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The RbCs electronic ground state revisited
C. Fellows, C. Amiot, J. Vergès
To cite this version:
C. Fellows, C. Amiot, J. Vergès. The RbCs electronic ground state revisited. Journal de Physique II, EDP Sciences, 1992, 2 (4), pp.939-946. �10.1051/jp2:1992149�. �jpa-00247683�
Classification Physics Abstracts
33.20E-32.50F-31.50
The Rbcs electronic ground state revisited
C-E- Fellows (~), C. Amiot (~) and J. Vergls (~)
(~) Laborat6rio de Espectroscopia e Laser, Universidade Federal Fluminense, Niter6i, RJ 24000, Brazil
(~) Laboratoire Aimd Cotton, CNRS II, Bitiment 505, Campus d'orsay 91405 Orsay, Cedex,
France
(Received 17 October 1991, accepted 10 December 1991)
Abstract. In this article the l~L+ electronic ground state is studied up to near the disso- ciation limit. Calculations have been carried out using previous experimental data and taking
into account the exchange energy that cannot be neglected. In this calculations, new values for
the multipolar van der Waals coefficients Cm have been obtained showing a good agreement with the previous theoretical ones.
1. Introduction.
Spectroscopic studies of the Rbcs molecule are quite scarce. Fewer are the studies treating
the dissociation energy of the electronic ground state of this molecule, the llL+ The first study has been published by Kat6 and I(obayashi [I] where the fluorescence has been induced
between the llL+ and the 31II electronic states using the 568.2 nm Kr+ laser line.
After that, Gustavsson et al. [2] have published a more complete study of this electronic
ground state using laser induced fluorescence combined with Fourier transform spectroscopy.
In this work very reliable molecular constants have been obtained and with these constants a RKR potential energy curve of the l~L+ electronic ground state was calculated up to u"
=
l19.
With this potential energy curve a new value of the dissociation energy for the electronic
ground state has been estimated. Nevertheless, the problem was that in the available data tile outermost turning point in"
= l19), corresponded to a maximum elongation of lo.679 1.
Following Leroy's criterion [3], in order to treat the external turning points in a multipolar expansion, V(r) = £~ Cn/r", without considering the exchange energy, the internuclear
distance must be greater than rjim
rjim = 2((r()~/~ + (r()~/~),
where A and B denote the outermost electron orbital radii for the two different atoms. The
ground state of Rbcs molecule dissociates into Rb5s~Si/2 + Cs6s~Si/2. ivith values for (i~)
940 JOURNAL DE PHYSIQUE II N°4
taken from Floese-Fischer [4] we have trim " 13.461. For this reason, in the article of Gus-
tavsson et al. [2], a simple calculation involving only the dissociation limit and the first term in the second order of the multipolar expansion (C6) has been performed providing "an upper
limit of the dissociation energy" for the electronic ground state of the Rbcs molecule.
In this work we have tried to calculate a more accurate value for the dissociation energy,
using the data from Gustavsson et al. [2] and taking into account the exchange term that cannot be neglected in the internuclear forces for r values lower than rjim.
In order to perform the calculations different analytical expressions for the exchange energy
have been used. The first one, most commonly used, is the single exponential form [5],
TX = Ae~~~, (1)
where Vex is the exchange energy, A and a are adjustable constants and r is the internuclear distance. The second one is a more complex form, proposed by Knox and Rudge [6], expressed by:
Vex = A'r~exp [(~° 7r, (2)
r
where (x means the exchange energy, A', b, ao and 7 are adjustable constants and r is the internuclear distance.
Finally, a damping function in(r) has been used in order to compensate the divergence of
£~ Cm/r" at smaller distances of r. In this case the multipolar expansion takes the form:
The damping function used in our calculations is the
same as the one used y
fair) = ii e-°(r-nP)in, (4)
where a and fl are adjustable constants, n is the multipole order in
= 6, 8, 10, and r is the internuclear distance.
2. Results.
The choice of the external turning points and the corresponding energy values to be used in the calculations have been done following Leroy's suggestion [3] that the term n + 2 in the
multipolar expansion is no longer valid for r < (2Cn+2/Cn)1/~ Using the theoretical values for Cio and C8 [9], we can obtain a lower limit of confidence for Cio. In the internuclear
distance range where r > 8.82 1, Cio has
a physical significance in the multipolar expansion.
It means that we have used in the calculations the values corresponding to u" greater than
108 [2]. The energy values and their corresponding external turning points are listed in table I. All calculations have been performed using this ensemble of experimental points. Firstly we
have performed our calculations using the truncated multipolar expansion, including exchange
energy, as follows:
V(r) = De § ~j §)
Vex, (5)
r r r
in a non linear least-square reduction.
E(cm-11
8 9 10 11 12 rli)
Fig-I. Contribution of Cm van der Waals coefficients and exchange energy calculated for r > 8.8
1, to the potential energy
curve. Vn
= De Cn/r" (n
= 6, 8, and 10) and Rx = De (Ae~~~). VT
lo
represents De (£CnIv" + I&x). The circles in the figure represent the experimental points.
n=6
Table I. Vibrational term-values Gin" ), outer turning points rid) used in the calculations.
109 3731.637 8.980
l10 3742.036 9.100
ill 3751.818 9.228
l12 3760.985 9.365
l13 3769.540 9.sll
l14 3777.487 9.699
lls 3784.832 9.839
l16 3791.586 10.023
l17 3797.758 10.222
l18 3803.364 10.440
l19 3808.422 10.679
In a first calculation we used equation (I) as the exchange energy. The values obtained for
C6, 8, lo A and a are shown in table IIa and the contributions of each term to the potential energy curve are represented in figure I. Secondly, the exchange term has been replaced by equation (2) and we have obtained the results listed in table lib and the contributions of each
term are shown in figure 2. In table TIC we can compare the values of A', b, ao and 7 obtained in our calculations with those obtained theoretically by Knox and Rudge [6]. Finally we have
used the damping expression of equation (3).
Here, some points must to be considered. As can be observed in figure I and figure 2, the contributions to the potential energy curve due to the exchange energy are not so different, in
spite of the different analytical forms (c.f. Eqs. (I) and (2)). Due to this fact, the choice of
942 JOURNAL DE PHYSIQUE II N°4
Elcm-11
8 9 lo 1i 12 rli)
Fig.2. Contribution of Cm van der Waals coefficients and exchange energy calculated for r > 8.8
A. I&x
= De (A'r~exp [(ao/r) 7r]). For the dashed curves, Vn = De [I e"(r nfl)]".Cn/r"
6
(n = 6, 8, 10 and 12) and I&x
= De (Ae~~~). VT represents De (~jf2n.C2n/r~"+Vex). The circles
n=3
in the figure represent the experimental points. Vn and VT have the same significance as in figure 1.
Table II.- Dissociation energy value De, van der Waals coefficients Cn, exchange energy parameters and damping function constants calculated as explained in the text (~).
a) b) c) d) e)
~ll 29.361x 10~ 29.345 x 10~ 29.361x 10~ 28.627 x 10~
~l 11.431x 108 11.526 x 108 11.430 x 108 11.402 x108
~l 4.380 x 10" 4.300 x10~° 4.380 x 10~° 4A31x 101°
~l 1.653 x 10~~ 1.680 x 10~~
x 2.008 x
1.611 1-fill
cm 4.591 x x
0.651 0.635 x
3.640 x 3.508 x
1.287 0.561
o.689 0.234
(~) The C12 value in column e is calculated using theoretical values of C6, C8 and Cio from reference [9] and the recurrence relation 6
[he form of the equation (I) in order to represent the exchange energy in the fitting, has been done simply by the lower number of adjustable parameters. Other facts should be mentioned.
As it has been observed by Weickenmeier et al. [7], the inclusion of higher second order terms
Cn/r" up to at least n
= 12 in equation (3) is necessary. For this reason the C12 term has
been calculated using the recurrence relation proposed by Tang et al. [9]:
c~~n+~~ =
~~(~+~~
)3c~~n-~~ j6)
using the values of C6, C8 and Cio listed in table IIa. The value of C~~
m 1.653 x 10~~ cm~l
l~~, calculated by equation j6), has been held fixed in the calculations.
So, in this final part of the calculations the truncated form has been used:
The results are shown in Table IId. The contribution of each term to the potential energy
curve is shown in figure 2 (dashed curves). In table lie the theoretical values of Tang et al. [9]
for the second order van der Waals coefficients are shown in order to be compared with ours.
3. Discussion.
At first let us consider the results in table IIa. It can be observed that the C6, 8, lo values are
in good agreement with those calculated by Tang et al. [9]. On the other hand the values of A and a, can be compared with those obtained by Weickenmeier et al. [7]. We can observe
that the values of A and a obtained here for the Rbcs electronic ground state are of the same order of magnitude of the A and a values obtained for the Cs2 electronic ground state (forCs2>
A = 1.955x10~ cm~~ and
a = 1.523 l~~).
Secondly, we can analyse the values reported in table IIb. The C6, 8, lo values reported in this column, in spite of being slightly different from the values listed in table IIa, can still be
compared with those reported by Tang et al. [9] (Tab. lie) showing a good agreement too. By
the way, some differences can be observed when the A', b, ao and 7 values are compared with those calculated by Knox and Rudge [6]. These differences can be explained by considering that, in their theoretical article, Knox and Rudge [6] have used this analytical form to fit the
exchange energy (or the energy gap between the ~L and the ~Z electronic states), for r values greater than 12.756 1. This fact explains why the A' and b are so different. On the other hand,
the values of ao and 7 are in same order of magnitude.
Finally the obtained values listed in table IId, corresponding to the damping function, can
be discussed. Taking as example the calculations performed in Cs2[7], we can observe that the values of a and p are quite the same (for Cs2 a
= 0.654 l~~ and p
= 0.2771). Quanti-
tative results of the fit using equation ii) are compiled in table III which shows the damped
and undamped contributions of the different En terms (En = fn(r) x Cn/r" for the damped
function and En = Cn/r~ for the undamped one), the exchange energy Eex = Ae~~~, and the dissociation energies De as calculated from equation ii) (damped function) and equation is) (undamped function).
Considering the De values listed in the last row of table III, for the damped and undamped functions, we can estimate the dissociation energy value for the l~L+ electronic ground state of the Rbcs molecule;
De = 3838.5 ~ 0.5 cm~~
944 JOURNAL DE PHYSIQUE II N°4
Table III. Vibrational quantum numbers u", damped (and undamped) contributions ofEn = C,i /r" f,i(r) (and E~ = Cn/r~ ) terms, exchange terms Eex = Ae~~~, and resulting dissociation
energies De.
U" E6 E8 Eio §12 Eex De
108 58.284 28.010 13.019 5.803 12.540
109 54.198 25.457 11.580 5.061 10.469
ill 46.261 20.666 8.964 3.751 7.021
l12 42.451 18.451 7.797 3.184 5.631
l16 28.504 10.899 4.070 1.479 1.951 3838.5
l17 25.384 9.348 3.366 1.181 1.416
l19 19.598 6.634 2.199 0.713 0.678 3838.2
This value is in a good agreement with that predicted by lent [10], using the RPC method,
where the estimated upper and lower limits for the dissociation energy were 3840 cm~~ and 3835 cm~~, respectively.
With the obtained dissociation energy value, the last bound vibrational quantum level of the l~L+ ground state can be estimated as follows. Considering Leroy's theory[11], we can
obtain a relation between the dissociation energy De, the vibrational quantum number u and the corresponding energy value Gin);
Gin) = De 11»D ») Hnl~"/~"~~~, Hn
= ~i/~
)
ji/~ 18)
where
n = 6, ~ is the reduced mass of the molecule, uD is a noninteger constant, and f is
a tabulated constant [3] ($
" 19.943 36 assuming units of energy, length and mass to be
cm~l, I, and amu, respectively). For n > 2, uD takes the physical significance of the effective vibrational quantum number at the dissociation limit. Using the value De = 3838.5 cm~~,
and plotting [De G(u)]1/~ = H6(uD u) as a function of u (Fig. 3), the value uD * 138.3 is derived. From the same figure an estimated value for C6 equal to 24.121x10~ cm~~ l~
can
[o~ GiiYi]'~~
6
<
2
0
106 116 126 136 ~
Fig.3. LeRoy-Bemstein [De G(v)]~/~ plot as a function of v for the last observed quantum numbers of the electronic ground state of the Rbcs molecule.
be obtained, being in reasonable agreement with the C6 values listed in table II. With the uD value obtained, being generally a noninteger one, the value of the last bound vibrational
quantum number for the l~Z+ electronic ground state can be predicted to be equal to 138.
4. Conclusion.
The direct fitting of the outer turning points of the l~ L+ potential energy curve in the range of 8.868 to 10.679 1, yielded experimental values of the Cn coefficients for the electronic ground
state of the Rbcs molecule.
The obtained coefficients show a good agreement with the theoretical values up to n = 12. The two analytical forms used here to describe the exchange energy are equivalent in the calculated range. For this reason we have used, in the final calculations with the damping function, the less complex one, represented by equation (I).
Using a damping function, to correct the divergence in the multipolar expansion, we have obtained a more precise value for the dissociation energy for the ground electronic state, cor-
recting the previous published one.
In spite of being a weak m~thod to determine long range parameters in potential energy
curve [12, 13], the method of direct fitting of the turning points is still the only way to obtain
interaction parameters in internuclear distance range where the exchange energy cannot be
neglected.
JOIJRNAL DE PHYSIQUE11 T 2, N' 4, APRIL 1W2 35
946 JOURNAL DE PHYSIQUE II N°4
References
ill Kat6 H. and Kobayashi H., J. Chem. Phys. 79 (1983) 123.
[2] Gustavsson T., Amiot C. and Verg+s J., Mol. Phys. 64 (1988) 279.
[3] Leroy R. J., Molecular Spectroscopy, Specialist Periodical Reports, I (Chemical Society Burling-
ton House, London, 1973) p. l13.
[4] Froese-Fischer Ch., At. Data 4 (1972) 301.
[5] Mason E. A. and Monchick L., Advances in Chemical Physics, J-O- Hirschfelder Ed., XII (Wiley,
New York, 1967) pp. 329-87.
[6] Knox H. O. and Rudge M. R., Mol. Phys17 (1969) 377.
[7] Weickenmeier H., Diemer U., Wahl M., Raab M., Demtr6der W. and Miller W., J. Chem. Phys.
82 (1985) 5354.
[8] Koide A., Meath W-J- and Allnatt A-R-, Chem. PAys 58 (1981) 105.
[9] Tang K. T., Norbeck J. M. and Certain P. R., J. Chem. Phys. 64 (1976) 3063.
[lo] Jenb F., J. Mol. Spectrosc. 143 (1990) 396.
[iii Fellows C. E., J. Chem. Phys. 94 (1991) 5855.
[12] Leroy R. J., J. Chem. Phys. 52 (1970) 2683.
[13] Davies M. R., Shelley J. C., Leroy R. J., J. Chem. Phys. 94 (1991) 3479.
