby localized states originating from the secondary valence band maximum at $\Gamma$ hybridize with continuum states from the primary valence band maximum at K/K$^{\prime}$. Resonant states have important consequences for electron transport as they can trap mobile carriers for several tens of picoseconds.

General formula of shakeup and shakeoff probabilities for arbitrary shells

SynopsisGeneral expressions for the shakeup and shakeoff probabilities following sudden change in central potential have been obtained for arbitrary atomic shells by the use of nonrelativistic hydrogenic wave functions. The final formulae are given in terms of Gauss-type hypergeometric functions. The obtained results are used to calculate the shakeup and shakeoff probabilities for outer-shell electrons following β−decay.

The Ground State (1s<sup>2</sup>2s)<sup>2</sup>S and the Low-lying Excited (1s<sup>2</sup><i>n</i>s)<sup>2</sup>S States Energy Calculations of Li-Like Ions Using Special Forms of the Hylleraas-Type Wave Functions

In this paper, we have applied the complex rotation method to the calculations energies of the ground state (1s22s) 2S and the low-lying excited (1s2ns)2S states with n = 3, 4 and 5; of lithium and its isoelectronic series. For the ground state, the calculations were made up to Z = 20 and for the low-lying excited states up to Z = 10. These energies calculations were made using new special forms of Hylleraas-type wave functions designed by combining incomplete radial hydrogenic wave functions and Hylleraas-type wave functions. Using a numerical calculation program, the values of the resonance energies are calculated. Our results are compared with the results of ab-initio calculations using Hylleraas type wave functions and with semi-empirical results by Screening Constant by Unit Nuclear Charge (SCUNC) formalism. Analysis of the present results is achieved by calculating the ratio and the difference between our values and those of other authors. The results obtained are in good agreement with those of the theoretical methods available in the literature. This agreement shows the adequacy of our wave function with small bases to satisfactorily describe the ground state and the low-lying excited states of the three-electron atomic systems.

Effects of built-in electric field on donor binding energy in InGaN/ZnSnN2 quantum well structures

InxGa1−xN/ZnSnN2 quantum well structures are studied in terms of a binding energy of a donor atom. 1s and 2p± impurity states are considered. The Schrödinger's and Poisson's equations are solved self-consistently. A hydrogenic type wave function to represent each impurity state is assumed. The calculations include band-bending in the potential energy profile introduced by the built-in electric field existing along the structures. The binding energy and the energy of the transition between the impurity states are represented as a function of the quantum well width, the donor position, and the indium concentration. An external magnetic field up to 10 T is included into the calculations to compute the Zeeman splitting. The maximum value of the transition energy is around 30 meV (nearly 7.3 THz) which occurs in a 15-Å In0.3Ga0.7N/ZnSnN2 quantum well. Being strong, the built-in electric field makes the transition energy drop quickly with the decreasing well width. For the same reason, the energy curves are found to be highly asymmetric function of the donor position around the well center. Compared to the bulk value, the transition energy in the quantum well structures enhances nearly two-fold.

Two-dimensional excitons in monolayer transition metal dichalcogenides from radial equation and variational calculations

Exciton energy spectra of monolayer transition metal dichalcogenides (TMDs) in various dielectric environments are studied with an effective mass model using the Keldysh potential for the screened electron–hole interaction. Two-dimensional (2D) excitons are calculated by solving a radial equation (RE) with a shooting method, using boundary conditions that are derived by applying the asymptotic properties of the Keldysh potential. For any given main quantum number n, the exciton Bohr orbit shrinks as becomes larger (m is the orbital quantum number) resulting in increased strength of the electron–hole interaction and a decrease of the exciton energy. Further, both the exciton energy and its effective radius decrease linearly with . The screened hydrogen model (SHM) (Olsen et al 2016 Phys. Rev. Lett. 116 056401) is examined by comparing its exciton energy spectra with our RE solutions. While the SHM is found to describe the nonhydrogenic exciton Rydberg series (i.e. the energy’s dependence on n) reasonably well, it fails to account for the linear dependence of the exciton energy on the orbital quantum number. The exciton effective radius expression of the SHM can characterize the exciton radius’s dependence on n, but it cannot properly describe the exciton radius’s dependence on m, which is the cause of the SHM’s poor description of the exciton energy’s m-dependence. Analytical variational wave-functions are constructed with the 2D hydrogenic wave-functions for a number of strongly bound exciton states, and very close exciton energies and wave-functions are obtained with the variational method and the RE solution (exciton energies are within a 6% of deviation). The variational wave-functions are further applied to study the Stark effects in 2D TMDs, with an analytical expression derived for evaluating the redshift the ground state energy.

Half theory. II. The application of fractional spherical harmonics to chemical bonding

In this paper, we use fractional hydrogenic wave functions as trial functions in a variational calculation of the ground state energies of the H2+ ion and the He atom. We find improved estimates over those obtained using the standard integer angular momentum atomic trial states. We will see that a benefit of using these fractional functions is the generation of a directional charge distribution, which may serve to orient the atoms in chemical bonds.