J. Plasma Fusion Res. SERIES, Vol.7 (2006) 187-189

Resonance States of Hydrogen Atoms Near Metal Surfaces
SUNO Hiroya∗ , KATO Daiji, KATO Takako and KIMURA Mineo1
National Institute for Fusion Science, Toki 509-5292, Japan
1 Kyushu University, Fukuoka 812-8581, Japan（Received: 4 October 2004 / Accepted: 12 September 2005）

Abstract
Resonant charge transfer between a metal surface and a single ion is investigated. A combination of thecomplex-absorbing potential (CAP) method and the basis splines method is used to calculate energy shifts
and lifetimes of a one-dimensional (1D) hydrogen atom located in the vicinity of an Al metal surface. Thismethod can be easily extended to the three-dimensional case.
Keywords:
resonant charge transfer, ion-surface interaction, numerical calculation
1．Introduction
Resonant charge transfer process between metal surfaces and ions plays an important role in a number of surface-diagnostic methods such as Auger electron spectroscopy, ion neutralization spectroscopy andsecondary-ion-mass spectroscopy. Several computational schemes have been developed for calculating
resonance-energy positions and widths associated withcharge exchanges between metal surfaces and oneelectron ions. The coupled-angular mode (CAM)
method was employed by Borisov et al. [1] to calculate
such resonance parameters, while Deutscher et al. [2]used the complex rotation method. A two-center basis
set expansion scheme was also developed by Kürpickand Thumm [3].
In this work, we use a combination of the complexabsorbing potential (CAP) method and the basis splines
method. A simple system of a one-dimensional hydrogen atom located near an Al metal surface is used to test
the eﬃciency of our numerical scheme. However, ourapproach can be easily extended to the realistic, threedimensional case. This paper is organized as follows.
In Sec. 2, we explain briefly our method. Results of ourcalculations are shown in Sec. 3. A summary is given
in Sec. 4.

0-0.01

Γ/2-0.02
-0.03

-0.04
-0.05
-0.06

-0.5

-0.4

-0.3
Er-0.2-0.1

Fig. 1 Diagram for positions Er and half-widths Γ2 of energies in the complex enery plane for resonancestates with a CAP Hamiltonian.

one iW(q), where W(q) is a slowly increasing potential
diﬀerent from zero only in the asymptotic region. Witha CAP Hamiltonian, resonance states appear, where resonance energies E = E r − iΓ/2 are obtained as discrete
complex eigenvalues of the Hamiltonian (see Fig. 1).
The CAP Hamiltonian of a one-dimensional hydrogen atom located at the origin of the x coordinate in thevicinity of a metal surface is written in atomic units asĤ = −2．Method
The complex absorbing potential (CAP) was introduced by Jolicard and Austin [4], and has been applied
previously to studies of photodissociation processes [5].
This method deals with a CAP Hamiltonian containing
a coordinate-dependent complex potential which consists of a real physical potential V(q) and an imaginary

1 d2+ V(x) − iW(x).
2 dx2

(1)

In the present work, we adopt a following quadratic potential for W(x),⎧⎪⎪x ≤ x0⎨ 0,W(x) = ⎪,(2)⎪⎩ γ(x − x0 )2 , x > x0

Corresponding author’s e-mail: suno@jamstec.go.jp∗ Present address: Earth Simulator Center, JAMSTEC, Yokohama 236-0001, Japan187

C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Resonance States of Hydrogen Atoms Near Metal Surfaces

where γ is a strength parameter and the semi-infinite
domain [x0 , ∞) represents the asymptotic region of thex coordinate.
As a finite basis for representing state vectors and operators we choose a set of fifth-order basis splines [6].
In such a basis, the Hamiltonian in Eq. (1) is represented as a complex symmetric matrix. This Hamiltonian matrix is diagonalized to determine a set of complex eigenvalues.

electron with the classical image charge of the nucleusof the atom:1.
(5)∆Ve−s =|2D − x|
The above expressions for V(x) in Eq. (3) correspond tothe case where the electron is outside the metal. On thesurface, the two potentials Ve−i and ∆Ve−s cancel eachother and they vanish inside the metal. V(x) for the 1Dhydrogen atom near the Al surface is shown in Fig. 2.
It is seen in Fig. 2 that this potential is characterized bya potential barrier leading to resonance states.
Figs. 3 and 4 present the resonance positions andwidths of energies for the four lowest states of the system under consideration. They are plotted as functionsof the distance D between the hydrogen atom and theimage reference plane. Except the third lowest state, theresonance positions become high as the ion approaches
the surface and the resonance widths increase as D de-3．Calculations
The method described above is applied here to a onedimensional hydrogen atom (located at the origin) nearan Al metal surface. The potential is given as a sum of
three interaction potentials for an atomic electron:
V = Ve−i + Ve−s + ∆Ve−s ,(3)where Ve−i is a pure Coulomb potential due to the nucleus of the atom, given in atomic units by
1
Ve−i = − .
|x|

0(4)

-0.1

Er (a.u.)Ve−s is an electron-surface potential and ∆Ve−s is a correction for Ve−s , due to the presence of the atomic nucleus. To describe the electron-surface interaction weuse the local potential by Jennings, Jones and Weinert[7]. It contains three parameters that we took from [7]corresponding to the Al surface: The first parameter isthe bottom of the metal conduction band V0 = 0.585
a.u. and the second one the size of the surface region
λ = 1.0. The third parameter, giving the position of thereference image plane, is incorporated in the definitionof the ion-surface distance. This distance D is measured
from the image reference plane. The third term ∆Ve−sin Eq. (3) is taken so as to describe an interaction of the-0.2
-0.3
-0.4
-0.5
05

102015
D (a.u.)2530Fig. 3 Resonance positions of the four lowest states asfunctions of D.

-2100-4Γ/2 (a.u.)

10-0.2V (a.u.)

D=10 a.u.
-0.4-610-810

D=7 a.u.
-0.6-1010-0.8-1-100-505x (a.u)

1015

205102015
D (a.u.)2530Fig. 4 Resonance widths of the states shown in Fig. 3 asfunctions of D. The symbols used to indicate numerical data for corresponding resonancs statesin Fig. 3 are also used in the same manner in theﬁgure.

Fig. 2 Potential V(x) of the 1D hydrogen atom near theAl surface for D = 7 a.u. and 10 a.u.

188

Resonance States of Hydrogen Atoms Near Metal Surfaces

ceases. The reason for this is due to an enhancement of
the tunneling eﬀect through the potential barrier. The
above behaviors are similar to those found in the 3D
case in Refs. [1,2,3,8].

[1]
[2]
[3]

4．SummaryIn this work, we have performed model calculationsof the resonance positions and widths of energies in a
one-dimensional hydrogen atom near an Al surface. We
have used a combination of the complex absorbing potential method with the basis splines method. Although
we have focused only on the 1D case, our approach canbe also extended to the 3D case. For the 3D case, we
expect to obtain qualitatively similar results and they
will serve as applications to realistic problems such as
surface-daignostic methods.

[4]
[5]
[6]
[7]
[8]

189

References
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(1998).
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121, 106 (1985).
T. Seideman, J. Chem. Phys. 98, 1989 (1993).
C. de Boor, A Practical Guide to Splines(Springer, New York, 1978).
P.J. Jennings et al., Phys. Rev. B 37, 6113 (1988).
J. Burgdörfer, in Review of fundamental processes
and applications of ions and atoms (1993) p.517.