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Total energies for the lowest configuration 5s 2 , 4d5s and 4d 2 of Sr, as a function of the cavity radius r c . This figure should be compared with figure 1: it shows that the onset of the 4d period is analogous to the onset of 3d and that the 4d orbitals are more compressible than 5s.
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The energy spectra of ground-state, ionized and excited multielectron atoms and ions of the 3d and 4d periods of the periodic table centred in impenetrable spherical confinement are detailed using Hartree-Fock configuration average calculations. It is shown explicitly for the first time that, owing to modifications in 3d and 4d orbital collapse, th...
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... turns out that this new ordering for confined atoms is precisely the one originally predicted by the aufbau principle, i.e. filling follows the order implied by the principal quantum numbers. To demonstrate how this happens, we first show, in figure 4, the case of Sr, at the beginning of the 4d row. Notice that Sr{4} has the lowest configuration 4d 2 , just as Ca{3} has 3d 2 in figure 3, and that the general behaviour of the configuration energies for Sr{r c } parallels the behaviour of the outermost configurations of Ca{r c }, crossing over from an s 2 to a d 2 ground state for r c between 4 and 5. ...
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... Confining a group of electrons within a potential well leads to the formation of a new quantum system, often referred to as artificial atom or quantum dot [4]. Spatial confinement can also be realized for atoms or molecules trapped in nanocavities, such as zeolite, fullerenes, helium droplets and so forth [8,17,18]. ...
An elegant calculation is carried out to investigate the effects of the non‐ideality of classical plasma on the energy levels of the hydrogenic atoms held in a spherical cage. Organized effect of the non‐ideal classical plasma is described by an analytical pseudopotential which contains the Debye length and non‐ideality parameter as parameters. Convergent results for the bound states are obtained variationally by utilizing a large trail function containing cosine term which automatically takes care of the requisite boundary conditions. For the plasma‐free case, our results are in excellent agreement with the most accurate results available in the literature. An inclusive study is made to explore the changes emerging in the energy levels due to the variation of the plasma parameters and cage size. Special emphasis is made on the determination of critical cage size precisely. The present study specifically reveals that the increasing plasma non‐ideality leads to the elongation of the critical cage size. Moreover, it is empirically found that the critical cage size for a given hydrogenic atom can be obtained from a scaling law.
... The electronic structure of atoms and molecules suffers important changes when these systems are immersed in environments different from the vacuum. For example, essential differences are observed in the total energy when an atom is under confinement [1][2][3][4][5][6][7][8][9][10]. Unfortunately, many computational codes have been designed to study the electronic structure of atoms in the vacuum and consequently they cannot be used when an atom is within an environment different from the gas phase, in particular, when the confinement is modeled by piecewise functions. ...
The finite element method (FEM) based on a nonregular mesh is used to solve Hartree-Fock and Kohn-Sham equations for three atoms (hydrogen, helium, and beryllium) confined by finite and infinite potentials, defined in terms of piecewise functions or functions with a well-defined first derivative. This approach's reliability is shown when contrasted with Roothaan's approach, which depends on a basis set. Therefore, its exponents must be optimized for each confinement imposed over each atom, which is a monumental task. The comparison between our numerical approach and Roothaan's approach is made by using total and orbitals energies from the Hartree-Fock method, where there are several comparison sources. Regarding the Kohn-Sham method, there are few published data and consequently the results reported here can be used as a benchmark for future comparisons. The way to solve Hartree-Fock or Kohn-Sham equations by the FEM is entirely appropriate to study confined atoms with any form of confinement potential. This article represents a step toward developing a fully numerical quantum chemistry code free of basis sets to obtain the electronic structure of many-electron atoms confined by arbitrary confinement.
... Spatially confined quantum systems have become the subject of increasing attention because of the wide variety of problems in physics and chemistry that can be modelled through them. Some of these problems are atoms trapped in cavities, in zeolite channels, in fullerenes, the electronic structure of atoms and molecules subjected to high external pressures, the behavior of the specific heat of a monocrystal solid under high pressure [1,2,3,4,5,6,7,8], etc. This growing interest is also due to the fabrication of quantum systems of nanometric sizes with potential technological applications such as in quantum wires, dots and wells [9,10]. ...
In this work we study the helium atom confined in a spherical impenetrable cavity by using informational entropies. We use the variational method to obtain the energies and wave functions of the confined helium atom as a function of the cavity radius $r_0$. As trial wave functions we use one uncorrelated function and four functions with different degrees of electronic correlation. We computed the Shannon entropy, Fisher information, Kullback--Leibler entropy, Disequilibrium, Tsallis entropy and Fisher--Shannon complexity, as a function of the box radius $r_0$. We found that these entropic measures are sensitive to electronic correlation and can be used to measure it. These entropic measures are less sensitive to electron correlation in the strong confinement regime ($r_0<1$ a.u.).
... Furthermore, many physical phenomena can be modeled by a confined quantum system for example: atoms and molecules subject to high external pressures, atoms and molecules in fullerenes, inside cavities such as zeolite molecular sieves or in solvent environments, the specific heat of a crystalline solid under high pressure, etc. A complete list of applications can be found in the articles and reviews on the subject [14][15][16][17][18][19][20][21][22][23][24][25][26]. ...
... Likewise, in the study of atoms inside fullerenes, the corresponding potential has been modeled effectively by the attractive spherical shell [18], the δ-potential [33] and a Gaussian spherical shell [34,35]. Moreover, a spatially confined system inside an impenetrable spherical barrier, in addition to the inverted Gaussian potential [36][37][38][39], was employed by Lumb et al to describe the emission of two photons in the hydrogen atom [40]. ...
... The FDM is a suitable method for solving the radial Schrödinger equation (14). In the present work, we used a fourth-degree approximation for the kinetic energy operator (18). The accuracy of the eigenvalues depends strongly on the corresponding step size h (equation (19)). ...
In this work, for the lowest states with angular momentum, l =0,1,2 the energies and eigenfunctions of the endohedrals H@C 36 and H@C 60 are presented. The confining spherically-symmetric barrier was modeled by an inverted Gaussian function of depth ω 0 , width σ and centered at r c , w(r)=-ω 0 exp[-( r-r c ) ² /σ ² ]. The spectra of the system as a function of the parameters (ω 0 , σ, r c ) is calculated using three distinct numerical methods: ( i ) Lagrange-mesh method, ( ii ) fourth order finite differences and ( iii ) the finite element method. Concrete results with not less than 11 significant figures are displayed. Also, within the Lagrange-mesh approach the corresponding eigenfunctions and the expectation value of r for the first six states of s, p and d symmetries, respectively, are presented as well. Our accurate energies are taken as initial data to train an artificial neural network that generates faster and efficient numerical interpolation. The present numerical results improve and extend those reported in the literature.
... As for example, much emphasis has been placed in the study of Shannon entropies [19][20][21][22], as well as the study of spectroscopy of confined atomic systems [18] and the use of various methods for solving these problems. A large list of applications can be found in reviews and books on the subject [1][2][3][4][5][6][7][8]24]. Only few articles have been published with a pedagogical point of view, addressing the study of the confined harmonic oscillator (CHO) and the confined hydrogen atom (CHA) [26][27][28][29][30][31][32]. ...
In this work we calculate the ground state energy of the hydrogen atom confined in a sphere of penetrable walls of radius Rc. Inside the sphere the system is subject to a Coulomb potential, whereas outside of it the potential is a finite constant V0. The energy is obtained as a function of Rc and V0 by means of the Rayleigh-Ritz variational method, in which, the trial function is proposed as a free particle wave function within a finite square well potential but including an exponential factor that takes into account the electron-nucleus Coulomb attraction. For an impenetrable sphere, , the energy grows fast as Rc approaches zero. On the other hand, when the height of the barrier V0 is finite, the energy increases slowly as Rc goes to zero. We also compute the Fermi contact term, nuclear magnetic screening, polarizability, pressure and tunneling as a function of Rc and V0. As expected, these physical quantities approach the corresponding values of the free hydrogen atom as Rc grows. We also discuss the pressure-induced ionization of the hydrogen atom. The present results are found in good agreement with those previously published in the literature.
... 囚禁原子的量子特性是原子和分子物理、化学物理中的一个前沿课题，引 起了理论和实验研究者的广泛关注 [1][2][3][4] 。囚禁原子的特性因量子囚禁效应而发生 变化，比如能谱、极化率、电离能、电子概率密度、化学反应性和跃迁频率等 均与自由空间中的原子有很大的不同。例如，囚禁在微腔中氢原子的能量简并 现象消失，能级结构出现重排，在某些激发态之间存在能量交叉。对此 Sen 和 Guerra 等人已经在实验上观测到囚禁的碱金属原子中的激发态电子可以从 s 态 过渡到 d 态 [5,6] 。另一方面，量子囚禁还会导致原子的电子概率密度出现局域-非 局域化现象 [7] 。基于以上这些显著的特性，囚禁原子模型已经被用于量子点、 量子阱或量子线中的类氢杂质态性质的研究 [8][9][10][11] ，并且在高压物理 [12] 、天体物 理 [13] 、半导体物理 [14] 等领域也被广泛应用。 随着量子信息理论的发展，各种量子系统的新奇现象都可以通过许多信息 理论方法来解决。量子系统的内部无序表现为电子概率密度的不均匀性，可以 利用不同的量子信息熵进行测量 [15] 。在众多信息熵中，香农信息熵被应用于许 多 不 同 的 领 域 [16][17][18][19] [23] ，并且应用该方法研究 了非平衡孤立系统的复杂度 [24] 。随后，Angulo 和 Antolín 计算了位置、动量中 的 LMC 复杂度 [25] 。Dehesa 等人研究了 D 维类氢原子系统在位置和动量空间中 的形状复杂度 [15] ，并提出量子体系的内部无序性与电子概率密度的不均匀性密 切相关。最近，Sangita 等人对囚禁在真空微腔中的氢原子的无序性和复杂度进 行了研究 [ (2) ...
The research on the disorder of quantum system plays a very important role in the field of quantum information, which has been widely concerned by many theoretical and experimental researchers. However, it is very difficult to study the disorder of atoms confined in microcavity due to their complex nonlocal space-time evolution characteristics. To solve this problem, we present a method to study the internal disorder of hydrogenic atoms confined in microcavity, that is, to characterize and investigate the disorder of the confined system by using the quantum information entropy and shape complexity of the system. The Shannon information entropy and shape complexity in position space and momentum space (Sr、Sp、C[r]、C[p]) are calculated and analyzed for different quantum states of hydrogenic atom in InN dielectric spherical microcavity, and pay special attention to explore the influence of quantum confinement effect on the disorder of the system. The results show when the radius of the spherical microcavity is very small, the quantum confinement effect is more significant, and a series of extreme points appear in the shape complexity curve of the system, which is caused by the combing interaction of information entropy and spatial inhomogeneity. With the increase of the radius of the spherical cavity, the effect of quantum confinement gets weakened, and the Shannon information entropy and shape complexity of the confined hydrogenic atom are similar to that of the hydrogenic atom in free space. Our work provides an effective method to study the internal disorder of a confined quantum. This work provides an effective method for the study of the internal disorder of confined quantum systems and provides a certain reference for the information measurement of confined quantum systems.
... As a very fascinating research area in the field of nano-science and nano-technology, quantum confinement has attracted great interest from both theoretical and experimental aspects during the past several decades. The physical proprieties of the atoms and molecules, such as the structure of the energy spectrum, binding energy, polarizability, hyperfine splitting and the diamagnetic screening constants may change due to the quantum confinement effect [1][2][3][4][5][6][7]. In view of its importance, the quantum confinement system has a wide variety of applications in the field of physics and chemistry, such as quantum dot, quantum well, quantum defects in solid, quantum circuit and quantum computer, etc [8][9][10]. ...
We study the quantum confinement effects on the thermodynamic property for the particle confined in a linear potential. The energy eigen-values and eigen wave functions of the particle subjected to a spatial confinement by a linear potential have been derived by solving the Schrodinger equation. Our calculation results show that the thermodynamic properties of this system, such as the average energy, the heat capacity and the chemical potential depend sensitively on the width of the quantum well and the gradient of the linear potential. For instance, in the heat capacity curve of the confined particle as a function of the thermodynamic temperature, there exist different regions. For smaller size of the quantum well, the quantum confinement effect on the particle becomes significant in comparison to the effect of the linear potential, the heat capacity curve of the current system approaches to the case of the particle in the one-dimensional (1-D) infinitely deep quantum well. For larger size of the quantum well, the effect of the linear potential plays an important role. In the low temperature region, the heat capacity curve is close to the particle in the linear potential confined in the half space, and the quantum confinement only affect the heat capacity at high temperature region. These results suggest that the thermodynamic property for the particle confined in a linear potential exhibits a phase transition modified by the width of the quantum well and the thermodynamic temperature. Our work provides some references for the experimental research on the thermodynamic property of the confined particle and has some practical applications in statistical physics, chemical physics, ion trap, etc.
... Under ambient pressure (1 atm), the experimental values are V 1 = 0.302 a.u., R 0 = 5.8 a.u., ∆ = 1.89 a.u., γ = 0.1 a.u. [89][90][91]. However, in present case we have assumed that, these values remain unaltered with change in pressure. ...
Almost a century ago, it was proposed that solid hydrogen could become metallic at sufficiently high pressure. Five years back, scientists observed an insulator to metal transition in liquid deuterium experimentally, at around 300 GPa. The present work discusses and demonstrates the metal-like behavior in elemental hydrogen under various plasma environments. At the onset, the Herzfeld criterion is invoked to examine such characteristics in such plasmas under multimegabar pressure. However, a thorough study using this condition can only explain the metal-like pattern in s-wave states under a shell-confined condition (the system is trapped inside two concentric spheres with inner and outer radii Ra,Rb). Moreover, using this criterion, it is not possible to explain such phenomena in (i) confined systems (involving all ℓ) and (ii) for ℓ≠0 states in a shell-confined environment. Here this criterion is modified to incorporate the environmental conditions (nuclear charge, screening constant, boundary conditions) by utilizing several independent and generalized scaling concepts. The present condition can interpret a metallic pattern in confined and shell-confined plasmas connecting ℓ≥0 states. Further, a different descriptor is proposed therefrom. The role of pressure in defining such a descriptor is also examined. Pilot calculations are performed using Debye Hückel, exponential screen Coulomb potential, and ion-sphere plasmas. The relevance of the shell-confined model in the context of plasmas is elaborated. Additionally, an attempt is made to investigate the metallic character in H-like systems embedded in fullerene under high pressure.
... In addition, atoms confined in zeolite traps, cluster, in fullerene cages can also be analyzed using the quantum confined model [11][12][13][14][15]. In these previous researches, people mainly focus on the study of the physical properties of confined atoms and molecules, such as the wave function, probability density, energy level structure, the electric dipole polarizabilities, hyperfine splitting constants, etc [16][17][18][19][20][21][22]. It has been shown that due to the quantum confinement effect, the physical properties of the confined systems are significantly changed. ...
The effect of the spatially inhomogeneous electric field on the quantum thermodynamic property of the particle confined in a quantum well has been investigated theoretically. The Schrodinger equation was solved for the particle subjected to a spatial confinement by an inhomogeneous electric field and energy eigen-values were obtained. Using the energy eigen-values, we have calculated the average energy, free energy, entropy, and heat capacity of this system. As an important parameter in the inhomogeneous electric field, the inhomogeneity gradient of the field plays an important role. Compared with the spatially homogenous electric field, some novel quantum effect appears. The energy eigen-value for the particle corresponding to each bound state increases as the inhomogeneity gradient increases, resulting in the increase of the average energy, free energy, and heat capacity; however, the entropy, which measures the disorder of the particle confined in the inhomogeneous field, gradually decreases with the increase of the inhomogeneity gradient. In addition, the confinement effect of the quantum well on the thermodynamic properties of this system has also investigated in great detail. The results show that the larger size of the quantum well is, the stronger the influence of inhomogeneous field becomes, which makes the thermodynamic properties of the particle in the spatially inhomogeneous field quite different from those in the homogeneous field. Our work may provide some references for the future research on the thermodynamic property of the confined atom and molecule in the inhomogeneous field and has some practical applications in statistical physics, chemical physics, and condensed physics, etc.
... Understanding the electronic properties of atoms under high-pressure conditions remains a formidable task for both theory and measurement alike. For the theorist, the description of atoms under confinement has been largely limited to simple semi-analytic models [1][2][3][4][5][6][7][8][9][10][11], which though interesting and useful, are only capable of providing essentially qualitative descriptions. For the experimentalist, there is problem of reaching significant pressures without the sample environment preventing easy access to the embedded material. ...
The electron and photon excitation of confined and compressed atoms is discussed. It is shown that the optical oscillator strength, the inelastic X-ray and electron impact cross-sections are all density-dependent. Ab-initio electronic structure calculations are presented for the 1 s 2 → 1 s 2 p ( 1 P ) transition in helium, which predict that the amplitudes for the inelastic scattering of electrons, optical absorption and the excitation energy are all significantly increased in bulk condensed phases of pure helium compared to those of an isolated atom. The enhancements increase with atomic density but are very similar for bcc and fcc structured bulk phases of the same density. These enhancements need to be incorporated into the analysis of scanning transmission electron microscope studies of helium nano bubbles embedded in different materials. The excitation energy and cross-section enhancements with increasing density arise from the contraction of the 2 p orbital on entering a condensed phase. For the bulk phases, but not the bubbles, the close agreement between the experimental and computed values of the energy shifts can be well described by a simple relationship.
