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\begin{document}
\author{D. Azinovi\'{c}, S. Milo\v {s}evi\'{c}, and G. Pichler}
\address{Institute of Physics, P.O.Box 304, Zagreb, Croatia}
\author{M.C. van Hemert}
\address{Department of Chemistry, Gorleaus Laboratory, University of Leiden, P.O.Box\\
9502, 2300 RA Leiden, The Netherlands}
\author{R. D\"{u}ren}
\address{Max-Planck-Institut f\"{u}r Str\"{o}mungsforschung, D-37018-G\"{o}ttingen,\\
Germany}
\title{LiAr, LiKr and LiXe excimers: Photochemical formation of the 3$%
^{2}\Sigma^{+} $-1$^{2}\Sigma ^{+}$ bands}
\date{\today }
\maketitle
\begin{abstract}
We investigated the photochemical formation of lithium-rare gas excimers in
the 3$^{2}\Sigma ^{+}$state through the reaction of Li$_{2}$(2(C)$^{1}\Pi
_{u}$) and the ground-state rare gas atom. Lithium-rare gas vapor mixture
was prepared in the heat-pipe oven. We populated the 2(C)$^{1}\Pi _{u}$
state of the Li$_{2}$ molecule using the XeCl excimer laser wavelength at
308 nm or the PTP dye laser wavelength at about 335 nm. The 3$^{2}\Sigma
^{+} $-1$^{2}\Sigma ^{+}$ transitions were observed with peaks at 414, 420
and 435 nm for LiAr, LiKr and LiXe, respectively. We estimated thermally
averaged rate constants for these photochemical reactions, which are (2.3$%
\pm $1.1) 10$^{-10}$cm$^{3}$s$^{-1}$ for LiAr, (6.9$\pm $3.2) 10$^{-10}$cm$%
^{3}$s$^{-1}$ for LiKr and (19$\pm $9) 10$^{-10}$cm$^{3}$s$^{-1}$ for LiXe.
{\it Ab initio }potential-energy curves and transition dipole moments for
LiKr were calculated applying the SCF MRDCI method. Available data for the
LiAr and LiKr excimers are presented, including potential-energy curves,
electronic transition dipole moments, and spectroscopic constants. Possible
photochemical formation of these molecules in the excited states is
discussed. We performed the quantum mechanical spectral simulations of the
LiAr and LiKr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ transitions, using {\it %
ab initio} potential-energy curves.
\end{abstract}
\pacs{PACS numbers: 33.20.Kf; 82.20.Pm; 34.20.Cf}
\newpage
\section{Introduction}
The alkali-rare gas (RG) molecular bands have been extensively studied for
the past thirty years which was systematized in the review paper published
by Rostas \cite{ros82}. Alkali-RG 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$
transitions, investigated by applying the laser induced fluorescence method
or electric discharge, were reported by Tam et al.\cite{tam75,tam76a,tam76b}%
, Webster and Rostas \cite{web78} and Wang and Havey \cite{wan84}.
In our previous work we studied the photochemical formation of the 2$^{2}\Pi
$-1$^{2}\Sigma ^{+}$ excimer bands of LiZn \cite{mil92}, LiCd \cite{hem92},
NaZn \cite{azi93}. The IIB atoms and rare gases have a closed outer
electronic shell. The structure and spectra of alkali-IIB and alkali-rare
gas molecules appear to be similar and one can compare the features of those
two groups of molecules. The potential well for the 2$^{2}\Pi $ state of
alkali-IIB excimers is deeper than the 3$^{2}\Sigma ^{+}$ state and the
wells overlap partly. In contrast, the wells of the alkali-RG 3$^{2}\Sigma
^{+}$ and 2$^{2}\Pi $ states do not overlap and it is possible to study
transitions from these states separately. After our investigation of
alkali-IIB excimers, we expected to observe the LiRG produced in a similar
photochemical reaction. Tam et al. \cite{tam76a} were the first to observe
photochemically produced alkali-RG bands. They excited the K$_{2}$ 2$^{1}\Pi
_{u}$ state by using the argon-ion laser line at 457.9 nm, which in
collisions with RG produced the excited KRG excimer and one free potassium
atom. Photochemical formation of the LiRG 3$^{2}\Sigma ^{+}$ state has not
been investigated so far. The 1$^{2}\Pi $ state of LiAr was photochemically
formed in reactive collisions of the Li$_{2}$(1(A) $^{1}\Sigma _{u}^{+}$),
excited by a HeNe laser at a wavelength of 632.8 nm, and Ar \cite{zha91}.
The potential-energy curves of alkali-RG were calculated by Pascale and
Vandenplaque using pseudopotential method \cite{pas74}. {\it Ab initio}
calculations of the potential-energy curves for LiRG are available for LiHe
\cite{jun88} and LiAr \cite{gu94,par97}. Ground state potentials were
determined experimentally by Buck and Pauly \cite{buc68} and recently by
Br\"{u}hl and Zimmermann \cite{bru95} for LiAr, and by Auerbach \cite{aue74}
for LiKr and LiXe. The first excited 1$^{2}\Pi $ states of LiAr and LiKr
were published by Scheps et al. \cite{sch75}.
This paper is organized as follows. In section II we present the experiment.
The experimental results (section III) are given separately for XeCl excimer
laser excitation, where we give the cross section measurements for
photochemical formation of LiRG relative to the collisional energy transfer
of Li$_{2}^{*}$+Li system, and PTP dye laser excitation. In section IV we
present results of the {\it ab initio} calculation of the LiKr
potential-energy curves and transition dipole moments. In section V, the
spectral simulation are performed using the {\it ab initio} potential-energy
curves for LiAr (\cite{gu94,par97}) and LiKr (this paper). In Section VI we
discuss the possibilities for photochemical formation of LiAr$^{*}$ and LiKr$%
^{*}$, compare the experimental results with the simulations and estimate
uncertainties in theoretical potential-energy curves and experimental rate
constants for photochemical reactions. Conclusions are given in section VII.
\section{Experiment}
The experimental arrangement is the same as in our previous paper \cite
{mil92}. The mixture of lithium, lithium dimer and rare gas was prepared in
a crossed heat-pipe oven. The lithium vapor pressure was varied in the range
from 5 to 20 Torr and rare gas pressure in the range from 5 to 700 Torr. The
temperature for the above mentioned range of lithium pressures was varied
from 900 K to 1150 K. When lithium vapor pressure was equal to the rare gas
pressure the heat-pipe oven was operating in the heat-pipe mode for lithium.
In that case the mixing with rare gas in the central part of the heat-pipe
oven was negligible and pure lithium and lithium dimer spectra were
observed. For the observation of LiRG bands it was necessary to have the RG
pressure higher than 30 Torr. The preparation of electronically excited Li$%
_{2}$ molecules in specific rovibrational levels of the 2(C)$^{1}\Pi _{u}$
state is achieved by means of the pulsed XeCl excimer laser lines at 308 nm
(LPX 105E) or using a pulsed dye laser (LPD 3002) working with PTP dye
(range: 330-350 nm). The horizontal laser induced fluorescence was rotated
by a Dove prism and focused to the vertical entrance slit of the
monochromator. The resolution of the system was about 0.1 nm. The signal
from the photomultiplier with S20 cathode was averaged by a boxcar averager
(PARC M162 and M164). The analog output from the boxcar was digitized by an
A/D converter and fed to a laboratory computer. The spectral response of the
system was determined by means of a calibrated tungsten-ribbon lamp and was
found constant in the region of interest (violet spectral region).
\section{Results}
\subsection{XeCl excimer laser excitation}
The XeCl excimer laser lines at 308 nm excite simultaneously the v'=13, J'=6
and the v'=19, J'=24 rovibrational levels in the Li$_{2}$ 2(C)$^{1}\Pi _{u}$
state with the excitation energies of 24897.07 cm$^{-1}$ and 26260.72 cm$%
^{-1}$, respectively. The 308 nm laser line should also excite energy levels
above the potential barrier of the double-minimum 2$^{1}\Sigma _{u}^{+}$
state, but they cannot be clearly identified since the spectroscopic
constants for the rovibrational levels above the barrier are not well known.
Identification of these transitions also requires a better spectral
resolution. The 3$^{2}\Sigma ^{+}$ state of the lithium-rare gas excimer is
populated in the photochemical process, given by
\begin{equation}
Li_{2}(2(C)^{1}\Pi _{u})+RG\rightarrow LiRG(3^{2}\Sigma ^{+})+Li
\end{equation}
Figures 1 a,b,c show the violet 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ bands
of LiAr (414 nm), LiKr (420 nm) and LiXe (435 nm), for excitation at 308 nm.
The temperature was T=1161 K, yielding a partial lithium vapor pressure of 9
Torr and a density of Li atoms of 7.5.10$^{16}$ cm$^{-3}$ and a Li$_{2}$
partial vapor pressure of 0.5 Torr and Li$_{2}$ density of 3.83.10$^{15}$ cm$%
^{-3}$, according to the Nesmeyanov tables \cite{nes63} and the ideal-gas
law. Argon, krypton and xenon pressures were 150, 300 and 200 Torr,
respectively. On all three spectra, lithium atomic lines were observed at
413.3 nm (5$^{2}$D-2$^{2}$P) and 460.3 nm (4$^{2}$D-2$^{2}$P) and their
intensities decreased at higher rare gas pressure in the heat-pipe oven. We
also observed the Li$_{2}$ diffuse band at 458 nm (2$^{3}\Pi _{g}$-1$%
^{3}\Sigma _{u}^{+}$) and the interference continuum at 452 nm (2$^{1}\Sigma
_{u}^{+}$-1(X)$^{1}\Sigma _{g}^{+}$), as indicated by vertical bars \cite
{li92}.
Intensities of the LiRG, Li$_{2}$ diffuse band and Li$_{2}$ interference
continuum show an interesting dependence on the rare gas pressure. These
intensities are determined as areas under the observed bands. The Li$_{2}$
interference continuum at 452 nm and the diffuse band at 458 nm overlap and
we first obtained the area under both bands together. The intensity of the Li%
$_{2}$ diffuse band was estimated from the intensity at its maximum (458 nm)
and its half width, which is about 8 nm and does not change with rare gas
pressure \cite{li92}. The interference continuum intensity is then the total
intensity minus the diffuse band intensity.
As an example, Figures 2 a,b,c show the LiXe band behavior with increasing
xenon pressure for 50, 100 and 246 Torr, respectively. Comparison of the
relative intensities of LiXe and Li$_{2}$ continuum bands versus Xe
densities is shown in Fig. 3. We compare the photochemical reaction of Li$%
_{2}$ with Xe (see Eq. (1)) with the collisional transfer of population
resulting in the Li$_{2}$ diffuse band at 458 nm:
\begin{equation}
Li_{2}(2(C)^{1}\Pi _{u})+RG\rightarrow Li_{2}(2^{2}\Pi )+Li
\end{equation}
By inspection of Fig. 3 we observe a slight decrease in the intensity at 452
nm, which indicates a small contribution of Li$_{2}$(2$^{1}\Sigma _{u}^{+}$%
)+Xe $\rightarrow $ LiXe(3$^{2}\Sigma ^{+}$)+Li reaction. However, in all
subsequent analysis we shall not take into account this reaction.
The photochemical reaction rate constant $k_{LiRG}$ giving LiRG* can be
evaluated from:
\begin{equation}
k_{LiRG}=\frac{\nu _{diff}}{\nu _{LiRG}}\frac{\alpha (\nu _{diff})}{\alpha
(\nu _{LiRG})}\frac{I_{LiRG}}{I_{diff}}\frac{\gamma _{LiRG}}{\gamma _{diff}}%
\frac{\Gamma _{diff}}{\Gamma _{LiRG}}\frac{[Li]}{[RG]}k_{diff}
\end{equation}
where $\nu _{diff}$, $\nu _{LiRG}$, $\alpha (\nu _{diff})$, $\alpha (\nu
_{LiRG})$, $I_{LiRG}$, $I_{diff},\Gamma _{diff},$ $\Gamma _{LiRG}$ are
frequencies, spectral responses, intensities at the band maxima and
radiative transition rates for the given transitions as given in more
details in Ref. \cite{azi96}. Note that in Ref. \cite{azi96} (Eq. 12) the
similar expression is given for the corresponding cross section $\sigma $(cm$%
^{2}$). All these values can be taken from the observed spectrum. The values
$\gamma _{LiRG}$ and $\gamma _{diff}$ are:
\[
\gamma _{LiRG}=\Gamma _{3^{2}\Sigma ^{+}-1^{2}\Sigma ^{+}}+\Gamma
_{3^{2}\Sigma ^{+}-2^{2}\Sigma ^{+}}+\Gamma _{3^{2}\Sigma ^{+}-1^{2}\Pi }
\]
\[
\gamma _{diff}=\Gamma _{2^{3}\Pi -1^{3}\Sigma _{u}^{+}}+\Gamma _{2^{3}\Pi
-2^{3}\Sigma _{u}^{+}}+\Gamma _{2^{3}\Pi -3^{3}\Sigma _{u}^{+}}+\Gamma
_{2^{3}\Pi -1^{3}\Pi }
\]
where $diff\equiv $ $2^{3}\Pi -1^{3}\Sigma _{u}^{+}$ and $LiRG\equiv
3^{2}\Sigma ^{+}-1^{2}\Sigma ^{+}$. The factor $\gamma _{LiRG}$/$\gamma
_{diff}$ can be estimated from the values of transition dipole moments for
transitions from the LiRG 3$^{2}\Sigma ^{+}$ state to all radiatively
coupled lower states \cite{gu94} and the values of transition dipole moments
for transitions from the Li$_{2}$ 2$^{3}\Pi _{g}$ state to all radiatively
coupled lower states \cite{rat87}. For LiAr this factor is 8.6 and for LiKr
and LiXe we could not find any calculated value. Therefore we assume in the
first approximation the same value of 8.6 also for LiKr and LiXe. The factor
[Li]/[RG] is the atom density ratio of lithium and rare gas. The rate
constant for collisional energy transfer (2) giving the Li$_{2}$ diffuse
band at 458 nm as obtained by Weyh et al. \cite{wey96} is:
\begin{equation}
k_{diff}=<\sigma _{diff}v(Li_{2},Li)>=(25\pm 7)10^{-16}(8kT/\pi \mu
)^{1/2}=(5.7\pm 1.6)10^{-10}cm^{3}s^{-1}
\end{equation}
The intensities of the LiXe band and Li$_{2}$ diffuse band and atom
densities of Li and Xe from the graph in Fig. 3 give k$_{LiXe}$=(19$\pm $9)
10$^{-10}$ cm$^{3}$s$^{-1}$, using the least square fit method.
Using the same method we obtain rate constants for photochemical formation
of LiAr and LiKr in the 3$^{2}\Sigma ^{+}$ state, which are k$_{LiAr}$=(2.3$%
\pm $1.1) 10$^{-10}$ cm$^{3}$s$^{-1}$ and k$_{LiKr}$=(6.9$\pm $3.2) 10$%
^{-10} $cm$^{3}$s$^{-1}$. The ratios of the rate constants for LiRG (3$%
^{2}\Sigma ^{+}$) and Li$_{2}$(2$^{3}\Pi _{g}$) formation are k$_{LiAr}$ / k$%
_{diff}$=0.4, k$_{LiKr}$ / k$_{diff}$=1.21 and k$_{LiXe}$ / k$_{diff}$=3.33.
The k$_{LiRG}$ values from LiAr to LiXe increase by a factor of about 8,
which implies that, for the heavier rare gas atom with deeper potential well
of the relevant 3$^{2}\Sigma ^{+}$ state, i.e. larger energy difference
between the Li$_{2}^{*}$ energy level and the minimum of the 3$^{2}\Sigma
^{+}$ state potential, the probability of the photochemical reaction (1)
rapidly increases, as compared to competitive collisional transfer processes
between the excited states of alkali dimers (2).
\subsection{Dye laser excitation}
The PTP dye laser covers the range from 330 nm to 350 nm. This wavelength
range enabled us to excite the Li$_{2}($2(C)$^{1}\Pi _{u})$ levels up to
v'=6, which has the energy of 23660 cm$^{-1}$. There is no evidence in the
spectrum around the laser line that we have excited any level in the inner
well of the double minimum Li$_{2}$ 2$^{1}\Sigma _{u}^{+}$ state, below the
potential barrier \cite{li93}. In order to study the dependence of the LiKr
band intensity on the exciting laser wavelength we measured the selected
wavelength excitation spectrum. Figure 4 presents the selected wavelength
spectrum taken at the LiKr band maximum (420 nm). The laser wavelength was
scanned from 335.2 to 335.4 nm. For large Franck-Condon factors in the Li$%
_{2}$ 1(X)$^{1}\Sigma _{g}^{+}\rightarrow $2(C)$^{1}\Pi _{u}$ rovibrational
excitation, identified in Fig. 4, we get maxima in the LiKr band intensity,
which proves the validity of the photochemical reaction (1).
\section{Potential-energy curves: The LiKr excimer}
We calculated LiKr potential-energy curves for the four lowest states of the
$^{2}\Sigma ^{+}$ symmetry and three of the $^{2}\Pi $ symmetry by using the
MOLCAS2 package \cite{mol91}. The basis set for Li (6s4p3d) was used as in
Ref. \cite{mil92}. For Kr we used primitive functions (21s16p10d) Ref. \cite
{par89}. We added additional exponents to this: 0.08 and 0.03 for s type,
0.04 and 0.0165 for p type, 0.229 and 0.0916 for d type functions. This
additional exponents were obtained from extrapolation of the s and p
exponents in a log plot. For d type functions the exponents were taken from
Ref. \cite{ric91}. As in Ref. \cite{ric91} even-tempered 4f type exponents
were added. With this basis set almost the complete Hartree-Fock total
energy was obtained at -2752.051 Hartree \cite{sch92}. Spherical coordinates
were used giving with the above basis set 210 primitives and 81 contracted
basis functions. The calculation was performed in the C$_{2v}$ symmetry. In
MRCI 28 electrons were frozen (7,3,3,1) in a1, b1, b2, a2 irreducible
representations and 22 virtual orbitals were deleted (12,5,5,0). In MRCI
calculations 11 electrons were correlated (1s and 2s of Li and 4s and 4p of
Kr). Nine main references were used defining a real space of 51709
configurations. Preceding to the MRCI either a SCF or a CASSCF calculations
were performed. First order relativistic corrections (mass-velocity and
Darwin term) were included as well \cite{mol91}.
Table \ref{tab1} presents the {\it ab initio} potential-energies for the
LiKr 1$^{2}\Sigma ^{+}$, 2$^{2}\Sigma ^{+}$, 3$^{2}\Sigma ^{+}$, 4$%
^{2}\Sigma ^{+}$, 1$^{2}\Pi $, 2$^{2}\Pi $ and 3$^{2}\Pi $ states. In Table
\ref{tab2} we give dipole moment for the ground state and transition dipole
moments for the LiKr $\Sigma $-$\Sigma $ transitions. Note that values and
dependence on R obtained for LiKr molecule are similar to those of LiAr \cite
{gu94,par97} which supports our assumptions in Sec. 3.1. Table \ref{tab3}
presents the spectroscopic constants for the LiKr 1$^{2}\Sigma ^{+}$, 1$%
^{2}\Pi $, 2$^{2}\Sigma ^{+}$, 3$^{2}\Sigma ^{+}$ and 2$^{2}\Pi $ states.
Table \ref{tab4} presents a comparison of the R$_{e}$ and T$_{e}$ taken from
the LiKr 1$^{2}\Sigma ^{+}$ and 1$^{2}\Pi $ {\it ab initio} potentials with
the values calculated by Pascale and Vandenplaque \cite{pas74} and with the
experimental values given by Auerbach \cite{aue74} and Scheps et al. \cite
{sch75}. From these data, we may conclude that the {\it ab initio}
calculation of the ground state gives too deep minimum, because the
experimental value is about 70 cm$^{-1}$. We believe that this is mainly due
to the basis set superposition error. However, the difference potentials
relevant to spectroscopy should be more accurate.
\section{Spectral simulation of the LiAr and LiKr 3$^{2}\Sigma ^{+}$-1$%
^{2}\Sigma ^{+}$ bands}
{\it Ab initio} potential-energy curves, transition dipole moments and
spectroscopic constants for LiAr are published by Gu et al. \cite{gu94} and
Park et al. \cite{par97}. The {\it ab initio} LiAr ground-state potential
given in Ref. \cite{gu94} is completely repulsive whereas calculation of
Ref. \cite{par97} shows shallow minimum at about 10 Bohr, closer to the
experimental values \cite{buc68,bru95}. The 3$^{2}\Sigma ^{+}$ state has a
shallow minimum and contains only 10 bound vibrational levels for J'=25.5.
Note that shallow outer well of the 3$^{2}\Sigma ^{+}$ potential, which is
obtained in calculation of Ref. \cite{par97}, does not affect spectral
formation of the considered LiAr diffuse band. We performed the quantum
mechanical simulation using the standard Numerov-Cooley method for the
bound-free transitions \cite{num33}. Rotational averaging was performed over
J'=0.5-40.5, assuming a Boltzmann distribution. The rovibrationally averaged
LiAr spectral simulations for the effective temperature of 800 K are given
in Fig. 5a. The calculated peak position is at 416.4 nm using potentials
from Ref. \cite{gu94} and 413.1 nm using potentials from Ref. \cite{par97},
which is in a good agreement with the measured peak position at 414 nm.
Prior to this simulations we performed another spectral simulations of the
LiAr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ transition using a different set
of the {\it ab initio} potential energy curves and transition dipole moments
as reported in Ref. \cite{azi94}. The calculated maximum of the LiAr band
was at 405 nm. The shape of the simulated LiAr band was similarly asymmetric
as in Fig. 5a but without the quantum oscillations on the blue wing of the
band.
Using the potential-energy curves given in Table \ref{tab1}, transition
dipole moments given in Table \ref{tab2} and spectroscopic constants given
in Table \ref{tab3} we performed quantum-mechanical spectral simulations of
the LiKr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ band. The emission profiles
for the range of rotational numbers from 0.5 to 150.5 and vibrational
numbers from 0 to 10 were obtained in which the rotational averaging
assuming the Boltzmann distribution for an effective temperature of 1000 K
was assumed. The rotationally averaged spectra are vibrationally averaged
assuming the Boltzmann distribution among 11 vibrational levels which give a
significant contribution to the spectrum. Figure 5b shows simulation of the
LiKr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ band with the maximum at 433 nm,
which is shifted by 710 cm$^{-1}$ from the experimental position at 420 nm.
Table \ref{tab5} gives the experimental positions of LiAr, LiKr and LiXe
bands observed in this work and in the work of Wang and Havey, as well as
the calculated positions of the LiAr and LiKr excimer bands.
\section{Discussion}
First we present the systematization of the possible photochemical reactions
populating LiAr and LiKr in their excited states. Table \ref{tab6} presents
a list of possible photochemical reactions populating the LiAr (upper part)
and LiKr (lower part) excited states. We give the energy defect, and the
positions of extrema in the difference potentials of the excited LiAr (LiKr)
product of the reaction, and the spectral position of the observed bands.
Because of the shallow minima of the LiAr excited states, usually we have to
excite very high rovibrational levels in the Li$_{2}$ excited states. The
other way of LiAr* formation is through the three-body recombination. These
reactions are presented as collisions of Li atom in the excited state with
two Ar atoms, giving LiAr in the excited state and the other argon atom in
the ground state. Such collisions were studied in several papers. Excitation
of the LiAr 1$^{2}\Pi $ state by exciting the lithium atomic 2$^{2}$P state
was studied by Scheps et al. \cite{sch75}. The formation of the LiAr 3$%
^{2}\Sigma ^{+}$ and 2$^{2}\Pi $ states by exciting lithium to 3$^{2}$P and 3%
$^{2}$D levels was studied by Wang and Havey \cite{wan84}. In the work of
Zhang and Ma, the lithium dimer was excited by a He-Ne laser to the high
rovibrational levels of the Li$_{2}$ 1(A)$^{1}\Sigma _{u}^{+}$ state and
they observed the photochemical reaction leading to the LiAr 1$^{2}\Pi $
state \cite{zha91}. Other photochemical reactions listed in Table \ref{tab6}
were not studied up to now.
Collisions of one lithium atom in an excited atomic state with two Kr atoms
are the only three-body recombinations observed so far. Excitation of the
lithium resonance line gives the LiKr band at 801 nm \cite{sch75}. Wang and
Havey studied the excitation of lithium 3$^{2}$P and 3$^{2}$D atomic states,
populating the LiKr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ states [6].
Photochemical reactions from the lithium dimer molecular states, as the
input channel, to the LiKr excited states, as the output channel, have not
been observed to date.
The observation of the relatively intensive LiRG 3$^{2}\Sigma ^{+}$-1$%
^{2}\Sigma ^{+}$ bands can be explained by analyzing the shape of the
potential-energy curves of Li$_{2}$ and LiRG, which are presented in Figs.
6a and 6b, respectively. The 2$^{2}\Sigma ^{+}$ state is repulsive, with a
shallow van der Waals well at large internuclear separations. Such a
repulsive 2$^{2}\Sigma ^{+}$ state has an avoided crossing with an
energetically higher 3$^{2}\Sigma ^{+}$ state, which has the same symmetry.
This avoided crossing is responsible for the increase of the transition
dipole moment of the 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ transition
(intensity borrowing). The 2$^{2}\Pi $ states in alkali-rare gas excimers
does not overlap with the 3$^{2}\Sigma ^{+}$ state and the transition dipole
moment for the 2$^{2}\Pi $-1$^{2}\Sigma ^{+}$ transition is about 3 times
lower than that for the 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$ transition.
Photochemical population of the 3$^{2}\Sigma ^{+}$ state is preferred here,
since for the photochemical population of the 2$^{2}\Pi $ state more energy
is needed for excited alkali dimers in the input channel of the reaction.
The 2$^{2}\Pi $ state can be achieved in three atom collisions, including
the alkali atom in the second excited P state or the first excited D state
with two rare gas atoms. Because of the shallow 3$^{2}\Sigma ^{+}$ states in
LiRG excimers, the higher laser photon energies are more probable to
populate the LiRG 3$^{2}\Sigma ^{+}$ state in reaction (1). Therefore the
excimer laser line at 308 nm gives more intensive LiRG bands than the dye
laser excitation at photon energies of 335 nm. The smallest excitation
energy in the Li$_{2}$ 2(C)$^{1}\Pi _{u}$ state which will enable
photochemical formation of the 3$^{2}\Sigma ^{+}$ state is 25544 cm$^{-1}$
and 24395 cm$^{-1}$ for LiAr and LiKr, respectively. According to Table \ref
{tab5} the calculated LiKr excited states are too deep by about 500 cm$^{-1}$%
, which moves T$_{e}$(LiKr) to 24900 cm$^{-1}$. The excimer laser line at
308 nm which populates Li$_{2}$(2(C)$^{1}\Pi _{u}$) with E=26260.7 cm$^{-1}$
provides for all LiGR molecules enough energy to start reaction (1). In the
case of dye laser excitation at 335 nm, E=23660 cm$^{-1}$ does not reach the
threshold for the reaction (1) for LiAr and LiKr. However, we observe the
reaction at high temperature and rare gas pressure because the collisional
energies give an additional 1000 cm$^{-1}$. The positions of maxima of the
LiRG bands at 414, 420 and 435 nm correspond to energies of 24154, 23809 and
22988 cm$^{-1}$ for LiAr, LiKr and LiXe, respectively. Spectral simulations
give maxima in the LiAr and LiKr bands at 416 and 433 nm, which correspond
to energies of 24038 and 23095 cm$^{-1}$, respectively. Using all these
values we can roughly estimate the error of the 3$^{2}\Sigma ^{+}$-1$%
^{2}\Sigma ^{+}$ difference potential, which is about 120 cm$^{-1}$ for LiAr
and 700 cm$^{-1}$ for LiKr.
The large uncertainties in the determination of the rate constants k$_{LiRG}$%
, which are about 50 \%, are mainly due to the uncertainty of the
collisional energy transfer cross section $\sigma _{diff}$ \cite{wey96},
which is 28 \%. The second large contribution is the uncertainty in the
intensity measurements of the Li$_{2}$ diffuse band at 458 nm, which is
about 15 \% because of the overlap with the Li$_{2}$ interference continuum
at 452 nm. The LiRG bands are not disturbed by other molecular transitions
and the error in the intensity determination is not higher than 5 \%. The
errors in the atom density ratios [RG]/[Li], frequencies and spectral
responses are not larger than few \%. The accuracy of factor $\gamma _{LiRG}$%
/$\gamma _{diff}$ for LiAr depends on the uncertainties of transition dipole
moments. For LiKr and LiXe we took the same value as for LiAr and, in these
cases, rate constants may be underestimated since the transition dipole
moments for LiKr and LiXe are slightly larger than for LiAr.
\section{Conclusion}
We investigated photochemical formation of the lithium-rare gas 3$^{2}\Sigma
^{+}$ states in the reaction given by relation (1). We worked with all rare
gases, from He to Xe, but we observed no LiHe and LiNe 3$^{2}\Sigma ^{+}$-1$%
^{2}\Sigma ^{+}$ bands in the spectrum. The LiAr, LiKr and LiXe 3$^{2}\Sigma
^{+}$-1$^{2}\Sigma ^{+}$ transitions were observed at 414, 420 and 435 nm,
respectively. We found that the rare gas pressure had to be at least about 2
times larger than the lithium vapor pressure if we want to observe the LiRG
band as a result of the above reaction. We estimated rate constants for this
photochemical reaction at T = 1160 K as (2.3$\pm $1.1) 10$^{-10}$ cm$^{3}$s$%
^{-1}$, (6.9$\pm $3.2) 10$^{-10}$ cm$^{3}$s$^{-1}$ and (19$\pm $9) 10$^{-10}$
cm$^{3}$s$^{-1}$ for LiAr, LiKr and LiXe, respectively, measured relatively
to the collisional energy transfer \cite{wey96}. The photochemical reaction
can populate only the 3$^{2}\Sigma ^{+}$ state of the Li-rare gas excimer
because the 2$^{2}\Pi $ state is energetically too high. Spectral
simulations of the LiAr and LiKr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$
transitions were calculated using the recently calculated ab initio
potential-energy curves. Using potentials of Ref. \cite{par97} spectral
simulation of the LiAr band gives peak at 413.1 nm, in better agreement with
experimental position at 414 nm, compared to the case when using potentials
of Ref. \cite{gu94}. The simulated position of the LiKr band is at 433 nm.
It is shifted by about 13 nm to the red, as compared with the experimental
value of 420 nm. The LiKr ground state {\it ab initio} potential is about
200 cm$^{-1}$ too deep when compared with the experimental potential. To
reproduce the experimental position of the LiKr band, we have to add about
800 cm$^{-1}$ to the 3$^{2}\Sigma ^{+}$ {\it ab initio} potential.
\section{Acknowledgments}
This work was financially supported by the Ministry of Science and
Technology of the Republic of Croatia. We also gratefully acknowledge
partial support from the Alexander von Humboldt Stiftung, Germany. One of us
(S.M.) is grateful to the Max-Planck-Institute f\"{u}r
Str\"{o}mungsforschung for the hospitality during his stay in G\"{o}ttingen.
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\newpage
\begin{figure}[tbp]
\caption{a) The LiAr excimer band at 414 nm. Argon pressure is 150 Torr and
the temperature 1161 K. b) The LiKr excimer band is at 420 nm. Krypton
pressure is 300 Torr and T=1161 K c) The LiXe excimer band is at 435 nm.
Xenon pressure is 200 Torr and T=1161 K. The Li-rare gas bands, lithium
dimer diffuse band at 458 nm and the interference continuum at 452 nm are
indicated by bars. * - the lithium atomic 4$^{2}$D-2$^{2}$P line is out of
scale.}
\label{fig1}
\end{figure}
\begin{figure}[tbp]
\caption{Dependence of the LiXe band and Li$_{2}$ band shapes and
intensities on the xenon pressure: a) P$_{Xe}$=50 Torr, b) P$_{Xe}$=100
Torr, and c) P$_{Xe}$=246 Torr. Temperature is 1161 K. Bars indicate the
positions of the LiXe band and Li$_{2}$ bands.* - line out of scale.}
\label{fig2}
\end{figure}
\begin{figure}[tbp]
\caption{Intensities of the LiXe band, Li$_{2}$ diffuse band and Li$_{2}$
interference continuum versus the atom density of xenon. The intensities are
the areas under the bands and the errors of intensities are up to 20 \% for
Li$_{2}$ bands and up to 10 \% for the LiXe band.}
\label{fig3}
\end{figure}
\begin{figure}[tbp]
\caption{Selective absorption spectrum on the LiKr excimer band maximum at
420 nm. The maxima correspond to the Li$_{2}$ 2$^{1}\Pi _{u}$-1$^{1}\Sigma
_{g}^{+}$ rovibrational transitions (v',v'',J'), identified using
spectroscopic constants for the Li$_{2}$ 2$^{1}\Pi _{u}$ and 1$^{1}\Sigma
_{g}^{+}$ states \protect\cite{kon84,ish91,sch85}.}
\label{fig4}
\end{figure}
\begin{figure}[tbp]
\caption{a) Spectral simulation of the LiAr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma
^{+}$ transition using the {\it ab initio} potential-energy curves from Ref.
\protect\cite{gu94} (dashed line) and Ref. \protect\cite{par97} (full line).
b) Spectral simulation of the LiKr 3$^{2}\Sigma ^{+}$-1$^{2}\Sigma ^{+}$
transition using the {\it ab initio} potential-energy curves from Sec. IV.}
\label{fig5}
\end{figure}
\begin{figure}[tbp]
\caption{a) The Li$_{2}$ 1$^{1}\Sigma _{g}^{+}$, 1$^{1}\Sigma _{u}^{+}$, 1$%
^{1}\Pi _{u}$, 2$^{1}\Sigma _{u}^{+}$ and 2$^{1}\Pi _{u}$ potentials
\protect\cite{kon84,ish91,sch85}. b) {\it Ab initio} potential-energy curves
of the LiKr excimer.}
\label{fig6}
\end{figure}
\newpage
\begin{table}[tbp]
\caption{Potential energies for LiKr molecule in hartree, interatomic
separation R is in Bohr.}
\label{tab1}
\begin{tabular}{llllllll}
R & 1$^{2}\Sigma ^{+}$ & 2$^{2}\Sigma ^{+}$ & 3$^{2}\Sigma ^{+}$ & 4$%
^{2}\Sigma ^{+}$ & 1$^{2}\Pi $ & 2$^{2}\Pi $ & 3$^{2}\Pi $ \\ \hline
3.25 & -0.4005085 & -0.3234644 & -0.2961731 & -0.2876754 & -0.3380897 &
-0.2920849 & -0.2772029 \\
3.5 & -0.4375632 & -0.3612275 & -0.3339143 & -0.3248739 & -0.3780887 &
-0.3292013 & -0.3156819 \\
3.75 & -0.4596559 & -0.3832798 & -0.3561417 & -0.3465490 & -0.4021021 &
-0.3505374 & -0.3382916 \\
4. & -0.4726528 & -0.3956805 & -0.3688309 & -0.3585256 & -0.4161919 &
-0.3621480 & -0.3510792 \\
4.25 & -0.4801896 & -0.4022790 & -0.3757338 & -0.3645550 & & & \\
4.5 & -0.4844912 & -0.4055250 & -0.3791750 & -0.3670242 & -0.4283643 &
-0.3697614 & -0.3611132 \\
5. & -0.4883810 & -0.4080389 & -0.3809768 & -03667541 & -0.4310826 &
-0.3689544 & -0.3618997 \\
5.5 & -0.4897088 & -0.4095498 & -0.3796883 & -0.3641447 & -0.4303854 &
-0.3653886 & -0.3595355 \\
6. & -0.4903606 & -0.4121060 & -0.3774106 & -0.3602489 & -04289024 &
-0.3617274 & -0.3567152 \\
7. & -0.4911732 & -0.4165793 & -0.3728927 & -0.3552986 & -0.4263529 &
-0.3564139 & -0.3523502 \\
8. & & & & & -0.4251961 & -0.3541763 & -0.3507252 \\
8.5 & -0.4920470 & -0.4216463 & -0.3701660 & -0.3536025 & & & \\
9.5 & & & & & -0.4248510 & -0.3532627 & -0.3505684 \\
10. & -0.4923555 & -0.4237854 & -0.3696169 & -0.3531456 & & & \\
11. & -0.4921555 & -0.4240683 & -0.3693427 & -0.3526411 & -0.4244894 &
-0.3527058 & -0.3505085 \\
12.5 & -0.4919990 & -0.4239981 & -0.3693398 & -0.3523484 & & & \\
15. & -0.4920005 & -0.4240205 & -0.3696144 & -0.3522318 & -0.4242991 &
-0.3524386 & -0.3505300 \\
18. & -0.4918726 & -0.4238290 & -0.3695808 & -0.3522761 & -0.4241580 &
-0.3522776 & -0.3504068
\end{tabular}
\end{table}
\newpage
\begin{table}[tbp]
\caption{Dipole moment for the ground state and transition dipole moments
for LiKr in atomic units.}
\label{tab2}
\begin{tabular}{lllll}
R & 1$^{2}\Sigma ^{+}$ & 2$^{2}\Sigma ^{+}-1^{2}\Sigma ^{+}$ & 3$^{2}\Sigma
^{+}-1^{2}\Sigma ^{+}$ & 4$^{2}\Sigma ^{+} -1^{2}\Sigma ^{+}$ \\ \hline
3.25 & 1.77776874 & 1.86186478 & 0.92245651 & 0.38236908 \\
3.5 & 1.79671425 & 1.90200787 & 0.91466456 & 0.36047910 \\
3.75 & 1.79152689 & 1.93262393 & 0.90494948 & 0.33517901 \\
4 & 1.76002886 & 1.96183103 & 0.89052320 & 0.30862715 \\
4.25 & 1.70208274 & 1.99678065 & 0.86668951 & 0.28140826 \\
4.5 & 1.61937599 & 2.04227469 & 0.82738405 & 0.25065849 \\
5 & 1.40927266 & 2.12598452 & 0.69529641 & 0.24336991 \\
5.5 & 1.14072158 & 2.25505190 & 0.50135385 & 0.23199599 \\
6 & 0.87099314 & 2.33929207 & 0.29932825 & 0.17813524 \\
7 & 0.42212540 & 2.38938202 & 0.06776487 & 0.03502065 \\
8.5 & 0.12787435 & 2.41388895 & 0.00413534 & 0.03975754 \\
10 & 0.03951016 & 2.41989614 & 0.00830578 & 0.06651151 \\
11 & 0.02049474 & 2.41730242 & 0.00368579 & 0.06967306 \\
12.5 & 0.01641412 & 2.41121085 & 0.00348032 & 0.06437032 \\
15 & 0.01526793 & 2.39976461 & 0.00479587 & 0.07530521 \\
18 & 0.01638541 & 2.39239426 & 0.00267635 & 0.08321014
\end{tabular}
\end{table}
\newpage
\begin{table}[tbp]
\caption{Calculated spectroscopic constants of LiKr (in cm$^{-1}$.)}
\label{tab3}
\begin{tabular}{llllllllll}
State & R$_{e}$(Bohr) & D$_{e}$ & T$_{e}$ & B$_{e}$ & $\omega _{e}$ & $%
\omega _{e}x_{e}$ & $\alpha _{e}$ & v$_{max}$ & E$_{exp}$($\infty )$ \\
\hline
1$^{2}\Sigma ^{+}$ & 9.36 & 273 & -273 & 0.105 & 9.9 & 0.85 & 0.012 & 5 & 0
\\
1$^{2}\Pi $ & 5.06 & 1411 & 13444 & 0.362 & 218.4 & 9.1 & 0.003 & 11 & 14855
\\
2$^{2}\Sigma ^{+}$ & 11.71 & 88 & 14767 & 0.068 & 4212 & 4.3 & 0.005 & 4 &
14855 \\
3$^{2}\Sigma ^{+}$ & 4.95 & 2509 & 24395 & 0.373 & 229.1 & 5.1 & 0.005 & 22
& 26904 \\
2$^{2}\Pi $ & 4.64 & 3806 & 26849 & 0.431 & 313.7 & 6.7 & 0.011 & 22 & 30655
\end{tabular}
\end{table}
\newpage
\begin{table}[tbp]
\caption{Comparison of the LiKr 1$^{2}\Sigma ^{+}$ and 1$^{2}\Pi $ excited
states with the previously published experimental \protect\cite{sch75},
\protect\cite{aue74} and calculated potentials \protect\cite{pas74}.}
\label{tab4}
\begin{tabular}{llll}
State & Reference & R$_{e}$(Bohr) & T$_{e}$(cm$^{-1}$) \\ \hline
1$^{2}\Sigma ^{+}$ & This work & 9.36 & -273 \\
& Auerbach \cite{aue74} & 9.04 & -67.9 \\
& Pascale et al. \cite{pas74} & 9.45 & -65.7 \\
1$^{2}\Pi $ & This work & 5.06 & 13444 \\
& Scheps et al. \cite{sch75} & 6.01 & 13800 \\
& Pascale et al. \cite{pas74} & 6.0 & 14300
\end{tabular}
\end{table}
\newpage
\begin{table}[tbp]
\caption{Maxima ($\protect\lambda $) of the LiAr, LiKr and LiXe excimer
bands compared with the observations ($\protect\lambda ^{*}$) of Wang and
Havey \protect\cite{wan84} and the calculations ($\protect\lambda ^{c}$). $%
^{a}$ using Ref. \protect\cite{gu94}, $^{b}$ using Ref. \protect\cite{par97}%
. }
\label{tab5}
\begin{tabular}{llll}
& LiAr & LiKr & LiXe \\ \hline
$\lambda $(nm) & 414 & 420 & 435 \\
$\lambda ^{*}$(nm) & 411 & 416 & 431 \\
$\lambda ^{c}$(nm) & 416.4 $^{a}$ & 433 & \\
& 413.1 $^{b}$ & &
\end{tabular}
\end{table}
\newpage
\begin{table}[tbp]
\caption{A list of photochemical reactions producing the excited LiAr and
LiKr excimers. We give the energy defect of the reaction, the spectral
positions of the listed transitions calculated from the extrema in
difference potentials and observed values (in nm), where available.}
\label{tab6}
\begin{tabular}{lrll}
Reaction & $\Delta $E(cm$^{-1}$) & Transition & Calculated position in nm,
comments \\ \hline
Li(2$^{2}$P)+Ar+Ar$\rightarrow $LiAr(1$^{2}\Pi $)+Ar & 79 & 1$^{2}\Pi
\rightarrow 1^{2}\Sigma ^{+}$ & 674, min at 4.2 Bohr, obs. at 790 \cite
{sch75} \\
Li$_{2}$(1$^{1}\Sigma _{u}^{+}$)+Ar$\rightarrow $LiAr(1$^{2}\Pi $)+Li & -9270
& 1$^{2}\Pi \rightarrow 1^{2}\Sigma ^{+}$ & 674, min at 4.2 Bohr, obs. at
740 \cite{zha91} \\
Li$_{2}$(1$^{1}\Pi _{u}$)+Ar$\rightarrow $LiAr(1$^{2}\Pi $)+Li & -2902 & &
\\
Li(3$^{2}$P)+Ar+Ar$\rightarrow $LiAr(3$^{2}\Sigma $)+Ar & 4807 & 3$%
^{2}\Sigma ^{+}\rightarrow 1^{2}\Sigma ^{+}$ & 416, min at 3.5 Bohr, obs. at
411 \cite{wan84} \\
Li(3$^{2}$D)+Ar+Ar$\rightarrow $LiAr(3$^{2}\Sigma $)+Ar & 5166 & & \\
Li(3$^{2}$P)+Ar+Ar$\rightarrow $LiAr(2$^{2}\Pi $)+Ar & 1837 & 2$^{2}\Pi
\rightarrow 1^{2}\Sigma ^{+}$ & 394, min at 3.5 Bohr, obs. at 380 \cite
{wan84} \\
Li(3$^{2}$D)+Ar+Ar$\rightarrow $LiAr(2$^{2}\Pi $)+Ar & 2196 & & \\
Li$_{2}$(2$^{1}\Pi _{u}$)+Ar$\rightarrow $LiAr(3$^{2}\Sigma $)+Li & -4064 & 3%
$^{2}\Sigma ^{+}\rightarrow 1^{2}\Sigma ^{+}$ & 416, min at 3.5 Bohr \\
Li$_{2}$(2$^{1}\Sigma _{u}^{+}$)+Ar$\rightarrow $LiAr(3$^{2}\Sigma $)+Li &
-4315 & 3$^{2}\Sigma ^{+}\rightarrow 1^{2}\Pi $ & 825, max at 5.2 Bohr \\
& & 3$^{2}\Sigma ^{+}\rightarrow 2^{2}\Sigma ^{+}$ & 1674, min at 4 Bohr \\
Li$_{2}$(2$^{1}\Pi _{u}$)+Ar$\rightarrow $LiAr(2$^{2}\Pi $)+Li & -7034 & 2$%
^{2}\Pi \rightarrow 1^{2}\Sigma ^{+}$ & 394, min at 3.5 Bohr \\
Li$_{2}$(2$^{1}\Sigma _{u}^{+}$)+Ar$\rightarrow $LiAr(2$^{2}\Pi $)+Li & -7285
& 2$^{2}\Pi \rightarrow 1^{2}\Pi $ & no extrema \\
& & 2$^{2}\Pi \rightarrow 2^{2}\Sigma ^{+}$ & 631, max at 14 Bohr \\ \hline
Li(2$^{2}$P)+Kr+Kr$\rightarrow $LiKr(1$^{2}\Pi $)+Kr & 1559 & 1$^{2}\Pi
\rightarrow 1^{2}\Sigma ^{+}$ & 744, min. at 4.5 Bohr, obs. at 802\cite
{sch75} \\
Li$_{2}$(1$^{1}\Sigma _{u}^{+}$)+Kr$\rightarrow $LiKr(1$^{2}\Pi $)+Li & -7876
& 1$^{2}\Pi \rightarrow 1^{2}\Sigma ^{+}$ & 744, min. at 4.5 Bohr \\
Li$_{2}$(1$^{1}\Pi _{u}$)+Kr$\rightarrow $LiKr(1$^{2}\Pi $)+Li & -1508 & &
\\
Li(3$^{2}$P)+Kr+Kr$\rightarrow $LiKr(3$^{2}\Sigma $)+Kr & 6539 & 3$%
^{2}\Sigma ^{+}\rightarrow 1^{2}\Sigma ^{+}$ & 433, min. at 3.5 Bohr , obs.
at 416 \cite{wan84} \\
Li(3$^{2}$D)+Kr+Kr$\rightarrow $LiKr(3$^{2}\Sigma $)+Kr & 6898 & & \\
Li(3$^{2}$P)+Kr+Kr$\rightarrow $LiKr(2$^{2}\Pi $)+Kr & 4085 & 2$^{2}\Pi
\rightarrow 1^{2}\Sigma ^{+}$ & 401, min at 3.5 Bohr, obs. at 392 \cite
{wan84} \\
Li(3$^{2}$D)+Kr+Kr$\rightarrow $LiKr(2$^{2}\Pi $)+Kr & 4444 & & \\
Li$_{2}$(2$^{1}\Pi _{u}$)+Kr$\rightarrow $LiKr(3$^{2}\Sigma $)+Li & -2332 & 3%
$^{2}\Sigma ^{+}\rightarrow 1^{2}\Sigma ^{+}$ & 433, min. at 3.5 Bohr \\
Li$_{2}$(2$^{1}\Sigma _{u}^{+}$)+Kr$\rightarrow $LiKr(3$^{2}\Sigma $)+Li &
-2583 & 3$^{2}\Sigma ^{+}\rightarrow 1^{2}\Pi $ & 1053, min at 4 Bohr \\
& & 3$^{2}\Sigma ^{+}\rightarrow 2^{2}\Sigma ^{+}$ & 1904, min. at 4.7 Bohr
\\
Li$_{2}$(2$^{1}\Pi _{u}$)+Kr$\rightarrow $LiKr(2$^{2}\Pi $)+Li & -4786 & 2$%
^{2}\Pi \rightarrow 1^{2}\Sigma ^{+}$ & 401, min at 3.5 Bohr \\
Li$_{2}$(2$^{1}\Sigma _{u}^{+}$)+Kr$\rightarrow $LiKr(2$^{2}\Pi $)+Li & -5037
& 2$^{2}\Pi \rightarrow 1^{2}\Pi $ & no extrema \\
& & 2$^{2}\Pi \rightarrow 2^{2}\Sigma ^{+}$ & 1324, min. at 4.5 Bohr
\end{tabular}
\end{table}
\end{document}