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5-28-2006
New stable multiply charged negative atomic ions
in linearly polarized superintense laser elds
Qi Wei
Purdue University, qwei@purdue.edu
Sabre Kais
Birck Nanotechnology Center and Department of Chemistry, Purdue University, kais@purdue.edu
Nimrod Moiseyev
Department of Chemistry, Technion-Israel Institute of Technology
Follow this and additional works at: hp://docs.lib.purdue.edu/nanodocs
is document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for
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Wei, Qi; Kais, Sabre; and Moiseyev, Nimrod, "New stable multiply charged negative atomic ions in linearly polarized superintense laser
elds" (2006). Other Nanotechnology Publications. Paper 31.
hp://docs.lib.purdue.edu/nanodocs/31
New stable multiply charged negative atomic ions in linearly polarized
superintense laser fields
Qi Wei and Sabre Kais
a兲
Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
Nimrod Moiseyev
Department of Chemistry, Technion-Israel Institute of Technology, 3200 Haifa, Israel
and Minerva Center for Non-linear Physics of Complex Systems,
Technion - Israel Institute of Technology, 3200 Haifa, Israel
共Received 6 March 2006; accepted 2 May 2006; published online 24 May 2006兲
Singly charged negative atomic ions exist in the gas phase and are of fundamental importance in
atomic and molecular physics. However, theoretical calculations and experimental results clearly
exclude the existence of any stable doubly-negatively-charged atomic ion in the gas phase, only one
electron can be added to a free atom in the gas phase. In this report, using the high-frequency
Floquet theory, we predict that in a linear superintense laser field one can stabilize multiply charged
negative atomic ions in the gas phase. We present self-consistent field calculations for the linear
superintense laser fields needed to bind extra one and two electrons to form He
−
,He
2−
, and Li
2−
,
with detachment energies dependent on the laser intensity and maximal values of 1.2, 0.12, and
0.13 eV, respectively. The fields and frequencies needed for binding extra electrons are within
experimental reach. This method of stabilization is general and can be used to predict stability of
larger multiply charged negative atomic ions. © 2006 American Institute of Physics.
关DOI: 10.1063/1.2207619兴
Singly charged negative ions in the gas phase are of
fundamental importance in atomic and molecular physics
and have attracted considerable experimental and theoretical
attention over the past decades.
1–8
With the advancement of
spectroscopic and theoretical methods, new atomic ions such
as Ca
−
and Sr
−
with small electron affinities 共about 40 meV兲
have been found to be stable.
9,10
However, the existence of
gas-phase doubly charged atomic negative ions has remained
a matter of some controversy.
6
In the sixties and seventies,
there were several experiments, which claimed the detection
of doubly charged atomic ions, but most of these observa-
tions have been shown to be artifacts, and no evidence of
atomic dianions were observed.
11,12
Theoretically, Lieb
13
for-
mulated an upper bound for the maximum number of elec-
trons, N
c
, that can be bound to an atomic nucleus of charge
Z, N
c
ⱕ2Z. This inequality gives the first proof that H
2−
is
not stable, which is in agreement with experiments
11
and
many ab initio studies.
3
There are many ab initio and density
functional calculations
2
of the electron affinities. Recently,
14
we have calculated the critical nuclear charges for atoms up
to N = 86, where N is the number of electrons, the results
clearly exclude the existence of any stable doubly negatively
charged atomic ions in the gas phase.
14,15
However, these
systems might be stable in very intense magnetic fields.
16–18
Small dianions such as O
2−
or CO
3
2−
are very common in
solution and solid-state chemistry, but are unstable in the gas
phase.
6
Thus, there is still an open question concerning the
smallest molecule that can bind two or more excess electrons
with both electronic, against electron detachment, and ther-
modynamic, against fragmentation, stability.
19
A number of
multiply-charged anions with relatively large size, more than
ten atoms, have been observed in the gas phase. However,
experimentally there are only a few stable small dianions,
6
consisting of less than ten atoms, including C
n
2−
共n =7–9兲,
20
S
2
O
6
2−
,
21
and most recently found, four penta-atomic dian-
ions, PtX
4
2−
and PdX
4
2−
共X=C1 and Br兲.
19
Extensive theoret-
ical work has been carried out on small gaseous multiply-
charged anions such as alkali-halides 共MX
3
2−
兲,
22
mixed
beryllium carbon dianions BeC
4
2−
and BeC
6
2−
,
23
Mg
2
X
4
2−
,
24
and small carbon cluster dianions.
25–27
On the other hand, it has been shown recently that su-
perintense radiation fields of sufficiently high frequency can
have large effects on the structure, stability, and ionization of
atoms.
28–33
One of the most intriguing results of Gavrila and
his co-workers is the possibility to have multiply charged
negative ions of hydrogen by superintense laser fields.
34
This
kind of stabilization phenomena has not been observed so far
by any experiment, due to the need for superintense radiation
fields. There are, however, experiments demonstrating light-
induced stabilization against photoionization when the atom
is initially prepared in a Rydberg state.
35
A classical interpretation for the stabilization which en-
ables an atom to bind many additional electrons has been
given by Vorobeichik et al.
36
They showed that for suffi-
ciently large value of
␣
0
=E
0
/
2
, where E
0
and
are the
amplitude and frequency of the laser field, the frequency
associated with the motion of the particle in the time-
averaged potential V
0
, is much smaller than the laser fre-
quency and, therefore, the mean field approach is applicable.
a兲
Author to whom correspondence should be addressed. Electronic mail:
kais@purdue.edu
THE JOURNAL OF CHEMICAL PHYSICS 124, 201108 共2006兲
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Moiseyev and Cederbaum have shown that the stabilization
effect takes place at increasing field strengths when first, the
photoionization rate decreases and second, electron correla-
tion and hence autoionization is suppressed.
38
For one-
electron atoms/ions, Pont et al.
37
have shown that by increas
-
ing
␣
0
, the electronic eigenfunctions of the “dressed”
potential of an atom in high intense laser field and the cor-
responding charge densities are split into two lobes located
around the end points of the nuclear charge, which is
smeared along a line. This phenomenon has been termed a
dichotomy of the atom. Within the framework of the dipole
approximation, the two charges are equal to half the atomic
nuclear charge and are separated by a distance R =
冑
2
␣
0
.
Transferring this approximation to the helium atom in strong
laser fields, it is described as a “hydrogen molecule” where
the distance between the two “hydrogen atoms” is controlled
by the field intensity. It is known in quantum chemistry that
the electronic correlation is reduced in the course of the
breaking of a chemical bond. Namely, atoms in high intense
linearly polarized laser fields behave like homonuclear di-
atomic molecules where the bond length can be controlled by
the laser field intensity. For sufficiently high laser intensity,
“dissociation” takes place due to the suppression of the elec-
tronic correlation and an atom with atomic number Z be-
haves in a high intensity laser field as two separate virtual
atoms each one of them associated with an effective atomic
number Z/2. For example, the helium atom in a sufficiently
strong linear laser field behaves like two virtual noninteract-
ing hydrogens and therefore can bind one or even two more
electrons since H
−
has a ground bound state. This idea stands
behind our present work. Here we carry out ab initio calcu-
lations for many electron atoms where the full electronic
correlation is taken into consideration. The interaction with
the laser field is taken into consideration by including the
exact expression of the dressed potential in our numerical
calculations.
A monochromatic field of electric field vector has the
following form: E共t兲= E
0
共e
1
cos
t+e
2
tan
␦
sin
t兲 with
e
j
共j =1,2兲 unit vectors orthogonal to each other and to the
propagation direction,
␦
=0 corresponds to linear polariza-
tion, and
␦
=±
/4 to circular polarization. The high-
frequency Floquet theory proceeds from the space translated
version of the time-dependent Schrodinger equation which
for N-electron atoms reads
28
兺
i=1
N
冋
1
2
P
i
2
−
Z
兩r
i
+
␣
共t兲兩
+
兺
j=1
i−1
1
兩r
i
− r
j
兩
册
⌿ = i
⌿
t
, 共1兲
where
␣
共t兲= 共
␣
0
/E
0
兲E共t兲 with
␣
0
/E
0
=1/共m
e
2
兲. This equa-
tion refers to a coordinate frame translated by
␣
共t兲 with re-
spect to the laboratory frame. By using the Floquet ansatz
one seeks to determine solutions to the following structure
equation
28
兺
i=1
N
冋
1
2
P
i
2
+ V
0
共r
i
,
␣
0
兲 +
兺
j=1
i−1
1
兩r
i
− r
j
兩
册
⌽ =
⑀
共
␣
0
兲⌽. 共2兲
Here V
0
, the “dressed” Coulomb potential, is the time aver-
age of −Z / 兩r +
␣
共t兲兩,
V
0
共r,
␣
0
兲 =−
Z
2
冕
0
2
d
兩r +
␣
共
/
兲兩
. 共3兲
For linear polarization, the “dressed” potential V
0
is
equivalent to that of a linear charge with a relative larger
charge density near the two end points and a smaller one
FIG. 1. The dressed Coulomb potential, −V
0
共r ,
␣
0
兲, for the He at
␣
0
=11.
FIG. 2. Electronic charge distribution for He
−
,He
2−
,
Li
−
, and Li
2−
in linearly polarized 共along the z axis兲
laser fields at their
␣
0
critical
=11,82,16, and 105 a. u., re-
spectively. Note that there is no overlap of the orbitals.
201108-2 Wei, Kais, and Moiseyev J. Chem. Phys. 124, 201108 共2006兲
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near the center. The length of the linear charge is 2
␣
0
.Ina
two-center coordinate system, V
0
has the following form:
V
0
共
␣
0
,r兲 =−
2Z
共r
A
· r
B
兲
−1/2
K
冋
冉
1−r
ˆ
A
· r
ˆ
B
2
冊
1/2
册
, 共4兲
where A and B are the two foci of the system 共two end points
of the linear charge兲. Z is the nuclear charge and K is ellip-
tical integral of the first kind. In Fig. 1 we show the potential
V
0
共
␣
0
,r兲 along the polarization field direction for He at fixed
value of
␣
0
=11 a . u. This functional form is typical for all
systems used in this report.
Since it is a two-center system, the standard basis sets of
elliptical functions are used here and have the following
form:
⌽共
,
,
兲
p,q,m
= 共
−1兲
p
q
关共1−
2
兲共
2
−1兲兴
m/2
e
−
␥
e
im
,
共5兲
where p, q, and m are non-negative integers, and
␥
is a
variational parameter which will be used to optimize the nu-
merical results, and
,
, and
are prolate spheroidal coor-
dinates with
=共r
A
+r
B
兲/2
␣
0
and
=共r
A
−r
B
兲/2
␣
0
.
Now we can proceed by using the self-consistent field
method to obtain the ground state energy and wave function
of a given atom with a nuclear charge Z in a laser field. Then
we find the critical value of
␣
0
for binding N-electrons to
such a given atom. As long as
⑀
共N兲
共
␣
0
兲⬎
⑀
共N−1兲
共
␣
0
兲, one of
the electrons on the N-electron ion autodetaches and there-
fore the atomic multiply charged negative ions are unstable.
In order to determine the stability of an atomic multiply
charged negative ion, we define
␣
0
critical
for which the detach-
ment energy D
共N兲
共
␣
0
critical
兲=0. The detachment energy is the
energy required to detach one of the N electrons from an ion
at a particular value of
␣
0
, D
共N兲
共
␣
0
兲=
⑀
共N−1兲
共
␣
0
兲−
⑀
共N兲
共
␣
0
兲.
Therefore we can find the critical value of
␣
0
for which
D
共N兲
共
␣
0
critical
兲=0. For values of
␣
0
larger than the
␣
0
critical
, none
of the N electrons will autodetach, and the N-electron atomic
multiply-charged negative ion supports a bound state.
We evaluated all the matrix elements by numerical meth-
ods. By self-consistent field methods we finally obtain the
ground state energies and wave functions of He
−
,He
2−
,Li
−
,
and Li
2−
which are shown in Fig. 2. Note that Li
−
does exist
in a field-free space; it was included only for comparison. We
start the self-consistent field calculation by fixing all elec-
trons with the same distance along the linear charge and it
takes only a few iterations to reach equilibrium. We used a
basis set of 81 basis functions which is accurate enough to
describe the ground state wave function of these systems. It
turns out that there is no overlap between the orbitals of
different electrons as seen in Fig. 2, so the spin exchange
term is not considered here. Finally we obtained the critical
laser parameters to make He
−
,He
2−
,Li
−
, and Li
2−
bound.
They are 11, 82, 16, and 105 a. u., respectively. That means
He
−
,He
2−
,Li
−
, and Li
2−
will be in bound states when the
␣
0
of the laser field is larger than their critical parameters. When
␣
0
of the laser field is large enough, these systems can bind
even more electrons.
The detachment energy as a function of
␣
0
for He
2−
and
Li
2−
is shown in Fig. 3. It is interesting to see the resem-
blance of the detachment energy curves to the potential en-
ergy curves for their equivalent diatomic molecules.
␣
0
maximum
is the
␣
0
with maximal detachment energy. The values are
listed in Table I, which are 26, 180, 42, and 250 a.u. and the
detachment energies at these points are 1.2, 0.12, 1.2, and
0.13 eV for He
−
,He
2−
,Li
−
, and Li
2−
, respectively. The fields
and frequencies needed for binding extra electrons are within
experimental reach. For example, when ultra-high-power
KrF laser 共5 eV photons兲 are used, the peak intensity in the
experiments should be I ⬇10
16
W/cm
2
共see Table I兲. The high
TABLE I. Critical parameters for stability of He
−
,He
2−
,Li
−
, and Li
2−
in superintense laser fields. The intensity
is determined by the following equation: I共W /cm
2
兲= 兩E
0
共a.u.兲兩
2
⫻3.509⫻10
16
, where E
0
=
2
␣
0
, we choose
=5 eV, see the text for more details.
␣
0
critical
共a.u.兲 I
critical
共W /cm
2
兲
␣
0
maximum
共a.u.兲 I
maximum
共W /cm
2
兲
Detachment
energy 共eV兲
He
−
11 4.8⫻10
15
26 2.7⫻10
16
1.2
He
2−
82 2.7⫻10
17
180 1.3⫻10
18
0.12
Li
−
16 1.0⫻10
16
42 7.1⫻10
16
1.2
Li
2−
105 4.4⫻10
17
260 2.7⫻10
18
0.13
FIG. 3. Negative of the detachment energy 共in a.u.兲 of the ground state of
He
2−
and Li
2−
in a linearly polarized high-frequency laser field as a function
of
␣
0
=E
0
/
2
, where E
0
and
are the amplitude and frequency of the laser
field. The maximum values of
␣
0
maximum
are given along with the detachment
energies.
201108-3 Multiply charged negative ions J. Chem. Phys. 124, 201108 共2006兲
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frequency field approximation holds when the field oscillates
much faster than the electrons,
e
/
L
1. On the basis of
semiclassical arguments we estimate the electron motion fre-
quency by calculating the electronic excitation of the atom in
the presence of the field,
e
=共E
1
−E
0
兲/ប. The excitation en-
ergy for He
−
is 1.3 eV and for Li
−
, 2.04 eV which is smaller
than the laser frequency 5 eV. For He
−−
and Li
−−
the
e
/
L
is much smaller than the He
−
and Li
−
. Therefore, our results
clearly show that we are indeed in the high frequency regime
and the electronic oscillations in the presence of the strong
laser field are much smaller than the laser frequency. At such
high frequency the time-averaged “dressed” potential, V
0
,is
the dominant term and therefore this approach is applicable.
When free electron lasers are used the frequency gets much
larger values and the superintense laser fields should be ap-
plied.
In summary, we predicted new stable multiply-charged
negative atomic ions in linearly polarized superintense laser
fields. This method of stabilization is general, within experi-
mental reach and can be used to predict stability of larger
multiply-charged negative atomic ions.
We would like to acknowledge the financial support of
the National Science Foundation and the Israel Science
Foundation. One of the author 共S.K.兲 thanks the John Simon
Guggenheim Memorial Foundation for financial support,
Professor R. N. Zare for his hospitality during a visit with his
group where part of this work has been done and Professor
Daniel Elliott for discussions.
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