Download Intensities of analogous Rydberg series in CF3Cl, CF3Br and in

Chemical Physics Letters 404 (2005) 35–39
www.elsevier.com/locate/cplett
Intensities of analogous Rydberg series in CF3Cl, CF3Br and in
those of their isolated atoms, Cl and Br
E. Mayor, A.M. Velasco, I. Martı́n
*
Departamento de Quı́mica Fı́sica y Quı́mica Inorgánica, Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena
s.n., E-47005 Valladolid, Spain
Received 15 December 2004; in final form 10 January 2005
Abstract
In the present Letter, we have studied the Rydberg spectroscopic behaviour of the CF3Cl and CF3Br isovalent molecules.
Absorption oscillator strengths for both molecules have been calculated with the molecular-adapted quantum defect orbital
(MQDO) approach. We have sought and found important similarities between the spectral intensities of analogous transitions in
these isovalent molecules. Further similarities with the spectra of their isolated halogen atoms has not only served the purpose
of assessing the quality of our calculations, but may also offer some relevant practical use.
Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction
The halogenated methanes are important compounds
in a variety of chemical processes, ranging from stratospheric ozone depletion [1] to dry etching of silicon
wafers for semi-conductor devices [2]. In addition,
mixed halogenated methane derivatives, such as CF3Cl
and CF3Br, have been used as a CF3 radical source in
many studies. One of the interests of their study is to
monitor the effect of reducing the strength of the effective potential created by the electronegative fluorine
atoms in CF4. This is accomplished by replacing one
fluorine atom with the more electropositive chlorine or
bromine atoms. The photoabsorption processes of the
above molecules has recently attracted attention, given
their aforementioned environmental and industrial
implications. The study of the photoabsorption of fluorinated compounds is also important because their dissociation products have played a role in the
development of high power HF chemical lasers [3].
*
Corresponding author. Fax: +34 983 423013.
E-mail address: [email protected] (I. Martı́n).
0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2005.01.028
In this Letter, the calculation and analysis of the
absorption in the discrete spectral region by CF3Cl
and CF3Br, as relevant representatives of the halogenated methanes, will be presented. Resonance, as well
as Rydberg transitions will be dealt with through the
molecular-adapted quantum defect orbital (MQDO)
method, that has proven to yield correct intensities for
Rydberg transitions in a variety of molecular species
[4–10], and that has been adopted in our calculations.
The correctness of our results will be tested not only
through a comparison with the experimental data available in the literature but also by performing a comparative analysis of the expected similarities in the intensities
of analogous transitions in both molecules. In addition,
we will extend the comparative analysis to the calculated
intensities for different multiplet transitions in their
respective isolated halogen atoms, Cl and Br. The intensities for these have been obtained through the relativistic quantum defect orbital (RQDO) method [11,12].
Similarities in the intensities of analogous electronic
transitions in all the four above species might be anticipated on the following grounds. First, CF3Cl and CF3Br
are isovalent analogues, i.e., they possess the same outer
36
E. Mayor et al. / Chemical Physics Letters 404 (2005) 35–39
electron structure in the ground state and, thereby, their
Rydberg spectra may be expected to exhibit important
resemblances [13,14] (in the absence of accidental features such as the perturbation of one of more states
belonging to the same Rydberg series by a valence state
of the same symmetry). In other recent studies, such as
the one involving a group of isoelectronic Rydberg radicals [15] we have also found that similarities in the spectral properties of analogous molecular systems can be
established, which are of great usefulness, not only for
allowing the prediction of the same type of properties
in other molecular species analogous that also happen
to exhibit analogies of the same type as those of the former, but also as a good tool for assessing the reliability
of our theoretical procedure. Not irrelevant to the above
similarities are the ones we have recently reported [16]
concerning the intensities of analogous transitions in
homologous atoms, that is, atoms belonging to different
rows of the periodic table but falling in the same group,
such as Cl and Br.
2. Method of calculation
The MQDO approach, formulated to deal with
molecular Rydberg transitions has been described in detail elsewhere [4]. A brief summary of this method follows. The MQDO radial wavefunctions are the
analytical solutions of a one-electron Schrödinger equation that contains a model potential of the form
ðc da Þð2l þ c da þ 1Þ 1
V ðrÞa ¼
;
2r2
r
1
2ð na da Þ 2
;
ð1Þ
ð2Þ
where T is the ionization energy. Both T and Ea are
expressed here in Hartrees.
The absorption oscillator strength for a transition
between two bound states, a and b, may be expressed
as follows:
2
f ða ! bÞ ¼ N ðEb Ea ÞQfa ! bgjRab j2 :
3
Rab ¼ hRa ðrÞjrjRb ðrÞi:
ð3Þ
In Eq. (3), N is the number of equivalent electrons in the
molecular orbital (MO) where the transition originates,
and Q{a ! b} referred to as the angular factors, result
ð4Þ
The present bound–bound transitions have all been considered to take place through the electric dipole (E1)
mechanism. The radial transition moments (4) within
the MQDO model result in closed-form analytical
expressions, which offer, in our view, an important computational advantage. The detailed algebraic expressions
are given in Ref. [7] as generalized for bound–bound
transitions in molecules.
The values of Q{a ! b} corresponding to the symmetry group C3v, to which the Rydberg states of both
CF3Cl and CF3Br belong, are collected in Table 1. In
this and the remaining tables we are employing a notation for the molecular Rydberg states that is commonly
found in the literature. The nl symbol of the atomic orbital to which the Rydberg MO can be correlated is followed by the symbol of the irreducible representation
to which the Rydberg orbital belongs within the molecular symmetry group (in parenthesis).
3. Results and analysis
In C3v symmetry, the electron configuration of the
eight outermost valence MOs of CF3Cl and CF3Br is
[17]
2
where a represents the set of quantum numbers and
symmetry symbols that define a given molecular state.
The analytical solutions of this equation are related to
Kummer functions. In Eq. (1), c is an integer chosen
to ensure the normalization of the orbitals and their correct nodal pattern. The number of radial nodes is equal
to n l c 1. The quantum defect, da, is related to
the energy eigenvalue of the corresponding state
through the following expression:
Ea ¼ T from the integration of the angular part of the transition
integral (plus the angular part of the dipole moment
operator), of which the radial moment is Rab
4
2
4
4
2
2
4
. . . ð3a1 Þ ð2eÞ ð4a1 Þ ð3eÞ ð4eÞ ð1a2 Þ ð5a1 Þ ð5eÞ ðX 1 A1 Þ;
where the numbering scheme is restricted to those MOs
formed from the valence atomic orbitals of C (2s,2p), F
(2s,2p), Cl (3s,3p) and Br (4s,4p), respectively. The ionization energies (IPÕs) for the 5e valence orbitals that
have been adopted in the present calculations, as determined by high resolution HeI and HeII photoelectron
spectroscopy (PES) [18,19], are equal to 13.08 and
12.08 eV, for CF3Cl and CF3Br, respectively. The 5e
orbital in these two molecules is, essentially, a lone pair
orbital centred Cl or Br [17]. The energy data chosen for
the Rydberg states have been the values measured by
Au et al. [20] for CF3Cl, and those reported by Suto
and Lee [21] for CF3Br. The energies of higher Rydberg
states than those available [20,21] have been extrapolated through the Rydberg formula, Eq. (2), on the
grounds that the quantum defect along an unperturbed
Table 1
Values of non-zero angular factors Q{a ! b} for C3v symmetry and
‘ = 0, 1
Q{npe(X1A1) ! nsa1(1E1 )} = Q{npe(1A1, 1E1) ! nsa1(1E1)}
= Q{nsa1(1E1) ! npa1(1E1)} = 1/3
Q{nsa1(1E1) ! npe (1A1, 1E1)} = 2/3
E. Mayor et al. / Chemical Physics Letters 404 (2005) 35–39
37
Table 2
Quantum defects corresponding to the different Rydberg series studied
in the present work for CF3Cl, CF3Br, and their isolated halogen
atoms (Cl and Br)
Table 4
MQDO absorption oscillator strengths for npe(1A1, E)–n 0 sa1(1E)
transitions for CF3Cl (n = 4) and CF3Br (n = 5)
Transition
CF3Cl
Transition
CF3Br
Rydberg state
4pe(1A1, E)–5sa1(1E)
4pe(1A1, E)–6sa1(1E)
4pe(1A1, E)–7sa1(1E)
4pe(1A1, E)–8sa1(1E)
4pe(1A1, E)–9sa1(1E)
4pe(1A1, E)–10sa1(1E)
4pe(1A1, E)–11sa1(1E)
4pe(1A1, E)–12sa1(1E)
0.2415
0.0290
0.0100
0.0047
0.0027
0.0016
0.0011
0.0008
5pe(1A1, E)–6sa1(1E)
5pe(1A1, E)–7sa1(1E)
5pe(1A1, E)–8sa1(1E)
5pe(1A1, E)–9sa1(1E)
5pe(1A1, E)–10sa1(1E)
5pe(1A1, E)–11sa1(1E)
5pe(1A1, E)–12sa1(1E)
5pe(1A1, E)–13sa1(1E)
0.2177
0.0295
0.0103
0.0050
0.0028
0.0017
0.0012
0.0008
ns(1E)
(n + 1)s(1E)
npe(1A1, 1E1)
(n + 1)pe(1A1, 1E1)
a
b
c
d
n=4
n=5
CF3Cla
Clb
CF3Brc
Brd
1.99
1.96
1.63
1.63
2.09
2.08
1.68
1.68
2.98
2.98
2.69
2.69
3.11
3.08
2.67
2.63
Au et al. [20].
Bashkin et al. [22].
Suto and Lee [21].
Moore [23].
Rydberg series is fairly constant. In this form, we have
been able to predict intensities for transitions which
have not been observed experimentally.
Table 2 displays the quantum defects for the different
Rydberg series of the four studied species. Those corresponding to the molecules were derived from experimental energy data [20,21]. For the atoms, the quantum
defects have been extracted from their tabulated energy
levels and ionization energies [22,23]. We may regard
Table 2 as composed of two groups of columns, one
comprising the quantum defect values of CF3Cl and
Cl and the other collecting those of CF3Br and Br.
Inspection of the two groups of columns reveals that
in each Rydberg series (characterized by the value of
the well defined orbital quantum number ‘ and by the
irreducible representation Ci to which the states belong),
important similarities in the magnitude of the quantum
defects of the two species (isolated halogen atom and
molecule) occur because, as mentioned above, the Rydberg orbitals of the two molecular species are essentially
atomic orbitals centered on Cl and Br.
In Tables 3 and 4, the MQDO absorption oscillator
strengths for the electronic transitions npe(X1A1) !
n 0 sa1(1E) and npe(1A1,,E) ! n 0 sa1(1E), with n = 3–4
and n = 4–5, all originating in the ground states of
CF3Cl and CF3Br, respectively, are collected. In addition we have calculated some transitions in which the
initial state is a nsa1 Rydberg orbital, i.e., the
nsa1(1E) ! npa1(1E) and nsa1(1E) ! npe(1A1,,E) transi-
Table 3
MQDO absorption oscillator strengths for npe(X1A1)–n 0 sa1(1E) transitions for CF3Cl (n = 3) and CF3Br (n = 4)
Transition
CF3Cl
Transition
CF3Br
3pe(X1A1)–4sa1(1E)
3pe(X1A1)–5sa1(1E)
3pe(X1A1)–6sa1(1E)
3pe(X1A1)–7sa1(1E)
3pe(X1A1)–8sa1(1E)
3pe(X1A1)–9sa1(1E)
3pe(X1A1)–10sa1(1E)
0.1191
0.0224
0.0087
0.0043
0.0024
0.0015
0.0010
4pe(X1A1)–5sa1(1E)
4pe(X1A1)–6sa1(1E)
4pe(X1A1)–7sa1(1E)
4pe(X1A1)–8sa1(1E)
4pe(X1A1)–9sa1(1E)
4pe(X1A1)–10sa1(1E)
4pe(X1A1)–11sa1(1E)
0.1450
0.0307
0.0110
0.0057
0.0032
0.0019
0.0013
tions for CF3Cl (n = 4, 5) and CF3Br (n = 5, 6). These
are collected in Table 5.
The only comparative oscillator strengths that we
have found in the literature correspond to the 3pe
(X1A1) ! 4sa1(1E) transition in CF3Cl. These are the
measurements, performed with a high-resolution dipole
(e,e) technique by Au et al. [20], equal to 0.1625 ± 0.032
and the value of 0.1516 ± 0.015, reported by Suto and
Lee [21]. A fairly good agreement of the presently calculated f-value, displayed in Table 3, with the comparative
data is apparent, and more so if we take into account the
estimated uncertainties in the experimental data. It
should be noted that the higher magnitude of the oscillator strength for the same transition reported by King
and McConkey (0.220 ± 0.088) [24] may in part be
attributed to the fact that their measurements, with a
zero angle electron energy-loss technique, were normalised to those of Gilbert et al. [25] and Jochims et al.
[26]. The accuracy of the earlier f-value equal to
0.1503, reported by Doucet et al. [27], is difficult to
establish, since both the extinction coefficient and the
bandwidth were determined graphically [27].
The clear similarities in the oscillator strengths corresponding to analogous transitions in the two molecules
under study allows us to establish the, at least qualitative, correctness of the present MQDO f-values, for
which no comparative data are available.
The aforementioned similarities could, a priori, have
been expected, on the grounds of the isovalent valence
shells of CF3Cl and CF3Br and the subsequent analogies
in the Rydberg states formed by excitations in the two
molecules. This is also in accord with the earlier predictions by Johnson and Hudgens [13,14], regarding the
Table 5
MQDO absorption oscillator strengths for nsa1(1E)–npa1(1E),
nsa1(1E)–npe(1A1, E) transitions for CF3Cl (n = 4, 5) and CF3Br
(n = 5, 6)
Transition
1
CF3Cl
1
4sa1( E)–4pa1( E)
4sa1(1E)–4pe(1A1, E)
5sa1(1E)–5pa1(1E)
5sa1(1E)–5pe(1A1, E)
0.29418
0.58837
0.41517
0.83033
Transition
1
CF3Br
1
5sa1( E)–5pa1( E)
5sa1(1E)–5pe(1A1, E)
6a1(1E)–6pa1(1E)
6sa1(1E)–6pe(1A1, E)
0.2587
0.51745
0.38095
0.76189
38
E. Mayor et al. / Chemical Physics Letters 404 (2005) 35–39
Table 6
Absorption oscillator strengths for analogous transitions in CF3Cl, CF3Br, and their isolated halogen atoms (Cl and Br)
2
2
3p P–4s P
4p 4P–5s 4P
4p 4D–5s 4P
4p 2D–5s 4P
4p 2S–5s 2P
4p 4S–5s 4P
4s 4P–4p 4(S,P,D)
4s 2P–4p 2(S,P,D)
5s 2P–5p 2(S,P,D)
5s 4P–5p 4(S,P,D)
Cl
0.1602
0.2486
0.2771
0.2862
0.3096
0.3085
0.9020
0.9370
1.2626
1.3514
CF3Cl
Transition
3p(X A1)–4s( E)
4p(1A1, E)–5s(1E)
0. 1191
0.2415
1
4p(X A1)–5s( E)
5p(1A1, E)–6s(1E)
0.1450
0.2177
4p P–5s P
5p 4P–6s 4P
0.1522
0.2753
4s(1E)–4p(1A1, E, 1A1, E)
0.8971
5s(1E)–5p(1A1, E, 1A1, E)
0.7761
5s 4P–5p 4(S,P,D)
0.9208
5s(1E)–5p(1A1, E, 1A1, E)
1.2455
6s(1E)–6p(1A1, E, 1A1, E)
1.1428
6s 4P–6p 4(S,P,D)
1.2265
Transition
1
1
analogies in the valence structure and expected electronic behaviour of the presently studied molecules.
Finally, we have compared the oscillator strengths for
analogous transitions in the two molecular species and
their isolated halogen atoms, given their analogies in
the electronic structure of the ground states. The results
are collected in Table 6. Because the atoms possess an
infinite, higher, symmetry than the molecules, the state
described by a given nl notation in an atom is symmetry-split in the molecule. On the other hand, there may
be several different terms and multiplets arising from a
given outer nl atomic electron configuration. In such
cases, the comparison should be made between the
resulting f-values of an nl–n 0 l ÔsupermultipletÕ in the
atom and the sum of the intensities of all nl–n 0 l transitions that comprise all the different irreducible representations involved in the molecule. A ÔsupermultipletÕ is
[28] the group of multiplets with different L-value but
the same spin multiplicity that arise from a given electron configuration in an atom.
A very similar spectroscopic behaviour of the halogen
atoms when they are free and when bound within the
molecules is clearly noticed (see Table 6). This behaviour
is probably due, at least partly, to the feature anticipated
by Gilbert et al. [25] and confirmed by the present calculation, concerning the large size of the atomic orbitals
entering the molecular Rydberg orbital. The former
are orbitals that belong to the atom around which the
electron that departs in the related photoexcitation process is mainly concentrated. In the present work we take
the similarities in the magnitude of the oscillator
strengths for equivalent transitions in the isolated halogen atom and its parent molecule, as a proof of the correctness of our MQDO calculations. In addition, a
rather good accord between the oscillator strengths for
the analogous multiplet transitions of the homologous
atoms Cl and Br, themselves, is observed (see Table 6).
This feature also helps to establish the reliability of these
data, on physical grounds [28,29].
To further analyze the correctness of our results, a final test on the present MQDO oscillator strengths has
been applied, based on the regularities complied with
by the intensities of the transitions along a non-per-
CF3Br
1
Transition
2
Br
2
turbed Rydberg series. The transitions originating from
a given state should display a systematic trend along the
different Rydberg series as the final state increases in energy. It has long been established [17] for all spectral series belonging to the hydrogen atom, as well as for others
that present a ÔhydrogenicÕ behaviour, that the square of
the radial transition integral (to which the oscillator
strength is proportional) should display the same trend
as the inverse of the third power of the quantum number
(n)3 of the upper state, as n increases with excitation
along the spectral series. For a non-hydrogenic system
(even though molecular Rydberg states exhibit a nearhydrogenic character, in particular in a high degree of
excitation), n should be replaced by its corresponding
Ôeffective quantum numberÕ n* = n d. Complying with
this behaviour at suffiently high values of n* is a qualitative proof of the correctness of the oscillator strengths.
This is more easily observed if the product f(n*)3 is plotted against the n* value for the upper state along a Rydberg series. A decreasing trend is expected for the first
few values of n*, after which the slope becomes practically unchanged with increasing n* [17]. This behaviour
is observed in the different Rydberg series presently
studied for CF3Cl and CF3Br, of each an example is
shown in Figs. 1 and 2, respectively.
1.5
1.3
fn* 3
Transition
1.1
0.9
0.7
0.5
0
1
2
3
4
5
6
7
8
9
n*
Fig. 1. Systematic trends of the npe(X1A1)d-(n + 1 n + 7)sa1(1E)
oscillator strengths along the Rydberg series in CF3Cl (n = 3) and
CF3Br (n = 4).
E. Mayor et al. / Chemical Physics Letters 404 (2005) 35–39
7
6
fn* 3
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11
n*
Fig. 2. Systematic trends of the npe(1A1, E)–(n + 1 n + 8)sa1(1E),
oscillator strengths along the Rydberg series in CF3Cl (n = 4) and
CF3Br (n = 5).
Additional intensities for a number of previously
unstudied Rydberg transitions in CF3Cl and CF3Br,
all of them having an excited state as lower state, comply
with the expected behaviour along a spectral series, as
we show in graph form (Figs. 1 and 2). We take this feature as a proof of the, at least, qualitative correctness of
our results.
We have sought and found important analogies between the spectral intensities of analogous transitions
in these isovalent molecules. Further similarities with
the spectra of their isolated halogen atoms has not only
served the purpose of confirming previously predicted
ground-state electronic structure of the two molecules,
but also assessing the quality of our calculations. These
may also offer some relevant practical use.
Acknowledgements
This work has been supported by the D.G.I. of the
Spanish Ministry for Science and Technology within
Project No. BQU2001-2935-C02, and by European
FEDER funds. E.M. also wishes to acknowledge his research grant awarded by the Spanish Ministry for
Education.
References
[1] F.S. Rowland, M.J. Molina, Rev. Geophys. Space Phys. 13
(1975) 1.
39
[2] J.W. Coburn, Plasma Etching and Reactive Etching, American
Institute of Physics, New York, 1982.
[3] S.J. Arnold, H. Rojeska, Appl. Opt. 12 (1973) 169.
[4] I. Martı́n, C. Lavı́n, A.M. Velasco, M.O. Martı́n, J. Karwowski,
G.H.F. Diercksen, Chem. Phys. Lett. 202 (1996) 307.
[5] A.M. Velasco, E. Mayor, I. Martı́n, Chem. Phys. Lett. 376 (2003)
167.
[6] A.M. Velasco, E. Mayor, I. Martı́n, Chem. Phys. Lett. 377 (2003)
189.
[7] I. Martı́n, C. Lavı́n, J. Karwowski, Chem. Phys. Lett. 255 (1996)
89.
[8] A.M. Velasco, I. Martı́n, C. Lavı́n, Chem. Phys. Lett. 264 (1997)
579.
[9] I. Martı́n, A.M. Velasco, C. Lavı́n, Int. J. Quantum Chem. 86
(2002) 59.
[10] E. Bustos, A.M. Velasco, I. Martı́n, C. Lavı́n, J. Phys. Chem. A
108 (2004) 1293.
[11] I. Martı́n, J. Karwowski, J. Phys. B: Atom. Mol. Opt. Phys. 24
(1991) 1539.
[12] J. Karwowski, I. Martı́n, Phys. Rev. A 43 (1991) 4832.
[13] R.D. Johnson III, J.W. Hudgens, Chem. Phys. Lett. 141 (1987)
163.
[14] R.D. Johnson III, J.W. Hudgens, J. Chem. Phys. 94 (1991) 5331.
[15] I. Martı́n, Y. Pérez-Delgado, C. Lavı́n, Chem. Phys. Lett. 305
(1999) 178.
[16] C. Lavı́n, A.M. Velasco, I. Martı́n, Quantum Systems in Chemistry and PhysicsBasic Problems and Model Systems, vol. 1,
Kluwer Academic Publishers, Dordrecht, 2000.
[17] J.C. Creasey, D.M. Smith, R.P. Tuckett, K.R. Yoxall, K. Codling,
P.A. Hatherly, J. Phys. Chem. 100 (1996) 4350.
[18] T. Cvitas, H. Gusten, L. Klasinc, J. Chem. Phys. 67 (1977) 2687.
[19] T. Cvitas, H. Güsten, L. Klasinc, I. Noradj, H. Vancik, Z.
Naturforsch. A 33 (1978) 1528.
[20] J.W. Au, G.R. Burton, C.E. Brion, Chem. Phys. 221 (1997) 151.
[21] M. Suto, L.C. Lee, J. Chem. Phys. 79 (1983) 1127.
[22] S. Bashkin, J.O. StonerAtomic Energy Levels and Gotrian
Diagrams, vol. 1, North Holland-American Elsevier, New York,
1975;
S. Bashkin, J.O. StonerAtomic Energy Levels and Gotrian
Diagrams, vol. 2, North Holland-American Elsevier, New York,
1978.
[23] C.E. Moore, Atomic Energy Levels; Nat. Bur. Stand. Ref. Ser. 35;
Nat. Bur. Stand., Washington, DC, 1971.
[24] G.C. King, J.W. McConkey, J. Phys. B: Atom. Molec. Phys. 11
(1978) 1861.
[25] R. Gilbert, P. Sauvageau, C. Sandorfy, J. Chem. Phys. 60 (1974)
4820.
[26] H.W. Jochims, W. Lohr, H. Baumgärtel, Ber. Bunsenges. Phys.
Chem. 80 (1976) 130.
[27] J. Doucet, P. Sauvageau, C. Sandorfy, J. Chem. Phys. 58 (1973)
3708.
[28] W.C. Martin, W.L. Wiese, Atomic, Molecular and Optical
Physics Handbook, American Institute of Physics, Woodbury,
NY, 1996.
[29] I. Martı́n, E. Charro, C. Lavı́n, Progress in Theoretical Chemistry
and Physics, Kluwer Academic Publishers, Dordrectch, 2000.