Electronic energy levels and wave-functions evolution from 3-D to 2-D of a hydrogen atom confined by two parallel planes

The compression of an atom produced by two planes induces a change in its electronic structure that evolves from a free atom in 3-D to a 2-D atom. This behavior is of importance in low-dimensional materials and high compression produced by an anvil cell. In this work, we study the evolution of the energy levels and electronic wave-functions of a hydrogen atom placed between two impenetrable planes as a function of the inter-plane separation through a numerical approach. As the inter-plane separation is reduced, the electron motion is restricted along the direction normal to the planes, similar to a particle in a box, while leaving the electron to move unrestricted along the planes, thus, breaking the spherical geometry of the H atom caused by the planes’ compression. The energy levels evolve from 3-D, described by nlm quantum numbers to a 2-D described by n′ml′ , where l′ is the quantum number for a particle in a box along the z direction and n′ is the principal quantum number of the 2-D atom radial direction. We evaluate the energy levels from 3-D to 2-D and the radial average distance 〈ρ〉 in cylindrical coordinates, as a function of the inter-plane separation D along the z-direction. We find that as the inter-plane separation is reduced, the angular momentum quantum number l merges to the z-component of the angular momentum and it produces two branches, a symmetric for l-even and one anti-symmetric for l-odd, connected to a particle in a box quantum number l′ along the z-axis with implications in the atom photo-luminescence, resulting from the symmetry of the system. Furthermore, states with l-odd merge with states with l-even, as they have the same energy and average distance when D → 0. We provide an Aufbau principle for it. Our results agree to the analytical solutions at the 3-D and 2-D limiting cases.


Introduction
The development of new materials during the last decades has given rise to important challenges in the understanding of nontrivial physical properties of quantum systems when subjected to spatial restrictions or extreme conditions.The interaction of confined particles with the environment that produces the confinement or volume boundary is usually simulated by means of suitable boundary conditions imposed on the wavefunction.These conditions lead to important quantum effects as the size of the confinement is reduced, such that the influence of shape on quantum confinement matters as much as size [1].The wealth of problems related to this phenomenon range from atoms and molecules confined within nanotubes, fullerene traps [2][3][4][5][6][7][8][9][10][11], zeolitic nano-channels [12,13] to the behavior of donor centers in quantum-well semiconductor structures [14][15][16][17][18][19][20].For example, the understanding of hydrogen-like donor impurity states in low-dimensional semiconductor structures is of importance in quantum dots and quantum-well wires [15][16][17][18][19][20].In this context, different models of quantum confinement have proved suitable for the understanding of the physical properties of such systems.For a review on the subject see, for instance [5,[21][22][23][24][25][26][27].
For the case of half-plane confinement, the behavior of a hydrogenic impurity at a semiconductor surface was carried out by Levine [43], more than five decades ago and laid down the nodal character of the wavefunction for an atom sitting on top of an infinitely planar boundary.The case of the behavior of atomic hydrogen in half-space with an impenetrable plane boundary under general Robin boundary condition was studied by Artykova et al [49] in the adiabatic approximation.The case of classical dynamics of a Rydberg hydrogen atom near a metallic surface in the presence of a uniform electric field has been considered by Iñarrea et al [50], while the behavior of a hydrogen atom interacting with the surface of a polar semiconductor has been studied by Elmahboudi and Lepine [51], where the interaction of the electron with the surface is described microscopically in terms of electron-phonon and electron-exciton interactions.MacMillen [52] studied the energy shift due to the compression of a hydrogen atom near a sharp planar surface of a semi-infinite substrate based on the phaseintegral method.
The case of the compression by a second surface, parallel to the first one, it would lead to a 2-D restriction for small interplane separation.For a 2-D system, several approaches have been implemented to study quantum confinement.LumbTalwar [53] solved the time-independent Schrödinger equation for a two-dimensional hydrogen atom under the combined effects of a static electric and a magnetic field with circular confinement.Aquino et al [54] studied the two-dimensional hydrogen atom confined within a circle of impenetrable walls.The same group studied the internal disorder of the confined two-dimensional hydrogenic atom numerically in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the statistical Crámer-Rao complexity measure [55,56].Chaos-Cador and Ley-Koo [57] studied the 2-D hydrogen atom confined in circles, angles, and circular sectors.Stevanovic et al [58] reported several interesting characteristics of the eigenspectrum of 2-D hydrogen atom confined inside an impenetrable circular boundary of radius ρ c .The mathematical justification to study two-dimensional atoms with the three-dimensional Coulomb potential has been provided by Tusek [59].Two dimensional materials with van der Waals bonding are known to exhibit a quantum confinement effect in which the electronic band gap is larger than the three-dimensional counterpart [60].The two-dimensional hydrogen problem was analytically solved by Yang et al [61] for electronic transitions, life-times, and hyperfine structure by considering both bound and continuum solutions.A review regarding quantum confinement effects in low-dimensional semiconductor systems is given by Yoffe [62], while Parfitt [63] provides a revision of the 2-D hydrogen atom.
Despite of all the above developments for the two limiting confining cases, 3-D and 2-D, the study of a hydrogenic atom compressed by two parallel planes, as a function of the inter-plane separation, has been scarcely analyzed due to the complex nature of the confining geometry.Its importance resides in that single metal atoms confined in 2-D materials have gained substantial attention as potential heterogeneous catalysts for various electrochemical applications [64].For example, quantum confinement effects induced by two planes are believed to be able to boost the photo-luminosity intensity [65], and just recently, the creation of a 2D solid helium via diamond lattice confinement has been achieved [66].
In this work, we study the energy-levels of a confined hydrogen atom with impenetrable confining planes as a function of the planes separation.A Finite-Differences (FD) approach is used to solve the Schrödinger equation in a numerical lattice.The FD approach for derivatives is one of the simplest methods to solve differential equations.One of its main strengths is that it can achieve high accuracy and stability as well as that it renders the differential equation in linear algebra form with a straightforward implementation.Our numerical approach is tested against the limiting known analytical cases when the separation planes is large or small.A discussion on the evolution of the lowest states, as the planes separation changes, is provided.
Our work is organized with the following layout.In section 2, an outline of the confinement model and the method used to solve the time-independent Schrödinger equation based on a FD approach and its numerical implementation is presented.In section 3, the results of this work together with their analysis and the mathematical details that support the numerical findings at the limiting cases are given and their discussion is provided.In section 4, the conclusions of this study are presented.Atomic units are used unless otherwise stated.

Atom placed between two planes
Let us consider a hydrogen-like atom of nuclear charge Z placed between two planes separated a distance D and located in the middle of the planes at the origin, as shown by the sketch of the geometrical configuration in figure 1.By placing the planes parallel to the xy-plane, a cylindrical geometry on the z-axis is achieved.The atom nuclear position is located at the origin of a cylindrical coordinate system and the electron-nuclear distance is given by r = (ρ, j, z).The nucleus is assumed clamped to the origin and the electron is allowed to move under the influence of both the nuclear Coulomb potential and the confining barrier V 0 , implied by the planes boundaries.The azimuthal symmetry of the problem allows us to separate the corresponding stationary Schrödinger equation in cylindrical coordinates.By defining the total wave-function as where ( ) e 2 is the azimuthal wave-function and m = 0, 1, 2, K, is the magnetic quantum number, one finds that ψ m (ρ, z) satisfies the 2-D equation Here, the confining potential V c (ρ, z) is chosen as a barrier of constant height at the upper and lower surfaces simulating an average constant field acting on the electron due to the surrounding medium.The confining potential is given by  for all ρ.The case V 0 → ∞ would correspond to an impenetrable boundary and it is the case discussed in this work.We have chosen the case of a infinitely hard barrier, i.e. a hard confinement boundary as a benchmark testing ground.Our first task is to find the eigenfunctions and eigenvalues of equation (2) under the constrictions implied by equation (3).Clearly, equation (2) is not separable and consequently we resort to its numerical solution by means of a Finite-Differences (FD) approach, as stated in the Introduction.

Finite-Differences approach and implementation
Since the atom is placed between two infinitely extending planes, the coordinate ρ is not limited.However, computationally this is impossible to achieve and we are forced to consider a large radial distance, which we denote R max (see figure 1).Note that, according to equation (3), the z-coordinate of the electron is bounded by the planes' surfaces, whose distance to the nucleus at the origin is given by ±D/2.This means that as the interplane separation is reduced, the electron becomes confined by the walls in this direction, while the xy-plane is Figure 1.Cylindrical geometry configuration for a hydrogen atom placed in the middle of two parallel planes (dashed areas).The planes separation is denoted by D and V 0 is the potential that represent the material of the planes.In our case V 0 → ∞ .See text for discussion.
unbounded.Consequently, our 3-D hydrogen atom will tend to a 2-D hydrogen atom when D → 0, as the electron is compressed by the impenetrable walls in the z-direction (V 0 → ∞ ).
The FD approach requires a numerical grid, whose size is defined by the large cylindrical box of radius R max and height D enclosing the atom, as shown in figure 1.In the particular case of impenetrable walls and the atom placed in the middle of the planes, the boundary conditions, A further simplification of our problem occurs by noticing that there is a reflection symmetry about z = 0, as well as an inversion symmetry.Consequently, there are even and odd solutions along the z-axis (see section 3).This symmetry will provide a second good quantum number along the z-axis.As the details of the implementation of the FD approach for the case of cylindrical confinement have been reported previously by the author [41,42], here only a summary is provided.Any electronic state of the system is labeled by the magnetic quantum number m.Once a radius R max and planes separation D are defined the original eigenvalue problem is converted into an equivalent linear algebra matrix approach as follows.Let the continuous variables ρ and z be discretized in a set of fixed coordinate positions ρ i and z j , which may or may not be uniformly spaced.Thus, we express the wave-function Ψ m (ρ, z) as ij m Y , where ρ → ρ i and z → z j are known at the ij-th point on a 2-D numerical grid, and where i = 0 and i = N ρ + 1 represent the boundary of the system for the coordinate ρ while j = 0 and j = N z + 1 for the coordinate z.Here, N ρ and N z are the number of points in the ρ and z grids, respectively.In our case, the boundary conditions that equation (2) fulfill are of the Dirichlet type, since for all m and Ψ m (ρ 0 , z) = 0 for m > 0. For m = 0, we have a Neumann condition at ρ 0 = 0 [41,42].Equation (2) together with the boundary conditions can be written in matrix form by transforming the problem to a 1-D vector array by for a total of N = N ρ × N z points.By using the FD definition for derivative and integration [67] centered in the mid-point, equation (2) becomes where H is a symmetric pentadiagonal banded matrix [41,42] with and L is related to the metric generated by the numerical grid (see [41] for details).Equation ( 4) is just a matrix eigenvalue problem.The total number of eigenvalues and eigenvectors depends on the number of grid points (dimension of the matrix).The diagonal elements of H include the evaluation of the confining potential V c (ρ i , z j ) for any state labeled by the quantum number m.The only limitation of the FD approach is the size of the numerical grid, which is given by the N grid points for a given choice of R max and D. That is, for a given R max , the larger the number of points N (denser grid points), higher memory requirements and thus more expensive computational resources are necessary to archive higher accuracy.
In this work, ρ and z have a logarithmic distribution in order to have a good wave-function cusp description at the nuclear position.For this, we require that the maximum distance from the origin to the first nearest point in the grid to be ;0.02a.u. in the ρ and z directions.The grid box used here is defined by a set of grid points ρ running from ρ 0 = 0 up to R 500 u., such that the total number of points is N ρ = 311.By taking advantage of the reflection and inversion symmetry of the problem, the z grid points are set from z 0 = 0 a.u. to z D 2 with a total number of points N z that varies as D changes while keeping the first nearest point in the z-grid from the nuclei ∼0.02 a.u., as in the ρ-grid.The grid chosen here is a compromise between the required accuracy and acceptable CPU time for all calculations reported here.The previous choice of grid renders a convergence up to the fifth digit in our calculations for the free H atom.For instance, for D R 2 max = , one requires N = 193442 grid-points, which is too large for a pentadiagonal matrix such that the CPU time can be up to several weeks in a 3 GHz computer.The eigenvectors and eigenvalues were obtained via tested ARPACK subroutines [68].
The calculations were carried out for different plane separations D for an impenetrable potential following the aforementioned numerical grid.The inter-plane distance D/2 starts at 0.05 and was varied from 0.1 to 10 a.u. in steps of 0.1 a.u., in steps of 0.2 from 10.0 to 20.0 a.u., in steps of 0.5 a.u.from 20.0 till 30.0, in steps of 1.0 a.u.from 30.0 till 40.0, and in steps of 2.0 from 40.0 till 100 a.u., and in steps of 4.0 from 100.0 till 160.0 a.u., while the barrier potential height, V 0 = ∞ a.u., is set automatically by imposing the boundary conditions Ψ m (ρ, z = ± D/2) = 0. Several additional D values were evaluated at the regions of avoiding energy crossings for a total of 388 inter-plane separations reported in this work.Here, we limit ourselves to the case m = 0, as the electronic properties of interest, reported in a follow up paper, only require transitions to states with Δm = 0 from an initial 1s hydrogen state.Finally, for D < 3 a.u., the grid in the z-coordinate had a fixed number of points N z = 50 to obtain a better description of the energy levels induced by the particle in a narrow well as D → 0.

Analytical solutions for a 2-D and 3-D hydrogen atom
Before discussing the numerical results, let us review the analytical solutions for the wave-functions and energy values known for the limiting cases when D → 0 (2-D atom) and D → ∞ (3-D atom).The presence of the planes defines a preferred direction or symmetry for the system.In this case, the z-axis becomes the axis of symmetry and is a well defined direction for the atom in the presence of the planes.For a free unconfined hydrogen atom, such a direction does not exist and averaged results over all the direction are required when reporting electronic properties.
Limit when D → 0 (2-D atom): In the case when the impenetrable walls have a small inter-planar separation, one can assume that z = ρ and the confinement potential can be separated, i.e.Equation (3) becomes, approximately as D → 0. Consequently, the Schrödinger equation is where l 1 ¢ = , 2, 3, .... is the quantum number for the excitation states along the z-direction.Note here that l¢ is not the angular quantum number l in 3-D.In the limit of and m becomes the total angular momentum quantum number, i.e. the angular momentum along the z-axis for rotations in the xyplane.This means that as the inter-plane distance is reduced, the electron rotation around the nuclei has only angular momentum along the z-axis, thus becoming confined in the plane.Furthermore, equation (7) shows even and odd solutions.If l¢ is odd, we have even parity wave-functions under reflection at z = 0 and if l¢ is even then we have odd parity wave-functions under reflection.
The solution for the radial component has been reported by Yang et al [61] for a 2-D hydrogen atom are the associated Laguerre polynomials [70] and the energy of the 2-D hydrogen atom is With this, the total electronic energy of the 2-D hydrogen atom is in the limit of small inter-plane separation as D → 0. Notice that as D → 0, the dominant term is the confinement induced by the planes and the energy increases inversely proportional to the square of D. Consequently, in quasi 2-D, any state is described by the quantum numbers n ml ¢ ¢.We will use this notation when describing the electronic states for D → 0. This choice follows the 3-D spherical coordinates (r,θ,j) →nlm, as n corresponds to the radial coordinate, l to the angular coordinate θ, and m to the axial angular coordinate j.Since in cylindrical coordinates (ρ, j, z) →n ml ¢ ¢ for each independent coordinate; n¢ for ρ, m for j, and l¢ for z.
Limit when D → ∞ (3-D atom): The case when D → ∞ is well known in the literature [71], since it corresponds to a free 3-D atom.The analytical solutions (in spherical coordinates) can be found in many quantum mechanical books, e.g.[69,71,72].As the transformation from spherical to cylindrical coordinates is trivial, as z r cos q = , r sin r q = , and j is the same, all the electronic properties can be expressed in terms of the cylindrical coordinates ρ, j, and z.
At this stage let us make some clarifying remarks on the labeling of states used throughout this work.As stated above, the cases D → 0 and D → ∞ have good quantum numbers and are properly described by the n ml ¢ ¢ and nlm notation.However, for finite D, the symmetry of the system is D ∞h such that the quantum number m defines the symmetry of the species.For m = 0, 1, 2, ..., the magnitude of the magnetic quantum number, gives rise to σ, π, δ, ... terms for the D ∞h group.Furthermore, the parity of the state with respect to inversion by l corresponds to g (gerade) and u (ungerade) states, for l = 0 and 1, respectively, at D → ∞ and similarly for l 1 ¢ = and 2 at D → 0. As the (x,y)-plane is perpendicular to the main symmetry axis z, the parity with respect to reflection operator for the D ∞h group is ( ) As stated in [38, 41] for a D ∞h group, it is customary to label the 1s and 2s states as σ g states, 2p, 3p −σ u −, and π u −states, 3d−σ g −, π g −,δ g -states and so on for intermediate D plane separation.Hence, the lowest states are labeled s s p p s d s d p s ¼ However, this notation loses the nodal information of the states.This is why the traditional nl notation is kept even for finite D values since in this work we show the evolutions of the state from 3D (nlm) to 2D (n ml ¢ ¢) states, as we present below.

Energy levels
Our results for the energy levels of a hydrogen atom, when placed in the middle of two planes, are shown in figure 2(A) for states with n 10, as a function of the planes separation, D. These results correspond to the case of a magnetic quantum number m = 0.At large planes separation, we observe that the energy levels tend to the unconfined 3-D hydrogen atom energies, E nlm = − Z 2 /2n 2 in a.u..However, as the planes separation is reduced, we start to observe some interesting behavior.Starting from the 1s energy level, in figure 2(B), we find that it is almost unaffected for planes separation with D > 10 a.u..For D < 10 a.u., it starts to rise, reaching a positive energy at D ∼ 2.3768 a.u..In this case, the electron is not delocalized as it remains in the state n 1 ¢ = along the ρ-direction and l 1 ¢ = in the z-direction [ground state, equation (11)].From equation (11), one finds that at , the energy becomes null (E = 0) in the limit D → 0. In our case, one obtains D = 1.5708 a.u.for the ground state, which is lower than our numerical result.This shows that both, the z-confinement and the Coulombic potential are still competing.For the n = 2 energy levels, we find that it breaks its degeneracy and rises faster for the 2p state than for the 2s energy level, as shown in figure 2(A).We encounter an energy level crossing between the 2p and the 3s energy level at D = 7.8662 a.u..The 2p energy level keeps increasing as the inter-plane distance is reduced with a crossing with the 3d state at D = 7.2332 a.u..The same occurs with the 3p energy level which shows a faster increase with a level crossing occurring between the 3p and the 3d states at D = 20.2463a.u.. Interestingly, for D > 20.0 a.u. the 3d has a slightly higher energy value than the 3p, becoming degenerated at very large values of D. The 3p energy level keeps increasing, as D diminishes, showing a crossing with the 4s state at D = 16.4020a.u., followed with a crossing with the 4d state at D = 15.1469a.u..A similar behavior is observed for the 4p energy level with a first crossing with the 4d energy level at D = 26.0048a.u.followed with a crossing with the 5s state at D = 23.1761.Due to the symmetry of the system and since parity is a good quantum number, one expects avoiding and crossing of levels.We observe that all states with l-odd of a given n level have crossings with l-even of higher n states and find avoided crossings between states with the same l parity.Thus, the effect of the planes confinement changes the ordering of the energy levels, even though the ρ-coordinate is unrestricted.In general, higher excited states (Rydberg states) are affected the first at large D planes separation consequence of the planes boundary in the z-direction.Notice that the 1s and 2s states do not show any level crossing.Energy levels of a hydrogen atom confined by two impenetrable planes as a function of the planes distance, D, when the atom is placed in the middle of the planes.The total energy of the confined atom for n 10 is shown in (A).Note that the energy of the l-odd states has a faster increase as D is reduced that leads to a second branch l 2 ¢ = , while the l-even energy states leads to a first branch l 1 ¢ = .The behavior of the 1s state, as a function of D is shown in (B).The energy for l-even states is shown in (C), while the energy for l-odd states is shown in (D) for up to n = 7.Our results show avoiding crossings between states with the same parity.The avoiding crossings are traced by following the color lines.See text for discussion.
To further understand the effects of the planar confinement, the behavior of the l-even states as a function of the inter-planes separation is shown in figure 2(C), where the avoiding crossings for states with the same l-even parity are observed.The energy levels for all the l-odd angular momentum quantum numbers are shown in figure 2(D).Notice that the energy level avoiding crossings are sharper in this case.In all the cases, the energy levels increase their energy and even becoming positive as the inter-plane separation is reduced.However, notice that these states are still bounded, as these states have a bound ρ energy component, i.e.E ρ < 0. The l-even and lodd energy levels show two branch regions as D diminishes.This branching is more evident in figure 2(A) in the D region around 10 and 30 a.u.for l-even and l-odd, respectively.In both figures 2(C) and (D), the results are shown for n 7, but the avoiding crossings occurs for all n > 1.
In order to understand the branching behavior at small inter-plane separation, figure 3 shows the behavior of the E z component of the energy for the hydrogen atom, obtained by subtracting the E ρ energy for a 2-D hydrogen atom at D = 0 from the total energy, i.e.E z = E − E ρ with E ρ given by equation (10) for its respective n¢ value.Recall that it only makes sense for small D, since for large planes separation, the potential is not separable.We have identified n¢, the principal quantum number in 2-D unambiguously by means of the mean value of the ρ-coordinate, 〈ρ〉, from equation (12) and figure 8 (see section 3.3 below).The labels identification is provided in table 1 and its deduction and discussion is given in section 3.3.With this, we obtain the behavior of the energy levels, as the z-confinement acts, as a function of the inter-planes separation, D. We find that states with l-even, i.e. s, d, g, .... converge towards l 1 ¢ = states (even z-states, gerade) for small inter-plane separation D, while states with l-odd (p, f, h, K) converge towards l 2 ¢ = (odd z-states, ungerade) as D → 0.
As observed, for inter-plane separations smaller than 10 a.u., the dominant contribution to the energy arises from the particle in an infinite well produced by the two-planes confinement.This means that the odd-parity of the l-states in the 3-D atom is conserved through the l¢-even parity on the z-excitation.The same occurs for even- parity through the l¢-odd parity .These results also confirm that for large inter-plane separation D, high excited states (Rydberg states) start to be affected the first by the 1-D infinity box confinement along the z-direction and the lower states require smaller inter-plane separations to feel the effect of the confinement, as they show a good  8) for l 1 ¢ = and l 2 ¢ = , which reveals the behavior of the hydrogen electron when confined by an infinity box along the z-axis as the hydrogen atom is compressed.These results show the two branches noticed in figure 2. See text for discussion.
Table 1.Label state identification for n¢, the corresponding 3-D nl states, and its averaged 〈ρ〉 value.Notice that the 3-D states are ordered as l l 1, 2 ¢ = ¢ = (l-even,l-odd).〈ρ〉 is in a.u.agreement to equation (8) at those large values of D. Thus, the size of the orbital, as determined from 〈ρ〉, is correlated with the effect induced by the inter-plane separation D.
The numerical energy values for n 5 are reported in table 2 for some representative values of the interplane separation D, distributed uniformly in a log scale for reference purposes.The numerical values for the case of the critical inter-plane separation for which a crossing between l-even and l-odd states occur, as observed in figure 2(A), are reported in table 3. Notice that the critical crossing of a state with a given n only occurs with other states with equal or larger n.Now, in order to understand the ρ behavior of the energy as as a function of D, we study the E ρ component to the electronic energy of a hydrogen atom by subtracting the z component to the total energy, i.e.

Wave-functions
The effects of the planes compression on the electronic wave-function is shown in figures 5, 6, and 7.The electronic wave-function, in the ρz-plane, for the six lowest states of the hydrogen atom for the inter-plane separation D = 20.0 a.u. is shown in figure 5. We observe that the behavior of the lowest orbitals is very close to the one of a free atom for n = 1, n = 2, and n = 3 (six lowest states).However, the 3p and 3d orbitals start to show the effect of the compression by narrowing the electron distribution along the z-axis, i.e. compressing the electron motion along the z-direction.Notice, from figure 2, that the energy levels of these states are still bound at this inter-plane separation.Now, the electronic wave-function for the case of an inter-plane separation of D = 8.0 a.u. is shown in figure 6.We notice that the n = 1 and n = 2 orbitals still resemble the 3-D atomic case.However, the n = 3 wavefunctions resemble a hydrogen atom in 2-D with only the radial ρ nodal structure for n¢ and the symmetric l 1 ¢ = (gerade) structure along z.This supports the conjecture that the z-confinement produced by the impenetrable planes is strongly affecting the electron motion along the normal to the planes, such that the electron behaves more like in an impenetrable box along this direction.This is confirmed by the behavior of the energy levels as shown in figures 2, 3, and 4.
The case for an inter-plane separation of D = 4.0 a.u. is shown in figure 7. We notice that only the ground state wave-function still resembles the 3-D 1s ground state, while the states with n > 1 show the behavior of a 2-D hydrogen atom, that is, they are described by equation (9) and would correspond to the states 201, 301, 401, 501, and 601 in the notation n ml ¢ ¢ for l 1 ¢ = , i.e. symmetric (gerade) state.Also, at these small inter-plane separation, we notice that only the state with symmetry l 1 ¢ = (gerade) appear (first branch in l¢) and the l¢-odd (ungerade)   states have very large energy levels such that electronic transitions into the l¢-odd states (second branch) will appear only in the extreme UV region.A discussion of these results will be reported in a follow up paper regarding dipole transitions for this system.

Average electronic mean orbital distance
The average mean orbital distance, 〈ρ〉, for a given excited state, is a measure of the localization of the electron.Before presenting the numerical results, let us calculate the orbital mean average distance, as obtained by using the analytical expressions for the eigenfunctions at the 2-D and 3-D limiting cases of section 2.3.
For the case when D → 0, the 2-D hydrogen atom average electron z-position within the planes becomes zero, 〈z〉 = 0, due to parity.For the ρ-direction, Yang et al [61] report the average electron position as Furthermore, notice that at D = 0 (2-D atom), 〈ρ〉 is independent of l¢, such that 〈ρ〉 merges for l¢-even and l¢-odd symmetry as D → 0. Equation ( 12) also shows why we stop reporting states up to n = 7 which fit very well within the numerical box with R 500 max = a.u.. Higher excited states require a larger box as D → 0, confirming that as D is reduced, the atom is squeezed into a large flat 'pancake' (or 'Mexican tortilla') for the excited states.
For a 3-D atom (D → ∞ ), the average electron position within the planes becomes 〈z〉 = 0 due to parity, similar to the previous case.Since the 3-D standard solutions are in spherical coordinates, then Table 2. Energy levels of a hydrogen atom confined by two impenetrable planes as a function of the inter-plane separation, D, for the lowest energy levels with n < 6.The notation at D = 0 is n ml    x y rsin is the usual radial average distance of the electron in spherical coordinates [71].Notice that for the 3-D case, 〈ρ〉 depends on the angular momentum quantum number l and m, as well as on n, in contrast to the 2-D case, which only depends on n¢ and m.The angular integration, I lm , for m = 0 gives We observe that in the 3-D case, the excited states are more compact around the atom (small 〈ρ〉) than in the 2-D case, as shown in figure 8(A).The figure shows the behavior of the electronʼs orbital mean distance along the ρ-coordinate as a function of the inter-planes separation for the lowest n 7 states of a hydrogen atom confined by two impenetrable walls.This shows that, in 2-D, the excited states become very diffuse (stretched out) along the ρ-direction.The limiting theoretical results when the hydrogen atom is free (unconfined) are shown at D = 400 a.u.(+ symbols), as given by equation (13), and for the 2-D hydrogen case at D = 0.1 a.u.(open blue circles), as given by equation (12), showing an excellent agreement to our numerical results.We find that as the inter-plane separation is reduced, the compression on the z-axis induces a stretching along the ρ-direction, as observed by the increase of the average 〈ρ〉 distance as D is reduced, for states with n > 2. For D < 10 a.u., a proper identification of the n ml ¢ ¢ states is achieved from the respective nlm states.The level identification is reported in table 1.For instance, the 1s and 2p states, for the 3-D case, converge into the n 1 ¢ = state of the 2-D atom.Furthermore, states with l-odd (ungerade) merge with states with l-even (gerade), i.e. they have the same energy and average distance, as D → 0. This is expected since there is no longer l angular momentum as D → 0, i.e. since L = − ∂ 2 /∂j 2 , thus m becomes the total angular momentum in 2-D, as there is no longer an azimuthal θ dependence on the electronic motion.The evolution of the energy levels from the 3-D states, nl, towards the 2-D states, n¢, can be better visualized in figure 9 which serves as a memory aid.
The figure shows the 3-D states in a ladder with the upper n = 1 state and descend by increasing the quantum number n, while showing the l state by increasing the steps towards the right.Interestingly, the 2-D states shown in table 1, are clearly visible by the diagonal red lines that connect the 3-D states that converge towards a n¢ state in 2-D (red number on the diagonal arrow, n¢).Clearly, n¢ increases towards the right by descending on the ladder.We observe that states with ns and (n + 1)p states converge into the same n¢.The same occurs for nd and (n + 1)f, ng and (n + 1)h and so on.In general, the 3-D states with nl and (n + 1)(l + 1) converge into a 2-D state n¢ with l = 0, 2, 4, etc..., which converge into the l 1 ¢ = (gerade) states while the (n + 1)(l + 1) converge into the l 2 ¢ = (ungerade) states.Since n m 1 ¢ -are the number of nodes in the 2-D wave-function and n − l − 1 are the number of nodes in 3-D, we observe that the two states that converge into an n¢ state are those which have the same number of nodes in 3-D, but the number of nodes in 2-D increases more than those in 3-D according as how n¢ is increasing into the ladder.This is an Aufbau principle to determine the ordering of the n¢ states in 2-D from the 3-D states.Recall that our results are only for m = 0 states, so we are missing the other m states that would lead to a Hundʼs type rule in 2-D for a multi-electron system.Work is in progress in this direction.The mean averaged orbital distance for an electron in the ρ direction as a function of the planes separation, D, for n 7 states of hydrogen confined by two impenetrable walls.In the limiting cases of 2-D (D → 0), we compare to the results of equation (12) for the ρ-coordinate (open circles) and in the limiting case of D → ∞ , we compare with the results of equation (13) (+ symbols).Notice how the l-odd (ungerade) states merge with l-even when D → 0 consequence of the compression in the z-coordinate.Frame (B) shows the results for l-even (s, d, g,K) states.Frame (C) shows the results for l-odd (p, f, h,K) states.Notice the behavior of the states around the avoiding crossings.See text for discussion.Table 4. Expectation value of the radial distance along the ρ-direction, 〈ρ〉, of a hydrogen atom confined by impenetrable planes as a function of the inter-plane separation, D, for the lowest energy levels with n < 6.In D = 0 and D = ∞ , we show the analytical results of equations (12)

6 )
allows a solution by separation of variables, ψ m (ρ, z) = χ(ρ)Z c (z).The solution in the z coordinate is just the one for one electron in a 1-D infinite box potential[69]

Figure 2 .
Figure 2. Energy levels of a hydrogen atom confined by two impenetrable planes as a function of the planes distance, D, when the atom is placed in the middle of the planes.The total energy of the confined atom for n 10 is shown in (A).Note that the energy of the l-odd states has a faster increase as D is reduced that leads to a second branch l 2 ¢ = , while the l-even energy states leads to a first branch l 1 ¢ = .The behavior of the 1s state, as a function of D is shown in (B).The energy for l-even states is shown in (C), while the energy for l-odd states is shown in (D) for up to n = 7.Our results show avoiding crossings between states with the same parity.The avoiding crossings are traced by following the color lines.See text for discussion.

Figure 3 .
Figure3.The z-contribution to the energy levels of a hydrogen atom confined by two impenetrable planes, i.e.E z = E − E ρ , as a function of the planes separation, D, when the atom is placed in the middle of the planes.The dashed lines are the 2-D theoretical results of equation (8) for l 1 ¢ = and l 2 ¢ = , which reveals the behavior of the hydrogen electron when confined by an infinity box along the z-axis as the hydrogen atom is compressed.These results show the two branches noticed in figure2.See text for discussion.
E ρ = E − E z through the previous identification of l 1 ¢ = for s, d, g, ... (l-even, gerade) and l 2 ¢ = for p, f, h, ... (l-odd, ungerade) states.Again, recall that it is only valid for small D, when the Schrödinger equation is separable.The results are shown in figure 4(A) for both l-even and l-odd.In the same figure, the analytical results for a 3-D hydrogen atom are presented at D = 400 a.u.(×symbols), while at D = 0.01 a.u., we show the analytical results of equation (10) for the 2-D hydrogen atom (open circles).The results for l-odd (p, f, h,K) states are shown in figure 4(B), while figure 4(C) shows the results for l-even (s, d, g, ...) states.We observe that for small D values, it tends towards the 2-D hydrogen atom.At intermediate D values, there is a competence between the infinitely well potential and the 3-D Coulombic interaction.The 1s, 2s, 2p, 3s, and 3p energy levels are shown in the inset, figure 4(D), where it is observed that the 1s energy level changes from -0.5 a.u. at D = 200.0a.u. to −2.0 a.u. as D → 0, while the 2p state converges to the 1s and the 3p converges to the 2s energy state.

Figure 4 .
Figure 4. (A) The ρ-contribution to the energy, E ρ = E − E z , as a function of D for a hydrogen atom compressed by two planes, for n > 2. Frame (B) shows the energy levels for l-odd, while (C) shows the energy levels for l-even states.Frame (D) shows the behavior of the 1s, 2s, 2p, and 3p states.The color lines indicate the l-symmetry, as in figure 2. The open blue circles at D = 0.1 a.u.are the analytical results of equation (10) while the×symbols are the 3-D results.See text for discussion.

Figure 5 .
Figure 5. Electronic wave-function for the first 6 lowest states of a hydrogen atom placed between two impenetrable planes as a function of ρ and z for the inter-planar separation of D = 20.0 a.u..These states still resemble the 3-D free hydrogen atom orbitals, as denoted by the upper labels.See text for discussion.

Figure 6 .
Figure 6.The same as figure5, but for the inter-planar separation of D = 8.0 a.u.. Except for n = 1 and n = 2, the state with n = 3 resembles the 2-D confined atom for the symmetric case l 1 ¢ = .This is emphasized in the labels in the upper part of the figure with the notation n ml ¢ ¢ for the 2-D case.See text for discussion.

Figure 7 .
Figure 7.The same as figure 5, but for the inter-planar separation of D = 4.0 a.u..All the states resemble a 2-D hydrogen atom wavefunction along the ρ-coordinate with l 1 ¢ = along the z-coordinate.The states are described by the upper labels in the notation n ml ¢ ¢.See text for discussion.

Figure 8 .
Figure 8. (A)The mean averaged orbital distance for an electron in the ρ direction as a function of the planes separation, D, for n 7 states of hydrogen confined by two impenetrable walls.In the limiting cases of 2-D (D → 0), we compare to the results of equation(12) for the ρ-coordinate (open circles) and in the limiting case of D → ∞ , we compare with the results of equation (13) (+ symbols).Notice how the l-odd (ungerade) states merge with l-even when D → 0 consequence of the compression in the z-coordinate.Frame (B) shows the results for l-even (s, d, g,K) states.Frame (C) shows the results for l-odd (p, f, h,K) states.Notice the behavior of the states around the avoiding crossings.See text for discussion.

Table 3 .
See text for discussion.The representative D values follow a uniform grid spacing in a log scale.Inter-plane separation, D (in a.u.), at which there is a crossing between l-even and l-odd states for the lowest excitations up to n 7.