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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. 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