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## Battaglia.nl

**Radiative association of LiH**„

**X ****1**⌺؉…

**from electronically excited lithium atoms**
*Department of Chemistry, The University of Rome, Citta` Universitaria, 00185 Rome, Italy*
Full quantum calculations are carried out for the collisional processes involving H atoms in their ground
electronic state and electronically excited lithium atoms, Li(1

*s*22

*p*). The channels that are being consideredare those leading to the formation of bound rotovibrational states of LiH(

*X *1⌺ϩ). The effects of both thespecific features of the involved electronic potential-energy curves and of the transition moments couplingbound and continuum states during the process are studied in relation to the low-energy behavior of thecorresponding cross sections. The role of the multitude of open channel resonances on the final rate constantsis also analyzed and discussed over a broad range of bath temperatures. ͓S1050-2947͑96͒05011-1͔
PACS number͑s͒: 33.80.Gj, 33.90.ϩh, 34.50.Bw

**I. INTRODUCTION**
fore suggested that the detection of LiH in our galaxy couldhave a great impact in the final estimate of Li abundance.

The LiH molecule has been widely studied in the last few
In order to further investigate the implications of such a
years because of the possible importance of its role in obser-
different environment one should be able to also evaluate the
vational cosmology ͓1–7͔. In the framework of the hot big
radiative recombination rates of initial atoms in different
bang model ͑see, for example ͓8͔͒, in fact, the first stellar
electronic states. To this end, we decided to carry out a fully
objects are supposed to be formed in a gas consisting of H,
quantum-mechanical treatment of the following process:
2H, 3He, 4He, and 7Li and hence, before a first generation
of stars, LiH and LiHϩ are the only available molecular spe-
cies that, given their high permanent dipole moment, can
1

*s*22

*p *͒ϩH͑1

*s*͒→LiH͑

*X *1⌺ϩ͒ϩ

*h*.

interact with cosmic background radiation ͑CBR͒ photons.

It has been shown ͓2,3͔ that the presence of the LiH mol-
Process ͑2͒ can, in fact, be relevant for the LiH molecule
ecule in the early universe, if its abundance were not too low,
formation in our galaxy, given the presence of an optical
could have produced a partial smearing of primary CBR
wavelength background radiation that can produce excited
anisotropies and could be used to detect primordial clouds.

lithium atoms in different electronic configurations.

Stancil

*et al. *͓5͔ have recently developed a model of the
It is also worth noting at this point that process ͑2͒ is
kinetics of the lithium chemistry in the early universe, and
much more efficient than process ͑1͒ because it implies free-
the primordial LiH abundance is found to be much lower
bound transitions between states belonging to different
than the one needed to produce relevant observational ef-
potential-energy curves, and therefore spontaneous emission
fects. Such a low LiH abundance is primarily due to the
has a greater probability of occurring. It would not have been
small value of the rate constants for the main LiH formation
available in the early universe because of the low radiation
mechanism in the early universe environment, i.e., the direct
Li͑1

*s*22

*s*͒ϩH͑1

*s*͒→LiH͑

*X *1⌺ϩ͒ϩ

*h*.

**II. THEORY**
Direct two-body association only becomes possible if the
This process implies a free-bound transition between states
residual energy is released radiatively through photon emis-
belonging to the same adiabatic potential-energy curve, and
sion. A fully quantum-mechanical photoassociation theory
therefore its probability of occurring is not very large ͓4,7͔.

can be formulated and used to calculate the cross section at
The results of Stancil

*et al. *seem to exclude any smearing
various collision energies

*E *from which one obtains, in turn,
of CBR anisotropies by LiH molecules and any possibility of
the rate coefficient at various bath temperatures

*T *͓10,11,7͔.

observing primordial clouds in which the density would be
Within the range of validity of the Born-Oppenheimer
too low to activate alternative molecular formation mecha-
͑BO͒ approximation, the wave function of the two colliding
nisms. Maiani

*et al. *͓9͔ have recently investigated the possi-
partners can be written as a superposition of partial waves
bility of detecting LiH molecules in giant molecular clouds
that are the solutions of the Schro¨dinger radial equation with
in galaxies, in which the chemistry should be rather different
a potential

*V*(

*R*) corresponding to one particular BO
from that acting in the early universe. Maiani

*et al. *͓9͔ there-
potential-energy curve. The cross section is given, in atomicunits, by ͓10,7͔
*Mailing address: Professor F. A. Gianturco, Dipartimento di
Chimica, Citta` Universitaria, 00185 Rome, Italy. Fax: ϩ39-6-

*E*,

*v*Ј,

*J*Ј

*JJ*Ј

*M v*Ј

*J*Ј,

*EJ*
49913305. Electronic address: FAGIANT@CASPUR.IT
where

*k*2ϭ2

*E*,

*p *is the statistical weight of the initial
potential-energy curve ͑PEC͒, is the emitted photon fre-

*JJ*Ј is the Ho¨nl-London coefficient, and

*v*Ј

*J*Ј͑

*R *͒͑

*R *͒

*f EJ R *͒

*dR*
is the matrix element of the transition moment (

*R*) be-tween the final bound state ͉

*v*Ј

*J*Ј͘ ͑normalized to unity͒ and
for the

*B *1⌸→

*X *1⌺ϩ transition. The notation

*J*max means
the initial energy-normalized partial wave ͉

*EJ*͘.

*J*max(

*v*), and in our calculations we have grouped together
Since we are dealing with radiative processes, the allowed
terms corresponding to the same value of the emitted photon

*J *and

*J*Ј values are related by dipole selection rules, i.e.,
frequency. As we have noted elsewhere ͓7͔, Eq. ͑8͒ is

*J*Јϭ

*J*Ϯ1 for ⌺→⌺ transitions, and

*J*Јϭ

*J*,

*J*Ϯ1 for ⌳→⌳Ј
slightly different from the one given in Ref. ͓11͔ because of
transitions with ⌳ and/or ⌳Ј 0. Hence the number of partial
this specific choice in our analysis. In our case all possible
waves is given, for each final vibrational state

*v*Ј, by
final values of

*J *are being considered in counting the contri-
butions to Eqs. ͑8͒ and ͑9͒ with the correct frequency weight-
max(

*v *Ј) ϩ 1 , where

*J *max(

*v *Ј) is the highest rotational level
for the vibrational state

*v*Ј.

ing. To further obtain the corresponding rates requires an
The BO potential-energy curves that correspond asymp-
additional integration over the Maxwellian distribution of the
totically to the Li(1

*s*22

*p*)ϩH(1

*s*) atomic partners are given
partner’s relative velocities at a given temperature

*T*, i.e., in
by

*A *1⌺ϩ,

*B *1⌸,

*c *3⌺ϩ, and

*b *3⌸. Of them, the only two
PEC’s that can combine radiatively with the LiH groundstate (

*X *1⌺ϩ) are the

*A *1⌺ϩ and the

*B *1⌸ ones. The statis-
͵

*E*͑

*E*͒

*e*Ϫ

*E*/

*kBTdE*. ͑10͒
tical weights are 1/12 for the former and 1/6 for the latter.

The Ho¨nl-London coefficients are given by ͓12͔

**III. DETAILS OF COMPUTATIONS**
**A. The molecular data**
*J*Јϭ

*J*Ϫ1→

*SJJ*Јϭ

*J*
The

*B *1⌸ adiabatic PEC employed in the present work
has been computed earlier by Boutalib and Gadea ͓13͔, who
for the

*A *1⌺ϩ→

*X *1⌺ϩ transition, and by ͓12͔
employed a nonempirical pseudopotential model for Li and afull configuration-interaction ͑CI͒ treatment of the valence
electrons. Core-valence correlation effects were taken into

*J*Јϭ

*J*ϩ1→

*SJJ*Јϭ ,
account following a core-polarization potential method ac-cording to the formulation of Foucrault

*et al. *͓14͔ that repro-duced very well the atomic lithium spectral properties.

The two ⌺ϩ PEC’s involved, i.e., the

*X *1⌺ϩ and the

*A *1⌺ϩ ones, together with the transition moment betweenthem have been computed by Berriche ͓15͔, who used the
pseudopotential approach of Boutalib and Gadea ͓13͔. He
employed a denser set of points to evaluate the full potential-
energy curves and introduced a further correction due to thevariation of the electron affinity of the

*X *1⌺ϩ state as a
for the

*B *1⌸→

*X *1⌺ϩ transition. Considering the distribu-
function of the internuclear distance for the H atom. A de-
tion over final vibrational bound states that are formed in the
tailed description of Berriche’s

*X *1⌺ϩ adiabatic PEC can be
recombination process, one notes that the total cross section
Finally, the transition moment

*B *1⌸→

*X *1⌺ϩ has been
taken from Partridge and Langhoff ͓16͔, who carried out
extensive CI calculations using a 22127␦ Slater-type or-
͑

*E*͒ϭ ͚

*v*͑

*E*͒,
bital ͑STO͒ basis set to describe the molecular target elec-
The adiabatic PEC’s and transition moments that we have
where

*v*(

*E*) is the partial cross section for the channel cor-
used are reported in Fig. 1. One clearly sees there that the
responding to the final vibrational state

*v*, and is given by ͓7͔
two electronically excited PEC’s are very different in shapeand can support a very different number of bound states: in
either state such levels are much fewer than those that exist
for the electronic ground state ͓6͔, a feature that will alsoaffect the present findings, as we shall report below. Further-
more, the behavior of the two transition dipole moments isalso very different: the ⌺-⌺ transition has a clear maximum
for the

*A *1⌺ϩ→

*X *1⌺ϩ transition, and by
at intermediate relative distances, while the ⌸-⌺ transition

RADIATIVE ASSOCIATION OF LiH(

*X *1⌺ϩ) FROM . . .

FIG. 1. Potential-energy curves and transition moments for the
FIG. 2. Partial cross sections for the ⌳ϭ0 ͑upper part͒ and for
processes considered in this work. Top diagrams:

*X *1⌺ϩ curve
the ⌳ϭ1 ͑lower part͒ processes. In both sets of curves the reported
from Ref. ͓6͔;

*A *1⌺ϩ curve from Ref. ͓15͔;

*B *1⌸ curve from Ref.

vibrational level refers to the final bound state of the

*X *1⌺ϩ mo-
͓13͔. Bottom diagram: ⌺-⌺ transition moment from Ref. ͓15͔; ⌸-⌺
simply goes through a largely flat region of slightly increas-ing values. The bearings of such differences on the finalcross sections will also be discussed in Sec. III B.

**B. The computed cross sections**
Starting from Eqs. ͑8͒ and ͑9͒, the corresponding partial
cross sections have been computed for 451 collision energyvalues between 10Ϫ4 and 10 eV. Radial wave functions forboth bound and continuum states have been generated usingthe Numerov method ͑see, for example, ͓17͔͒ and the nu-merical stability of the results has been tested with the use ofvarious point grids during propagation.

In Fig. 2 we have reported some of the partial cross sec-

*B *1⌸→

*X *1⌺ϩ. The

*X *1⌺ϩ state has 24 ͑

*v*
tional levels; the ground state

*v*ϭ0 evidently is not the mostprobable final state for the radiative recombination event.

Radiative association processes seem, therefore, to producemost probably vibrationally excited bound molecules, as wealso found in our previous work ͓7͔. This is due primarily toFrank-Condon factors, which are larger for the highest vibra-tional levels of the final molecule, and to the peculiar behav-ior of the transition moments discussed in Sec. III A. In anyevent, both processes appear to occur with fairly large partialcross sections and with fractional probabilities that, in thecase with ⌳ϭ0, cover quite a broad range of values at lowcollision energies. We think that this is once more the mani-festation of the pronounced maximum in the transition mo-
FIG. 3. Total cross sections for the ⌳ϭ0 ͑upper part͒ and for the
⌳ϭ1 ͑lower part͒ processes. In both diagrams some of the

*J *values
One also notes that shape resonances are much more
reported refer to the dominant contribution to the resonances shown
present in the process

*A *1⌺ϩ→

*X *1⌺ϩ than in the

TABLE I. Computed rate coefficients for the ⌳ϭ0 and ⌳ϭ1
FIG. 4. Rate coefficients for the ⌳ϭ0 ͑upper part͒ and for the
⌳ϭ1 ͑lower part͒ processes. Both quantities are reported in a log-log scale as a function of temperature.

Condon factors are greater for the free-bound transitions than
for the bound-bound ones. Moreover, the

*B *1
transition is favored by dipole selection rules, which allow
also transitions with ⌬

*J*ϭ0. Furthermore, the different fea-
tures of the two transition moments also indicate that the one
associated with the ⌸-⌺ transition extends over a broad
range of distances and therefore can efficiently couple sev-eral continuum states with the localized final bound states.

*B *1⌸→

*X *1⌺ϩ one: the

*B *1⌸ state is much less bound than

**C. The computed rate coefficients**
the

*A *1⌺ϩ one, so it has only two pseudobound states, theenergies and

*J *values of which are reported in the lower part
Rate coefficients for the two processes have been com-
of Fig. 3, together with the total cross section for the ⌳ϭ1
puted by numerical quadrature of Eq. ͑10͒ for a set of tem-
process. The pseudobound states of the

*A *1⌺ϩ PEC, on the
peratures ranging from 5 to 7000 K. They are reported in
other hand, are much more numerous, as is also shown in the
upper part of Fig. 3, and it is not possible to label each of
Rate coefficients for the ⌳ϭ1 process turned out to be in
them with its

*J *value because of the marked spectral conges-
tion exhibited as the collision energy increases. Energy and

*J*values of pseudobound states have been computed using the
1

*T *͒ ϭ 1.86ϫ 10Ϫ14

*T *Ϫ0.34 cm3 sϪ1,
method described by Le Roy and co-workers ͓18,19͔ andallow us to locate the open-channel resonances quite pre-
while the

*A *1⌺ϩ→

*X *1⌺ϩ process shows rather different
features in its temperature dependence. In particular, for a
The results shown in Fig. 3 also confirm what was men-
specific

*T *value of about 750 K we observe a rather marked
tioned before, i.e., that the final total cross sections are gen-
change in the slope and in its sign: a slow increase with
erally greater for the ⌳ϭ1 process than for the ⌳ϭ0 one.

temperature at low

*T *and a fast decrease with

*T *as the tem-
This is due to the features of the two electronically excited
perature goes above 1000 K. A possible microscopic expla-
PEC’s: the

*A *1⌺ϩ PEC is strongly bound, so Franck-Condon
nation for such a behavior could be had by observing again
factors should be greater for transitions involving initial and
the resonant effects in the total cross sections of Fig. 3. The
final bound states rather than initial continuum and final
presence of a large amount of shape resonances ͑see the up-
bound states, especially for a system like the present one
per part of Fig. 3͒ in the 10Ϫ3/0.6-eV energy range counter-
where the

*A *and

*X *minimum energy radial positions are not
acts the cross-section decrease with increasing collision en-
greatly displaced with respect to each other. On the other
ergy. In fact, cross sections for the ⌳ϭ1 process, in which
hand, the

*B *1⌸ PEC is weakly bound, so in this case Franck-
the shape resonances are almost absent, turn out to decrease
RADIATIVE ASSOCIATION OF LiH(

*X *1⌺ϩ) FROM . . .

faster in the 10Ϫ3/1-eV energy region than the ones for the
magnitude greater than the ones for the

*X *1⌺ϩ→

*X *1⌺ϩ pro-
⌳ϭ0 process. At higher collision energies, where resonances
cess ͓4,7͔, a very interesting result for its astrophysical im-
are essentially wiped out and not present, cross sections start
again to fall off with increasing collision energy, as shown
We have also studied the influence of the involved
by the upper range of computed total cross sections between
potential-energy curve and transition moment features on the
1 and 10 eV of energy. Hence the corresponding high

*T *rates
final results. Molecular formation turns out to be most prob-
decrease with the temperature increase at roughly similar
able in excited vibrational states, and the

*B *1⌸→

*X *1⌺ϩ
rates for both processes. In other words, the different nature
process is the more efficient. Shape resonances have a great
of the PEC for the ⌳ϭ0 case indicates that in this instance
influence on the final behavior of rate coefficients, especially
the larger contribution of the resonant enhancements effec-
in the low

*T *regions of the thermal bath of the process.

tively delays to higher

*T *values the decreasing behavior of
In general, one can say here that the use of a fully
the recombination rates that is seen instead, even at very low
quantum-mechanical treatment of such microscopic pro-

*T *in the case of the ⌸-⌺ process. This qualitative explanation
cesses, i.e., the use of realistic PEC’s of accurate transition
is well supported by the temperature value at which rate
moments and the fully

*ab initio *evaluation of the various
coefficients for the ⌳ϭ0 process show a maximum: 750 K is
channel wave functions ͑bound and continuum͒ allows one
equivalent to ϳ65 meV, i.e., a collision energy value that is
to generate from first principles, and totally in an

*ab initio*
well inside the region populated by shape resonances, as
fashion, the final quantities that need to be employed to pro-
shown by the partial cross sections of Fig. 2.

duce a more global modeling of the complex kinetics for thelithium chemistry under different environments.

**IV. CONCLUSIONS**
In this paper we have carried out full quantum-mechanical

**ACKNOWLEDGMENTS**
calculations for direct radiative association of the LiH mol-ecule starting from lithium atoms that are electronically ex-
The financial support of the Italian National Research
cited. Rate coefficients for such a process can be used to
Council ͑CNR͒ and of the Italian Ministry for University and
estimate LiH abundance in our galaxy, once the fraction of
Research ͑MURST͒ is gratefully acknowledged. We thank
electronically excited lithium in the interstellar medium can
Dr. H. Berriche for having made his results available to us,
and Professor F. Melchiorri and Professor A. Dalgarno for
As was expected, the process studied turns out to be much
making us aware of the problems of lithium chemistry.

more efficient than the one involving atoms in their ground
P.G.G. also thanks Dr. M. Capone for fruitful discussions
state: rate coefficients are found to be at least five orders of
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Source: http://www.battaglia.nl/paolagorigiorgi.org/Papers/rate2.pdf

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