CHINESE JOURNAL OF PHYSICS
AUGUST 1993
VOL. 31, NO. 4
Quantum States of a Confined or Half-Confined Hydrogen Atom
C. Yang* and B. I. Wang
Department of Physics, Tamkang Unzversity,
Tamsui, Taiwan 137, R.O.C.
(Received February 15, 1993)
A calculational method is developed to solve the Schrodinger equation of a confined or “h alf-confined” hydrogen atom, i.e., a hydrogen atom with wavefunction which
vanishes at a given closed or opened surface. The method is based on the use of the
.
general solution (a linear combination of linearly independent particular solutions) of
the differential equation and the numerical selection of the combination coefficients and
the energy to satisfy the boundary condition. This method is applied to investigate
the electronic energy and the electric dipole moment of a hydrogen atom confined in
a box with an impenetrable wall (e.g., in an interstitial position or vacancy inside a
solid) or “half-confined” near a vacancy (defect) on a solid surface in dependence on
the distance from the surface. The electron probability density at the atom’s nucleus
is also calculated. The application of this method to the cases of penetrable surfaces is
discussed.
I. INTRODUCTION
Many physical and chemical processes taking place at solid surfaces, such as catalysis,
adsorption, etching etc., are closely connected with conditions of the surface as well as the
quantum states of the atoms and molecules participating in these processes (see, e.g., [1,2]).
In Ref. [3] the behavior of hydrogen interacting with solid surfaces is reviewed in detail.
The influence of the solid surface upon atom states is theoretically investigated in Refs.
[4-321. In all these papers (except Ref. [20]) the surface is considered as an impenetrable
wall for atomic electrons. In [20] a partially penetrable spherical surface is considered.
In earlier works [4-221 the simplest case of an atom confined in the center of a spherical
box was studied. The influence of the box radius on energy levels [4,19,20,24,25], eigenfunctions [20,24,25], polarizability [4,20], pressure [4,18-201, nuclear magnetic screening constant
[11,20] and hyperfine splitting [l&20] h ave been studied. The cases of the hydrogen atom
[4-l&20,22] a n d a t oms of many electrons [19,21] h ave been considered. It has been found
that the energy levels, pressure, magnetic screening and hyperfine splitting increase (the
polarizability decreases) monotonically and nonlinearly with decreasing of the box radius.
481
@ 1993 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
482
.
QUANTUM STATES OF A CONFIKED OR HALF-CONFINED HYDROGEN A T O M
VOL. 31
In Refs. [30,31] formulae for the energy shift and the normalized wavefunction close
to the origin are obtained for a nonrelativistic partical bound in a smooth spherically
symmetric single-well potential enclosed in a large impenetrable sphere.
Later Ley-Iioo et al. have studied the cases of nonspherical boxes, namely, boxes
defined by pairs of confocal, coaxial and mutually intersecting pa.raboloidal walls [24] and
prolate spheroidal boxes [25]. The hydrogen atom is supposed being located at the focus.
Because of the absence of spherical symmetry nonvanishing anisotropic components of the
hyperfine splitting are found. In the case of prolate spheroidal boxes the Hc and HeH++
molecular ions with the nuclei located at the foci have been also considered.
In Refs. [18,20,22,24-261 a comparison with atomic hydrogen trapped in interstitial
sites of o-quartz and corresponding discussions have been made.
The cases of a.toms “half-confined” near opened surfaces ha.ve been considered in Refs.
[2i-29,321.
Shari et al. [27] obtained an exact solution to the Schrodinger equation for a hydro
genie system in a. half space (i.e., in the presence of an impenetrable plane surface). It was
found [27-291 that 1) when the atom approaches the surface its energy increases, and that
2) the ground-state wavefunction is no longer spherically symmetric.
You et al. [32] obtained an exact solution to the Schrodinger equation of a hydrogen
atom when the nucleus is located at the focus of an upwardly opened axially symmetrical
paraboloidal impenetrable surface. It was found that the absence of spherical symmetry of
the ground-state wavefunction leads to a non-zero electric dipole moment pointing toward
the surface.
Works [4-7,11,15,18,20,22-321 are based on analytic methods of direct solving the
hydrogen-atom Schrodinger equation under the boundary condition that the wavefunction
vanishes at the given surfa.ce. With such techniques it is very difficult or impossible to
treat ca.ses of surfa.ces of arbitrary forms and arbitrary pa,rtially penetrable surfaces. The
perturbation theory [9,10,12,14-171 and the variation method [8,13,21] have been also used
to treat the problem of a confined atom? but only for some specified cases.
In the present pa.per we suggest a method of solving the Schrodinger equa,tion of
a hydrogen a.tom confined or “h alf-confined ” near a.n arbitrary impenetrable or partially
penetra.ble surface. This method is applied to cases of axially symmetrical surfaces. The
energy and the electric dipole moment of the atom in the ground state and in some lowlying
excited states have been determined. The electron proba.bility density at the nucleus also
was calculated.
C. YANG AND B. I. WANG
VOL. 31
485
These are linear equations in ulm. We solve these equations (i.e., find the coefficients
aim and the corresponding value of 6,) by means of a subroutine SQRLS [35]. Then we vary
E until 6, achieves its absolute minimum. We regard this energy value as the eigenvalue.
When the nucleus is located on the axis of symmetry of the surface S, nz is a good
quantum number. This means that we may consider only the axially symmetric wavefunctions (m = 0) in (8) (at 1 east for the ground state).
IV. TEST OF THE METHOD
For a hydrogen atom located at the center of an impenetra.ble sphere, the above described method and the exact solution [4-G] give the same result: in this case the Schrodinger
Eq. (1) allows sepa.ration of variables in spherical coordinates.
We use our method to treat two cases [27,32] o f a L‘llalf-confined” hydrogen atom in
which exact solutions are known. To do this, first of all ue must determine the maximum
orbital number L in the sum (8). Generally speaking, the bigger L is, the higher is the
accuracy. However, L is restricted from above by the required computer memory volume
and the calculation time. We choose this number by controling the coefficients aim with
l a r g e l(1 = L,L - 1,L - 2;. G): they must make practically no contribution to (8) and,
consequently, to the state energy E and the value of 6, (10).
Then we must determine the radius R of the limiting sphere CR. In classical mechanics, the ma,ximum distance R,, of an electron (with energy E) from a nucleus in the
Coulomb field is l/[E]. Evidently, we must take R > R,l. For a hydrogen atom at the
center of a sphere of radius R = 8no the ground-state energy En differs from its value (-0.5
a.u.) at R + co approsimately to 3 . 10e5. In this case R,/ = ‘2no. With this accuracy we
choose R > 4R,l (or R > 4/]E]) in our calculations.
Another problem is the choice of the number N of points on the surface S f CR that
we should take for the calculation of 6,. -4s in the case of L, the value of N is limited by
the memory volume and the calculation time. We take N = 300 and increase this value in
cases of “bad” surfaces, e.g., surfaces with sma.11 wells (see $5).
For an impenetrable plane surface the ground-state energies Eo calculated by means
of our method (M = 0, L = 19, R = 12 - 40, N = 300) for various distances dz from the
atom’s nucleus to the plane are given in Table I as well as the results of Ref. [29]. The
coincidence is quite good.
For the case of an impenetrable paraboloidal surface 5’ of focal distance <u
2=
x2ty2
x0
co
--
2
(14)
with the atom’s nucleus at the focus, the ground-state energy and the electric dipole moment
486
QUANTUM STATES OF .4 CONFINED OR HALF-CONFINED HYDROGEN ATOM
VOL. 31
TABLE I. Ground sta.te energy Eu of a half-confined hydrogen atom at a, distance dZ from
an impenetrable plane surface (d, are in au, Eo in 2 Ryd)
.
dz
Eo
E O(exact)
4.00
3.00
2.00
1.00
0.20
0.01
-0.498685
-0.492648
-0.460621
-0.308858
-0.140000
-0.125000
-0.498692
-0.492679
-0.460870
-0.308659
-0.140108
-0.125000
are calculated both with our method and from the exact solution of Ref. [32] (Table II).
The coincidence is also very good.
In both ca.ses the minimum of 6, as a function of E is very sharp. The uncertainty
AE in calcula.ting E (the “r esolution” of the method) may be estimated from the energy
deviation at which 6, has increased, for example, by a factor three. We find IAE/EoI is of
the order of 10e5 - 10T6 or less, while the value of 6, is of the order of 10e6 or less.
It should be mentioned that in our method we use an exact solution of the Schrodinger
Eq. (9) and vary the parameters arm and E to satisfy the boundary condition (2) as
accurately as possible. So this is not a traditional variational method and the obtained
approxima.te values of E must not be necessarily larger than the exact ones.
TABLE II. Ground-state energy Eo and electric dipole moment of a hydrogen atom located
at the focus of an axially symmetrical paraboloidal surface of focal distance co
((0 are in au, Eo in 2 Ryd, dipole moments in eat)
CO
8.0
6.0
4.0
3.0
2.0
1.8
1.6
Eo
EO(exact)
dipole
dipoleeXaCt
-0.497675
-0.487223
-0.431219
-0.338163
-0.125000
-0.OG74023
-0.0174170
-0.4976747
-0.4872231
-0.4312189
-0.3381674
-0.1250000
-0.06740629
-0.01742440
0.04100
0.1299
0.3649
0.6345
1.581
2.341
5.009
0.0409116
0.129952
0.364930
0.634748
1.58123
2.34104
5.01179
C. YANG AND B. I. WANG
VOL. 31
487
V. HYDROGEN ATOM CONFINED IN A SPHERICAL BOX
OR HALF-CONFINED NEAR A SOLID PLANE
SURFACE VACANCY
Now we apply our calcula.tion method to cases in which esact solutions are unknown.
First we consider a hydrogen atom at some arbitrary point inside an impenetrable
spherical box of radius Ro. This is a model of a hydrogen atom confined, for example, in
an interstitial position within some solid in which atoms or ions have closed-shell structure.
By use of our method we can investigate the quantum state of the atom with the nucleus
located not necessarily at the center of the sphere.
The calculation shows that the ground-state energy increases quite slowly with the
distance d, of the atom’s nucleus from the sphere center at small d, < 1 and rapidly when
d, becomes greater (d, ry Ro) (Fig. 1). In other words, oscillations of a hydrogen atom
-0.5
0
1
2
3
4
5
6
dz
FIG. 1. The ground-state energy Eo (in 2 Ryd) of a hydrogen atom confined in an impenetrable
spherical box versus the dista.nce d2 (in a~) from the atom’s nucleus to the center of the
box. Letters A-D correspond to different box radii Ro = R.5,4,3 (in (10).
488
QUANTUM ST.4TES OF A CONFINED OR HALF-CONFINED HYDROGEN ATOM
VOL. 31
in a spherical box are not harmonic ones. When the atom’s nucleus is not at the center
of the box (i.e., when d, # 0) the ground-state electron probability density distribution is
not spherically symmetric: the electron is partially “pushed away” from the surface. So
a nonzero electric dipole moment pointing toward the surface appears and increases with
d, (Fig. 2). Thus the atom oscillating in a box is accompanied by an alternating dipole
moment and consequently generates electromagnetic waves.
A similar situation should take place when a hydrogen atom moves in a planar or
axial channel of a monocrystal oscillating perpendicularly to the direction of motion.
Next we consider a case when a hydrogen atom is “half-confined” near a vacancy
(defect) of a plane solid surfa.ce. We a.pproximate the vacancy by mea.ns of a well of axially
symmetric paraboloidal form with a focal distance (0. The well’s depth d is equal to its
maximal radius pu (i.e., (0 = d/2). The atom is located at the axis of the paraboloid at a
distance d, to the plane.
.
4
1
I
I
6
FIG. 2. The ground-stat,e dipole moment (iu euo) of a hydrogen atom confined in an impenetrable
spherical box versus the distance d, from the at,om’s nucleus to t,he center of the box,
Letters A-D lneau t.he same as in Fig. 1.
VOL. 31
C. YAKG AND B. I. WANG
489
-0.2
-0.3
-0.4
-0.5
-3
-2
-1
0
1
2
3
dz
FIG. 3. The ground-state energy E0 (in 2 Ryd) of a hydrogen atom half-confined near a plane solid
surface with a paraboloidal vacancy versus the distance dz (in ~0) from the atom ’s nucleus
to the plane surface. The depth d of the vacancy is equal to its radius po. A--p0 = 0,
B-p0 = 2, C-p0 = 3, D--p0 = 4. po in a~.
The calculation shows that the ground-state energy increases when the atom approaches the plane, but more slowly than in the case of an ideal plane surface without well.
The bigger and deeper the well is, the lower is the ground-state energy for the same distance
d, to the plane (E’ig. 3). This means that the atom is stabler with a nearby vacancy than
without. For example, if the dimension (depth and radius) of the vacancy is 3 au and the
hydrogen atom is in the plane (d, = 0) the energy difference is about 0.63 Ryd.
Again, the ground-state wavefunction is not spherically symmetric (Fig. 4). When
the atom approaches the surfa.ce, the ma.simum of the atom’s electron density moves away
from the surface and becomes broader. In the limiting case when there is no well on the
plane surface (d = 0) the ground state is 2p [27-291, i.e., the normalized wavefunction is
l
xqr) = -T esp( -r/2) cos 6,.
46
(15)
490
QUANTUM STATES OF A CONFINED OR HALF-CONFINED HYDROGEN ATOM
VOL. 31
F I G . 4 . The curves of constant values of the ground-stat.e electron probability densit.y [Al* (q(r)
is the normalized wavefunction) of a hydrogen atom half-confined at d2 = 1 from a plane
solid surface with a vacancy of radius po = 2 (d, and po in ~0). The depth d of the vacancy
is equal to its radius. Figures at the curves denote different values of (Q(r)/’ : 1 - 0.2,
2 - 0.1, 3 - 0.06, 4 - 0.02, 5 - 0.006, G - 0.002 (in ~0~). S denotes the surface on which
1Q(r)12 = 0.
The maximum of the square modulus of this function is located at the point (r = 2,
0 = 0). Its magnitude is about 0.01077 whereas the ma.ximum value of the squa.re modulus
of the ground-state wavefunction of an unconfined hydrogen atom (far from the surface),
as it is well-known, is 7r-l z 0.3183.
The absence of spherical symmetry of the electron wavefunction when the atom
approaches the surfa.ce results in a nonzero electric dipole moment which increases with
decrease of d, (Fig. 5). In the limiting case of a plane surface without vacancy (d = 0) the
maximum dipole moment in the ground state (at d, = 0) can be easily determined from
(15) a.nd is equal to 3.75 eno. In the presence of a va,cancy, the maximum dipole moment
(at dZ = -d) is larger because of the infIuence of the well.
VOL.
31
C.YANGANDB.I.WANG
4
I
I
491
I
I
I
0
1
2
3
0
75
a2
.-0
1
0
-3
-2
-1
3
FIG. 5. The ground-state electric dipole moment (in eao) of a hydrogen atom half-confined near a
plane solid surface with a vacancy versus the distance d, (in ~0) from the atom ’s nucleus
to the plane. Letters A-D mean the same as in Fig. 3.
The hyperfine splitting [3G] given by the Fermi contact term is proportional to the
electron density ]!P(O)]* at the nucleus (!P(T ) is the normalized wvavefunction). In Fig. G
this quantity 2rersu.s the distance d, is given for the case of plane surface without vacancy
(d = 0). We find the dependence is not monotonic: with the decrease of d, the electron
density at the nucleus increases at first and then decreases quite rapidly because of the
repulsive role of the surface for the atom’s electron. .4t a distance d, < 0.5~ the electron
density at the nucleus is very small.
The You ’ s model [32] corresponds to the case d + d, = (b/2, d + 00, dZ + -cm
(in our notation). Because of the infinite depth of the well in this model, the ground-state
energy and the dipole moment are noticeably larger than in the model of plane surface with
a paraboloidal well of a finite depth d at the same distance of the hydrogen nucleus from
the well bottom d + d, = 6. For example, at d = 2, <u = 1 the ground-state energy in
You’s model is positive (i.e., the electron is not in a bound state) while in the second model
the electron is in a bound state and the dipole moment is approximately equal to 2.5 eao.
-”
-
492
QUANTUM STATES OF A CONFINED OR HALF-CONFINED HYDROGEN ATOM
0.5
I
I
-2
-1
I
I
P(O)l’
0.4
I
VOL. 31
1’
0.3
0.2
0.1
0.0
-3
FIG. 6. The ground-state electron density IQ(O)l’ (in a03) at the nucleus of a hydrogen atom
half-confined near a plane solid surface with a vaca.ncy uers7l.s t.he distance d, (in Q) from
the atom’s nucleus to the plane surface. Letters A-D mean the same as in Fig. 3.
VI. EXCITED STATES
The first electronic exited state of a free hydrogen atom has a 4-fold degeneracy
(without regard to the spin degeneracy). For the hydrogen atom confined or half-confined
near a solid surface the degeneracy is removed partially or completely, depending on the
symmetry of the surface.
The foregoing calculational method ($3) is applied to calculate the excited-state
energies and wavefunctions as well. In both cases considered in $4, results a.gree quite well
with the exact solutions. For the hydrogen atom confined in a spherical box, excited states
with negative E exist at R, >_ 6.2~.
The energy and the dipole moment of the first excited state in the case of the plane
surface with a paraboloidal vvell are sl~own in Fig. 7. Examples of the electron probability
density distribution in this exited state are given in Fig. 8 for the case that the distance of
L...
C. YANG AND B. I. WANG
VOL. 31
I
I
I
I
I
493
1 '1.5
I
I
4.0
L
.-::
-0
3.5
3.0
1
I
I
I
I
I
I
I
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-0.125
3.5
dz
FIG. 7. The first excited-state energy and dipole moment of a hydrogen atom half-confined near
a plane solid surface with a vacancy. po = 3. (E in 2
Ryd; dipole in eao; d, and po in a~).
The depth d of the vacancy is equal to its radius.
the hydrogen nucleus from the well bottom d + d, = 5~x0. The distribution is spherically
unsymmetric. It is also unsymmetric with respect to the plane .z = 0 perpendicular to
the axis of symmetry. Again the electron is “pushed away” from the surface. There is a
density maximum at the distance - 3an from the nucleus and this distance increases when
the nucleus approaches the surface.
VII. PENETRABLE SURFACES
The calculational method of 53 may be applied to cases of partially penetrable surfaces
as well. Behind the penetrable surface the electron’s wavefunction is not identically equal
to zero. So at the surface the usual boundary conditions of continuity of the wavefunction
e(~) and its gradient C@(T) should take place.
494
QUANTUM STATES OF A CONFINED OR HALF-CONFINED HYDROGEN ATOM
VOL. 31
a
6
4
2
0
-2
- 4
-6
-a
-10
-10
I
-a
1
-6
I
-4
I
-2
I
0
I
I
2
4
6
a
I
I
10
FIG. 8. The curves of constant values of the ground-state electron probability density lQ(r)12 (Q(r)
is the normalized wavefunction) of an excited-state hydrogen atom half-confined near a
plane solid surfa.ce with a vacancy. Figures at the curves denote different values of j\sl(r)l’:
1 - 0.01, 2 - 0.02, 3 - 0.006, 4 - 0.002, 5 - 0.0006 (in a,“). dt = 2, d = p. = 3 (in ac). S
denotes the surface on which [Q~(r)l = 0.
If we find the general solution @ Iln;)( T of
) the Schrodinger equation in the space benind
the surface (spa.ce II), we may write down the bounda.ry conditions a.s following
@‘(T)ls=
#(T)\s,
(16)
~@)(T)l,= vq(;)(T)ls,
@(T)l CRI = 0 ’
(17)
(18)
qjn;)(T)I CRII = 0 ’
(19)
where *j”‘(
T
)is the electron wavefunction in the space 1 where the atom is located. S
denotes the penetrable surface. CR1 and CR11 denote the parts of the large limiting sphere
in the spaces I and II respectively. If the surface S is a closed one, the sphere part CR1 is
absent.
Then we must vary the combination coefficients in
E to minimize the “d issatisfaction degree”
q(;)(T)
and
ME’
and the energy
- _._
VOL.
31
C. YANG
AND B.I.WANG
495
under the normalization condition (11). The combination coefficients and the energy which
will be found will give the sought wavefunction and the energy eigenvalue.
ACKNOWLEDGMENTS
We thank Drs. J. H. You, Y. Shan, W. Y. Chen, and C. M. Wei for fruitful discussions.
We also thank C. L. Sie and 2. T. Guo for assistance. We are grateful to the National Science
Council of the Republic of China for support under contract number NSCSl-020S-M032-01.
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