Asymptotic properties of the three-Coulomb-center problem eZ
1
ZZ
V. Yu. Lazur,
1,
*
M. V. Khoma,
1,†
and R. K. Janev
2,3,‡
1
Department of Theoretical Physics, Uzhgorod National University, Uzhgorod, Ukraine
2
Macedonian Academy of Sciences and Arts, MK-1000 Skopje, Macedonia
3
Institut fuer Plasmaphysik, Forschungszentrum Juelich GmbH, EURATOM Association, Trilateral Euregio Cluster,
D-52425 Jülich, Germany
Received 12 December 2005; published 23 March 2006
An analytic study is presented of the asymptotic properties of the three-Coulomb-center problem eZ
1
ZZ. The
electron energies and wave functions of this system, where e designates an electron and Z, Z
1
are bare nuclei,
are calculated asymptotically exactly for large distances L between the fragments of a quasimolecular system.
The electron exchange interaction between eZZ and Z
1
fragments is also calculated asymptotically exactly and
used to estimate the electron-capture cross section in slow collisions of Z =1 and Z
1
=2,3,4 systems.
DOI: 10.1103/PhysRevA.73.032723 PACS numbers: 34.10.+x, 34.20.-b, 34.70.+e
I. INTRODUCTION
Solutions of the Schrödinger equation with two- and
three-Coulomb-center potentials are of considerable interest
from the point of view of various problems related to few-
body systems, particularly when considering their collision
dynamics in an adiabatic approximation. In molecular phys-
ics these systems play the same fundamental role as the hy-
drogen atom in atomic physics. The results obtained on the
two-center Coulomb problem eZ
1
Z
2
have found numerous
applications in the physics of slow ion- atom- molecule
collisions, the spectroscopy of complex chemical com-
pounds, etc. 1–3.
The three-center eZ
1
Z
2
Z
3
Coulomb problem has received
much less attention in the past than the two-center Coulomb
problem. The reason for this is certainly related to the fact
that, while in the case of the two-center Coulomb problem
the Schrödinger equation allows a separation of variables in
prolate spheroidal coordinates due to the higher dynamical
symmetry of the system, such a separation of variables is
not possible in the case of the three-center Coulomb prob-
lem. Studies of the eZ
1
Z
2
Z
3
system have so far been limited
to the use of approximate analytical methods only 4–6.
The question of the existence of a variable separation op-
erator , commuting with the Hamiltonian of an n- n 2
center Coulomb system, was studied for n =4 in 7 and in
the general case in 8. It has been shown 8 that in all cases
when there is an operator that commutes with the Hamil-
tonian = L + O, where O is an operator depending on inter-
center separations and the charges the problem is reduced to
the one- and two-center problems. That means that the
Schrödinger equation of the eZ
1
Z
2
Z
3
system is not separable
in any orthogonal coordinate system and, therefore, its solu-
tion necessarily deals with partial differential equations. This
fact substantially complicates all specific calculations of
adiabatic electronic wave functions molecular orbitals
MO’s and energies potential energy surfaces PES’s for
a given system. The lack of a separation of variables in the
eZ
1
Z
2
Z
3
problem introduces complexity even in approximate
analytical treatments of the problem. These treatments usu-
ally address the asymptotic properties of the system at large
and small intercenter distances. However, for many physical
problems, knowledge of these properties is highly useful and
sometimes sufficient for their adequate description.
The aim of present article is to undertake an asymptotic
study of the discrete spectrum of the eZ
1
Z
2
Z
3
system with
Z
2
= Z
3
= Z. The electronic Hamiltonian of this system de-
pends on three coordinates: Q
1
, Q
2
, and Q
3
further desig-
nated by the symbol Q, which determine the configuration
of the nuclear triangle Z
1
ZZ. These coordinates are chosen to
represent the distance L of nucleus Z
1
to the center of mass
of identical charges Z + Z, the distance R between the identi-
cal nuclei, and the angle between the vectors R and L see
Fig. 1. However, as a matter of convenience of calculations,
in each considered part of configuration space we shall in-
troduce and use the most natural coordinates that facilitate
the asymptotic solution of the eZ
1
ZZ problem. The results
obtained will then be written in L, R, and coordinates, in
which they acquire a more transparent physical meaning.
The article is organized as follows. In the next section,
after a brief summary of some of the known properties of the
eZ
1
ZZ system, asymptotic formulas for the energies of the
eZ
1
ZZ system in the form of power expansions in L are
presented. In Sec. III, the method of constructing the
*Electronic address: lazur@univ.uzhgorod.ua
†
Electronic address: khoma@email.uz.ua
‡
Electronic address: r.janev@fz-juelich.de
FIG. 1. Geometry of the quasimolecule eZ
1
ZZ and the notation
used.
PHYSICAL REVIEW A 73, 032723 2006
1050-2947/2006/733/03272315/$23.00 ©2006 The American Physical Society 032723-1