CHINESE JOURNAL OF PHYSICS
VOL. 33, NO. 5
OCTOBER 1995
The Hydrogen Atom Near Partially Penetrable Surfaces
C. Yang, C. L. Sie, and C. W. Liu
Department of Physics, Tamkang University,
Tamsuz, Taiwan 251, R.O.C.
(Received July 4, 1995; revised manuscript received August 22, 1995)
The quantum states of a hydrogen atom (or hydrogen-like ion) located in a closed
or opened vacuum space near a partially penetrable surface are investigated. The
electronic potential energy beyond the surface is assumed to be constant. So the electron
has a finite possibility to penetrate the surface. It is found that in the case while the
electronic potential energy Vo beyond the surface is negative the number of electron
bound states in the atom may be reduced. While V. is quite low, the atom has no
bound states at all. If Vo is close to the eigenenergy of atom, the eigenenergy reaches a
minimum at some distance between the atom’s nucleus and the surface. If Vo is positive,
the energy of the hydrogen atom is smaller near a projection of the surface at the same
nearest distance from the atom’s nucleus to the surface. If Vo is negative, the hydrogen
atom energy may be smaller near a pit (vacancy) or a channel.
The electric dipole moment, the diamagnetic screening constant, the polarizability,
and the hyperfine splitting of the atom are calculated for the ground states.
PACS. 03..65.Ge - Solutions of wave equations: bound states.
PACS. 31.15.+q - General mathematical and computational developments.
PACS. 73.20.Hb - Impurity and defect levels, energy states of absorbed species
I. INTRODUCTION
The quantum states of a hydrogen atom (or hydrogen-like ion) located near a closed
or opened impenetrable surface have been studied in last decades by many authors (see brief
surveys in Ref. [I,21 and the references cited there). But an absolutely impenetrable surface
should be considered only as a limiting case. In order to approach the physical reality one
must investigate the influence of a partially penetrable surface upon the atomic states. This
problem was discussed in Ref. [3] in the case of a plane boundary of a semispace and in Ref.
[4] in the special case of spherical surfaces with the hydrogen atom in its centre and with
a constant finite positive (repulsive) electronic potential energy beyond it. In the present
paper we investigate much more common cases when the atom is located at an arbitrary
position near a surface of arbitrary form and beyond the surface the electronic potential
energy may be both positive and negative.
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@ 1995 THE PHYSICAL SOCIETY
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THE HYDROGEN ATOM NEAR PARTIALLY PENETRABLE SURFACES
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II. THE SCHRijDINGER E Q U A T I O N A N D V A R I A B L E S S E P A R A T I O N
We consider a hydrogen atom near a solid surface S of arbitrary form that divides
the space into two parts I and II. The atom is located in the part I (e.g., vacuum) and
the potential energy VI(T) of the atomic electron in this part of space is
The distances and energies are measured in the atomic units Ug = tL’/rnz and e2/ag = 2
Ryd, respectively. In part II (inside the solid) we suppose
If Vo -+ +m the solid surface is impenetrable for the atomic electron and the surface
repulses the electron. If V. is finite and positive, the surface remains repulsive and it takes
some positive work to move the electron from the vacuum (part 1) into the solid (part II).
If Vo < 0, the electron inside the solid has lower energy than in the vacuum far from the
nucleus. This case may serve as a model for the metal and IV01 corresponds to the work
function (see, e.g., Ref. [5], chapter 18). In this case the surface is attractive to the electron
if the nucleus is sufficiently far and is repulsive if it is close by. In both cases of finite Vo
the surface is penetrable in the sense that the atomic electron has a finite probability to be
found inside the solid.
In this simple model the Coulomb interaction between the electron of the hydrogen
atom and the atoms (both electrons and nuclei) of the solid is thoroughly described by
VII(T). So it is not necessary to consider any other interaction, e.g., image potential (see
Ref. [l]-[5]).
The task of this paper is to find the eigenvalues E and the corresponding eigenfunctions of the hydrogen atom Schrodinger equation
(+ -
V(T) +
E) Q(T) = 0
)
(outside the solid)
(inside the solid)
(3)
(4)
with the boundary conditions of continuity of the wave function and its gradient component
normal to the surface:
dl)(T)ls=
@)(T)ls ,
(5)
(6)
L-_.-
C. YANG, C. L. SIE, AND C. W. LIU
VOL. 33
dl)(‘)lc,=
0)
e)(T)lcR=
533
0.
In Eqs. (5)-(7) Q.(‘)(T) is the electron wave function in the space part I where the atom
is located (outside the solid), !@(I’)( P ) is the electron wave function in the space part II
(inside the solid), CR denotes a sphere of some large radius R .
The solution of the Schrodinger equation both inside and outside the solid may be
written in the form
(8)
l,m
(cl = 1,11) where
X,(0,4> = p;“(cos 0)
exp(im$) ,
(9)
1 = 0,1,2;.., m = O,fl,f2,...,il,a~~’ are arbitrary constant coefficients, P;“(cos 0) are
the associated Legendre functions.
Outside the solid the electronic wave function !P(I)(T) must be finite at T = 0. SO
(see, e.g., Ref. [S])
R{‘)(r) = (~IT)‘exp(-~lr)F(-LI’
+ I+ 1,21+ 2,2k~),
ICI=&%!?,
(10)
(11)
where F(cr,y,z) is the confluent hypergeometric function (see, e.g., [7], Chapter 13):
(a)k = C+Y + 1). . . (a + k - 1))
(cY)o = 1.
(13)
Inside the solid the wave function 9 (“)(T) must vanish at T -+ co. So we must take
the solution
R!“)(r) = (Icur)-‘-’ exp(-kllr)F(-l, -21,2kllr),
(14)
kII = d2(vo - E),
(15)
where the confluent hypergeometric function reduces to a polynomial of degree 1. We
consider the case of bound states E < V, (and E < 0, if the surface is an opened one).
In the special case when the surface is a sphere and the atom’s nucleus is located at
the centre, we may consider only one term in the linear combination (8) with some definite
number 1 [4] b ecause of the momentum conservation. But in a spherically nonsymmetrical
case we must take into account the whole linear combination (8) and choose the combination
coefficients ai:) and the energy E to satisfy the boundary conditions (5)-(7).
534
THE HYDROGEN ATOM NEAR PARTIALLY PENETRABLE SURFACES
VOL. 33
III. SPHERICALLY SYMMETRICAL CASE
The case of a partially penetrable spherical surface with the hydrogen’s nucleus in
its centre was investigated by Ley-Koo et al. [4] for VO 2 0. In that case the surface
repulses the atomic electron so its eigenenergy increases monotonously as the sphere radius
decreases. But if Vo < 0 and the sphere radius r. is large, so the electronic potential energy
in the space part 11 (inside the solid) is lower than the Coulomb energy
(T 2 To>
(16)
the surface is attractive. If the sphere radius is small
e2
(17)
To < IhI
the surface becomes repulsive. Therefore, the dependence of the electronic energy E upon
TO may be nonmonotonous.
In the spherically symmetrical case, if we take a state with some definite number 1, the
boundary conditions (5) and (6) can be written as the equality of logarithmic derivatives
d In R;‘l( To)
din Ri”)(To)
=
(18)
dr
dr
Values of eigenenergy E calculated from (18) in dependence of TO for several Vo < 0 are
given in Fig. 1. In fact, for VO = -0.5, -0.4 ( in e2/ug) the dependence is nonmonotonous:
there are minima at ~~ N 2 (in a~). At sufficiently low Vo bound states of the hydrogen
atom may vanish at all: the surface attracts so strongly that the electron moves away
from the atom into the space part II. The critical values V,(cr) for which the ground state
eigenenergy is equal to the potential energy outside the sphere:
E = V,“’
(19)
are ah0 given VeTsus ~~ in Fig. 1 (dashed curve A). The dashed curve B corresponds to
VO for which there is only one bound state in the atom (excited bound states are absent).
We find both dashed curves A and B are nonmonotonous. Approximately at VO < -0.648
there are no bound states in the atom at any TO.
IV. GENERAL CASE
In a general case when the boundary conditions are not spherically symmetrical (e.g,
the atom is located at an arbitrary position inside a cavity or at some distance from an
opened surface), one must match the wave function (8) with many terms of the linear
combination for o = 1, II and its normal gradient component at the surface. We introduce
a “dissatisfaction degree” (see Ref. [l])
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C. YANG, C. L. SIE, AND C. W. LIU
535
FIG. 1. Dependence of the energy E of a hydrogen atom located at the centre of a spherical surface upon the sphere radius ~0 for various values of Vo. a : VO = -0.5; b : VO = - 0 . 4 ;
c : VO = -0.3; d : Vo = -0.2; e : V. = -0.1; f : V. = 0.0; g : Vo = -0.125; h : Vo = - 0 . 1 0 0 ;
i : Vi = -0.050. Dashed curves A and B indicate the critical Vo values versus TO for
which there are not bound states or there is only one bound state (excited bound states
are absent), correspondingly.
Js pM(T) - !D(“)(T)~2dS
J IV)12dV
+ Js
J
ss(af$,E) =
V
Iv,dl)(T)
V
(20)
v,d")(r)12ds
Iv,i@(r)12dv
’
where S denotes the surface and V denotes the whole space where the electronic wave
function
does not vanish (i.e., inside the large sphere CR). We vary the combination
coefficients c~i$ and the energy E to minimize 6s(a[z’,E).
Q(T)
For a given E the minimum of 6s( a{:) , E) takes place if the coefficients a[:’ satisfy
the system of equations
~&~‘, E)
aa{;)
=
O.
(21 >
We suppose the volume integrals in the denominators of Eq. (20) depend on a{:’ and
E weakly. For the first approximation we put these integrals and their ratio
-_-
C. YANG, C. L. SIE, AND C. W. LIU
VOL. 33
537
From Figs. 2-5 we see that the dependence of these quantities upon the nearest distance 2 between the atom’s nucleus and the surface is quite different for different electronic
-0.38
E
-0.35
i
-8.40
i
-0.45
-0.50
-8.51
2.w
4.86
6.‘20
Z
FIG. 2. Dependence of the ground state energy E of a hydrogen atom located at distance 2
from a plane surface upon 2 for various values of Vl. a : Vo = -0.49; b : Vo = -0.40;
c : Vo = -0.30; d : Vo = 0.00; e : V, = 1.00; f : Vo = 5 . 0 0 .
B.zB-
P
-0.w-
-B.zB-
FIG. 3. Dependence of the ground state electric dipole moment P upon 2 in the plane surface case.
The letters a - f mean the same as in Fig. 2.
.,.
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THEHYDROGENATOMNEARPARTIALLYPENETRABLESURFACES
VOL. 33
0.332
a0.25-
0.26-
0.15
I
0.l0l....,,,..,,,..,....,..~~..~~~,~...~
6.03
0.00
4.00
2.@a
z
FIG. 4. Dependence of the ground state hyperfine splitting u upon 2 in the plane surface case. The
letters a - f mean the same as in Fig. 2.
0.40-
0.20,,...,....1.,.,,.,,.,,,,,.,.,.,,,..,
4.00
6.02
0.BB
2.m
z
FIG. 5. Dependence of the ground state nuclear magnetic shielding A upon Z in the plane surface
case. The letters a - f mean the same as in Fig. 2.
VOL. 33
C. YANG, C. L. SIE, AND C. W. LIU
539
potential energy Vo. While V, > 0 these curves are similar qualitatively to those of the
impenetrable cases [1,2]. Namely, with decreasing of 2 the absolute values of E and P
increase monotonously (P is negative, i.e., the electric dipole is directed from the atom’s
nucleus toward the surface) and the values of cr and A increase at first and then decrease.
These phenomena can be understood if one recognizes that at Vo > 0 the electron is “pushed
away” from the surface. With decreasing of 2 the electron density near the atom’s nucleus
becomes greater at first and then smaller.
But at Vi < 0 the curves are quite different. Because of the attraction of the electron
by the surface at large 2 and repulsion at small 2 the energy E depends on 2 nonmonotonously. At sufficiently low Vo the electric dipole moment P becomes positive, i.e.,
the dipole is directed from the surface toward the atom’s nucleus. The values of g and A
change with 2 monotonously.
For comparison the corresponding curves for the cases while the plane surface has a
pit (vacancy) or a projection are calculated as well (Figs. 6-13). We find that these curves
are similar qualitatively to those for the plane case. However, quantitative comparison
shows that at the same V. >_ 0 and the same shortest distance to the surface the hydrogen
atom energy is lower near a projection. But at Vi < 0 the energy may be lower near a pit
(vacancy) or a channel.
FIG. 6. The same as in Fig. 2 for a plane surface with a pit. The pit radius is 1 ag, the pit depth
is 0.5 ag. The letters a - f mean the same as in Fig. 2.
THE HYDROGEN ATOM NEAR PARTIALLY PENETRABLE SURFACES
VOL. 33
FIG. 7. The same as in Fig. 6 for a plane surface with a projection. The projection radius is 1 a~,
the projection height is 0.5 ag.
-0.40
Yi. ,,,,,,,, ,,,,,,,,,, ~ ,,,,,,,,,,,
0.03
2.Qa
4.00
6.00
z
FIG. 8. The same as in Fig. 3 for a plane surface with a pit. The pit radius is 1 a~, the pit depth
is 0.5 a~. The letters a - f mean the same as in Fig. 3.
---
VOL. 33
C. YANG, C. L. SIE, AND C. W. LIU
541
1-0.4A-&_-__
-0.20
4.00
2.m
6.M
z
FIG. 9. The same as in Fig. 8 for a plane surface with a projection. The projection radius is 1 a~,
the projection height is 0.5 ag.
;
0.25-
&/
0.201 f
0.15
0.10,....,,.,,,,,,,,,...,~.....,,.,~~~~~
6.0(3
4.m
0.00
2.00
z
FIG. 10. The same as in Fig. 4 for a plane surface with a pit. The pit radius is 1 UB, the pit depth
is 0.5 ag_ The letters a - f mean the same as in Fig. 4.
,.
ti__ __
542
THE HYDROGEN ATOM NEAR PARTIALLY PENETRABLE SURFACES
VOL. 33
0.30-
o0.25-
0.20-
0.15-
FIG. 11. The same as in Fig. 10 for a plane surface with a projection. The projection radius is 1
ag, the projection height is 0.5 ag.
0.40
FIG 12. The same as in Fig. 5 for a plane surface with a pit. The pit radius is 1 ag, the pit depth
is 0.5 ag. The letters a - f mean the same as in Fig. 5.
C. YANG, C. L. SIE, AND C. W. LIU
VOL. 33
543
0.x?-
0.Z-
I
0.267-m..,..,...,..~,,,,,,,I,,1,,..,,,
4.00
6.00
0.00
2.e0
z
FIG. 13. The same as in Fig. 12 for a plane surface with a projection. The projection radius is 1
ag, the projection height is 0.5 ag.
The electron probability density distribution is shown in Figs. 14-16. We find that the
electron has finite probability to penetrate through the surface and its density distribution
is substantially deformed by the surface.
l.ea
_, m
1::
surface __:
.__.” . _. -_.. __.’ _..’ ,/-~~..~Y.~acuum
,:’ .’ ,.’
) .“) “.\. . . . . _ . . ‘.. ,.
_’
i
: i ‘f : r
iL&l/ .i‘ : i
: :., ;. ,.
‘. _
i :
,:.; :: :
FIG. 14. The ground state electron probability density distribution of a hydrogen atom located at
1 aB from a plane surface. V. = 5.00. The figures at the curves mean IS(r)
544
THE HYDROGEN ATOM NEAR PARTIALLY PENETRABLE SURFACES
l.e0
VOL. 33
solid
_- surface.-+q%:;
:
_l.y.T-:
:
:
:
,.
:
:
:
:
.
-_
..
'.
.I
:
.
. . . . . *.
0.83
'_
:
:
:
:
...
-3.@a- ..,
!
_.
'0.01
..
.__ . . ...'.
0.033
:.
_'
_'
:
vacuum
.._ ..'
8:8al
-5.633
I I I I I I I I II1 I I I \ I I I *I>. I I I I I. 7
3.00
-1.W
1.00
-3.Fsa
-. .
FIG. 15. The ground state electron probability density distribution of a hydrogen atom located at
0.25 ag from a plane surface with a pit. The pit radius is 1 ag, the pit depth is 0.25 a&3.
Vo = 5.00. The figures at the curves mean IQ(r)]’
3.00,
channel
solid
solid ,_..... . ..__
__..
‘.
.
_
_:’
._
_ .... . . . . . ___,_
_’
:
,.'
._
._
l.oB-e_surfa&_+_ ..- _ _
:
;'
,;.
:
;
:
_
:
I
,;
_
:
;
;
:
:
_--.-.-...,,
i
\
...
.
:
.
'
:
:
FIG. 16. The ground state electron probability density distribution of a hydrogen atom located at
1 aB from a plane surface with a perpendicular channel. The channel diameter is 2 aB.
V, = -0.49. The figures at the curves mean jQ(r)12.
You [13] suggested a physical mechanism of strong body adsorption of solid palladium
to hydrogen ‘H and/or deuterium 2D atoms. He mentioned that due to the deformation
of the electronic wave function of the ‘H atom or the 2D atom a nonzero electric dipole
i-.
VOL. 33
C. YANG, C. L. SIE, AND C. W. LIU
545
moment results [14] which attracts and collects the free electrons of the metal Pd thus
making a strong attraction of the atom to the metal surface.
From our calculation we find out that in the case of negative Vo the hydrogen (and
deuterium) atom energy near a channel or inside it may be lower than that at large distances
from the surface. It means that there is an additional mechanism of attraction of the atom
to the surface near crystal channels perpendicular to the surface. It seems that our results
are in agreement with You’s.
ACKNOWLEDGMENTS
We thank Y. Zh. Xu for asistance. We are grateful to the National Science Council
of the Republic of China for support under contract number NSC83-0208-M032-010.
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_