Confined Hydrogen Atom Research Articles

Effect of confining potential on information entropy measures in hydrogen atom: extensive and non-extensive entropy

A spherically confined hydrogen atom has been considered using two different confined potentials such as modified Kratzer and non-spherical oscillator potentials. First, the Schrodinger equation in $$ r $$ -space has been solved and the exact analytical wave functions and energy levels have been obtained. Then, the Renyi entropy and Shannon entropy in $$ r $$ -space have been calculated. In addition, we have numerically calculated the wave functions in $$ p $$ -space by performing Fourier transform. The calculations have been done for different quantum numbers $$ n $$ and $$ l $$ and also quantum size $$ r_{0} $$ . The Bialynicki–Birula–Mycielski inequality is also tested. It is found that this inequality depends on non-extensive parameter $$ q $$ .

Fisher information for endohedrally confined hydrogen atom

Abstract Electron localization and delocalization of endohedrally confined hydrogen atom has been investigated employing Fisher information theory. Confinement has been modeled using a spherical Gaussian-type potential. B-spline bases expansion method was used to solve the Schrodinger equation to obtain the required energy eigenvalues and the corresponding wave functions. Changes in energies with depths of potential are explained using Hellmann-Feynman theorem. The behavior of Fisher information against the confining potential depths and positions are demonstrated. Moreover, our results show that Fisher information is an effective way to measure the localization of valence electrons.

Confinement of hydrogen atom with Dirac equation

AbstractThe problem of a particle confined in a spherical cavity is studied with the Dirac equation. A hard confinement is obtained by forcing the large component to vanish at the cavity radius. It is shown that the small component cannot vanish simultaneously at this radius. In the case of a confined hydrogen atom, the energies are given by an implicit equation. For some values of the radius, explicit analytical expressions of the energy exist like in the nonrelativistic case. Very accurate energies and wave functions are obtained with the Lagrange‐mesh method with few mesh points. To this end, two differently regularized Lagrange‐Jacobi bases associated with the same mesh are used for the large and small components. The importance of relativistic effects is discussed for hydrogen‐like ions. The validity of this definition of hard confinement is discussed with a soft‐confinement model studied with the R‐matrix method.

Persistent currents and induced magnetization in presence of external magnetic field and transition probabilities in presence of combined laser pulse and external magnetic field for a confined hydrogen atom

In this work we propose to study an atom confined in a potential which consists of a Hulthén potential plus a ring-shaped potential. The atom is further subjected to a spherical confinement. The time-independent Schrödinger equation (TISE) of the system is solved numerically. Exact energy levels are obtained. The persistent current and induced magnetic field of such a confined atom is evaluated. Finally, the atomic system in this potential and confinement is subjected to short electromagnetic pulses, which are shown to induce enormous currents.

The application of scattering technology in the study of polyoxometalate solutions

Polyoxometalates(POMs), a large group of well-defined nanoclusters, are formed by linking early-transition-metal oxide polyhedrons through shared corners, edges and planes. POMs are widely used in various fields, such as catalysis, single molecular magnets, photoelectric materials, proton conductors, magnetic materials, and biomaterials, due to their abundant compositions and structures. However, how to design and synthesize POMs with specific structure and function remains a challenge for researchers. In-depth studies of POMs solution behavior are required to solve this problem. Scattering techniques, using microwaves, (near)infrared, visible light, ultraviolet light, X-rays, and neutrons as probe, are employed to investigate the structure and dynamics of materials. By detecting the interactions between the probe and the particles, physical properties, such as particle size, shape and internal structure, can be determined. This article focuses on the application of laser light scattering (LLS), small angle X-ray scattering (SAXS), and small angle neutron scattering (SANS) in the study of polyoxometalate solutions. By LLS, researchers discovered the self-assembly of POM macroanions, for example, researchers find the supramolecular blackberry structure formed by {Mo154} macroions in aqueous solution. Meanwhile the self-assembly processes and the self-recognition behaviors were determined, in mixed dilute aqueous solutions, the clusters {Mo72Fe30} and {Mo72Cr30} self-assemble into different blackberry structures of the Cr30 and Fe30 type. SAXS are employed to study POMs morphology and solution behavior, determine the counterion distribution around POMs in solutions, and probe the interactions among POMs in solutions. The effect of Rb+ on the assembly process, and the effect of solvent polarity on the assembly process, all of these can determined by SAXS. Moreover, the kinetic behaviors of confined hydrogen atoms in POMs and the morphology of POMs in hybrid materials can be obtained through SANS. Researchers study the difference between the mean square displacement measured in fully hydrogenated and partially deuterated {Mo72V30} by SANS. These studies are instructive to the design of POMs structure and function. However, there are still many basic problems of POM need to be solved. For example, the correlation between structure and properties of POM. And in the preparation of POM, how do the reducing agent, pH and catalyst work? Solving these basic issues requires numerous chemists effort. Scattering techniques play a key role in the study of polyoxometalate solutions since the structure and morphology information of nanoscale molecules can be obtained. With its unique advantages, scattering techniques will promote the development of POM.

Information‐entropic measures in free and confined hydrogen atom

AbstractShannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.

Derived properties from the dipole and generalized oscillator strength distributions of an endohedral confined hydrogen atom

We report the electronic properties of a hydrogen atom confined by a fullerene molecule by obtaining the eigenvalues and eigenfunctions of the time-independent Schrödinger equation by means of a finite-differences approach. The hydrogen atom confinement by a C60 fullerene cavity is accounted for by two model potentials: a square-well and a Woods–Saxon. The Woods–Saxon potential is implemented to study the role of a smooth cavity on the hydrogen atom generalized oscillator strength distribution. Both models characterize the cavity by an inner radius R0, thickness Δ, and well depth V0. We use two different values for R0 and Δ, found in the literature, that characterize H@C60 to analyze the role of the fullerene cage size and width. The electronic properties of the confined hydrogen atom are reported as a function of the well depth V0, emulating different electronic configurations of the endohedral cavity. We report results for the hyper-fine splitting, nuclear magnetic screening, dipole oscillator strength, the static and dynamic polarizability, mean excitation energy, photo-ionization, and stopping cross section for the confined hydrogen atom. We find that there is a critical potential well depth value around V0 = 0.7 a.u. for the first set of parameters and around V0 = 0.9 a.u. for the second set of parameters, which produce a drastic change in the electronic properties of the endohedral hydrogen system. These values correspond to the first avoided crossing on the energy levels. Furthermore, a clear discrepancy is found between the square-well and Woods–Saxon model potential results on the hydrogen atom generalized oscillator strength due to the square-well discontinuity. These differences are reflected in the stopping cross section for protons colliding with H@C60.

Fine structure in the hydrogen atom boxed in a spherical impenetrable cavity

AbstractThe spectrum of the hydrogen atom confined in a spherical impenetrable box of radius Rc has been investigated by many authors up to date, but not at the level of relativistic corrections. It is well known that, as Rc diminishes, all energy levels and the pressure increase very rapidly, whereas the polarizability goes to zero. In this report, we have computed the relativistic corrections that underlie the fine structure of the confined hydrogen atom, as a function of Rc. Such corrections correspond to relativistic kinetic energy, spin‐orbit coupling and the Darwin term, which are calculated in the frame of time‐independent perturbation theory, for which, use was made of the exact confined hydrogen atom wave functions. We show that for a confinement radius of 0.5 au the relativistic corrections increase up to three orders of magnitude with respect to those corresponding to the free atom. As Rc decreases, the kinetic energy correction and the spin‐orbit coupling for become negative whereas their absolute value and the Darwin term, which is positive, increase very rapidly.

Various complexity measures in confined hydrogen atom

Several well-known statistical measures similar to LMC and Fisher-Shannon complexity have been computed for confined hydrogen atom in both position (r) and momentum (p) spaces. Further, a more generalized form of these quantities with Rényi entropy (R) is explored here. The role of scaling parameter in the exponential part is also pursued. R is evaluated taking order of entropic moments α,β as 23,3 in r and p spaces. Detailed systematic results of these measures with respect to variation of confinement radius rc is presented for low-lying states such as, 1s-3d,4f and 5g. For nodal states, such as 2s,3s and 3p, as rc progresses there appears a maximum followed by a minimum in r space, having certain values of the scaling parameter. However, the corresponding p-space results lack such distinct patterns. This study reveals many other interesting features.

Sum rules and the role of pressure on the excitation spectrum of a confined hydrogen atom by a spherical cavity

Sum rule relations over the excitation spectrum of a quantum system contain information about both the energy spectrum and eigenfunctions of the system in a compact form, particularly regarding closure relations. In this work, the effects of pressure induced by a spherical cavity on an atomic hydrogen impurity on the dipole oscillator strength (DOS) sum rule, Sk, and its logarithmic version, Lk, are studied by means of a numerical approach based on a finite-difference solution to the Schrödinger equation. Pressure effects are accounted for by means of a spherical cavity of radius R0 immersed in a medium characterized by a penetrable potential height V0. The DOS sum rules Sk and Lk are investigated as a function of these cavity parameters and thus directly related to the impurity static pressure and surrounding material. One finds that the sum rules are fulfilled within the numerical precision for low pressure conditions. However, when the barrier height is large or infinite (a non-penetrable cavity), the sum rule, for positive k, differs from its closure relation. One finds that this occurs for a cavity radius au, corresponding to a pressure such that the first p-state that contributes to the sum rule has positive energy and it is due to the fact that the spherical confinement cavity potential dominates over the Coulombic interaction for the hydrogenic impurity. Thus, as pressure increases, the excitation spectrum approaches that of a particle confined by a spherical cavity while the ground state is slightly affected by the cavity and more closely resembles a hydrogenic atom. Therefore, the sum rule over the excitation spectrum tends to a particle confined by a spherical cavity, while the closure relation gives that of a confined hydrogen atom in the ground state. For negative k, low excitations are the most important and this behavior is not presented. As the sum rule is the static dipole polarizability, the results are compared to available data in the literature, showing excellent agreement. This behavior in the sum rule and oscillator strength in electron–impurity excitations affects optical transitions of importance in semiconductor nanostructures.

Benchmark values of Shannon entropy for spherically confined hydrogen atom

AbstractThe information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.

Endohedrally confined hydrogen atom with a moving nucleus

We studied the hydrogen atom as a system of two quantum particles in different confinement conditions; a spherical-impenetrable-wall cavity and a fullerene molecule cage. The motion refers to the center of spherical cavities and the Schrödinger equation is solved by means of a generalized Sturmian function expansion in spherical coordinates. The solutions present different properties from the ones described by the many models found in the literature, where the proton is fixed in space and only the electron is considered as a quantum particle. Our results show that the position of the proton (i.e. the center of mass of the H atom) is very sensitive to the confinement condition, and could vary substantially from one state to another, from being sharply centered to being localized outside the fullerene molecule. Interchange of the localization characteristics between the states when varying the strength of the fullerene cage and mass occurred through crossing phenomena.

The confined hydrogen atom: a linear variational approach

We study the size effect on the confinement of a hydrogen atom in a spherical box of impenetrable walls. We compute the energy of the ground and a few excited states as a function of the box radius Rc. To obtain the energy eigenvalues and eigenfunctions we utilize the linear variational method via a basis set of free-particle-in-a-box wave functions. For small values of Rc convergence is attained with a small number of basis set functions, whereas for Rc ≥ 5.0 au, it is necessary to use over 100 terms in the expansion.

Polarizability of off-center spherically confined hydrogen atom

The non-relativistic static and dynamic dipole polarizabilities of the ground and lowest excited states of the hydrogen atom, off-center in a spherical cavity, were investigated using B-Splines. For this purpose, the off-center Coulomb interaction, commonly evaluated by a multipole expansion was obtained in a different and faster approach. In the nearly centered case, the results obtained with a spherically symmetric description were recovered. The convergence tests performed showed the robustness of the numerical code used for these new calculations, and enabled us to carry out computations of the polarizabilities of the ground and lowest excited states of the off-center Hydrogen atom, as as function of the cavity radius and the distance from the center.

Confined H(1s) and H(2p) under different geometries

In this paper the diffusion Monte Carlo method is applied to the confined hydrogen atom with different confinement geometries. This approach is validated using the much studied spherical and cylindrical confinements and then applied to cubical and squared ones, for which data are not available, as new applications of the method relevant to solid state physics. The energy eigenvalues of the ground state and one low-lying excited state are reported as a function of the characteristic confinement length.

High-harmonic generation and spherically confined hydrogen atom

The dynamics of a hydrogen atom in a spherical box under the action of a strong infrared femtosecond laser is numerically investigated. The spherical box is introduced to model collisions of ionized electrons with neighboring atoms to model their influence on high-order harmonic generation. The physical situation appropriate for such application is considered in detail. The presence of tunneling in the process is investigated. In the strong field limit this consideration is extended including a dressing approach in the calculation. The harmonic spectrum is calculated for a small radius of the sphere assuming a high gas pressure. The intensities of the harmonic lines are reduced and the envelope of the spectrum is strongly modified with respect to the plateau of harmonics. The inclusion of the dressing approach in the calculation can significantly modify this result. The situation is changed if a kicking electric field is added at the moment when ionization probability has local maximum. The electron kinetic energy is drastically increased, and as a consequence of the dynamic Stark shifts in the kicking field, and an unusual electron recombination is possible. The odd harmonics of a very high-order and a somewhat different form of plateau of harmonics appear. The intensities of so obtained single-atom harmonic emission are dramatically enhanced. To check the reliability of the model with an impenetrable spherical box, high-pressure argon gas is also considered in relation to the single atom high-order harmonic generation.

Laser-induced excitation and ionization of a confined hydrogen atom in an exponential-cosine-screened Coulomb potential

The energy spectra of spherically confined hydrogen atom embedded in an exponential-cosine-screened Coulomb potential is worked out by using the Bernstein-polynomial method. The interaction of short laser pulses in the femtosecond range with the system is studied in detail. The effect of shape of laser pulse, confinement radius, Debye screening length as well as different laser parameters on the dynamics of the system has been explored and analyzed.

Photoionization of confined hydrogen atom by short laser pulses

The photoionization of hydrogen atom confined by a spherical barrier in ultrashort laser pulses is studied. The spectrum of confined hydrogen atom is worked out by employing finite basis set method based on B-spline as well as B-polynomial representation of wavefunction. Effect of confinement on spectra is studied and the effect of pulse parameters and confinement on photoexcitation and ionization is particularly stressed on.

Electronic structure of a cylindrically confined hydrogen atom by the B-splines method: energy levels and dipole polarizability

The non-relativistic static and dynamic dipole polarizabilities of a hydrogen atom experiencing a cylindrical confinement are investigated. Two methods based on B-splines are used for computations of energies and wavefunctions. The first method is a variational-based method while the second one proceeds by a fit of the non-separable Coulomb potential in the product form. The computed energies compare very well with previous computations. They converge, as well as the dipole polarizability, to the exact unconfined free atom values with increasing basis size. The fit approach is found to be advantageous, as it helps to reduce the computational time without loss of accuracy.

Elucidating Fermi's golden rule via bound-to-bound transitions in a confined hydrogen atom

We demonstrate an effective method for calculating bound-to-continuum cross-sections by examining transitions to bound states above the ionization energy that result from placing the system of interest within an infinite spherical well. Using photoionization of the hydrogen atom as an example, we demonstrate convergence between this approach for a large volume of confinement and an exact analytical alternate approach that uses energy-normalized continuum wavefunctions, which helps to elucidate the implementation of Fermi's golden rule. As the radius of confinement varies, the resulting changes in physical behavior of the system are presented and discussed. The photoionization cross-sections from a variety of atomic states with principal quantum number n are seen to obey particular scaling laws.