1.Introduction

This invited review article is written against the background of W. Jaskolski's 1996 Physics Report on Confined many-electron systems^{1}, the fifteen chapters of Theory of Confined Quantum Systems in Advances in Quantum Chemistry (AQC) 2009 Vols. 57^{2}^{-}^{10} and 58^{11}^{-}^{16}, with S.A. Cruz as Special Editor, and the nine chapters of Electronic Structure of Quantum Confined Atoms and Molecules^{17}^{-}^{25} edited by K. A. Sen in 2014. It is interesting to contrast^{1} a review of the state-of-the-art in that year by a single author, with the two sets of reviews of specific topics in the field, twenty two and twenty six years later, illustrating advances and the diversification of the field in the respective periods.

The readers can get an idea on the physical systems and confinement conditions in each review from their respective titles in the references. Notice that ^{2}^{-}^{8} are restricted to one and two electron atoms,^{9} and^{11} to many-electron atoms, while^{10} treats confined atoms as open systems,^{12} reports on the photoionization of atoms encaged in spherical fullerenes while^{13} a Density Functional Theory Study of Molecules confined inside fullerene and fullerene-like cages,^{14} deals with the Spectroscopy of Confined atomic Systems: Effect of Plasma,^{15} characterizes the Energy Level Structure of Low-dimensional Multi-electron Quantum Dots, and^{16} is about Engineering Quantum Confined Silicon Nanostructures: Ab-Initio Study of the Structural, Electronic and Optical properties. We may add that in^{2}^{,}^{3}, ^{5}^{-}^{8} and ^{12} the atoms or molecules are confined inside spheres, while in^{4} the confining boundaries are conoidal, and in^{9} they are spherical, prolate spheroidal, and plane.

In the Monograph, its Editor pointed out in the Preface that, *it deals with the simplest among the quantum confined model systems, recording the significant developments in the field subsequent to the two published volumes of Advances in Quantum Chemistry* . The titles and authors can be identified in^{17}^{-}^{25}, recognizing common topics and Contributors in^{17}^{-}^{20} and^{24} from the previous set of reviews; here we spell out the other titles^{21} “Confined Quantum Systems Using the Finile Element and Discrete Variable Representation Methods",^{22} “Bound and Resonant States in Confined Atoms",^{23} “Spatial and Shell-Confined One Electron Atomic and Molecular Systems: Structure and Dipole Polarizability", and^{25} “Study of Quantum Confinement of _{2} Molecule with Monte Carlo. Respective Role of the Electron and Nuclei Confinement", illustrating new methods of solution, new physical effects, and new ways of modeling the confinement.

Concerning the use of the words superintegrability and symmetry breakings in the title of this contribution, the reader may find in^{4}, and in^{26} of^{17} these respective uses for the one-electron hydrogen atom and quantum dot, in their free and confined configurations. In fact, the Schrödinger equation for the hydrogen atom can be separated and integrated in spherical, spheroconal, prolate spheroidal and parabolic coordinates; and for the harmonic oscillator modeling one-electron quantum dots in cartesian, cylindrical and also spherical and spheroconal coordinates. That situation extends to the respective quantum systems confined by natural boundaries corresponding to fixed values of the respective coordinates. For the free systems, their solutions in the different coordinate systems share the same degeneracies; due to their respective O(4) and SU(3) symmetries. In contrast, the systems confined in the successive natural boundaries exhibit the corresponding symmetry breakings due to the changes in the boundary conditions, with different energy eigenvalues and eigenfuctions, and reduced degeneracies. The title and contents of^{18} also illustrate these effects.

In the case of the hydrogen atom, the superintegrability is subject to the condition of the position of the nucleus at the center of the sphere, at one of the foci in the prolate spheroid, at the common focus of the paraboloids, each one with their respective set of constants of motion; if the nucleus occupies other positions, the separability of the Schrödinger equation does not hold any longer: the original constants of motion, energy eigenvalues and eigenfunctions. cease to be such.

In going from the one-electron systems to the many-electron counterparts, the presence of the additional electron-electron Coulomb interaction breaks the respective O(4) and SU(3) symmetries. Upon confining them, there are additional symmetry breakings dependent on the characteristics of the confinement.

In the process of planning the writing of this contribution, the author has considered the alternatives of reviewing the field at large a la Jasholski, or following the example of the AQC and Monograph individual reviews. Given the time constraints of the invitation, the decision has been to make a conceptual and connecting review, which is intended to serve the readers as a guide to become familiar with the variety of works in the literature illustrating the progress in recent years; and also sharing the recent results of our own work, and the problems which we are currently investigating.

This article is organized as follows: Section 2.1 identifies some connections between the different reviews in AQC dealing with the same physical systems one-electron, two-electron and many-electron atoms, and harmonic oscillators, under the same or different confinement conditions, taking into account their respective methods of solution, and their results, on specific properties. Section 2.2 starts by analyzing the further advances in the common topics in the reviews of the Monograph and those of AQC, and goes on to the remaining four new topics. Section 3 includes a selection of references from the literature illustrating recent progress in the field, taking into account the novelty of the confined quantum systems, the confining environments and their modelings, the properties of such systems, and their connections with other fields of materials science. In Section 4, the Author updates the progress in his own and collaborators’ works, and gives a preview on current investigations. Section 5 consists of a discussion on the reviewed recent progress.

2.Background Reviews

This section includes brief comments about the Reviews in AQC2009 and in the Springer Monograph 2014, in 2.1 and

2.2 respectively. The first set identifies the key concepts and methods in each Review, recognizing some connections and differences among some of the Reviews, thus establishing the points of reference and the threads guiding the writing and reading of this contribution. The second set follows the same ideas, connecting first the contributions with common topics or Authors from the previous set, and going on with the new ones.

2.1Theory of Confined Quantum Systems 2009

This section identifies some common features and differences in the reviews in Advances in Quantum Chemistry^{2}^{-}^{16} in order to illustrate the diverse ways of modeling and analyzing the confined quantum systems, and how the results on some of their properties compare. The identification follows the order of the references covering the atoms with increasing number of electrons. Their common features and differences are some of the elements to follow in the successive sections in order to recognize the further advances and recent progress.

*2.1.1*

“Properties of Confined Hydrogen and Helium Atoms"^{2} analyzes the solutions for the Schrödinger equation for the hydrogen atom confined in a spherical box with the boundary condition of vanishing at the radius of the box R, as well as the polarizability of the atom. In this way model wave functions are constructed for the entire range of ∞; interpolating between those of the free atom for

Table 1 provides a comparison of the energies of the confined hydrogen atom in the state ^{2}^{,}^{10}. The agreement between the first two entries is quite good for all states and radii, while the differences of the third entry increase as the radius is decreased and the excitations are higher.

The helium atom confined in a spherical box is analyzed in terms of the effective Hamiltonian of Eq. (3.65) using screening factors ^{29} and ^{39}.

There are also two sections on the hydrogen atom, centered in a sphere and at a focus in prolate spheroidal coordinates, confined by a centered harmonic oscillator potential, respectively.

*2.1.2*

“Exact Relations for Confined One-Electron Systems"^{3} deals with the spherical confinement of the hydrogen atom in Sections 1-6, the isotropic harmonic oscillator in Section 7, and information theoretical uncertainty-like relations in Section 8. Its bibliography is very detailed and complete, proving very useful for the reader. Additionally, some of its results are very graphical and their validity goes beyond the specific topic. Since its Section 2 has a good overlap with the previous chapter, we go on to its Section 3 on “Commutation Relations and Hypervirial theorems". In order to connect some of the basic concepts, we start by pointing out the difference between the boundary conditions of the Dirichlet, Neumann and Robin types of vanishing wavefunction, vanishing normal derivative of the wavefunction and logarithmic derivative of the wavefunction equal to a real-valued function *P* and the same functions. While the constants of motion commute with the Hamiltonian, a dynamical quantity *B* is calculated and the proper boundary conditions of the eigenfunctions are taken into account. The section under consideration goes on to consider scaling transformation and to obtain the Kirkwood-Buckingham relation, including illustrations of their applications. Sects 4) “Energy and Region Modifications", 5) “The system in an external potential", and 6) “On Mean Values and Other Properties of Confined Systems" illustrate specific results for the different properties of the confined systems for the different conditions of confinement.

Section 7 explains the degeneracy of the confined D-dimensional harmonic oscillator and also the density at the equilibrium point as investigated in its Ref.^{105}. From the properties of the confluent hypergeometric functions, the interdimensional state degeneracy under the transformation given by

Section 8 defines the one-electron Shannon information entropy ^{109}. The total entropy as the sum of ^{110}. Figure 5 illustrates *R* for the confined hydrogen-like atoms.

*2.1.3*

The hydrogen atom confined in semi-infinite spaces limited by Conoidal boundaries^{4} is different from the other chapters in the open versus closed regions of confinement and the respective boundaries of confinement. As already mentioned in the Introduction, most of the chapters deal with confinement inside a sphere or spheroid, and one of them in a semi-infinite space with a plane boundary. This chapter reports on a series of works with paraboloidal, hyperboloidal, and circular conical boundaries confining the hydrogen atom. The review led to recognize the complementary possibilities of confinement by dihedral angles and by elliptical cones. Exact solutions of the Schrödinger equation in spherical, spheroidal, paraboloidal and spheroconal coordinates for the free hydrogen atom are known; the presence of the natural boundaries with a fixed value of the respective coordinates lead also to exact solutions for the hydrogen atom confined in the corresponding closed or open space. The latter exhibit symmetry breakings in comparison with the free system; the confinement by open boundaries has served to study surface effects on atoms and molecules.

*2.1.4*

“The Hydrogen and Helium Confined in Spherical boxes"^{5} overlaps in contents and methodologies with the previous chapters. Here we emphasize the review of the complementary methods of solutions and the higher accuracies attained in the numerical calculations. Its bibliography is also very complete and recommendable. For the hydrogen atom the exact solutions are products of the factors removing the singularity at the origin and at infinity, and the regular Kummer hypergeometric functions. Perturbation theory for very small boxes using the free particle confined in a spherical box as the non perturbed system and the nucleus-electron Coulomb energy as the perturbation leads to a series of powers of the confining radius for the energy of the ground state. The method of linear variational functions applied in its Ref.^{12} uses the superposition of ns hydrogen orbitals with ^{13} used the basis of the spherically confined isotropic harmonic oscillator; he also reports the variational boundary perturbation theory, on its Ref.^{86}, using B-splines functions. Other variational methods include the variational boundary perturbation theory Refs.^{14}^{,}^{15}, and variational trial functions with a linear cut-off factor (^{16}^{-}^{18}.

Concerning the increasing accuracy of the computed results, Tables 5-8, Figs 1-2, with their associated discussions and references are very illustrative and impressive.

The investigations on the helium atom confined in a spherical box started fifteen years later after those on the hydrogen atom. Ref.^{49} did it by using a Hylleraas trial wave function for the ground state ^{101}, ^{103}, ^{105}, ^{108}, ^{117} and ^{118} followed in the successive decades. Table 10 illustrates the comparison of their respective results for ^{105} was one of the pioneering applications of the Diffusive Monte Carlo type, including those for ^{102}, ^{104}, ^{16}, ^{18}, and others illustrating simple variational calculations.

*2.1.5*

“Exact solution for Confined Model Systems Using Kummer Function"^{6} shows applications to the hydrogen atom in 3 and 2 dimensions, and to the Harmonic Oscillator in 2 dimensions; as well as to the constant potential, for the free particle inside an impenetrable spherical wall, and for the exterior solutions for soft confinement. Section 2 presents the different forms of the Kummer confluent hypergeometric functions in their M and N forms, regular near the origin and in the U form regular at infinity; including their relationships Eq. (20), other alternatives Eq. (23), for the first two; and also the last one, Eq. (41) and its alternatives Eq. (58). The derivative of M Eq. (19) and that of U Eq. (43) lead to the same type of functions with their parameters shifted by one unit are also important. The analysis of the exceptional solutions, and the examples of taking the limits to arrive at two independent solutions Eqs.(39) and (40), are also illuminating. The applications to the K-dimensional hydrogen atom in its free configuration lead to its polynomial eigenfunctions and eigenvalues Eqs. (65) and (66), and likewise for the harmonic oscillator, Eqs. (70). In the case of the constant potential, the situation for the solution in Eq. (73) is different, because the M function does not truncate for the good behavior at both

Section 3 defines the confinement in terms of boundary conditions, including examples of exterior confinement, Eqs. (75) and (76). Another exterior confinement with reduction of the dimensionality of the system, Eq. (77), describes a 3-dimensional system which is separable in its transverse and axial components, Eqs. (78) and (79); the transverse potential

The boundary condition of

Sections 4, 5 and 6 give examples of the different confinements for the successive potentials.

*2.1.6*

“Perturbation theory for a Hydrogen Atom Confined Within an Impenetrable Spherical Cavity" (7) is implemented for large, small and intermediate values of the radius of the cavity evaluating accurately the energies, oscillating strengths, dipole polarizabilities and nuclear shielding factors for the different eigenstates. The asymptotic expansions for the energy, Eqs. (24), for the eigenfunctions in terms of the oscillator strengths, Eq. (28), were developed by the Author in its Ref. ^{25}. Table I shows the comparison of the precise numerical and asymptotic values of the energies of the states 1s, 2s and 2p for ^{4}^{,}^{40}, illustrating their improved agreement for larger radii. On the other hand, the same Author in its Ref. ^{31} obtained expressions for the oscillator strength ^{12}, ^{29}, and ^{33}^{-}^{37}. The small size boxes are based on perturbation theory taking the free electron in the spherical box as the unperturbed system and the nucleus electron coulomb energy as the perturbation Tables 4, 5, 6 show the comparisons of the small-box perturbation and precise numerical energies for the ns, np lower energies and for the 3d, 4d, 5d and 4f, 5f, 6f eigenstates for the successive ranges of the confining radius and the respective estimated radius of convergence. Table 7 shows the perturbation theory expansion coefficients for the expansion of energies in powers of the confining radius for the 1s, 2p, 3d, 4f and 5g levels to order 31.

We go on to discuss the intermediate box size perturbation theory implemented around the radial nodes of the free hydrogen atom radial eigenfunctions. The Author in its Ref. ^{25} derived the first order approximations to energy of Eqs. (89)-(91) in the vicinity of the indicated radius. Higher order corrections can naturally be also evaluated, as illustrated by Eqs. (98)-(99). Correspondingly, Eqs. (100)-(101) give the corresponding expansions for the polarizabillity and nuclear shielding factors, and Tables (14) and (15) illustrate their numerical values, compared to precise values for the different radius of confinement.

*2.1.7*

“Comparative Study Between the Hartree-Fock and Kohn-Sham Methods for the Lowest Singlet and Triplet States of the Confined Helium Atom" ^{8} investigated the same physical system and states as in 2.1.4, in two alternative formalisms. Section 2 contains the basic Hartree-Fock equations for the energy of the helium atom in its ground state, considered as a close-shell system, and for its first excited state considered as an open-shell system with the difference that the latter contains the additional exchange integral contribution. The Kohn-Sham (K-S) equations are also presented for the respective states, and with the choice of the OEP-KLI exchange-correlation functional for the open shell system. The Hartree-Fock (H-F) and K-S Eqs. (15) and (18) for the closed-shell system are the same. The corresponding comparison of Eqs. (17) and (19) for the open-shell system shows a difference. The results of the Table II of a previous work, in the common Ref. ^{19} here and ^{117} in 2.1.7, show that for the closed-shell system there are discrepancies between the HF and KS-SIC-OEP-KLI methods. The aim of^{8} is two-fold: 1) recompute the ground state results by using an optimized basis set and 2) compute the lowest triplet HF state by the HF method, and compare its results with those of the KS method. In this way a reliable comparison between HF and KS is made for the confined He atom in both states.

Table 1 shows the total energy and the exchange energy for the singlet state as functions of the confining radius in KS-LDA, KS-LDA-SIC and HF including a comparison for the latter in Ref. ^{4}. The comparison of the respective entries in the last three shows a fairly good agreement among them, while those of the first one are different in the first decimal figures.

Table 2 shows the correlation energies estimated as the difference between the precise values of the energy, based on wave function expanded in a 40 Hylleraas basis set, and the HF energies of this work and its Refs. ^{30} and ^{31}, as functions of the confining radius. The first two entries show that the correlation energy does not show appreciable changes in the third decimal figure. The Authors conclude that these results can be used as bench marks for other correlated methods.

Table 3 contains the corresponding results for the confined helium atom in the lowest triplet state. Their respective entries differ significantly compared with the ground state.

The Authors conclude that this work shows the importance of exponent optimization in the Roothan approach to solve the H-F equations for the singlet and triplet ground states of the confined helium atom. The comparison of the HF results and the KS-SIC-OEP-KLI results exhibits that the latter is inadequate for the description of the unpaired electrons.

*2.1.8*

“Thomas-Fermi-Dirac-Weizsäcker Density Functional Formalism Applied to the Study of Many-electron Atoms"^{9} provides references on calculations using ab-initio Hartree-Fock ^{67}^{,}^{63} and Dirac-Fock ^{61} procedures, density functional theory^{4}^{,}^{60} and Thomas-Fermi-Dirac-Weizsäcker (TDF𝜆W) statistical atomic model^{64}^{,}^{65}, for confined atoms. The first three have been restricted to confinement in a spherical box with an impenetrable wall, while the fourth one has been applied also to spherical boxes with penetrable walls, prolate spheroids with impenetrable walls, and an open space with a plane hard wall boundary. Section 2 describes the details of the calculations for confinement by closed boundaries with illustrative results for different atoms: 1) Spherical hard box. Figure 2 for ground state energy of Neon versus HF values^{4} in very good agreement. Table 1 for Carbon and Neon, also compared with^{4}, and values reported in^{64} including values of optimized orbital parameters ^{65} using exact electron-electron interaction versus Poisson solution. This type of confinement allows the escape of the electrons through the box when the energy equals the height of the barrier ^{46}^{,}^{65}. Table 4 shows the corresponding informations for oxygen, silicon, and germanium for a barrier of height

Section 3 deals with the confinement in the presence of a plane hard wall boundary using prolate spheroidal coordinates, corresponding to the equatorial plane ^{46}^{,}^{75}. This model is important for the atom-surface interactions.

*2.1.9*

“Confined Atoms Treated as Open Systems" ^{10} is conceptually methodologically, and content-wise different from the other chapters. The system of interest is bounded by surfaces it shares with the atoms of the confining material: matter and momentum can be transfered across the surfaces. This is a problem of quantum mechanics of a proper open system

Section 4 deals with calculations of the pressures in terms of the surface virial. Calculation of the pressure in terms of the surface virial is illustrated in 4.3 for atomic pressures in diatomic molecules, 4.4 pressure exerted on atom confined in adamantane cage and 4.5 compression of hydrogen molecules in a neon vise.

In 4.3 the diatomic molecules

Table I gives the properties for the AB representative molecules: binding critical point charge of A, surface virial of A and B, volumes of A and B, and pressures of A and B. Data for the diatomic hydrides AH ground states, A=Li, Be, B, C, N and F are illustrated in Fig. 2. The zero-flux interatomic surfaces with the proton at the fixed position labeled H. The bond critical point for Li is located 1.67 a,u, from the proton, and are increasingly smaller for the other atoms. Table 2 illustrates the same properties as in Table 1. Figure 3 contains the atomic pressures for the successive molecules; the highest one on H is in BH. Data for the diatomic fluorides AF (A=Li,Be,B,C,N,O) in the ground state are shown in Fig.4 for the atomic pressures in the successive molecules; the highest one on F is in the polar molecule CF. Table 3 illustrates the same data as Tables 1 and 2 for the fluorides.

In 4.4 data on atoms and ions

In 4.5 The system used to introduce the quantum definition of pressure consists of a linear chain of hydrogen molecules compressed between a pair of Ne atoms. Figure 7 illustrates the case of five

*2.1.10*

“Modelling Pressure Effects on the Electronic Properties of Ca, Sr, and Ba by the Confined Atoms Model"^{11} uses the confinement inside a sphere with an impenetrable wall and the spin-polarized version of Density Functional Theory. The methodology described in Sec. 2 covers the pressure as the negative rate of change of the total atomic energy with respect to the volume, Eq. (2), and the SP-DFT framework, in which the total energy changes according to its dependence on the number of electrons N and the spin number N 𝑠 as the natural variables, Eq. (3). For the atoms under consideration, in the free configurations there are singlet-triplet transitions for the valence electrons ^{32} gives the ^{33} in the LDA ^{34} is used in this work. The results are presented and discussed in Sect.3 under the following headings:

1) Transition pressures for Ca, Sr and Ba. Figure 1 illustrates the orbital energies 3p, 3d and 4s for the confined Ca atom decreasing as functions of the increasing radius of confinement, the last two ones show a crossing around ^{14}^{)} and ^{15}. The comparison is favorable for Ca, but not for the heavier atoms.

2) Electronic transitions as functions of pressure. According to Fig. 3 the possibilities of excitations energies are three:

*2.1.11*

“Photoionization of Atoms Encaged in Spherical Fullerenes"^{12} discusses in its Section 2 The Modeling of Doped Fullerenes. In its first subsection on Preliminaries, it is explained that the atom is confined with its nucleus at the center of the cage and that the radius of the cage is significantly larger than the radius of any

On the other hand, the cages may be neutral, charged or multilayered. Their following subsections describe modelings, successively.

2) The

3) The

4) The

with a, b, c,...either 1 or 0 for the successive layers.

5) The _{
c
} . It is also the limiting situation of the

*2.1.12*

“Spectroscopy of Confined Atomic systems: Effect of Plasma"^{14} reviews the experimental techniques, Theoretical Development, Interpretation of Spectral Properties and Atomic Data of atoms under Liquid Helium. In the Introduction it is recognized that atoms or ions embedded in a plasma also form a class of confined systems. Their atomic structural properties may change drastically depending on the coupling strength of the plasma with the atomic electrons.

The coupling constant of a plasma is defined as the ratio of the average Coulomb energy to the average kinetic energy of the plasma particles. For plasma particles obeying classical statistics, the average kinetic energy per particle is

Concerning the theoretical modeling of the plasmatic confinement in the respective coupling limits, Eq. (10) describes the Debye screening model potential as a Coulomb potential with an exponentially radial decreasing factor, and Eq. (17) the ion-sphere model for the interaction of an ion with a bound electron as the superposition of the electron-nucleus Coulomb energy and the electron-(Z-1)electron charge uniformly distributed in the volume of a sphere of radius 𝑎 electrostatic repulsion energy.

*2.1.13*

“The energy level structure of Low-dimensional Multi-electron Quantum Dots"^{15} reviews the Computational Methodology, Quasi-one-dimensional Quantum Dots and Quasi-two-dimensional Quantum dots investigations.

In the Introduction, Quantum dots are identified as confined quantum systems of a finite number of electrons bound in a fabricated nanoscale potential, 1-100 nm, with a discrete energy-level structure following Hund’s rules. Their properties can be changed by controlling the size and/or shape of the fabricated potential. The energy-level structure and optical properties of quantum dots and atoms are qualitatively different because of the differences in their respective confining harmonic and Coulomb potentials, apart from their sizes and dimensionalities.

The computational methodology based on the quantum chemical molecular orbital theory allows the calculations of the ground or low-excited states of multielectron quantum dots for a specific value of the strength of the confinement. The results vary depending on the strength of the confinement, due to a strong variation of the relative importance of the electron-electron interaction with respect to that strength; consequently, it is necessary to develop a unified method of interpreting the complicated energy-level structure of quantum dots for the whole range of the strength of confinement.

Previous studies allowed the identification of the polyad quantum number, defined by the total number of nodes in the leading configuration of the configuration interaction, CI, wavefunction, as approximately conserved for harmonic oscillator quantum dots.

In the present contribution the interpretation of the energy-level structure of quasi-one-dimensional quantum dots of two and three electrons is reviewed in detail by examining the polyad structure of the energy levels and the symmetry of the spatial part of the CI wave functions due to the Pauli principle. The interpretation based on the polyad quantum number is applied to the four-electron case, and is shown to be applicable to the general multi-electron cases. The qualitative differences in the energy-level structure between quasi-one-dimensional and quasi-two-dimensional quantum dots are briefly discussed by referring to differences in the structure of their internal space.

Electronic Structure of Quantum Confined Atoms and Molecules 2014

In this section, the further advances in the period 2009 to 2014 are illustrated by connecting and comparing the respective pairs of reviews ^{1}^{-}^{18}, ^{4}^{-}^{17}, ^{5}^{-}^{19}, ^{2}^{-}^{20}, ^{11}^{-}^{24} with common coauthors on related topics; and also by describing and analyzing the other reviews in the monograph ^{21}^{-}^{23}, ^{25}.

*2.2.1*

“Surface Effects in the Hydrogen Atom Confined by Dihedral Angles" ^{17} presents the updating of ^{4}, reporting mainly the results of the described confinement which is illustrated here in two parts: with the titles of the articles reviewed in its Section 2, and its main body in Section 3 on the topic of the title.

In fact, its Refs. ^{20} Comment on the electron in the field of an electric dipole moment, ^{21} Ground-state energy shift of He close to a surface and its relation with the scattering potential: a confinement model, ^{22} The hydrogen atom in a semi-infinite space with an elliptical cone boundary, ^{23} Lamé spheroconal harmonics in atoms and molecules, ^{24} Rotations of asymmetric molecules and the hydrogen atom in free and confined configurations, ^{25} Ladder operators for Lamé spheroconal harmonic polynomials, ^{26} Ladder operators for quantum systems confined by dihedral angles, ^{27} Complete pure dipole spheroidal electrostatic fields and sources, ^{28} Surface current distribution on spheres and spheroids as sources of pure quadrupole magnetic fields were reviewed in Sections 2.1-2.5, illustrating further advances previewed in ^{4}.

The exact solutions for the hydrogen atom confined by dihedral angles were constructed in spherical, prolate, spheroidal and parabolic coordinates. The energy levels share common degeneracies identified in Sect. 3.1 and illustrated in Table 1 in terms of the respective quantum labels of their eigenstates

*2.2.2*

“Symmetry Reduction and Energy Levels Splitting of the One-Electron Atom in an Impenetrable Cavity" ^{18} is methodologically connected with ^{3}. The Reviewer singularized this chapter since the Introduction to illustrate the ideas of superintegrability and symmetry breaking. Here we elaborate some more on the basic idea of symmetry, which in quantum phenomena manifests itself in the degeneracy of the eigenstates with a common eigenenergy; correspondingly, the reduction of the symmetry leads the reduction of the degeneracy with the consequent splitting of the energy levels. Concerning the confinement in polyhedral cavities, the reader’s attention is directed to: Sect. 2.3, Fig. 1 Energy levels for hydrogen at the center of the cavity formed by joint symmetrical truncation of both the cube with the edge of 4𝑎𝑢 and the octahedron; and Fig. 2 Energies of isotropic harmonic oscillator with the force constant

*2.2.3*

“The Confined Hydrogen Atom Revisited"^{19} is connected with the first part of ^{19}, and the second Author describes the first one as a continuation of the second. It is based on two of their own recent References, reviewed in Sec. 3 Variational and Perturbative Treatments of the Confined Hydrogen Atom with a Moving Nucleus, and in Sec. 4 Shannon and Fisher Entropies for a Hydrogen Under Soft Spherical Confinement, illustrating the elements of the novelty in their respective contents.

In Sec. 3, the conclusions in the moving nucleus versus the fixed nucleus modeling are the following. The increase in the kinetic energy due to the moving nucleus is comparatively larger than the energy difference between the Coulomb interactions for each scheme and all the values of the confining radius. The average nuclear position compared with the average electron position is closer to the origin by a factor of 34% for

In Sec. 4, the Shannon and Fisher entropies were calculated in the position (

*2.2.4*

“Variational Perturbation Treatment for Excited States of Confined Two-Electron Atoms" ^{20} is connected with the second part of ^{5}, and ^{8}. The Authors report Hylleraass-Scherr-Knight variational perturbation theory (HSK-VPT) calculations of the energies of the 1s 2s

*2.2.5*

“Confined Quantum Systems Using the Finite Element and Discrete Variable Representation Methods" ^{21} is a review of its Refs. ^{9}^{,}^{11}^{,}^{17}^{,}^{50}^{,}^{58}^{,}^{62}^{,}^{63}^{,}^{68}^{,}^{74}^{,}^{76}^{,}^{89}^{,}^{90}^{,}^{91}^{,}^{117}^{,}^{124} by the same Authors, in which they have developed and applied the methods in the title to a variety of quantum systems under different situations of confinement. Section 4 illustrates some of these specific applications showing numerical results matching favorably the most accurate in the literature.

4.1 One-dimensional harmonic oscillator confined by a modified Wood-Saxon potential, Eq. (5.27) and Fig. 1, was investigated in ^{11} in order to simulate the confinements in 1) The Zicovich-Wilson square tangential potential, Eq. (5.28) and in 2) A constant infinite barrier. The intensity and slope parameters of the Wood-Saxon potential are optimized to an optimum fit to the respective potentials, and the eigenenergies were calculated using 300 equality spaced points between [-5,5] and [-4,4] to build the Discrete Variable Representations. Table 1 shows the comparison of the results from ^{11} and ^{10}, and Table 2 those fron ^{11} and those from ^{94} and ^{95}.

4.2 Confined Hydrogenic Atom/Ion. This section illustrates of the theoretical and computational methodology based on the variational formalism for the bounded states and the p-version of the Finite Element Method, applied to the hydrogen atom confined in 1) an impenetrable spherical box and 2) in a

This case was investigated in ^{9} and its results for the energies of the 1s and 3s states, respectively, as functions of the confining radius are illustrated in Table 5 compared with those by Varshni, Zicovich-Wilson, Saha, Joslin and Aquino, and in Table 5.4 compared with Varshni, Goldman and Aquino. Table 5.5 illustrates the results on the polarizability for the 1s state compared with those of Dutt, Banerjee, Saha and Laughlin.

The endohedral confinement of the hydrogen atom was modeled in ^{17} by adding a Gaussian potential attractive wall to the Coulomb potential, Fig. 5.6, fitting its parameters to approximate the potential of Connerade et al ^{15}. Figure 5.7 shows the energies of the 1s, 2s, 3s and 4s states of H@ C 60 from the p-FEM calculations compared with those from ^{15}, as functions of

The plasmatic confinement of ions is modeled by Debye exponentially screened Coulomb potential, Eq.(5.38), in the weakly coupled plasma ^{117} obtained with self-consistent FEM are illustrated in Table 5.6 for the ground state energy (^{19}. Figure 5.10 illustrates the energy levels, -E, against the radius R of the ion-sphere for the hydrogenic ion, 2s, 3s, 2p, 3p ,3d, and also the 1s without a surrounding.

Section 4.3 contains the results of the energy spectrum for two types of quantum dots: 1) an impurity located in a parabolic quantum dot, and 2) a two-electron quantum dot. The spectrum is computed using the DVR method.

The Hamiltonian includes the kinetic energy, the electron-nucleus Coulomb energy, the harmonic oscillator energy and a Wood-Saxon potential. Table 5.8 illustrates the optimized Wood-Saxon parameters and binding energies for the quantum dot using the DVR method ^{11} compared to Varshni’s ^{118} exact values.

The Hamiltonian includes the kinetic energy and the harmonic oscillator potential for the two electrons confined in the quantum dot with an effective mass m*, and the Coulomb e-e repulsion in the medium with dielectric constant 𝜖, Eq. (5.42). The motions of the center of mass and the relative one between the electrons can be separated. The first one for the anisotropic confinement has the spectrum of Eq.(5.43) for the corresponding transverse and axial oscillators

The relative motion equations (5.44-5.47) were solved in ^{50} using a variational scheme based on the DVR method, using the spherical harmonics and radial function expansions. The calculations included 30 spherical harmonics and 100 radial basis functions. The solutions for the latter were obtained from 2500 DVR basis functions equally spaced in appropriate intervals of the pairs of transverse and axial frequency parameters. The energy spectra have a precision of at least 6 significant digits.

For the isotropic case, the center of mass energy spectrum is degenerate, Eq. (5.48), with the combination

Table 5.9 illustrates the three dimensional two-electron quantum dot energies for different values of the confining parameter and different configurations (NL,nl) from ^{50} based on DVR method and compared with results from ^{120}. Figure 5.11 illustrates the corresponding energy spectrum of the same two-electron quantum dot ^{50}.

Figure 5.12 illustrates the relative motion energy levels of anisotropic two-electron quantum dots for ^{50}. Figure 5.13 illustrates the total energies ^{50}.

*2.2.6*

“Bound and Resonant States in Confined Atoms" ^{22} is an investigation of the bound and autoionizing resonant states of two-electron Quantum Dots (QD) with

Section 3 presents the succesive results and discussions. 3.1 One-electron QD state. 3.2 Two-electron QD Bound states. 3.3 Two-electron Resonant States. Section 4 contains concluding remarks.

*2.2.7*

“Spatial and Shell-Confined One Electron Atomic and Molecular Systems: Structure and Dipole Polarizability" ^{23} investigates comparatively the confinement of the hydrogen atom in an impenetrable spherical box and in a spherical shell

The results are presented, illustrated and favorably compared with those in the literature in the following order. 3.1 Electronic energies and polarizabilities of hydrogen molecular ion spatially and endohedrally confined in spheroidal boxes. 3.2 Energies and polarizabilities of endohedrally confined centered Hydrogen atom. 3.3 Energies of spatially confined centered hydrogen atom. 3.4 Energies of spatially and endohedrally confined off-center hydrogen atom. 3.5 Dynamic polarizabilities of spatially confined centered and off-centered hydrogen atom. 3.6 Energies and polarizabilities of cylindrically confined hydrogen atom.

*2.2.8*

“Density Functional Theory Applied on Confined Many-Electron Atoms" ^{24} is connected with ^{9} and ^{11}, some of those connections allow us to concentrate here on Section 3 Modified Thomas-Fermi Model and Section 4 Kohn-Sham Model.

In Section 3 the starting point recognizes the unphysical divergent behavior of the electron density at the position of the nucleus in the Thomas-Fermi Model. The modification that Parr and Gosh proposed, Eq. (8.26) as a restriction in the minimization process, combined with the cusp condition on the derivative of the electron density and the electron density itself at the nucleus Eq. (8.27), lead to the well-behaved electron density of Eq. (8.28). The authors develop their own method to solve the equation in terms of the series expansion in Eq. (8.34). Table 8.1 shows the lower coefficients from their recurrence relations. Table 8.2 shows the values of the electron density at the nucleus for the noble gases

In order to estimate the pressure according to the Slater and Krutter proposal, the Poisson equation is sufficient, and Eq. (8.41) expresses the proportionality of the derivative of 𝑄(𝜔) with respect to 𝑄(𝜔) at the surface of the sphere

In Section 4, the Kohn-Sham formalism with different exchange-correlation functionals is applied to Na and K atoms under central spherical and impenetrable confinement. In Section 4.1, Table 3 illustrates the numerical values of the total energies for

*2.2.9*

“Study of Quantum Confinement of ^{25} incorporates the nuclear motion in contrast with the familiar Born-Oppenheimer approximation with nuclei in fixed positions; additionally, the calculations are based on the variational and diffusive versions of Monte Carlo. Sections 2.1 and 2.2 deal with the theory for both molecules starting from their respective 1. Schrödinger equations, with three and four particles and their interactions, including the confining potentials for electrons and nuclei. By using Jacobi coordinates, the motion of the center of mass is removed and the Schrödinger equations in terms of relative positions and reduced masses are obtained, Eqs. (9.5) and (9.30). 2. Trial wavefunctions for the free molecular ion as the product of the nuclear vibrational and rotational function

For the lowest vibrational rotational level,

When the nuclei are fixed, their separation R is a parameter as in the B-O approximation. The variational optimization of

When the nuclei are allowed to move,

3. Trial wavefunctions for the confined

The trial wave function for the free molecule

involves the same nuclear vibrational rotational function with their respective coordinates, and a Jastrow correlation factor, Eq. (9.35),

Section 2.3 and 2.4 describe the implementation of the Variational and Diffusion Monte Carlo calculations for Dirichlet boundary conditions.

Section 3 presents the results and discussion for 1. Hydrogen atom located at one of the foci confined by a hard spheroidal surface, with comparisons with its Refs. ^{20} and ^{9}. 2. Clamped nuclei H 2 + molecular ion by a spherical surface compared also with ^{20} and ^{9}. 3. Confined three-body

3.Recent Progress in the Literature up to January 2018

The literature in this section has been selected with the criteria of elements of novelty in some of the key words in the Abstract, which have been illustrated in Sec. 2 for the reviews in Advances in Quantum Chemistry, and also in the two types of review in the Monograph. The recent progress in the background of the set of those reviews is presented also by going from the simpler to the more complex confined quantum systems. Additionally, the new electromagnetic toroidal interactions and the new force approach to the old radiation reaction problem are also included.

3.1.Hydrogen-like Atoms

The common confined element in the twenty references reviewed here is the hydrogen atom or hydrogen-like systems. The readers may appreciate the recent progress along this line by comparing with ^{2}^{-}^{7} and ^{17}^{-}^{19}.

“Tunable Excitons in Biased Bilayer Graphene" ^{26} involves electron-hole excitations between the two layers. They open a pathway for possible nanoelectronic and nanophotonic devices operating at room temperature. The binding energy of the excitations may be tuned with the external bias going from zero to several tens of milielectron volts. The novel strong excitonic behaviors are associated with a one-dimensional Van Hove singularity joint density of states and a continuously tunable band gap based graphene layer. The excitonic electronic structure is markedly different from that of 2D hydrogen atom.

“Confinement approach to pressure effects on the dipole and the Generalized Oscillator Strengths of atomic hydrogen" ^{27} is based on the constant barrier outside model. The Schrödinger equation is solved by a finite differences method for the fixed values of the spherical radius and the barrier height. The GOS momentum transfer distribution for

“The hydrogen atom confined in both Debye Screening potential and impenetrable spherical box: ^{28} used the linear variational method based on B-splines basis functions to calculate the ground state and some low lying state energies and oscillator strengths. Comparisons with results in the literature for confined and free configurations are presented and discussed.

“High-harmonic generation and spherically confined hydrogen atom" ^{29} presents the investigation of the dynamics of the atom under the action of strong infrared femtosecond laser. The spherical box is introduced to model collisions of ionized electrons with neighboring atoms to model their influence on higher harmonic generation. The physical situation appropriate for such application is systematically analyzed.

“Static polarizability of an atom confined in a Gaussian potential" ^{30} uses the finite basis set method based on B-splines to calculate the energy spectrum, and the optical properties: dipole, quadrupole and octupole oscillator strengths and polarizabilities as functions of the depth and width parameters of the confining potential. The maximum uncertainty in the reported numerical results is estimated to be about

“Energy-level structure of the hydrogen atom confined by a penetrable cylindrical cavity" ^{31} reports the bound-state energy spectrum and its evolution for the atom located along the axis of the cavity, with penetrable and impenetrable boundaries based on the solution of the Schrödinger equation using a finite difference approach. New results are presented for a nuclear centered position for the penetrable case as the barrier height and cavity size change. Then special attention is given to the energy level evolution of states from the nuclear centered position (centered states), up to the cylinder cap (cap states), while the corresponding state evolution for intermediate nuclear positions (intermediate states) remains consistent with node conservation and symmetry. The energies of a given state increase as the nucleus moves up. The evolution as the barrier height and cavity size are reduced consists of the progressive extinction of the bound states in the order of cap, intermediate and centered states. A predominance of cavity size over barrier-height effects on the energy level shift is found.

“Endohedrally confined hydrogen atom with a moving nucleus" ^{32} studied the hydrogen atom as a system of two quantum particles in different confinement situations: a spherical impenetrable wall cavity and a fullerene molecule cage. The motion is referred 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 in the literature, in which the proton is fixed in space and the electron is a quantum particle. The position of the proton is found to be very sensitive to the confinement condition.

“Confinement effects on the electron transfer cross section: a study of ^{33} analyzes the pressure effects on the target and the physical process of interest by using the model of spherical confinement of the target at the center. The electron transfer probability is obtained by a time dependent solution of the Schrödinger equation by means of a finite difference approach and the Crank-Nicolson propagation method. Results are presented for the benchmark system

“The hydrogen atom confined by one and two hard cones" ^{34} studies the bound states of the system is a semi-infinite space. The solutions when the nucleus is at the apex of the conical boundaries and the electron is confined outside one or two symmetric of such boundaries can be evaluated exactly. The third case assumes the position of the nucleus as the origin of the coordinate system and the apex of one conical boundary at a distance h along its axis; the solution can be evaluated by the variational Monte Carlo Method. The latter is applied to calculate the force in an Atomic Force Microscope as the negative of the rate of change of the energy with respect to h.

“Benchmark Calculation of Radial Expectation Value ^{35} recognizes that for those physical systems numerous physical quantities have been established with very high accuracy. However, the expectation values in the title of practical importance in many applications has significant discrepancies among calculations by different methods. In this work the basis expansion method with cut-off Slater-type orbitals is used to investigate the two confined systems. Accurate values for several low-lying bound states were obtained by carefully examining the convergence with respect to the size of the basis. A scaling law for

“Sum rules and the role of pressure on the excitation spectrum of a confined hydrogen atom by a spherical cavity" ^{36} investigates the effects of pressure induced by a soft spherical cavity on a centered atomic hydrogen impurity, on the dipole oscillator strength sum rule

“Monotonicity in confined system problems" ^{37} is concerned with the changes in the electron density when an atom is confined at the center of an impenetrable confining sphere. The work presents four simple theorems that facilitate understanding the behavior of the electron density of the ground state with variations in the potential or in the radius of the impenetrable spherical cavity.

“Exact solution for the hydrogen atom by a dielectric continuum and the correct basis set to study many-electron atoms under similar confinements" [38] introduces a new model of confinement, alternative to the polarization potential of Jortrer and Coulson, and an alternative basis set to construct the exact solution of the corresponding Schrödinger equation, distinguishing between the solute and the solvent. Results are reported for the total energy, electron density, charge in the dielectric medium and Shannon entropy, as functions of the radius of confinement and of the dielectric constant 𝜖 ^{80}, where the extreme values correspond to Teflon and water, respectively.

The two-electron atom and ions

“Fine structure in the hydrogen atom boxed in a spherical impenetrable cavity" ^{39} reports the relativistic corrections of the kinetic energy, the spin-orbit coupling and the point interaction Darwin term to the non relativistic familiar system based on the nucleus-electron Coulomb interaction only. The first one is negative; the second one couples the orbital and spin angular momenta to a total one with

“Confinement of Atoms with Robin’s Condition: Spontaneous spherical symmetry breaking in atomic confinement" ^{40} and Spontaneous spherical symmetry breaking in atomic confinement ^{41} are commented together, pointing out their use of the “not going out" Robin’s boundary condition, and their identification of a symmetry breaking effect.

The boundary condition corresponds to the logarithmic derivative of the wave function at the confining surface being a real function

“Fisher information in confined hydrogen-like ions" [42] presents an investigation of Fisher information for the CHA-like systems in conjugate r and p spaces, compared with the, I, case of the free atom. Systematic results on I as a function of the confining radius are presented, with emphasis on the non-zero-(

“Derived properties from the dipole and generalized oscillator strength distributions of an endohedral confined hydrogen atom" ^{43} uses two alternative potentials, square-well and Woods-Saxon, to model the concentric spherical shell

“Relativistic two dimensional H-like model atoms in an external magnetic field" ^{44} are investigated on the basis of Klein-Gordon and Dirac equations solved by the power expansion method. For a given magnetic field B the approximate solutions are determined by imposing boundary conditions at a finite radius R. For particular values of B, exact polynomials solutions are obtained. The dependence on the effective coupling constant

“Atoms confined by very thin layers" ^{45} provides a mathematical justification for the interest in the two-dimensional atoms with the three-dimensional Coulomb potential. The Hamiltonian of an atom with N electrons and a fixed nucleus between two parallel planes is considered in the limit when the separation between the latter tends to zero. It is shown that the Hamiltonian converges in the norm resolvent sense to a Schrödinger equation acting effectively in

3.2.Diatomic Hydrogen-like Ion and Neutral Molecules, and Two-electron Atoms

This section reviews seven articles on molecules and one on He-like atoms.

The article “^{46} has the background of ^{14} and the works of its second author ^{3}^{,}^{20}. The embedding in a hot dense plasma is modeled by multiplying the Coulomb potential interactions with a Debye screening exponential radial factor with a decay distance D. The analysis of the problem is made within the Born-Oppenheimer approximation and in prolate spheroidal coordinates with the protons at the positions of the foci. The solution is based on the variational trial function of Eq. (6) for the ^{11}, ^{10}, ^{25}, and ^{26}; the energies and the polarizabilities are consistently lower from the other reported values. Table 4 illustrates the expectation values for the potential energy,

“The Effect of Confinement on the Electronic Energy and Polarizability of a Hydrogen molecular Ion" [47] is based on the simplest variational trial function consisting of the linear combination of two 1𝑠 hydrogen atom orbitals, reporting numerical values of the

This Reviewer included in ^{4} and ^{17} sections on comments to works in the literature, inviting the readers to do their own readings of the respective articles. Here, he reiterates such an invitation and makes his own specific comment: According to the Variational Principle, a variational result for the energy below the exact value is not better, it is wrong.

In “The hydrogen molecule and the ^{50}, a variational approach has been proposed for the nonseparable Schrödinger problem of the molecules in the title, in their ground states, confined by soft cavities and the nuclei not in the foci positions. The role of barrier height potential and cavity size and shape on the ground state energy of both confined molecules has been analyzed showing their importance in defining equilibrium bond lengths and energies by allowing full nuclear relaxation, consistent with the confinement conditions. The variational wave function for the

Figure 2 illustrates the variations of the total energy for H 2 + 1𝑠 𝜎 𝑔 with the nuclear separation for barrier potential height and eccentricity

Table I provides the numerical values of the total energies for

Table 2 illustrates the total energies

Figure 3 illustrates a) the equilibrium electronic energy, and b) the equilibrium bond length, as functions of the size of the cavity

This model adds more flexibility for the treatment of the electronic and vibrational properties of one and two-electron diatomics when subjected to spatial confinement allowing for a more realistic comparison with experiment.

“Spherically confined

“Quantum states of confined hydrogen plasma species: Monte Carlo calculations" ^{52} implements the diffusion Monte Carlo method with symmetry-based state selection to calculate the excited states

In “Ritz variational calculation for the singly excited states of compressed two-electron atoms" ^{53}, a detailed analysis on the effect of spherical impenetrable confinement on the structural properties of two-electron ions in S-states has been performed. The energy values of ^{5}^{,}^{8}^{,}^{20}.

3.3.Many-electron Atoms

The articles to be reviewed in this subsection have antecedents and common Authors in ^{8}^{,}^{11}^{,}^{24} and illustrate some of their recent advances from their previous reviews.

The article “Implementation of the electron propagator to second order of GPUs to estimate the ionization potentials of confined atoms" ^{54} considers the confinement in spherical boxes with impenetrable walls, and The Electron Propagator Technique in the second order approximation EP2 to evaluate the energy to remove one electron from an orbital, Eq. (2), in the Hartree-Fock formalism. The use of the GPUs for the faster evaluations of two-electron integrals and four index integrals Transformations to implement EP2, were incorcoprated in the original MEXICA-C code via CUDA Kernels 1 and 2, respectively, Fig. 1. The calculations of the ionization potential were performed for the free atoms with ^{30}, including the predictions of Koopman’s Theorem and the EP2 rules. The comparisons of the results of ^{30}, the calculations from MEXICA and the experimental values in eV ^{31} show average derivatives of 0.50 to KT and 0.36 and 0.34, respectively. The authors attribute their smaller derivation to the use of the Gaussian functions in ^{30} to their own use of Slater type orbitals. The calculations for the confined atoms are reported for helium in Table 2, beryllium in Table 3, and neon in Table 4, for boxes of different radius; their general trends including those of other references is that they diminish as the radii became smaller, including vanishing values for different radius in the successive calculations. The differences between EP2 and KT values are also included; for He differences exhibit negative values for large radii and positive values for the smaller ones, for Be they are all positive, and for Ne they are all negative for the

The other three articles: “Roothaan’s approach to solve the Hartree-Fock equations for atoms confined by soft walls: Basis set with correct asymptotic behavior" ^{55}, “Solution of the Kohn-Sham equations for many-electron atoms confined by penetrable walls" ^{56}, and “Electron-density delocalization in many electron atoms confined by penetrable walls: A Hartree-Fock study" ^{57} assume spherical confinement with a constant potential barrier outside, accounting for the soft and penetrable walls, as well as the radial basis sets inside and outside with the correct behaviors for very small and very large distances from the nucleus, taken from the common Ref. ^{27}, ^{4} and ^{30}. The progress in these three articles, compared with the counterparts for impenetrable confinement in their previous works cited and reviewed in the first half of this section, are described next.

In ^{55} the atoms investigated were H, He, Be, Ne and K with the following conclusions, taking as a reference that for impenetrable confinement in which the energies increase monotonically as the radius diminishes: A) The shell structure is drastically modified, and a delocalization effect may occur for a certain barrier height, B) inner orbital energies do not necessarily go up when the radius diminishes, C) the ionization of many electron atoms is confirmed for a finite potential, and D) the asymptotic behavior of the electron density changes with the height of the potential.

In ^{56} the Kohn-Sham formalism was applied to study five closed shell atoms: He, Be, Ne, Mg and Argon for two heights U=0, 0.5 of constant confining potential barrier, using four exchange-correlation functionals. Tables 1 and 2 illustrate the results for the free atoms including HF values for comparison of the total energies evaluated with Gaussian and STO-CAB functions and the respective exchange correlation functionals. Table 3 shows the comparison of the exchange energy of He in the two situations of confinement from HF and KS in two of the exchange-correlation functionals. Table 4 includes the exchange and correlation energies, and their difference, for Mg in the two situations of confinement and for the fourth exchange- correlation functional. Table 5 shows the Highest Occupied Molecular Orbital energies for Mg in the two situations of confinement, comparing the results from HF and KS with the four functionals. The comparisons indicate the need to explore other exchange and correlation functionals.

In ^{57}, HF calculation for the atoms H, He, Li, Be, N, Na, Mg, P and Ar confined in a sphere with a constant potential barrier of heights 0 and 0.5 were performed. The corresponding results were the basis to evaluate the Shannon entropy of the respective atoms as a function of the radius of confinement. The results for H are illustrated graphically in Fig. 1 for a barrier height of 0.5 displaying the Coulomb potential and constant potential energies as well as the radial distribution function for A) with a radius 0.65, B) with a radius 1.1, and C) with a radius 4, and horizontal dotted lines for the corresponding orbital energies of 0.4908, 0.1583 and -0.4945, respectively. In B) the electron density is localized between 0 and 3.5, while in C) a minor localization is recognized. Figure 2 displays the orbital energies in squares and Shannon entropies in circles as functions of the confining radius, in the respective intervals [-0.5,0.5], ^{2}^{,}^{7} and [0,5]. While the energy increases monotonically as the radius is reduced, the entropy starts at about 4 for a radius of 5 decreases to a minimum of less than 3 for a radius of about one, and shows a marked increase to close to 7 as the radius is reduced to 0.65; the first interval corresponds to localization of the electron and the second one to its delocalization as the orbital energy approaches the height of the barrier. Table 1 shows the HF total energy and the occupied orbital energy for Li, Be, N, Na, Mg and P in the three situations of confinement for five different values of the radius. Table 2 shows the corresponding information for the noble gases He, Ne and Ar. The readers may ascertain the systematic changes of decreasing energies for the confined atoms as the radius of confinement increases approaching their free atom values for a radius of 30, independently of the barrier height; and also their increasing values for a fixed radius and increasing barrier heights. The changes in the orbital and total energies are more pronounced for Li and Na, diminishing successively for Be and Mg, N and P, and He, Ne and Ar. The radial distribution defined in Eq. (16) is illustrated in Fig. 3 for Mg, zero barrier height and confining radii of 2, 3 and 30, as functions of the radial distance; the respective curves show intersections. Specifically, in the last pair the second one for its own confining radius goes below the curve for the free atom and vanishes for a distance of about 8 exhibiting a localization. On the other hand, for the confinement at the radius 2 something different happens: Table 1 shows that the orbital energy is approaching zero for smaller distances and Fig. 3 shows that the first curve goes above that of the free atom at a radial distance of around 6, exhibiting the delocalization effect like in the case of H. Fig. 4 shows the Shannon entropy versus radial distance for Mg confined in constant potential barriers of heights 0, 0.5 and

Table 3 gives the values of the Shannon entropies for the free atoms, with positive values for the lighter ones and negative values for the heavier ones, taking decreasing values for the heavier elements. Figures 5 illustrate the Shannon entropies of confined H, Li and Na referred to the entropy of the free H for A) 0, B) 0.5 and C)

3.4.Many-electron Molecules

This section reviews four articles involving the rotational properties of molecules and four more on the electronic and vibrational properties of molecules.

“Orientation of the absorbed dipolar molecules: A conical well model" ^{58} is a theoretical investigation on the orientation of single and two coupled polar molecules irradiated by a single laser pulse under a conical-well model. The orientation of a single hindered rotor shows a periodic behavior. In particular, the amplitudes of the oscillation are sensitive to the degree of the alternation of the field. Crossover from field-free to hindered rotation is observed by varying the hindering angle for different heights of the conical walls. For a small hindering potential and angle, the time-averaged orientation differs greatly from that behavior under strong dipole-dipole interaction. Entanglement induced by the dipole-time averaged entropy increases monotonically as the hindering angle is increased. To the competition between the confinement effect and dipole interaction is found to dominate the behavior of the coupled-rotor system.

“Anticrossing-mediated entanglement of the adsorbed polar molecules" ^{59} studies the entanglement of two adsorbed polar molecules in static electric fields. The concurrence is estimated to quantify the entanglement. The absorbed molecules reveal a significant rotational characteristic, such as anticrossing features, due to the influences of the electric field and quantum confinement. Numerical results demonstrate that these rotational properties dominate the amount and profile of concurrence. At zero temperature, an enhaced concurrence is obtained near the anticrossing in the ground state. Additionally, the analysis of the effect of the temperature on the concurrence shows that the temperature-dependent concurrence displays a suppresed behavior especially at the anticrossing.

“Orientation and heat capacity of horizontally absorbed molecules in electric fields" ^{60} presents the evaluation of the energy spectrum and the wave function to probe the rotational characteristic of the molecule. Numerical results indicated that the electric field and the effect of quantum confinement lead to anticrossing behaviors in the energy levels. The orientation reveals a stepped feature due to the anticrossig ground state. Moreover, the heat capacity displays two peaks near the anticrossing. By means of comparison, each peak of the heat capacity corresponds to a particular degree of orientation.

“Inelastic electron tunneling mediated by a molecular quantum rotator" ^{61} presents a basic theoretical framework of Inelastic Electron Tunneling (IET) that explicitly takes into consideration quantum angular momentum, focusing on a molecular

“The effect of spatial confinement on the noble gas HArF molecule: structure and electronic properties" ^{62} presents a systematic study on the dipole and hyperpolarizabilities of the confined argon hydroflouride molecule. Detailed analysis of the confinement induced charges in the structures of HArF is also presented. In order to render the influence of chemical compression on the properties in question, a two-dimensional harmonic oscillator potential, mimicking a cylindrical confinement, was applied. By comparing the results obtained for HArF and HF, the effect of Ar insertion on the above properties was discussed. A hierarchy of ab initio methods, including HF, MP2, CCSD and CCD(T), has been employed to investigate the effect of orbital compression on the electron correlation contributions to the studied electric properties. It was observed that the external confining potential modifies the electronic contributions to the dipole moment and (hyper) polarizabilities of HArF. In particular the first hyperpolarizability of the confined molecule is remarkably smaller than that of the free molecule.

“On the nonlinear electrical properties of molecules in confined spaces: From cylindrical harmonic potential to carbon nanotube cages" ^{63}. The effect of spatial confinement on the linear and nonlinear electric properties of LiF, LiH, HF and HCl is analyzed based on the results of ab initio quantum chemical calculations. Central to this study is the comparison of different models of confinement. The harmonic oscillator potential of cylindrical symmetry as well as a more sophisticated model, based on the supermolecular approximation, are applied in order to establish the correspondence between the real chemical environment and their approximate representation in the form of an analytical potential. In the case of super molecular approach, the molecular cages are represented by carbon nanotubes and nanotube-like Helium clusters. The results of calculations show that spatial confinement strongly influences the electric properties of the investigated molecules.

“About diverse behavior of the molecular electric properties upon spatial confinement" ^{64} reports on the influence of spatial restriction on the static electric properties of two

“Vibrational nonlinear optical properties of spatially confined weakly bound complexes" ^{65}. This study focuses on the theoretical description of the influence of spatial confinement on the electronic and vibrational contributions to (hyper)polarizabilities of two dimeric bonded systems, namely,

3.5.Quantum Dots

This section has the antecedents of ^{15}^{,}^{22} reviewing the progress in six recent articles.

“On the symmetry of three identical interacting particles in a one-dimensional box" ^{66} studies the quantum mechanical physical system in the tittle with two-particle harmonic interactions. The symmetry of the system is described by the point group

“The effects of intense laser on nonlinear properties of shallow donor impurities in quantum dots with Woods-Saxon potential" ^{67} presents a theoretical study of the title using the matrix diagonalization method and the effective mass approximation. This methodology enables confinement energies by varying the two parameters in the potential. The intense laser effects are included via the Floquet method, modifying the confining potential associated to the heterostructure. From the computed energies and wave functions, the optical absorption coefficients and the respective index between the ground

“Impurity position effect on optical properties of various quantum dots" ^{68} is an investigation of the effect in the tittle for a pyramid and a cone like quantum dots. First, the energy levels and wave functions in the presence of the impurity are calculated using the finite element method. Then, the influence of the impurity location on the refractive index changes and absorption coefficients for the two quantum dots was studied. It was found that: 1) there is a maximum value for the optical properties at a special impurity position, and 2) the values of the optical properties of a cone like quantum dot are larger than those of a pyramid quantum dot of the same volume and height. The conclusion is that their impurity location plays an important and considerable role in the electronic and optical properties of both quantum dots.

“Calculation of the hyperfine interaction in spherical quantum dot" ^{69} presents the results of unperturbed wave functions and energy eigenvalues of the ground and excited states of spherical quantum dot,

“Computation of hyperfine energies of hydrogen, deuterium and tritium quantum dots" ^{70} uses the same QGA and Roothan HF methodology as the previous entry by the same Authors. The results show that in the medium and strong confinement regimes the hyperfine energy and the hyperfine constant are strongly affected by dot radius, impurity charge, electron spin orientation, impurity spin and impurity magnetic moment. Additionally, for all dot radii, the hyperfine splitting and hyperfine constant of the confined hydrogen and tritium atoms are approximately equivalent to each other, and they are greater than those of the confined deuterium atom.

“Type-II quantum-dot-in-nanowire structures with large oscillator strength for quantum gate applications" ^{71} presents a numerical investigation of the exciton energy and oscillator strength in the quantum dots in the title. For a single quantum dot, poor overlap of the electron part and the weakly confined hole part of the excitonic wave function leads to a low oscillator strength compared to type-I systems. In order to increase the oscillator strength, a double quantum dot structure is proposed featuring a strongly localized exciton wave function and a corresponding four fold relative enhancement of the oscillator strength, paving the way towards efficient optically controlled quantum gate applications in the type-II-nanowire system. The simulations are performed using a computationally efficient configuration-interaction method suitable for handling the relatively large nanowire structure.

3.6.Confined atoms treated as open quantum systems

This subsection borrows the title of ^{10} and illustrates its basic idea with the series of articles ^{72}^{-}^{78} on “Endohedral confinement of molecular hydrogen", “The atomization process of endohedrally confined hydrogen molecules", “Pressure and size effects in endohedrally confined hydrogen clusters", “Thermal behavior of a 13-molecule hydrogen cluster under pressure", “Thermodynamic states of Nanoclusters at low pressures and low temperature: the case of ^{12} considering hydrogen fullerene cages instead of the familiar ones with Carbon, and also different species of confined systems and different processes. The main difference between the series and the last reference is in their methodologies: while the latter models the doped fullerenes with the spherical

In ^{72}, the first problem investigated involved a single

In ^{73}, the confinement of thirteen

In ^{74}, the investigation consist of a systematic study of zero-temperature structural and energy properties of _{4} through C_{5} and C_{6} as the cluster grows from

In ^{75}, the thermal behavior of a thirteen molecule hydrogen cluster under pressure was investigated using a combination of trajectory and DFT simulations. The analysis is based on characteristic descriptors: caloric curve, root-mean-square bond length fluctuation, pair correlation function, velocity auto correlation function, volume thermal expansion, and diffusion coefficients. The discussion is focused on the peculiarities of the transition ordered-to-disordered states as exhibited by the cluster under different pressures, 65 to 75 GPa, and temperatures, 10 to 600 K . These ranges correspond to the conditions of an envelope of Jupiter’s interior. The results for the successive descriptors are illustrated in Figs. 1-5 and in numerical values in Section IV. In the Section V of Conclusions, it is stated that “The results point out a smooth transition of the molecular cluster from an ordered solid state to an intermediate thermodynamic state as a precursor of a possible liquid state of the cluster". It is the confinement of the cluster which extends the persistence of the intermediate state of temperatures (above 100 K and up to 500 K). They suggest the existence of hydrogen clusters in Jupiter’s envelope.

In ^{76}, an equation of state is presented based on a confinement model of finite-size systems. The temperature and pressure of the system are obtained from the positions and velocities of the enclosed particles via a number of molecular dynamics simulations. The pressure has static and dynamic contributions, extending the Mie-Grüneisen equation of state to include weakly interacting anharmonic oscillators. The model consist of thirteen

In ^{77}, “Endohedrally encapsulated hydrogen clusters doped with inert helium

In ^{78}, the “evolution of the vibrational spectra of the isolectronic hydrogen clusters

The results for a low pressure 20 GPa compare well with measurements for the crystal. For moderate pressures 184 GPa the molecules self-assemble into a stable cluster. For high pressures 520 GPa molecular dissociation appears due to a high population in an antibinding orbital. The Figures illustrate the systematic changes in the different properties of the confined molecular/atomic system. These results are of interest in the search to achieve the metallic state of hydrogen under pressure.

3.7.Connections with Reviews in other fields of Materials Science

This section is limited to Reviews in surface science, and on toroidal dipole moments and interactions in metamaterials and in nanophotonics, in which some of our works have been cited.

The Review Article “Auger neutralization and ionization processes for charge exchange between slow noble-gas atoms and solid surfaces" ^{79} in Progress Surface Science, in its Ref. ^{147} cites our work: “Ground-state energy shift of He close to a surface and its relation with the scattering potential: a confinement model" ^{80}. Both articles have also the respective common Refs. ^{152}^{-}^{153} and ^{3}^{-}^{4} by Wethekam and Winter reporting evidence on the ground state evolution of the He atom close to an

The familiar electric multipole ^{81}. Any three-dimensional vector field may be expressed in terms of its longitudinal component; and two transverse components: one toroidal and one poloidal with parallel circle lines and closed lines in meridian planes, respectively. They may be expressed as gradient, angular momentum and rotational of the angular momentum operators acting on scalar Debye potentials, which may be chosen as multipole ^{81} is cited in the Reviews below.

A selected sample of toroidal dipole moment interactions and their novel physical effects is illustrated by the titles of Refs. ^{82}^{-}^{91} for metamaterials: “Observation of ferrotoroidic domains" ^{82}, “On the aromagnetism and anapole moment of anthracene nanocrystals" ^{83}, “Gyrotropy of a metamolecule: wire on a torus" ^{84}, “Toroidal dipolar response in a metamaterial" ^{85}, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial" ^{86}, “Resonant Transparency and Non-Trivial Non-Radiating Excitations in Toroidal Metamaterials" ^{87}, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials" ^{88}, “Planar superconducting toroidal metamaterial: a source for oscillating vector-potential?" ^{89}, “Toroidal circular dichroism" ^{90} and “Nonradiating sources, dynamic anapole, and Aharonov-Bohm effect" ^{91}.

“Theory and applications of toroidal moments in electrodynamics: their emergence, characteristics, and technological relevance" ^{92} is a review article in Nanophotonics. Its sucessive sections illustrate the place of the toroidal moments: in Sec. 2 Families of multipoles in electrodynamics: 1) Multipole expansions for potentials and fields, 2) Classification of multipoles according to symmetry rules, under space and time inversions, in electric 𝐸 associated with polarization

4. Reviewer’s and Collaborators’ Work

This section updates the work of the Reviewer and his collaborators in two lines of research: One- electron diatomic molecular ions confined by dihedral angles and Many-electron atoms in the same situation of confinement in 4.1; Theory of Angular Momentum in Bases of Lamé Spheroconal Harmonics and its applications in 4.2. It also contains in 4.3 a preview of ongoing investigations along those two lines, and on toroidal moments and their interactions, as well as on the Force Approach to Radiation Reaction.

4.1.Confinement of Atoms and Molecules in Dihedral Angles

Reference ^{17} in its Section 1.4.3 and 1.4.2 anticipated the investigations on the one-electron diatomic molecular ions and the filling of electronic shells in atoms under dihedral angle confinement. The two next subsections describe the updating of results on both problems.

4.1.1.The H

The authors of ^{17} were invited to participate in the Workshop on Concepts of Mathematical Physics in Chemistry in honor of Professor Frank Harris in December 2014. In the oral presentation preliminary results on the ^{93}.

The updating of the chapter under consideration starts by calling the attention of the reader to its references ^{1}^{-}^{12} dealing with the hydrogen molecule and molecular ion confined in prolate spheroidal boxes, and the last one in cylindrical harmonic confinement; ^{13}^{-}^{14} on the hydrogen atom and harmonic oscillators confined in dihedral angles; and the remaining ones ^{15}^{-}^{25} on the exact solutions for the free molecules, including the heteronuclear HeH 2 + , in the Born-Oppenheimer approximation. Naturally, the Schrödinger equations for the respective molecules are common for the free ones and for the ones confined by natural boundaries in confocal prolate spheroids or hyperboloids, or dihedral angles. The separability of the equation and the integration of the ordinary differential equations in the respective coordinates are also common. It is the change of boundary condition in the chosen confining geometry which determines the changes in the eigenvalues of the energy and the other constants of motion, and in the eigenfunctions. Our works in [8-9] involved confinement in impenetrable and penetrable prolate spheroidal boxes, respectively, with rotational invariance around the axis of the spheroid and

For the molecular ions confined in dihedral angles, in the homonuclear case ^{14}. For the homonuclear molecule, the matrix becomes tridiagonal because the entries in the first diagonals above and below vanish. For chosen values of the nuclear separation p and the electronic energy parameter

Equation (6) in the spheroidal coordinates ^{18}. The matrix is tridiagonal and nonsymmetric, the eigenvectors contain the coefficients in the Taylor series and the eigenvalue is

The eigenfunctions of the molecule must involve common values of

Figure 2 illustrates the difference between the free molecular ion and the quasi-free molecular on with

The free ion states are

The total energy of the molecule is the sum of its electronic energy and the Coulomb nuclear repulsion energy. Figures 3 follow from Figures 1 by adding the positive value and inversely proportional to the nuclear separation of the latter. Remember that the horizontal scale is logarithmic. Additionally, the energy curves for the extra angles of confinement

The reader may appreciate the similarity of the curves and their systematic changes with the nuclear separation and with the confining angle for each state: they show a well defined minimum which shifts to larger nuclear separations and higher energies as the angle diminishes from its larger value below to smaller ones above. The differences between the curves for A) the ground state and B) the excited state of the same parity are quantitative with the minima more to the right and higher. Remember also the different vertical scales. Since the nuclear energy approaches zero for large separations, the total energy curves approach the dissociation energies asymptotically, from below as the electronic energy does. Table 1 illustrates the variations of the total energy

Each molecular energy curve is also characterized by the position of its minimum

Figure 4 also follows from Fig. 2 by adding the Coulomb nuclear repulsion energy, for the ground and excited states of the molecule for

Figures 5 and 6 also show the comparisons between the hyperboloidal wave functions of the free molecular ion

The confined molecular ion also acquires an electric dipole moment in the equatorial plane ^{17}: 1.27307, 3.75000, 9.2807 which are slightly larger.

Concerning the

The results for the ground (001) and first excited (011) states are reviewed next in the same order as in the homonuclear molecular ion. Figure 7 shows their electronic energies for

Figure 8 shows the total energies for the same states and the same situation of confinement on a vertical scale in the interval [-1.7,1], in the interval of nuclear separations ^{20} in a linear scale. Both exhibit the divergent Coulomb nuclear repulsion energy for vanishing separations. The ground state total energy curve is monotonically decreasing and is identified as an antibinding state. The excited state exhibits a shallow minimum at about

Figure 9 is the set of total energy curves in the interval

Bates and Carson in ^{21} of the chapter under review studied the exact wave function of HeH 2+ in its free configuration in the Born Oppenheimer approximation, identifying for the first time the metastable character of its (101) excited state in 1956. References ^{22} and ^{23} by Ben-Itzhak et al reported the experimental evidence for the existence of the

Figure 10 illustrates the electronic hyperboloidal coordinate function for the ground state of the

In contrast, Fig. 11 with the corresponding eigenfunction for the excited state shows the common nodes at

Figures 13 A and B, and 14 A and B compare the hyperboloidal eigenfunctions for the A) free and B) confined in

4.1.2.O(2) Symmetry Breaking in Dihedrally Confined Atoms and Consequent Modifications of the Periodic Table

The authors also participated in the International Colloquium of Group Theoretical Methods in Mathematical Physics G31 in June 2016, with the contribution having the same title as this subsection. The written contribution for the proceedings ^{94} is focused on the evaluation of the electron-electron Coulomb repulsion multipole component matrix elements between hydrogenic wave functions needed for Hartree-Fock calculations for the atoms in dihedral angle confinement.

The introduction of this contribution describes the symmetry breaking, in the hydrogen atom confined in dihedral angles as reported in ^{17}: i) it acquires an electric dipole moment, ii) its Fermi contact term in the hyperfine structure vanishes, iii) its Zeeman effect in first order perturbation theory vanishes and degenerate state perturbation theory is needed for its analysis, iv) the degeneracies of the energy levels D as functions of the confining angle are identified, with the consequent anticipation that v) the filling of the electron shells in the successive atoms and periodicity in their properties are different form those of the free atoms. Quantitative answers about the latter require Hartree-Fock calculations for the atoms confined in dihedral angles, specially for those with a nucleus with a charge Z, and 𝑍 and

The hydrogenic wave functions under dihedral confinement ^{93} are presented in Eqs. 1 in their azimuthal sine and cosine dependence and

The electron-electron Coulomb repulsion is given in Eq. (4) in its multipole expansion form involving their harmonic radial

In Ref. ^{26} of ^{17}, we reported the identification of Ladder operators for quantum systems confined by dihedral angles, which also have their counterparts for the hydrogen molecular ion in the same situation of confinement. There, it is illustrated how multiplication by

Here, the Legendre polynomials within the multipole expansion with integer associativities ^{95}, we have identified that multiplication by

The matrix elements of

4.2.Angular Momentum Theory in Bases of Lamé Spheroconal Harmonics

In the same Workshop in honor of Professor Frank Harris, the authors of the contribution with the title of this subsection and its written version ^{96} a review of the key concepts of angular momentum theory in bases of spheroconal harmonics. Such an alternative had already been contemplated in ^{4} for the rotations of free asymmetric molecules and free quantum systems with central potentials, and their counterparts confined in elliptical cones as natural boundaries in spheroconal coordinates. The implementations of the latter for asymmetric molecules, the free particle, the hydrogen atom and the harmonic oscillator were reported in the 50^{th} Sanibel Symposium and its Proceedings, as well as in an invited review chapter in AQC61. The corresponding references were reviewed in ^{17} Subsection 1.2.3. The updating of this line of research starts with ^{96}, includes the application in ^{97} of interest in magnetic resonance imaging and neutral atom traps, and ^{98} reporting a new class of shift operators and recurrence relations for individual Lamé polynomials at the G31.

4.2.1.Concepts of Mathematical Physics in Rotations of Asymmetric Molecules

The main sections in ^{96} had the headings: 2. Symmetries in the rotations of Asymmetric Molecules and 3. Actions of Operators of Angular Momentum and Linear Momentum on Rotational Eigenstates. In this updating the review of ^{98} is limited to a characterization of the Lamé spheroconal harmonics as common eigenfunctions of the square of the angular momentum and asymmetry distribution Hamiltonian operators, Eq. (7), commuting with each other, Eq. (8). In this way, the review of Sect. 3 can be done more in detail with emphasis on the structure of the respective operators, their relationships, and their matrices in spheroconal harmonic bases.

The readers interested in the mathematical details are referred to the Appendix, which includes the cartesian and spheroconal coordinate transformations Eqs. (A1), the linear momentum operators Eqs. (A8), the angular momentum operators Eqs. (A12), and the simultaneous separations of the eigenvalue equations (A18) of the square of the angular momentum and (A19) of the asymmetry distribution Hamiltonian, provided the asymmetry coordinate parameters

The respective eigenvalue equations can be separated into ordinary Lamé differential equations (A21) in the respective elliptical cone coordinates, because the operators commute with each other. The spheroconal harmonics are products of Lamé functions in both elliptical cone coordinates Eq. (A24) with matching asymmetry parameters

The spheroconal coordinates

The spheroconal harmonics of lower order and monomial forms are identified in the following table, using only the symbols of the functions for

l | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 |

A | 1 | dn | cn | sn | cnsn | dnsn | dncn | dncnsn |

B | 1 | sn | cn | dn | cndn | sndn | sncn | sncndn |

r^{
l
} AB |
1 | x | y | z | yz | xz | xy | xyz |

The derivatives of each Jacobi elliptic integral involves the products of the other two with the coefficients

The spheroconal coordinates in Eq. (A4) are defined in terms of Jacobi elliptical integral functions connected by Eqs.(A2) and derivatives in Eqs.(A3). They are useful to obtain the displacement

The successive actions of the cartesian components of the radial position vector, linear momentum and angular momentum vectors on the singularity removing factors are in Table 4 for [AB], in Table 5 and 6 for the terms with the derivatives with respect to 𝜒 1 and to 𝜒 2 respectively, and in Table 7 and 8 also for the corresponding operations, including the distinction of their actions on the successive factors depending on one variable or the other, as well as the resulting species [AB] with definite cartesian parities. In Tables 5 and 6 the reader may notice the differences in the entries in the respective rows for 𝜒 1 and 𝜒 2 , which nevertheless lead to the same values of [AB]. Additionally, they coincide with the respective entries of their counterparts in Table 4. The reason for this coincidence is that both 𝑟 and 𝑝 are vectors and share the same negative parity, they can only connect rotational eigenstates of different parity and angular momentum with the selection rule

Since the numbers of non zero entries in the matrices for the explicit lower values of

From the individual matrices

4.2.2.Family of Lamé spheroconal quadrupole harmonic current distributions on spherical surfaces as sources of magnetic induction fields with constant gradients inside and vanishing asympotically outside

Constant gradient magnetic fields play key roles in magnetic resonance imaging and in neutral atom traps. Reference ^{28} in ^{17} reported our work on spherical and spheroidal quadrupole harmonic current distributions, on the respective surfaces, and the constant gradient magnetic induction fields in their interiors, as well as the exterior asymptotically vanishing companion fields.

Our work with the title of this section ^{97} is an example of the application of the Lamé spheroconal harmonics, providing an alternative family of solutions from the same problem. We start out by pointing out that of the five quadrupole harmonics

The solution of the equations of magnetostatics, in order to identify the quadrupole harmonic current distributions and their constant gradient magnetic induction fields, take into consideration the solenoidal character of the latter and Ampere’s law. Thus, the magnetic induction field can be written as the rotational of the magnetostatic vector potential. The potential itself is also solenoidal and we write it as the generator of rotations operator

The potential does not have a radial component, and its components along the unit vectors

The magnetic induction field inside and outside the spherical boundary are evaluated as the rotational of the magnetostatic vector potential. Their radial components are different because they are directly inherited from their respective potentials, and they share the same dependence in

In Eqs. (9) for the magnetic induction field inside and outside the sphere there appears the common scale factor in the square form in the denominator leading to the linear dependence 𝑟 and

The current field lines on the spherical surface follow from Eqs. (11) via the condition of tangentiality of Eq. (12) and its integration leading to Eq. (13), which is equivalent to the constancy of the scalar potential from which the magnetostatic vector potential was constructed.

While Eq. (9) uses the unit vectors

Figure 1 illustrates the quadrupole harmonic current distributions on the spheres for different values of the asymmetry parameter. The extreme values

Table I corrected in an Erratum [97] illustrates the values of the gradients in the

4.2.3.Shift Operators and Recurrence Relations for Individual Lamé Polynomials

The authors’ oral contribution to the Group 31 Meeting had the title “Review of the development and application of the spheroconal theory of angular momentum”. It consisted of a presentation of our then recent results of the previous two subsections, as well as our preliminary results of our then ongoing development on the topic with the title of this subsection. The written contribution for the Proccedings ^{98} is focused on reporting our new results on this topic.

Our previous work has been focused on the spheroconal harmonics as eigenfunctions of the square of the angular momentum and the asymmetry distribution operators. While in ^{8} of the reference under review, an article by the common authors in “SIGMA Ladder Operators for Lamé Spheroconal Harmonic Polynomials” were identified as the angular momentum operators

The following Table illustrates the changes in the species of the corresponding polynomials under the derivative operation:

The reader may notice in the first row the four species with an even number of factors, followed by the other four with an odd number; as well as the exchanges under the derivative in the second row. The shifting actions of the derivative operators are proven by mathematical induction. Eq. (6) illustrates its connecting the three

Equation (7) illustrates the actions of the derivative on the five Lamé polynomials with ℓ=2, the three from the previous case and the two of species 1 with

The generalizations of the raising action of the derivative on the ^{1} in ^{1} and

The multiplication of two individual Lamé polynomials with labels

In the generalization in the ket notation, implying orthonormality and completeness of the basis, the product of two of them can be written as the superposition of the connected states

as allowed by the addition of their angular momentum operators

4.3.Preview of Ongoing Investigations

This section outlines some of the problems which we have been investigating along the lines of 4.3.1 the Hydrogen atom confined in a dihedral angle, 4.3.2 applications of the angular momentum theory in bases of Lamé spheroconal harmonics, as well as 4.3.3 toroidal moments and interactions, and 4.3.4 radiation reaction in the force approach. Preliminary reports about their formulations, advances and results have been presented in the VIII Workshop on the Structure and Dynamics of Matter and Optics, and the LIX and LX Mexican National Physics Meetings.

4.3.1.SO(2) Symmetry Breaking in the Hydrogen Atom Confined in Dihedral Angles: manifestation in the Zeeman Effect

This problem was formulated in Sect. 1.3.5 of ^{17}, but at that moment time did not allow the implementation of its solution. Equation (1.17) in that section describes the interaction between the magnetic moment of the electron with its orbital and spin contributions and a uniform magnetic field in the direction of the edge of the confining dihedral angle. While in the free hydrogen atom, the z-component of the angular momentum is a constant of motion leading to the familiar ^{17}. The diagonal matrix elements of the z-component of the angular momentum in the basis of eigenfunctions

Equations (1.18) and (1.19) give the general forms of the matrix elements of

The spin contributions to the magnetic moment are readily incorporated via its diagonal elements with eigenvalues

The evaluation of the radial and polar angle eigenfunction overlaps can be done numerically, but we are also exploring alternative ways by using ladder and/or shift operators.

4.3.2.Applications of Angular Momentum Theory in bases of Lamé Spheroconal Harmonics

In 4.2.2 an application of Lamé spheroconal quadrupole harmonics in Magnetostatics has already been reviewed. Here we describe three additional applications under investigation, two of them for electromagnetic fields, and the third one for scattering in quantum mechanics.

Indeed, the first one is about Lamé Vector Spheroconal Harmonics, our contribution to the Meeting on Selected Topics of Mathematical Physics in honor of Professor Natig Atakishiyev as a tool to construct solutions of Maxwell’s equations for electromagnetic fields with a harmonic time dependence. These solutions are alternatives to the familiar vector spherical harmonics ^{99}, with the advantage that they come in families depending on the value of the asymmetry distribution parameter

The second application was presented in the LIX Congreso Nacional de Física Session MID12 with the title: “Complete Electromagnetic Multipole Expansion in bases of spheroconal harmonics". The starting point is the multipole expansion of the scalar outgoing wave Green function for the Helmholtz equation in the complete and orthogonal bases of spheroconal harmonics multiplied by the respective spherical Bessel functions in the radial coordinate

The third application: “Scattering by Asymmetric Targets in Spheroconal Harmonic Bases", M1E035, LX Congreso Nacional de Física, is explained next. It is an alternative to the familiar quantum theory of scattering in spherical harmonics, sharing the same radial spherical Bessel functions in the expansions of the plane wave and of the outgoing wave Green function for the free particle, as well as being eigenfunctions of the square of the angular momentum. The difference is that the respective harmonics are also eigenfunctions of the asymmetry distribution Hamiltonian and parity operators versus the z-component of the angular momentum. Correspondingly, the concept of phase shifts and their values are also common, but the angular distributions for the different angular momentum eigenstates

4.3.3.Electromagnetic Green Function Toroidal Moment Expansion in Circular Cylindrical Coordinates

The work in this section was reported in the contributions: “Electromagnetic multipole expansion in circular cylindrical geometry" Session M2D08 in the LIX Congreso Nacional de Física, “Normal TM modes in a toroidal cavity” in VIII Workshop on the Structure and Dynamics of Matter and Light, and “Unified treatment of toroidal antennas and resonant cavities", M1E10 in the LX CNF.

Our previous work: Complete electromagnetic multipole expansion including toroidal moments, cited by the groups that have identified toroidal dipole moments and interactions in metamaterials ^{85} and nanomaterials ^{92}, was done in spherical coordinates. Since the toroids with rectangular cross sections are more common, our work has been focused in the alternative geometry.

The first contribution started from the outgoing wave Green function in circular cylindrical coordinates as a Debye potential. A poloidal current on the surface of a toroid with inner and outer radii

The second contribution follows from the first one upon demanding the boundary conditions for the Transverse Magnetic modes in the resonant cavity namely the vanishing of the tangential components of the magnetic induction field outside the toroidal surface. In fact, the magnetic induction field is discontinuous at such a boundary without any normal components; the discontinuity is determined by the tangential surface current consistent with Ampere’s law. The boundary condition and the vanishing external electromagnetic field imposes the boundary conditions on the scalar multipole Debye potentials, which in turn define the respective resonant frequencies.

The combination of the previous paragraphs connects with the third contribution about a unified treatment of both antennas and resonant cavities, for each toroidal multipole.

(2)4.3.4.Analytical Solutions for the Radiation Reaction in the Force Approach

The radiation reaction problem was analyzed by Abraham and Lorentz starting from the Larmor formula for the power radiated by a charged particle with a mass

Its solutions involve several nonphysical features: 1) run away solutions exponentially divergent

The force approach to the radiation reaction was proposed by G.V. López two years ago with the following conditions: it must depend on the applied force, vanishing if the applied force vanishes; in such a case the particle does not accelerate and does not radiate ^{100}.

The identification of the radiation reaction force in the force approach also starts from the Larmor formula for the radiated power with the acceleration written as the applied force divided by the mass. It also imposes the conservation of energy and momentum, distinguishing between the cases of linear motion and circular motion. In the first case, the energy lost by the particle in a small time interval is

and equal to the work done by the radiation reaction force while producing the small displacement

For the circular motion including relativistic effects the radiated power includes the square of the time dilation factor in the denominator, Eq. (13), and the resulting radiation reaction force has the same characteristics as in the previous paragraph, and is also inversely proportional to the square of the time dilation factor, Eq. (15).

In both cases, if the applied force vanishes, the radiation reaction vanishes, the particle does not accelerate and does not radiate, satisfying the imposed conditions, and consistent with the observations.

The specific examples analyzed in ^{100} were those of relativistic motion in a straight line under a constant force, Eq. (8); and of circular motion for the charge under a constant magnetic field Eqs. (17-19) and their cartesian components Eqs. (20a,b).

For the linear motion, for Eq. (9) the first order derivative of the velocity with respect to time equation is separable with singularities in

For the circular motion no analytic solutions of Eqs. (20a,b) were constructed. The graphical solutions in Fig. 2 illustrate the differences between the trajectories without and with radiation reaction for increasing values of the applied force evaluated by numerical integration.

Our own analysis of the linear and circular relativistic motion equations has allowed us to construct exact solutions for the entire range of velocities

For the linear case, to begin with we have identified some typographical errors in the solutions of Eqs. (11a) where the first factor in the numerator is simply 1, instead of the binomial, and in Eq. (11b) where the binomial in the denominator of the first factor is missing the exponent 2 in the second term. We prefer to plot directly the variations of the speed for the motion with the radiation reaction included as given by the correct form of Eq. (10), by choosing the values of

In the circular motion under a uniform magnetic field, the applied force is proportional to the velocity, and so is the radiation reaction, apart from the dependence in the time dilation factor. The equations of motion (20a, 20b) also admit analytic solutions, which we are in the process of completing for the velocities and the trajectories.

The availability of analytical solutions allows us to formulate a physical interpretation of the radiation damping effects, which will hopefully contribute to test the force approach. Its author has also developed it further in ^{101} for any type of motion and in an electronic undulator application ^{102}.

5. Discussion and Concluding Remarks

This section includes summarizing and connecting discussions about the contents of the successive sections in the article. It also contains some concluding remarks about the recent progress in the field.

Section 2 includes descriptions and illustrations of the contents in the chapters in Advances in Quantum Chemistry and the Monograph, as points of the reference to appreciate the further advances in the period between their publications, and on to the subsequent recent progress from the literature.

Section 3 is presented in its successive Sections: 1. Hydrogen-like atoms ^{26}^{-}^{45}, 2. Diatomic Hydrogen-like Ion and Neutral Molecules and Two-electron Atoms ^{46}^{-}^{53}, 3. Many-electron atoms ^{54}^{-}^{57}, 4. Many-electron Molecules ^{58}^{-}^{65}, 5. Quantum Dots ^{66}^{-}^{71}, 6. Confined Atoms as Open Quantum Systems ^{72}^{-}^{78}, and 7. Connections with Reviews in other fields of Materials Science ^{79}^{-}^{92}. Some comments about them are appropriate as a guide to the reader to appreciate the recent progress. There is still a great proportion of investigations for the simpler systems, as the natural trend from the background reviews. However, there are also qualitative changes as illustrated next.

In 1, we can follow some of the keywords in the titles: tunable excitons, biased bilayer graphene, dipole oscillator strengths, generalized oscillator strengths, Debye confinement, high harmonic generation, Gaussian potential confinement, penetrable cylindrical confinement, endohedral confinement, moving nucleus confinement, confinement effects on electron transfer cross sections, confinement by cones, benchmark calculations, sum rules, monotonocity, confinement in dielectric continuum, symmetry breaking, Fisher information, relativistic two-dimensional hydrogen atom, confinement by very thin layers. Hopefully, this list will stimulate the readers to study the original references of their specific interests. It is also illuminating to follow the set of references ^{27}^{,}^{31}^{,}^{33}^{,}^{36}^{,}^{43} with a common Author.

In 2, seven of the references deal with the hydrogen molecular ion and the eighth with the helium-like atoms. Debye embedding is investigated in ^{46}. References ^{47}^{-}^{49} involve confinement in an impenetrable prolate spheroidal box and a variational solution with the simplest trial wave function in the original article, a Comment about it, and the response. Reference ^{50} investigates the molecular ion inside padded prolate spheroidal cavities with arbitrary nuclear positions. References ^{51}^{,}^{52} by the same authors investigate the spherically confined molecular ion in the

The investigation of the singly excited states ^{53}.

In 3, the set of references ^{54}^{-}^{56} with common authors have the antecedents of ^{8}^{,}^{11}^{,}^{24} on many-electron atoms. The work in ^{54} on the implementation of the electron propagator to second order GPUs to estimate the ionization potentials of (hard sphere) confined atoms was anticipated in ^{24}. The other works, “Roothan’s approach to solve the Hartree-Fock equations for atoms confined by soft walls: Basis set with correct asymptotic behavior" ^{55}, “solution of the Kohn-Sham equations for many-electron atoms confined by penetrable walls" ^{56}, and “Electron density delocalization in many-electron atoms confined by penetrable walls: A Hartree-Fock study" ^{57}, share the spherical confinement with a constant barrier outside; this feature is crucial for the analysis of ionization potentials. See also their connections with ^{38}.

In 4, the first four references deal with rotations of molecules, and the last four with the confinement effects on the structure and electric and optical properties of a variety of molecules.

References ^{58}^{-}^{60} with a common Author also share the orientation of adsorbed polar molecules in a conical well model, in anti-crossing mediated entanglement, and in electric fields and their effects on heat capacity, respectively.

Reference ^{61} investigates the inelastic electron tunneling mediated by a molecular quantum rotator.

References ^{62}^{-}^{65} have common Authors, investigating the “effect of spatial confinement on the noble gas HArF molecule: structure and electric properties", “On the nonlinear electric properties of molecules in confined spaces-from cylindrical harmonic potential to carbon nanotube cages" [The latter is investigated as a supermolecule including the confined molecules and the cage], “About diverse behavior of the molecular electric properties upon spatial confinement", and “Vibrational nonlinear optical properties of spatially confined weakly bound complexes".

In 5, Ref. ^{66} investigates the symmetry of three identical interacting particles in a one dimensional box, with harmonic oscillator interactions using group theoretical methods.

Reference ^{67} investigates the effects of intense laser on nonlinear properties of shallow donor impurities in quantum dots with the Wood-Saxon potential.

Reference ^{68} investigates the impurity position effect on optical properties of various quantum dots, pyramidal and conical specifically.

References ^{69}^{,}^{70} by the same authors implement Calculation of hyperfine interaction in spherical quantum dot, and computation of hyperfine energies of hydrogen, deuterium and tritium quantum dots, respectively.

Reference ^{71} investigates Type-II quantum-dot-in-nanowire structures with large oscillator strength for optical quantum gate applications.

In 6, the subsection borrows the title of ^{10} and illustrates its basic idea with the series of articles ^{72}^{-}^{78} on “Endohedral confinement of molecular hydrogen", “The atomization process of endohedrally confined hydrogen molecules", “Pressure and size effects in endohedrally confined hydrogen clusters", “Thermal behavior of a 13-molecule hydrogen cluster under pressure", “Thermodynamic states of Nanoclusters at low pressures and low temperature: the case of ^{12} considering Hydrogen fullerene cages instead of the familiar ones with Carbon, and also different species of confined systems and different processes. The main difference between the series and the last reference is in their methodologies: while the latter models the doped fullerenes with the spherical

In ^{7}, the connections of some of our works with other fields of Materials Science are illustrated. Specifically, the Review Article “Auger neutralization and ionization processes for charge exchange between slow noble-gas atoms and solid surfaces" ^{79} in Progress Surface Science, in its Ref. ^{147} cites our work: “Ground-state energy shift of He close to a surface and its relation with the scattering potential: a confinement model" [80]. Both articles have also the respective common Refs. ^{152}^{-}^{153} and ^{3}^{-}^{4} by Wethekam and Winter reporting evidence on the ground state evolution of the He atom close to an

On the other hand, the series of papers on Metamaterials and the Review Article in Nanophotonics, and their novel physical effects ^{82}^{-}^{91}, cited our article: “Complete electromagnetic multipole expansion including toroidal moments" ^{81}.

Section 4 contains the updating of the Reviewers’ and collaborators’ works along the lines of 1) Confined Atoms and Molecules in Dihedral Angles and 2) Development and Applications of Spheroconal Theory of Angular Momentum. New results in the first one were reviewed for the

The same section contains a Preview of on going investigations: 1) Zeeman effect in the Hydrogen atom confined in dihedral angles. 2) Vector spheroconal harmonics needed for the analysis and construction of solutions of Maxwell’s equations and their applications; quantum theory of scattering using spheroconal harmonics to incorporate asymmetries in the target. 3) Multipole Toroidal moments and interactions in circular cylindrical geometry, and their applications in radiating and resonant cavity systems with a unified treatment. 4) Radiation Reaction in the Force Approach.

The penultimate paragraph in the Introduction included the intention to make this a conceptually connecting review. Here the attempts to implement that intention are underlined.

In Section 2.1 the concepts of confinement in quantum systems, from simple to complicated, were identified and connected for the chapters in AQC. Section 2.2 illustrated the subsequent extensions in the chapters in the Monograph with common Authors, identifying further advances in the elapsed period; and also in the other chapters incorporating their new topics, concepts and methods.

The recent progress in Section 3 is guided by the same idea following the conceptual connections and extensions in the Literature for the successive types of quantum systems. In 3.1 for the hydrogen-like atoms the titles and their keywords may help the readers to decide which references to study more in depth according to their interests. For some references, specific connections have been pointed out, but each reader may identify others. In 3.2 the seven articles on the hydrogen-like diatomic molecules have the connection with ^{25} and references therein, as well as with ^{93}. Section 3.3 illustrates the recent progress for many-electron atoms from ^{11}^{-}^{24} to ^{54}^{-}^{57}, and specifically in going from impenetrable to penetrable spherical confinement. Section 3.4 on many-electron molecules involves investigations associated with different degrees of freedom: rotational motions in the first four with new forms of confinement and new physical effects; and in the last four, with common Authors, electronic structure, and also connected with [10] and ^{13}. Section 3.5 on Quantum dots contains one work on the symmetry of three identical particles in a box; three works on optical properties: in donor impurity quantum dots under an intense laser field, in impurity pyramidal and conical quantum dots depending on their positions, and in Type-II quantum-dot-in-a wire structures; and two works on hyperfine structure in a spherical quantum dot, and in hydrogen isotope quantum dots. Section 3.6 contains a series of seven articles on hydrogen cages confining hydrogen molecules, with common Authors, treated as supermolecules and connected with ^{10}, ^{12} and ^{63}. Most important, the successive investigations report results on atomization, clusterization, metallization of ^{80} and ^{81}.

Concluding conceptual and connecting remark: the interactions among the components of matter at the atomic, molecular, nanomaterial and metamaterial levels, are basically electromagnetic and due to their respective basic electric and magnetic individual properties. It is these properties and those interactions that determine the structure and the physical processes in their aggregates. The identification of new such properties and their new associated interactions leads to novel physical and chemical phenomena. Confinement of the aggregates, by other neighboring aggregates or by geometrical design, bring in additional elements of novelty in those phenomena or are responsible for them.