Theoretical study of the absorption spectra of the lithium dimer
H.-K. Chung, K. Kirby, and J. F. Babb
Institute for Theoretical Atomic and Molecular Physics,
Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138
Abstract
For the lithium dimer we calculate cross sections for absorption of radi-
ation from the vibrational-rotational levels of the ground X 1+g electronic
state to the vibrational levels and continua of the excited A 1+u and B 1u
electronic states. Theoretical and experimental data are used to character-
ize the molecular properties taking advantage of knowledge recently obtained
from photoassociation spectroscopy and ultra-cold atom collision studies. The
quantum-mechanical calculations are carried out for temperatures in the range
from 1000 to 2000 K and are compared with previous calculations and mea-
surements.
PACS numbers: 33.20.-t, 34.20.Mq, 52.25.Rv
Typeset using REVTEX
1
I. INTRODUCTION
The absorption spectra of pure alkali-metal vapors at temperatures of the order 1000 K
can be a rich source of information on molecular potentials and properties. Achieving a high
vapor pressure of lithium in experiments requires higher temperatures than the other alkali-
metal atoms, but there are some data from heat pipe ovens [1- 3] and from a specialized
apparatus [4]. In addition to the atomic lines the spectra exhibit gross molecular features
attributable to transitions between bound levels of the ground electronic state and levels of
the excited singlet states and weaker features arising from analogous triplet transitions.
Theoretically, the envelope of the alkali-metal molecular absorption spectra can be quan-
titatively reproduced using semi-classical models [5- 9] and ro-vibrational structure [7] and
continua [10] can be reproduced from quantum-mechanical models. In this paper we cal-
culate quantum-mechanically absorption spectra for the X 1+g-A 1+u and X 1+g-B 1u
transitions in Li2. Although both semi-classical [7] and quantum-mechanical [7,11] calcula-
tions have been performed and compared [7] previously for these transitions of Li2, recent
improvements in the molecular data prompt the present comprehensive study.
From photoassociation spectroscopy and cold collision studies performed in the last few
years as well as recent theoretical work there have been significant critical tests of and
improvements to the molecular potentials [12- 14], particularly at long-range [15- 17], and
transition dipole moment data [15] available for Li2, as well as to the value of the lifetime
of the Li 2p state [18,19]. This paper presents calculations of the spectra over the full range
of wavelengths where absorption in the X 1+g-A 1+u and X 1+g-B 1u bands is possible.
We calculate the satellite feature profiles at various temperatures, identify and explore the
influence of quasibound states and the contributions of bound-bound versus bound-free
transitions, calculate partition functions, and calculate lifetimes for the A 1+u and B 1 u
ro-vibrational levels.
II. QUANTUM THEORY OF ABSORPTION CROSS SECTION
In the quantum-mechanical formulation an absorption cross section from a vibration-
rotation state of the lower electronic state (v00; J00; 00) to the vibration-rotation state of the
upper electronic state (v0; J0; 0) is
00
0J00 8ss3 2 SJJ0000
oevv00J0000() = ______|<OEv00J0000|D(R) |OEv0J00>|g( - ij) _________ (1)
3hc 2J00+ 1
2
00
where g( - ij) is a line-shape function of dimension -1 , SJJ0000is the H"onl-London factor
and ij |Ev0J00 - Ev00J0000| is the transition frequency [20,21]. In this study, g( - ij) is
approximated by 1=, with the bin size. For a bound-free transition, the absorption
cross section from a bound level of the lower electronic state (v00J0000) to a continuum level
of the upper electronic state (ffl0J00) can be written as
00
0J00 8ss3 2 SJJ0000
oefflv00J0000() = ______|<OEv00J0000|D(R) |OEffl0J00>|_______ (2)
3hc 2J00+ 1
where the continuum wave function OEffl0J00is energy normalized. Free-bound or free-free
transitions are not considered since the temperatures studied here are not high enough for
these types of transitions to be important within the singlet manifold.
The radial wave function can be obtained from the Schr"odinger equation for the relative
motion of the nuclei
!
d2OE(R)_ 2 2 J(J + 1) - 2
+ ___2E - ___2V (R) - ________________ OE(R) = 0; (3)
dR2 h h R2
where V (R) is the rotationless potential energy for the electronic state, = 6 394:7 is the
reduced mass of the 7Li atoms, and E is for bound states the eigenvalue EvJ measured
with respect to the dissociation limit associated with the wave function OE(R) = OEvJ (R).
Similarly, for continuum states E is the relative kinetic energy of the colliding atoms EfflJ
associated with the energy-normalized wave function OE(R) = OEfflJ(R).
The total absorption cross section at frequency can be obtained by averaging over
initial vibration-rotation levels (v00; J00; 00) with a relevant weighting factor and summing
over all possible bound-bound and bound-free transitions with frequencies between and
+ [7] yielding
"
X 0 00
oe() = Z-1l oevvJ00J0000!J00(2J00+ 1) exp [-(De + Ev00J0000)=kT ]
J0J00v0v00 #
X Z 0 00
+ dffl0oefflvJ00J0000!J00(2J00+ 1) exp [-(De + Ev00J0000)=kT;] (4)
J0J00v00
where !J00is a statistical factor due to nuclear spin with the values [I=(2I + 1)] = 3_8for even
J and [(I + 1)=(2I + 1)] = 5_8for odd J, for 7Li2 with I = 3_2. With the zero of energy taken
to be the potential minimum, the partition function Zl of the lower state with dissociation
energy De is
X
Zl = !J00(2J00+ 1) exp [-(De + Ev00J0000)=kT ]; (5)
J00v00
assuming thermodynamic equilibrium. The resulting cross sections can be used to model
the dimer absorption spectra in the quasistatic limit.
3
III. MOLECULAR DATA
The adopted molecular potentials of the X 1+g, A 1+u and B 1u states are shown in
Fig. 1. The ground X1+g state potential was constructed using the approach of Ref. [22].
We adopted the recommended [23] potential obtained by Barakat et al. [24], who applied
the Rydberg-Klein-Rees (RKR) method to measured energies. The data were connected
smoothly to the long-range form
C6 C8 C10
V (R) = - ___- ___ - ____ + Vexc(R); (6)
R6 R8 R10
where Vexc(R) is the exchange energy [22] and the coefficients C6, C8, and C10 have been
calculated in Refs. [15,25- 27] and we use atomic units in this section. We adopted the
coefficients C6 = 1 393:39, C8 = 83 425:8 and C10 = 7:372 x 106 from Ref. [26]. The
two regions were connected at R = 23:885 a0 yielding a value of 8516.95 cm -1 for the
dissociation energy of the X1+g state of 7Li2, in satisfactory agreement with the accepted
value of 8516.61 cm -1 [23]. The form a exp (-bR) was used to extrapolate the potential
at short-range where the constants were determined to smoothly connect to the innermost
RKR points. The resulting X 1+g potential yields an s-wave scattering length of 33:6 a0,
in excellent agreement with the accepted [28] value of 33 2 and a sensitive test of the
assembled data.
For the excited A1+u state the RKR potential of Ref. [17] was adopted and smoothly
connected at about R = 140 a0 to the long-range form
C3 C6 C8
V (R) = ___ - ___ - ___; (7)
R3 R6 R8
with coefficients C3 = -11:000 226 and C6 = 2 075:05 from Ref. [26] and C8 = 2:705 x
105 from Ref. [15]. For R < 3:78 a0 the data were connected to the short range form
a exp (-bR). The dissociation energy for the A1+u state is determined to be 9 352:194 cm -1
in our calculation, in good agreement with the experimental value [29] of 9 352:5(6) cm -1
given in Ref. [30].
The potential for the B1u state has a hump with maximum at R 11 a0 and various
determinations of the hump location and height have been summarized in Refs. [31] and [30].
We adopted the IPA (Inverted Perturbation Approach) potentials from Hessel and Vidal [32]
for R < 9:35 a0 and the ab initio potential of Schmidt-Mink et al. [30] for 10:5 < R < 30 a0
with one additional point at R = 11:2 a0 fixing the barrier maximum energy at 512 cm -1
above dissociation. At R = 30 a0 the data were connected to the long-range form of Eq. (7)
by shifting down the data by 0:3 cm -1 from 10:5 < R < 30. The values of the coefficients
4
used in Eq. (7) were C3 = 5:500 113 and C6 = 1406:08 from Ref. [26] and C8 = 4:756 x 104
from Ref. [15]. The potential energy data for R < 9:35 a0 were fixed using the B 1u state
dissociation energy of 2 984:8 cm -1 , which we determined using the experimental value for
Te of 20 436:32 cm -1 [32], the atomic asymptotic energy of 14 904:0 cm -1 , and the X 1+g
dissociation energy of 8 516:61 cm -1 [23]. Finally, the data in the range 9:35 < R < 10:5 were
smoothly connected using cubic splines. For R < 4:254 61 a0 the data were extrapolated
using the short range form a exp (-bR).
Transition dipole moments for the X 1+g-A 1+u and X 1+g-B 1u transitions are avail-
able from ab initio calculations [33,34,30] and for X-A transitions from measured lifetimes of
A state levels [35]. For the electronic transition dipole moments D(R), we adopted ab initio
calculations of Ratcliff, Fish, and Konowalow [33] connected at R = 35 a0 to the long-range
asymptotic form
b
D1 (R) = D0 + ___: (8)
R3
The value of the coefficient D0 was 3:3175 for both X-A and X-B transitions and the
coefficient b was 283:07 or -141:53 [15] for, respectively, X-A or X-B transitions. For both
transitions, we multiplied D1 (R) calculated using the above coefficients by a constant such
that the value D1 (35) was identical to the corresponding ab initio value from Ref. [33]
to provide a smooth connection between short and long-range forms for D(R). The X-A
dipole moment function that we adopted is consistent with that derived by Baumgartner et
al. [35] from experimental measurements. There is no experimentally-derived dipole moment
function for the X-B transition.
IV. RESULTS
Bound and continuum wave functions were calculated using the Numerov method to
integrate Eq. (3). For the X and A states, eigenvalues were generally in good agreement
with the Rydberg-Klein-Rees values used as input to the potentials constructed. [There
is an apparent misprint for the energy of the v00= 9 level of the X1+g state in Ref. [24].
We used 3 098:641 2 cm -1 , consistent with Ref. [29].] The constructed B 1u state potential
can reproduce the rotationless IPA energies tabulated by Hessel and Vidal typically to
about 0:1 cm -1 , with the greatest discrepancy 0.15 cm -1 for the v0 = 13 value. Calculated
frequencies of B 1u-X 1+gtransitions were also compared with calculations of Verma, Koch,
and Stwalley [36] and the agreement was good, within 0.1 cm -1 . Four quasi-bound levels of
5
the B 1u state were found. For the rotationless potential their calculated eigenvalues are
143.91, 276.7, 391.74 and 483.88 cm -1 above the dissociation limit for v0 = 14 to v0 = 17.
The calculated term energies of vibration-rotation states in the X 1+g state were used
to compute partition functions using Eq. (5). The maximum vibrational and rotational
quantum numbers in our calculations are 41 and 123, respectively, for the X 1+g state.
In the harmonic approximation for ro-vibrational energies, the partition function can be
calculated using the simple expression
X123
Z"l ZR Zv !J00(2J00+ 1) exp [-hcBeJ00(J00+ 1)=kT ]
J00=0
X41
x exp [-he(v00+ 1=2)=kT ]; (9)
v00=0
which assumes that the term energy can be described by the first terms of the power se-
ries with respect to vibrational and rotational quantum numbers. Using constants e=c =
351.3904 cm -1 and Be = 0:672 566 cm -1 [32] the partition function from Eq. (9) was calcu-
lated and it is compared with the partition function calculated from Eq. (5) for the X 1+g
state as a function of temperature in Fig. 2. The anharmonicity of the potential for higher
vibrational levels accounts for the differences between the two results with increasing tem-
perature. For J > 2 the X 1+gstate supports quasibound vibrational levels. The expression
Eq. (5) for Zl does not specify whether quasibound states are to be included or not in the
summations. We evaluated Zl with and without the quasibound levels to ascertain their
importance and the results are shown in Fig. 2. The effect of the additional levels becomes
increasingly significant with higher temperature. For the present study covering tempera-
tures between 1000 and 2000 K there is not a significant distinction between Zl, Z"l, and the
result with the inclusion of the quasibound states.
The molecular fraction can be calculated using the expression
[NLi2]=[NLi]2 = (QLi2=Q2Li) exp (De=kT ); (10)
where the atomic partition function QLi is 2(2ssmLikT =h2)3=2, with the electronic partition
function for the atom well-approximated by the spin degeneracy of 2 for the temperatures
studied in the present work, and the molecular partition function QLi2is (2ssmLi2kT =h2)3=2Zl,
with the electronic partition function for the X 1+g state taking the value 1. The molecular
fraction Eq. (10) is plotted in Fig. 2. The absorption coefficient k() can be obtained if the
atomic density is known from
k() = [NLi2]oe(): (11)
6
A. Lifetimes
Lifetimes of the various ro-vibrational levels of the A 1+u state were measured [35] and
calculated, see for example, Refs. [37,38,22]. We calculated spontaneous emission transition
probabilities and lifetimes of rotational-vibrational levels of the A1+u state in order to test
the adopted transition dipole moment. The spontaneous emission rate from a bound state
(v0J00) to a bound state (v00J000) is
64ss43 2
A(v0J00; v00J000) = ________g |<OEv00J000|D(R) |OEv0J00>|; (12)
3hc3
where the electronic state degeneracy is
(2 - ffi0;0+00 )
g = _______________ (13)
2 - ffi0;0
and we have neglected change in the rotational quantum number. The total spontaneous
emission rate from the upper level (v0J00) can be obtained by summing over all possible
transitions to bound and continuum states
X X Z
A(v0J00) = A(v0J00; v00J000) + dffl00A(v0J00; ffl00J000); (14)
v0000 00
where A(v0J00; ffl00J000) is the spontaneous emission probability to a continuum energy ffl00
with partial wave J0. The lifetime is
o = 1=A(v0J00): (15)
Lifetimes of levels v0J0 of the A 1+ustate were measured by Baumgartner et al. [35]. The
A 1+u state is affected by indirect predissociation via the a 3+u and 1 3+u states [39,40].
The measured lifetimes om of vibration-rotation levels thought to be unaffected by indirect
predissociation taken from Ref. [35] and corresponding calculated lifetimes oc are presented
in Fig. 3 along with calculated term energies expressed relative to the potential minimum of
the A state. The energies are plotted in the order of the values listed in Table 1 of Ref. [35]
and correspond to a range of values of (v0; J0) from (0,15) to (24,25). The agreement between
oc and om is good within the experimental precision of 2 percent [35]. The quasi-bound
levels and continua of the X 1+g state inside or above the high centrifugal potential barriers
are found to be important in calculating lifetimes of high J levels. For instance, by including
transitions to three quasi-bound levels and the continuum states the calculated lifetime of the
A 1+u(v0 = 20; J0 = 50) level changes from 22.7 ns to 19.3 ns which is close to the measured
lifetime of 18:66 0:37 ns [35]. The calculated and measured lifetimes agree well up to about
7
4 500 cm -1 in agreement with the findings of Ref. [30]. From approximately 5 000 cm -1 and
higher the experimental lifetimes slightly decrease relative to theory by about 0.8 ns or five
percent, as demonstrated in Fig. 3. We investigated whether the 21+g state might supply
an additional spontaneous decay channel, but the theoretical value Te = 20 128 cm -1 [30] for
this state appears to place its minimum at around 6 000 cm -1 relative to the minimum of the
A 1+u state, apparently ruling this mechanism out. The reason for the significant downturn
of experimental lifetimes for higher term energies is currently not understood; however, the
overall excellent agreement between our calculated lifetimes and the measurements gives
us confidence in our molecular data and calculational procedures. We compare selected
examples from the present results with calculations by Watson [37] and by Sangfelt et al. [38]
in Table I. The calculations of Sangfelt et al. are larger than ours, probably because
theoretical transition energies were used. A simple rescaling using experimental energies, as
pointed out by those authors, yields lifetimes in good agreement with the present work. The
dipole moment function calculated by Watson [37] is in good agreement with that adopted
in the present study and cannot account for the shorter lifetimes obtained in that study. We
present a more extensive tabulation of A 1+u lifetimes in Table II covering the same values
tabulated in Table VII of Ref. [38].
Lifetimes for levels of the B 1u state have been calculated by Uzer, Watson, and Dal-
garno [41], Uzer and Dalgarno [39], and Sangfelt et al. [38] and there appear to be no
experimental data. In Table III we compare our calculated lifetimes of selected B 1u lev-
els with available calculations for higher values of J0. The present results lie between the
calculations of Uzer et al. [41] and those of Sangfelt et al. [38]. The lifetimes calculated by
Uzer et al. are longer than ours because their transition dipole moment function, calculated
using a model potential method, is smaller than the ab initio dipole moment of Ratcliff
et al. [33] adopted in the present study. The dipole moment function calculated by Sangfelt
et al., on the other hand, is in good agreement with that adopted in the present work. As
those authors pointed out, and as our results illustrate, the utilization in their calculation of
calculated excitation energies which were larger than experimental energies yielded lifetimes
that were too short. In Table IV we present a more extensive tabulation of B 1u lifetimes
covering the same values tabulated in Table VIII of Ref. [38].
B. Absorption cross sections
Absorption spectra arising from molecular singlet transitions at the far wings of the
atomic 2p line at 671 nm consist of a blue wing due to X 1+g-B 1u transitions and a
8
red wing due to X 1+g-A 1+u transitions. Calculations for bound-bound (bb) and bound-
free (bf) absorption cross sections were carried out separately using Eq. (4) with =
8 cm -1. The results for the total (the sum of bb and bf) absorption cross sections at
temperatures of 1000 K, 1500 K and 2033 K are given in Figs. 4-6. The ratios of the
peak cross sections of the X 1+g-B 1u wing to those of the X 1+g-A 1+u wing are higher
at lower temperatures. As temperature increases, absorption spectra spread out from the
peak spectral regions and there emerges near 900 nm a satellite feature arising from the
minimum [42] in the X 1 +g-A 1+u difference potential and the maximum of the transition
dipole moment function [43,33]. We also show the bf contributions to the cross sections
on each plot. The bf component contributes mainly to the extreme blue part of the blue
wing and increases in magnitude significantly as the temperature increases. It is found that
transitions to quasibound and continuum levels of the B 1u state contribute significantly
to the total absorption spectra in the case of X 1+g-B 1u transitions, apparently because
there is less vibrational oscillator strength density in the discrete part of the spectrum for
the B state compared to the A state. Transitions into quasibound states of the A 1+u or
B 1u states have been included in the results for the total cross sections in Figs. 4-6.
Theoretical quantum-mechanical calculations for absorption cross sections from the
X 1+g state to the A 1+u state over the spectral range 600-950 nm were carried out by
Lam et al. [7] using a constant dipole moment of 6.58 D at 1020 K, for which bf transitions
are not significant. Our calculations in Fig. 4 are about a factor of 10 less than the result
shown in Fig. 5 of Ref. [7], but agree well both in overall shape and in details of finer struc-
tures. We repeated the calculations using the constant dipole moment of Ref. [7] for both
classical and quantum-mechanical cross sections and although these two results agreed with
each other, they were also a factor of 10 less than the result shown in Fig. 5 of Ref. [7].
Thus it appears to us that there may be a mislabeling of the vertical axis in Fig. 5 of Lam et
al. [A similar calculation that we performed [44] for Na 2 at 800 K is in complete agreement
with Fig. 4 of Ref. [7].]
Calculations of absorption spectra at 2033 K over the spectral range 450-750 nm pre-
sented in Fig. 6 are in good agreement with the measured values of Erdman et al. [4] and
with quantum-mechanical calculations performed by Mills [11]. The experimental study of
Erdman et al. [4] involved an investigation of molecular triplet states [4] and did not explore
the satellite feature at 900 nm. The calculations over the range 450-750 nm by Mills [11]
included triplet molecular transitions and are not directly comparable with the present re-
sults. Nevertheless since the singlet transitions dominate the absorption we find excellent
qualitative agreement with the calculations presented by Mills.
9
ACKNOWLEDGMENTS
We thank R. C^ote for generously sharing assembled data and A. Dalgarno for helpful
discussions. We also are grateful to A. Gallagher, W. Stwalley, and M. Fajardo for helpful
correspondence. This work is supported in part by the National Science Foundation under
grant PHY97-24713 and by a grant to the Institute for Theoretical Atomic and Molecular
Physics at Harvard College Observatory and the Smithsonian Astrophysical Observatory.
10
TABLES
_TABLE_I._Comparison_of_calculated_lifetimes_in_ns_for_ro-vibrational_levels_of_the_A_1+u_state.___*
*__
___________________________________________________________________________________________________*
*__
_v0__________J0___________Watson_[37]____________Sangfeldt_et_al._[38]__________This_work__________*
*__
7 15 16.8 19.25 18.55
9 5 16.9 19.29 18.65
20 8 17.3 19.04
___________________________________________________________________________________________________*
*___
TABLE II. Lifetimes in ns for ro-vibrational levels of the A 1+u state calculated as described
in_the_text._______________________________________________________________________________________*
*___
v0__________________________J0_=_0__________________________J0_=_9___________________________J0_=_1*
*5_
0 17.74 17.77 17.82
1 17.87 17.90 17.94
2 17.98 18.01 18.06
3 18.09 18.12 18.17
4 18.20 18.23 18.27
5 18.30 18.33 18.37
6 18.39 18.42 18.46
7 18.48 18.51 18.55
___________________________________________________________________________________________________*
*___
11
TABLE_III._Comparison_of_calculated_lifetimes_in_ns_for_ro-vibrational_levels_of_the_B_1u_state.___*
*__
___________________________________________________________________________________________________*
*__
v0__________J0___________Uzer_et_al._[41]_________Sangfeldt_et_al._[38]_________This_work__________*
*__
0 15 8.3 6.83 7.66
5 9 8.5 7.20 7.95
___________________________________________________________________________________________________*
*___
TABLE IV. Lifetimes in ns for vibrational-rotational levels of the B 1u state calculated as
described_in_the_text._____________________________________________________________________________*
*___
v0__________________________J0_=_1__________________________J0_=_9___________________________J0_=_1*
*5_
0 7.65 7.65 7.66
1 7.70 7.71 7.72
2 7.76 7.76 7.77
3 7.81 7.82 7.83
4 7.88 7.89 7.90
5 7.94 7.95 7.97
6 8.02 8.03 8.04
7 8.10 8.10 8.12
___________________________________________________________________________________________________*
*___
12
FIGURES
FIG. 1. Adopted potentials for the X 1+g, A 1+u, and B 1u electronic states.
13
FIG. 2. Comparison at various temperatures of the partition functions Z"l, calculated using
Eq. (9) and experimentally determined spectroscopic constants, curve A, and Zl, from Eq. (5)
and numerically determined eigenvalues, curves B and C. Inclusion of quasibound states in the
calculation of Zl results in curve C as discussed in the text. Curve D represents the molecular
fraction [NLi2]=[NLi]2, Eq. (10), as a function of temperature.
14
FIG. 3. Comparison of our calculated lifetimes (diamonds) and measured lifetimes (circles)
from Ref. [35]. The error bars indicate the quoted experimental uncertainty of 2 percent. The
levels given are those that were measured, ordered by increasing energy, as listed in Table 1 of
Ref. [35].
15
FIG. 4. Total absorption cross sections from X 1+g-A 1+u and X 1+g-B 1u transitions in-
cluding bound to bound and bound to free transitions at a temperature of 1000 K. The satellite
feature near 900 nm does not appear at this temperature and bound-free absorption (dotted curve)
is insignificant. The inset presents a magnified view of the bound-free contribution.
16
FIG. 5. Total absorption cross sections from X 1+g-A 1+uand X 1+g-B 1u transitions includ-
ing bound to bound and bound to free transitions at a temperature of 1500 K. As the temperature
increases, the absorption spectra are distributed over a wider spectral range and the ratio of the
peak cross sections between X 1+g-B 1u and X 1+g-A 1+ubands decreases. The satellite feature
near 900 nm and bound-free absorption (dotted curve) are noticeable at this temperature. The
inset presents a magnified view of the bound-free contribution.
17
FIG. 6. Total absorption cross sections from X 1+g-A 1+uand X 1+g-B 1u transitions includ-
ing bound to bound and bound to free transitions at a temperature of 2033 K. At this temperature
the satellite feature near 900 nm is now prominent and bound-free absorption (dotted curve) con-
tributes significantly at 450 nm. The inset presents a magnified view of the bound-free contributio*
*n.
18
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