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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.64 no.4 México jul./ago. 2018

 

Research

The energy correction due to a finite size nucleus of the hydrogen atom confined in a penetrable spherical cavity

N. Aquino1 

R.A. Rojas1 

H.E. Montgomery2 

1Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, 09340 Ciudad de México. México

2Chemistry Program, Centre College, 600 West Walnut Street, Danville, KY. USA 40422.

Abstract

We have computed accurate values for the ground state energy of a hydrogen atom confined by a finite spherical barrier of height V 0 as a function of the confinement radius R c . We consider the nucleus as a sphere with a uniform charge distribution instead of as a point particle. The contribution to the ground state energy due to the finite nuclear size is computed as a function of the confinement radius, R c , and the height of the barrier, V 0 , using time-independent perturbation theory. For an impenetrable cavity with R c = 0.5 au, we found that this energy correction is fifty times higher than the corresponding value for the free hydrogen atom. For a finite value of V 0 , we found that the maximum of the energy correction is reached at a value R CMAX , which is very close to the position at which the electron density is most compact around the nucleus. This is confirmed though evaluation of the Shannon entropy in configuration space.

Keywords: Confined hydrogen atom; finite nucleus correction; Shannon entropy.

PACS: 67.63.Gh; 67.80.fh

1. Introduction

Eighty years ago, Michels et al. 1 used the confined hydrogen atom (CHA) as a model to study the change in the polarizability of a hydrogen atom subjected to high external pressure. In this model, the nucleus of the hydrogen atom was clamped at the center of an impenetrable sphere of radius R c , while the electron could move within the included volume. In this system, ionization is not possible. The way to account for ionization is to allow the walls to be penetrable. This penetrable model was successfully used by Ley-Koo and Rubinstein2 to explain the ionization of a hydrogen atom trapped in alpha-quartz3-4.

Many applications have been developed from these models and they have been generalized to cavities with different geometries. This model has subsequently been applied to a wide range of physical problems 1-13. Observable properties of the systems such the energy spectrum, transition frequencies and probabilities, polarizability and the behavior of atoms trapped in fullerenes, etc., are changed by spatial confinement. Reviews and books on those topics are available 5. Recent experimental studies show that the electron capture nuclear decay rate is increased under compression 14-16. A partial explanation of this effect was given using the model of many-electron atoms confined in an impenetrable spherical cavity 17. With the advent of technology to construct atomic scale confinements, the study of confined systems has become increasingly relevant.

In most of the works on the properties of atoms and molecules, either free of any confinement or confined in cavities, it is assumed that the nuclei are points with charge and mass but without extent. The inclusion of a nucleus of finite size in the free hydrogen atom is accompanied by a shift in the electron energy 20-26. The magnitude of this shift is very small compared to the energy of the hydrogen atom with a point nucleus. Until now, no one has studied how this energy shift changes for the hydrogen atom, confined in spherical penetrable cavities, when a nucleus of finite size is considered.

Pyarelal and Bhatnagar 20 proposed a model of the hydrogen atom with an impenetrable nucleus of finite size. In that model, the wave functions must vanish at the surface of the nucleus. The reduction in the volume available for movement of the electron produces an increase in the energy of the electronic states. This problem has an exact solution, but the model is unrealistic.

A more realistic model consists of a spherical nucleus of radius r 0 , with an uniform distribution of charge. For hydrogen-like atoms with a small nuclear charge, Z, it is well-known that the Schrödinger equation adequately describes those systems 19. An exact solution to this problem was found by Ley-Koo et al. 19, in which they studied the muonic atoms, free of any confinement, with Z up to 90. In this work, we will adopt the model of a spherical nucleus with an uniform charge distribution, and we will use first-order perturbation theory to calculate the correction to the ground-state energy of the hydrogen atom confined in spherical penetrable and impenetrable cavities, as a function of the radius of confinement R c .

The objectives of this work are twofold: the former is to establish benchmark values for the energy of the ground state of the hydrogen atom confined in a spherical penetrable cavity, the second is to calculate the energy shift of this system when considering a spherical nucleus of finite volume with an uniform distribution of charge.

The organization of this work is as follows: in Sec. 2, we solve CHA with a point nucleus in a spherical, padded cavity. In Sec. 3 we use first-order perturbation theory to compute the energy correction due to inclusion of a nucleus of finite size. In Sec. 4 we show the results of the calculation of the energy correction due to a finite size nucleus, obtained for both penetrable and impenetrable confinements. Finally, in Sec. 5 we give our conclusions.

2. The CHA in a spherical padded cavity

In this section we give a brief description of the solution of the hydrogen atom confined in a spherical padded cavity. A detailed explanation can be found in Refs. 2, 7 and 9.

In atomic units (m=e==1), the Schrödinger equation for a hydrogen atom at the center of a sphere of radius R c and confined by a constant potential V 0 is

-122+V(r)Ψ(r,θ,ϕ)=EΨ(r,θ,ϕ), (1)

where the potential is

Vc=*20c-1r,0rRcV0,r>Rc. (2)

Equation (1) can be solved using separation of variables

ψ(r,θ,ϕ)=R(r)Yl,m(θ,ϕ) (3)

where Yl,m(θ,ϕ) is a normalized spherical harmonic and R(r) is a radial function composed of Ri(r), the wavefunction inside the sphere and Re(r), the wavefunction external to the sphere.

The Schrödinger equation for the inner region 0rRc

-121r2ddrr2ddr+ll+12r2-1r X Rir=ERi(r) (4)

whose solution is given by 8-9:

Ri(r)=Ae-ρ/2ρlM(-β+l+1,2l+2,ρ), (5)

where M(a,b,r) is the confluent hypergoemetric function 18 and

β=1-2E,ρ=2r/β. (6)

The external region, r>Rc, is described by the Schrödinger equation

-121r2ddrr2ddr+ll+12r2+V0 X Rir=ERi(r) (7)

which can be written as

d2dy2+2yddy-l(l+1)y2-1Re(r)=0, (8)

where

y=kr,k2=2μ(V0-E)/2, (9)

and whose solution is given by 2

Re(y)=By-l-1e-yM(-l,-2l,2y). (10)

Hereafter, we restrict our attention to states with l=0 because these are the only states which have non-zero value of Ri(r) when r=0.

The eigenvalues are determined by the requirement that the total wavefunction must be continuous with continuous first derivatives at r=Rc. This is most easily accomplished by matching logarithmic derivatives at R c , resulting in the equation

ddr[lnRi]|Rc-ddr[lnRe]|Rc=0. (11)

The normalization constants A and B are found from the requirements that

Ri2Rcβ=Re(kRc).

and the normalization condition

0r0(Ri)2r2dr+r0(Re)2r2dr=1. (12)

As mentioned above, the zeroes of the logarithmic derivative equation are the eigenvalues of the problem. The first zero corresponds to the ground state energy. With this value, we construct the wave function in each region. To find the zeroes of the equation we used Mathematica 9 with the command FindRoot with 50-digit precision variables. The eigenvalues obtained through this procedure are shown in Tables I, II, III, V as a function of the box radius R c and the potential height V 0 . These results are in complete agreement with previous calculations 2,7,9. The results are shown with 15 digits after the decimal point.

TABLE I CHA ground state energy E10 (1) , and first-order correction due to the finite nucleus, E10 (1) (Rc), as a function of the confinement radius Rc for V0 = 0. Also shown is the ratio between the energy correction of the confined system, E10 (1) (Rc), and the correction of the free hydrogen atom E10. Energies are in hartrees and distances are in bohrs. 

R c E 10 E101 (10-10) E101(Rc) E101
0.75 0.002551608753406 0.342781939506244 0.345
0.8 -.017424391031037 0.808141286815622 0.814
0.9 -0.067406311452319 1.325196851197549 1.335
1.0 -.125000000000000 1.544741614031751 1.556
1.1 -.180067083549980 1.621160850903080 1.633
1.2 -0.229179151514072 1.627437259807232 1.639
1.5 -0.338167417956141 1.507712255705773 1.519
2.0 -0.431218889241793 1.287458459987721 1.297
2.5 -0.470393522970229 1.151246460119175 1.160
3.0 -0.487223082818398 1.075285232620169 1.083
3.5 -0.494519692585279 1.034355146835889 1.042
4.0 -0.497674689400819 1.012980250520460 1.020
4.5 -0.499025598864902 1.002187518743767 1.009
5.0 -0.499596671366575 0.996918364180960 1.004

TABLE II CHA ground state energy E10(1), and first-order correction due to the finite nucleus, E10(1) (Rc), as a function of the confinement radius Rc for V0 = 5. Also shown is the ratio between the energy correction of the confined system, E10(1) (Rc), and the correction of the free hydrogen atom E10. Energies are in hartrees and distances are in bohrs. 

R c E 10
0.4 4.827691517791768 6.456668836007435 6.506
0.5 3.907609648085746 10.240494963124124 10.319
0.6 2.982382648801240 9.598999388792230 9.672
0.8 1.666642892127310 6.980485974979947 7.034
0.9 1.229573168163173 5.926377707785826 5.972
1.0 0.893377387585463 5.081081669583943 5.120
1.5 0.028688218476904 2.781656875168397 2.803
2.0 -0.273954162644265 1.889596355389960 1.904
2.5 -0.397386813501905 1.473407522212738 1.484
3.0 -0.452163079706601 1.256864293020748 1.266
3.5 -0.477542879189045 1.137865941478808 1.146
4.0 -0.489507801753420 1.071253920964084 1.079
4.5 -0.495153894267392 1.034166451405600 1.042
5.0 -0.497794417047201 1.013927192430064 1.021

TABLE III CHA ground state energy E10 (1), and first-order correction due to the finite nucleus, E10 (1)(Rc), as a function of the confinement radius Rc for V0 = 10. Also shown is the ratio between the energy correction of the confined system, E10 (1)(Rc), and the correction of the free hydrogen atom E10. Energies are in hartrees and distances are in bohrs. 

R c E 10 E101 (10-10) E101 (Rc)E101
0.3 9.846148432090808 9.930542047590157 10.007
0.5 5.639829938566619 17.491618217286550 17.626
0.6 4.097374126091258 13.822727567461880 13.929
0.7 2.995752750577570 10.917393851422899 11.001
0.9 1.620350180030345 7.179161687379589 7.234
1.0 1.185777102316168 5.992750426662855 6.038
1.5 0.116618456909048 3.048420555184699 3.071
2.0 -0.240385375166330 1.999717840968987 2.015
2.5 -0.382937517117798 1.527421154803173 1.539
3.0 -0.445545908145495 1.285799849400771 1.295
3.5 -0.474425028203553 1.080287038138810 1.162
4.0 -0.488027526391939 1.080287038138810 1.088
4.5 -0.494454829661503 1.039165583170746 1.047
5.0 -0.497468459482318 1.016626789889568 1.024

TABLE IV CHA ground state energy E10 (1), and first-order correction due to the finite nucleus, E10 (1)(Rc), as a function of the confinement radius Rc for V0 = 10. Also shown is the ratio between the energy correction of the confined system, E10 (1)(Rc), and the correction of the free hydrogen atom E10. Energies are in hartrees and distances are in bohrs. 

R c E 10 E101 (10-10) E101 (Rc)E101
0.5 14.747970030350280 54.536609722874516 54.956
0.6 9.527707806146348 33.740936784485875 34.000
0.7 6.469926127251262 22.720335291772468 22.895
0.8 4.543380181009424 16.278628704347120 16.403
0.9 3.262189536240119 12.229690823293298 12.323
1.0 2.373990866103664 9.538267646127938 9.611
1.5 0.437018065247256 3.969102513288022 3.999
2.0 -0.125000000000000 2.355893659642013 2.374
2.5 -0.334910185427921 1.695773616320432 1.708
3.0 -0.423967287733454 1.374211998517624 1.384
3.5 -0.464357128440197 1.202793634385484 1.212
4.0 -0.483265302078022 1.107644656981758 1.116
4.5 -0.492205427798878 1.054376061163177 1.062
5.0 -0.496417006591452 1.024902478009865 1.032

3. Finite nucleus size correction

The Hamiltonian of the hydrogen atom with a finite nucleus confined by a spherical penetrable wall is given by

H=-22+Vr(r), (14)

where

Vr(r)=*20l12r0rr02-3,0<r<r0-1r,r0<r<RcV0,Rc<r<, (15)

The first two terms in Eq. (15) are the usual terms of a free hydrogen atom with a finite size nucleus 6,19,21-23, and 𝑟 0 is the radius of the hydrogen atom, a proton. The third term is introduced to confine the hydrogen atom in a spherical cavity of radiusR c .

We define the potential

Ṽ(r)=*20l-1r,r<r<R00,r0<r<Rc. (16)

Adding and subtracting Ṽ(r) to Eq. (14), and grouping terms we obtain the

H=-22+Vc(r)+H'(r). (17)

This Hamiltonian can be written in the more familiar form

H=H0+H' (18)

From Eq. (17) we immediately identify the first two terms as the Hamiltonian of a hydrogen atom confined in a penetrable spherical cavity, as analyzed in previous section. We note that forH 0 , the unperturbed Hamiltonian, the eigenfunctions ψnlm0 and eigenvalues Enl0 are well-known.

The perturbation is given by (6, 21-23)

H'=*20c12r0rr02-3+1r,r<r0,0,r0<r<. (19)

As mentioned above, we are only interested in states with l=0, because the electron density is non-zero at the origin. The correction of the energy to first-order is given by

En0(1)=ψn00|H'|ψn00. (20)

A straightforward calculation gives the following expression. The eigenfunctions ψn00 are an orthonormal set of wave functions with the form

ψn00(r,θ,ϕ)=Rn0(r)Y00(θ,ϕ). (21)

Substituting (21) in (20) we obtain

En0(1)=12r00r0|Rn0(r)|2rr02-3+2r0rr2dr. (22)

In the region rr0, taking in account that for a proton, r01,

Rnl(r)Rnl(0). (23)

Equation (22) then becomes

En0(1)=r0210|Rn0(0)|2. (24)

In this work we will use the most recent measured proton radius r0=0.8335 femtometers (1.575086726×10-5 bohrs) 24.

4. Shannon entropy

Claude E. Shannon in 1948 introduced the so called Shannon entropy in his paper “A Mathematical Theory of Communication” 27. The quantum version of Shannon entropy 𝑆 𝑟 , in configuration space is defined as

Sr=-ρ(r)lnρ(r)dr, (25)

where ρ(r) is probability density of the electron, in atomic units.

The Shannon entropy in quantum computation means the absolute limit of the best possible lossless compression of any communication, under some particular constraints 33. On the other hand, the Shannon entropy has a wide variety of applications in Physics and Chemistry 34. Usually, it is interpreted as the uncertainty associated with the particle position, which is related with the degree of localization (delocalization) of the particle. In chemistry, the Shannon entropy is associated with the delocalization of an electron in aromatic compounds 35.

Panos et al. 36, computed and plotted S r as a function of the atomic number, Z, for many-electron atoms. They found that the curve Sr(Z) has relative minimum values for the closed shell atoms as He, Ne, Ar and Kr. They interpreted this fact as the electronic density of those atoms is more compact (localized) than their near neighbors. Recently, this interpretation for the Shannon entropy was used successfully in the study of many-electron atoms confined in soft spherical cavities 37,38. In this work we will adopt this interpretation for the Shannon entropy.

5. Results

In Tables I, II, III, we show the corrections to the ground-state energy of CHA, taking into account a nucleus with finite size. The correction to the energy is small compared with the energy of the unperturbed confined atom. As the confinement radius R c grows, E10(1) approaches the value of the first-order correction of the free hydrogen atom, E10free(1)=0.9923592777×10-10 hartrees, according to Eq. (24).We can see that there is a change in the value of the energy correction that depends on R c andV 0 . This behavior can be seen more clearly in Fig. 1, where we have plotted the ratio between the correction of the ground state energy of CHA due to the finite nucleus, relative to the correction for the free hydrogen atom. It should be noted that this ratio is independent of the value of r 0 and depends only on the ratio of the wave functions evaluated at the origin

FIGURE 1 Ratio E10(1) (Rc)=E10free(1) 10free of the energy correction of CHA to the free hydrogen atom as a function of Rc. 

E10(1)E10free(1)=|R10(0)|2|R10free(0)|2, (26)

where Rfree(r) and R10(r) are the radial wave functions of the free hydrogen atom and the CHA, respectively.

For a spherical cavity with impenetrable walls, the correction to the energy grows rapidly as R c tends to zero because by reducing R c the electron is closer to the nucleus without the possibility to escape. In Fig. 1 we can see that for a value of Rc=1, the energy correction E10(1) to the ground state of the CHA is 10 times greater than in the free hydrogen atom.

This value is in complete agreement with that calculated by Goldman and Joslin 6. They found that for very small values of 𝑅 𝑐 and very excited states, the correction of the energy, E10(1), can be several orders of magnitude greater than the value for the free atom.

For a fixed value of the barrier heightV 0 , the situation is quite different. The energy correction E10(1) (R c ) grows as R c decreases, and it reaches its maximum value at a confinement radius that we call R cmax . As R c continues to decrease, the energy correction, E10(1)(Rc), decreases also, approaching zero as R c approaches a critical radius, at which the total energy of the electron is equal to the height of the barrierV 0 .

Intuitively, one expects that the energy correction E10(1) be higher in a small size cavity, in which the electron is closer the nucleus, i.e. a cavity in which the electronic density is more compact. One way to quantify the compactness of the electronic density is by means of the Shannon entropy 27-38, as was mentioned above. This idea is supported in the interpretation of localization-delocalization associated with the Shannon entropy, S r . A small value of S r , means that the electron density is more localized around the nucleus 30-32,36-38. At R c , where the entropy curve, Sr(Rc), has a minimum value, the electron density is more compact around the nucleus, and thus the contribution to the energy correction E10(1)(Rc) is greater. In Figs. 2 to 5 we plot the energy correction E10(1)(Rc) and the Shannon entropy Sr(Rc) for a few values of V 0 as a function of R c . In Fig. 2 we plot together E10(1)(Rc) and Sr(Rc), as a function of the confinement radius, R c , for V0=. We see that when R c decreases, S r also decreases. This means that the electron density around the nucleus increases and there is an increase in E10(1). This behavior continues as R c diminishes.

FIGURE 2 Energy correction E10(1) and Shannon entropy Sr as a function of Rc for V0 = 0 hartrees. 

In Figs. 3, 4, 5 we show the energy correction E10(1)(Rc) and the Shannon entropy S r for fixed values of V 0 . In all figures, we see that the curve of the energy correction E10(1)(Rc) reaches a maximum value at a confinement radius Rcmax which depends on the value of V 0 . We can also see that the curve of the Shannon entropy Sr(Rc) reaches its minimum value at a confinement radius Rc*, 𝑖.𝑒., in a spherical box of radius Rc* the electron density is more compact around the nucleus. From Figs. 3-5 we can see that Rcmax<Rc*. This result can be interpreted in the following way. As R c decreases, the value of S r also decreases, and the density becomes more compact up to a maximum value at Rc*. As Rc continues to decrease, Sr begins to increase and the electron starts to be delocalized, the wave function increases its value at the origin and inside the barrier. If Rc continues decreasing, the value of the wave function at the origin decreases quickly, but it grows fast inside the barrier. This process continues until ionization take place.

FIGURE 3 Energy correction E10(1) and Shannon entropy Sr as a function of Rc for V0 = 1 hartrees. 

FIGURE 4 Energy correction E10(1) and Shannon entropy Sr as a function of Rc for V0 = 5 hartrees. 

FIGURE 5 Energy correction E10(1) and Shannon entropy Sr as a function of Rc for a spherical impenetrable of radius Rc. 

6. Conclusions

In this work we calculated the energies and wave functions, with high numerical precision, for the ground state of the hydrogen atom confined in a penetrable spherical cavity. We also calculated the energy correction due to a nucleus of finite size for the CHA as a function of R c and V 0 . For finite barriers, the curves of the energy correction reach a maximum value, while the curves of the Shannon entropy as a function of Rcreach a minimum value. The maximum of the energy correction is always close to the position at which the Shannon entropy has its minimum value, i.e. at the position in which the electron density is most compact.

Acknowledgments

One of us (RAR) is grateful to Universidad Autónoma Metropolitana for providing the funds through a PhD studies scholarship.

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Received: January 17, 2018; Accepted: March 18, 2018

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