Franck–Condon analysis of the SF6- electron photodetachment spectrum

Chemical Physics Letters 445 (2007) 84–88 www.elsevier.com/locate/cplett

Franck–Condon analysis of the SF 6 electron photodetachment spectrum Raffaele Borrelli Dipartimento di Chimica, Universita` di Salerno, I-84084 Fisciano, Salerno, Italy Received 14 May 2007; in final form 27 July 2007 Available online 6 August 2007

Abstract The electron photodetachment spectrum of the SF 6 anion is analyzed in terms of the Franck–Condon factors between the anion and the neutral species. It is shown that the most intense progression can be easily assigned to the excitation of the m1 symmetric stretching mode, belonging to the a1g representation, while the second one, which appears shifted by 440 cm1 from the first, is caused by a simultaneous excitation of the m1 and m4 (t1u) modes. The results are consistent with an Oh symmetry of the SF 6 anion. Ó 2007 Elsevier B.V. All rights reserved.

1. Introduction It is well known that neutral SF6 can easily give an electron attachment reaction producing the SF 6 molecular anion [1]. The latter is a long lived species, with a lifetime of ca. 10 ls [2,3], which is involved in many technological applications, mainly in etching plasma or as a gaseous dielectric [4,5]. The ease of formation of the SF 6 anion and its stability has attracted considerable interest on the chemistry and physics of this species [6–8]. Particular attention has been paid to the determination of the adiabatic electron affinity of the neutral molecule, and to the study of the changes in molecular structure and vibrational frequencies which follow electron attachment [1,9–14]. Both these aspects are of fundamental importance to understand the origin of the stability of the anion. While the bare molecule has been thoroughly characterized by a number of spectroscopic [15–17] and diffraction [18,19] techniques very little is known about its anion. From the analysis of the photoelectron spectrum of the ˚ has been SF anion a S–F bond length of 1.717 A obtained, however no direct experimental information is currently available on the structure of SF 6 [20]. Ionization

E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.07.086

techniques combined with infrared spectroscopy have been used to characterize several anions and cations obtained from SF6, allowing for an identification of some of the vibrational frequencies of these species [13]. Nevertheless, the spectra are rather congested and only one frequency have been tentatively assigned to SF 6. Theoretical analyses provide a significant insight to the characterization of this anion, although not a unique answer. The adiabatic electron affinity (AEA) of SF6 can be almost exactly reproduced by highly correlated methods (i.e. MP2, CCSD) employing large basis sets [12]. These methods predict a stable octahedral (Oh) equilibrium geometry [21,12]. On the other hand DFT based methods give quite unsatisfactory results for the AEA, and, furthermore, some functionals predict a C4v symmetry [11]. Recently a vibrationally resolved photoelectronic spectrum of SF 6 has been reported [8]. This spectrum shows a very long progression with a fairly constant spacing of about 750 cm1, which has been assigned to the excitation of the sole a1g mode of the neutral SF6, which is simply given by the symmetric combination of the six S–F stretches. Alongside this progression a second one is found whose peaks are shifted by about 440 cm1 from the first. This vibrational progression has not yet found a clear explanation, since it does not belong to any of the fundamental frequencies of SF6. Several mechanisms have been

R. Borrelli / Chemical Physics Letters 445 (2007) 84–88

proposed to account for its observation, including the possibility that the anion is actually formed with a significant population in a low lying electronic state with nearly the same geometry of the ground state. However, no evidence of such a low lying electronic state is given by any electronic structure calculation method. The assignment of this band is further complicated by the possibility of a lower, C4v, symmetry of the anion with respect to the neutral SF6 [11]. In this Letter, we will use recent methodologies for the calculation of Franck–Condon (FC) integrals [22,23], combined with high level ab initio calculation of equilibrium geometries and vibrational normal modes, to assign the vibronic pattern of the photoelectronic spectrum of SF 6. That allows a reliable assessment of both the experimentally observed vibronic pattern [8], and of the molecular geometry of SF 6. 2. Computational details Theoretical calculation of SF 6 have shown that MP2, CCSD(T), and some DFT based methods give very similar equilibrium geometries, and predict a stable minimum with an octahedral symmetry [10,12]. In particular, it is found that upon electron attachment the molecule undergoes an ˚. increase of the S–F distance of approximately 0.2 A In the present work, equilibrium geometries of both neutral and anion species of SF6 have been calculated at MP2 level using a 6-311G(2df) basis set. Both the anion SF 6 and the neutral SF6 are considered to belong to the Oh point group symmetry, hence the molecular geometry is completely determined by the S–F bond length. Vibrational normal modes were obtained at the same level of theory but using a 6-311G(2d) basis set. As can be seen from Table 1 our results are in good agreement with the available experimental data, and with previous high level theoretical calculation [12]. However, it must be noticed that the AEA calculated at the MP2/6-311G(2df) level is only 0.08 eV, which is very far from the experimental value of 1.0 eV.

Table 1 Equilibrium geometry and normal mode frequencies of SF6 and SF 6 calculated at MP2/6-311G(2df) and MP2/6-311G(2d) level, respectively Symmetry

m1 m2 m3 m4 m5 m6 ˚) r(S–F) (A

a1g eg t1u t1u t2g t2u

Neutral

Anion a

Theory

Exp.

Theory

781 660 980 614 529 346 1.564

774 643 948 615 524 348 1.565

634 461 739 332 280 224 1.711

Exp.

620b 683c

Oh symmetry point group have been imposed in the optimization of both species. a Ref. [32]. b Ref. [13]. c Ref. [8].

85

This is very likely due to the lack of diffuse functions in the basis set which are important to obtain an accurate energy of the anion. The GAUSSIAN 94 package [24] has been used to perform all the calculations. The MOLFC package program [25] has been used for the computation of the FC pattern in the photoelectronic spectrum. 3. Results As well known [26,23] the starting point for the analysis of a vibronic spectrum is the determination of the Duschinsky normal modes transformation QA ¼ JQN þ K

ð1Þ

where, in the case we are dealing with, QA and QN are the normal modes of vibration of the anionic and neutral species respectively. K is the vector of normal mode displacement, and J is the Duschinsky transformation matrix, defined as K ¼ TyA ðnN  nA Þ;

J ¼ TyA TN

ð2Þ

where TA, TN are the mass-weighted normal coordinates in the Cartesian representation of the anionic and neutral species and nA ; nN their equilibrium geometries, respectively. The axis switching matrix is a unit matrix for a system with Oh symmetry [27]. Due to its high symmetry SF6 has doubly and triply degenerate vibrations (see Table 1) which demands a certain attention in the application of Eq. (2), and in the interpretation of the final results. Indeed, it is well known that normal modes of vibrations of a degenerate representation are defined up to a unitary transformation matrix, that is, any linear combination of degenerate normal vibrations is a normal mode of the system itself. Hence, it is always possible to choose the normal modes of a degenerate representation of two electronic states in such a way that they transform exactly into each other, i.e. in such a way that the relative block of the Duschinsky matrix is a unit matrix. Doktorov et al. have clearly discussed this point in the derivation of analytical formulas of the FC integrals for degenerate vibrations in terms of Wigner’s D functions [28]. SF6 and SF 6 have 15 normal vibrations, nine of which belong to t2u, t2g, e2g and a1g symmetry representations, and six to the t1u (see Table 1). The resulting transformation matrix is block factorized since only vibrations of the same symmetry can mix each other. Any mixing effect between degenerate vibrations of the same representation, which could result by the immediate application of Eq. (2) would be only apparent. Indeed, as already pointed out, the corresponding block of the Duschinsky matrix can always be transformed into an identity matrix by a proper rotation of the normal modes. In this sense it is always possible to eliminate the Duschinsky effects between degenerate vibrations. Of course the same holds for the triply degenerate vibrations of t2u

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Table 2 Duschinsky matrix J and adimensional normal mode displacements K in the Cartesian coordinate representation relative to the t1u modes and the a1g mode

m3

J m3

K m4

0.978

0.167 0.123 0.123 0.167

0.783 0.587 0.587 0.783 m4

0.207

0.978 0.207

0.978 0.207

m1

m1

0.207

0.978 1.000 8.327

and t2g symmetry. On the other hand the six t1u modes can show a true Duschinsky effect. Indeed, they form two sets with frequencies 614 and 980 cm1 in SF6, and 332 and 739 cm1 in SF 6 , respectively and any mixing between vibrations with different frequencies cannot be removed by simply rotating the normal modes. In Table 2 the calculated Duschinsky matrix, and the displacement of the equilibrium positions relative to the modes belonging to the a1g (m1) and t1u (m3, m4) representation are reported. The difference of the S–F bond length between the neutral and the anionic species reflects into a very large displacement of the m1 vibration, i.e. of the symmetric combination of the S–F stretchings, ca. 8.3 in adimensional coordinates. Because of this large displacement a very long progression of the totally symmetric mode is expected to appear in the calculated spectrum. Of course only the totally symmetric m1 vibration shows a shift in the equilibrium position. A mixing of the m3 and m4 modes can also be observed in the Duschinsky matrix. As already pointed out this is a true mixing effect. A fundamental parameter for a correct description of the photoelectronic spectrum of SF 6 is the adiabatic electron affinity of SF6, in that it provides the position of the origin of the spectrum, i.e. of the 0–0 transition. In this work we will use the widely accepted experimental value of 1.0 eV [14]. The calculated electron photodetachment spectrum is shown in Fig. 1. The spectrum has been obtained by letting the SF 6 be in its ground vibrational state and allowing for excitations of all the 15 normal modes of the neutral species. About 109 FC integrals have been calculated and convoluted to obtained the final spectrum (see caption of Fig. 1 for details). The 0–0 transition is not visible and, as explained above, is located at 1.0 eV. The overall bandwidth is about 2 eV, with an onset located at about 1 eV from the origin. The same spectrum but on a narrower spectral region, between 2 and 3.4 eV, is reported in the inset of Fig. 1. Two distinct and sharp vibrational progressions can be clearly identified in the spectrum. The most intense progression is characterized by a 781 cm1 spacing between peaks, and is generated by the excitation of the a1g mode. The maximum peak of this progression falls at

Intensity (arbitrary units)

Mode

—4

4.5

x10

—4

4

4

3.5

3

3

2

2.5

1

2 1.5 1

x10

18

10

11842 0

0 2

2.5

0

3

0—0 (AEA)

0.5 0 0

1

2 3 Energy (eV)

4

5

Fig. 1. Electron photodetachment spectrum of SF obtained with 6 equilibrium geometries and vibrational frequencies calculated at MP2 level. The spectrum (solid line) is obtained by convoluting the Franck– Condon lines with a Lorentzian function of 70 cm1 width. The inset shows an enlargement of the figure in the range 2.0–3.4 eV.

29 934 cm1 (3.71 eV), which is quite close to the experimentally measured value of 3.46 eV, corresponding to the maximum of the electron photodetachment cross section [29]. The peak is associated to the 0 ! 28 transition, which is very close to the 0 ! 27 assignment obtained from a fit of the experimental data [8]. Alongside this progression a less intense one whose peaks appear shifted at higher energies by about 440 cm1 with respect to the former can be observed. This second progression is generated by the simultaneous excitation of the m1 mode, and of the triply degenerate m4 mode. Our analysis indicates that the peaks of this progression can be labeled as 1n0 420 (Here we adopt the standard notation labeling a transition as X ji , where X refers to the normal mode numbering, see Table 1, and i and j to the initial and final quantum numbers, respectively.). Each peak involves a two quanta transition of the m4 mode, as this is a non-totally symmetric vibration, and can exhibit only two quanta excitations, of the type 0 ! 2, 0 ! 4 etc. From our calculation 0 ! 4 and higher order transitions have too low an intensity to be observed. The maximum of this progression is located at 30 381 cm1 2 (3.76 eV), and correspond to the 128 0 40 transition. The positions of the calculated vibronic lines match almost perfectly those of the experimental spectrum [8], allowing us to assign the observed satellite progression to 1n0 420 type transitions. The position of the peaks of this progression can be misleading, and has probably been one of the causes of its troublesome assignment. If we consider a generic peak of the m1 mode progression corresponding to the transition 1x0 , where x is an integer number, the peak falling at higher energy by 440 cm1 will correspond to 2 the 1x1 0 40 transition. Indeed, its position will be 2 Æ 614  781 = 447 cm1 from the 1x0 peak. Clearly the 1x0 420

R. Borrelli / Chemical Physics Letters 445 (2007) 84–88

transition will fall at 2 Æ 614 = 1228 cm1 from the 1x0 . This situation is clearly visible in Fig. 1. This fact has probably hidden the simple nature of the progression to other authors [8]. It must be noticed that the predicted and observed intensities are only in a qualitative agreement. The experimental spectrum shows a quasi continuum with two vibrational progressions put on the top of it. This continuum is almost absent in the theoretical spectrum, where, in fact, the vibronic lines are very sharp. Furthermore, the calculated relative intensities of the two progressions are significantly different whereas those of the experimental one seem to be very similar. A main source of this disagreement could derive from the accuracy of the normal mode frequencies of the anion SF 6 obtained from MP2 calculations. Indeed, the 0 ! 2 transition of the m4 mode derives its intensity exclusively from the change in its vibrational frequency upon electron detachment from the anion. The higher is the difference the higher will be the peak intensity. The vibrational spectrum of the argon–SF 6 complex in the gas phase gives a frequency of the m3 of about 680 cm1 [8], and a very similar result has been obtained from IR measurements in liquid neon [13]. Computational analyses based on DFT methods, using a B3LYP [30,31] functional with a DZP++ basis set, predict frequencies of 112 cm1 and 660 cm1 for the m4 and m3 modes respectively [13], which are significantly lower than those predicted by MP2 calculations. This would suggest that the m4 mode, which is clearly the most important in our analysis, should have a significantly lower frequency than that calculated at MP2 level. Of course, this would have no effect on the spacing of the vibrational pattern of

1.4

x10

—4 —4

x10

Intensity (arbitrary units)

1.2

1

118 0

0.8

1

0.6

118 42 0 0

0.8 0.4

the spectrum which depends on the frequencies of the neutral SF6. Fig. 2 shows the electron photodetachment spectrum calculated by using the values 680 cm1 and 110 cm1 for frequencies of the m3 and m4 (t1u) modes of SF 6. The overall agreement between the theoretical and the experimental spectra of Ref. [8] is significantly improved. In particular, the relative intensities of the two progressions are much more similar to the observed ones, and, furthermore, a broad continuum shows up in the calculated spectrum which is indeed experimentally observed. This continuum is generated by a number of combination bands involving two and four quanta excitations of the m4 mode, and has a minor contribution from two quanta excitations of the m3 mode. 4. Conclusions The theoretical electron photodetachment spectrum of the SF 6 anion obtained in the Franck–Condon approximation is in reasonable agreement with the experimental one, and allows a quite accurate interpretation of the observed vibronic pattern. The theoretical spectrum shows essentially two progressions involving excitations of the m1 (a1g) and m4 (t1u) normal modes. In particular these progressions can be labeled as 1n0 , and 1n0 420 . The positions of the theoretical absorption lines are very close to those actually observed [8], although their intensities are only in qualitative agreement. The agreement between theoretical and experimental intensities can be improved if the calculated frequencies of the m4 and m3 modes, i.e. the two t1u symmetry modes, are lowered to about 110 and 680 cm1 respectively. Indeed, there are experimental and theoretical evidences that these modes could have a frequency lower than that obtained at the level of calculation used here. Notwithstanding the discrepancies in the relative peak intensity the almost exact match of their position strongly support our interpretation of the photoelectronic spectrum of SF 6 . Furthermore, according to our results the observed electron photodetachment spectrum is consistent with an octahedral symmetry of the anion.

0.2

0.6

Acknowledgments

0 2

0.4

2.5

3

The author thank Prof. Andrea Peluso for his helpful suggestions. The financial support of the University of Salerno is gratefully acknowledged.

0.2 0

87

0

1

2 3 Energy (eV)

4

5

Fig. 2. Electron photodetachment spectrum of SF obtained with 6 equilibrium geometries calculated at MP2 level, and modified vibrational frequencies of the t1u modes of the anion. The spectrum (solid line) is obtained by convoluting the Franck–Condon lines with a Lorentzian function of 70 cm1 width. The inset shows an enlargement of the figure in the range 2.0–3.4 eV.

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