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An (e, 2e) spectroscopic invistigation and a green's function study of the ionization of chloro- and bromo-ethylene
Volume 101, number 45
CHEMICAL
PHYSICS LETTERS
AN (e, 2e) SPECTROSCOPIC INVESTIGATION OF THE IONIZATION
-28 October
1983
AND A GREEN’S FUNCTION STUDY
OF CHLORO- AND BROMO-ETHYLENE
R. CAMBI, G. CIULLO, A. SGAMELLOTTI, Diparrimento di chimica. Uniwrsit~ di Pen@.
F. TARAIWELLI
06100 Perugin. Ifaly
and
R. FANTONI, A. GIARDINI-GUIDONI,
I.E. MCCARTHY * and V. DI MARTIN0 **
ENEA. Centro Ricerche Energin Frascati, 00044 Ftaswti. Rome. Italy Received 1 August 1983
A Green-s function study and an (e,2e) spectroscopic investigation have been carried out on the CzH-&l and CzHSBr molecules. The complete valence ionization spectra of both molecules are discussed_ An analysis of the experimental and theoretical electron momentum distributions is also presented, allowing an unambiguous assignment of the spectral features.
1. Introduction
The application of the one-electron Green’s function to photoelectron spectroscopy has been discussed in detail by Cederbaum and Domcke 1I] _The position of the poles of the Green’s function give the electronic levels of the ion, while the pole strengths provide the ratios of the cross sections for ionizations leading to the corresponding energy levels in the high-energy limit. The kinematics of dipole (e,Ze) spectroscopy is arranged to simulate the photoelectron reaction. The scattering process is characterized by large impact energies and low momentum transfer, the scattering angle being = zero. Under these conditions, the ionization involves distant collisions and the cross section can be taken as proportional to the optical oscillator strength, as in the case of photoabsorption. On the other hand, binary (e,2e) spectroscopy (used in the present work) is a large momentum-transfer coincidence technique, in which the (e,2e) cross section is a function of the momentum (q) of the ejected electron before ionization. It is now well known that binary * On leave from The Flinders University of South Australia. ** Guest. 0 009-2614/83/0000-0000/$03.00
0 1983 North-Holland
(e,2e) spectroscopy gives the low-momentum range for each ion state at an arbitrary total energy. This means that we can choose an energy where the highener=v limit is valid, i.e. where the plane-wave approximation for the continuum orbitals holds, and measure both the distribution of the ion states and their corresponding low-momentum profiles; these are usually well described by the spherically averaged square of the appropriate molecular orbital multiplied by a spectroscopic factor [2,33. The high-energy limit, where the experimentally determined structural information is independent of the total energy, is reached at least at 1000 eV. The analysis of (e,2e) experiments by the Green’s function method has been successful in several recent applications to small molecules [4-61. Here we report an investigation of the considerably larger systems C,H,Cl and C,H,Br. In section 2 we point out the direct connection of the Green’s function to the (e,2e) cross section_ In section 3 the theoretical results obtained for CZH3Cl and C2H3Br are compared with the experimental data; a complete investigation of the electronic structure of C,H,Br performed by means of binary (e,2e) spectroscopy is presented, including the momentum distribution curves, and preliminary results on C,H,Cl are also reported. 477
CHESIICAL PHYSICS LETTERS
Volume 101, number 4.5 2. Relationship of the one-particle the (e,2e) cross section
Green’s function
to
Cederbaum and Domcke [I] have considered in great detail the application of the one-particle Green‘s function to ionization processes_ The total cross section P(w) for ionization is expressed in terms of the target Hartree-Fock orbitals 4 and one-particle photoionization amplitudes:
where 4 is a continuum orbital for the ejected electron. The spectroscopic application of the binary (e,2e) reaction is optimal in non-coplanar symmetric kinematics. Here we cannot distinguish the ionized electron from the scattered one_ We label them B and A respectively. and treat B in the formalism in such a way as to bring out the analogy with photoionization. Antisymmetrication with respect to A and B is implicit when necessary. Apart from initial- and final-state degeneraties, we have for the transition probability P(w) in the (o.2e) reaction p(w) =
x
(3r)4(k,k,/ko)
cF
I(FI(~,I~I~~,I~~)I~~(~
=
(k&jItlQ/iO)’
(4)
r^ being the electron-electron I matrix. introducing the notation Z, for the appropriate sum and average over initial- and final-state degeneracies, we have = (3+4(x-
P(o)
k /srx- )
AB
0
X CT* 7 Im[GI,(m ant Bn Bt
-w.
- i$]S(w
-EA
(5) -EB)
where
Expressions (5) and (6) are identical (apart from the kinematic constant) to eqs. (Z-4) of ref. [I]. The Green’s furlction (6) is, in general, not diagonal_ It has been shown [I] that, if we assume it is ahnost diagonal, we may write the approximation P(w) = (27r)4(~,kB/XO)
-15,
-Q.
(3
where the labelling of electron energies and momenta is as in refs. [2,3] _The N-electron molecular ground state is I*$> and the observed ior]-electron B state is IF>. Instead of a dipole operator, we connect these states through an operator (X-,ITIX-,) for particular I;lomenta of electrons 0 (incident) and A. The operator T contains all the complexity necessary to describe the reaction fully. A great advantage of the (e.2e) reaction is that we can perform the whole spectroscopic application in the energy region where it is valid ro approximate IF) by the product of a plane wave and the final ion I*r-l>. In non-coplanar symmetric kinematics there is a large body of experimental information [2.3] confirming the validity of the plane-wave impulse approximation where. for a particular momentum k,, the secondquantized foml of*the (e,2e) amplitude is
(3
478
TBI
28 October 1983
X oG I-r,,l’Pt(x)s(w
-E,
- EB),
(7)
where Pt(h) is the pole strength for ionization from orbital & to the final state h. We note that (4) factorizes into the product of a half-off-shell t-matrix element and the momentum-space orbital (q I+$ for
q =k, -k,
-k,.
(8)
Performing the sum and average over the electron spin states. the spherical average for rotational degeneracy and averaging over vibrations by calculating 4 at the average internuclear spacing. we obtain the following expression for the (e,2e) cross section leading to a particular ion state X: aA(kA ,kB) = (2~)~ (kAkB/ko) x &e&=&h
Ibld#,
(9)
where by JdQ we indicate integration over the angular dependence of q_fee is the half-off-shell Mott scattering cross section,
VoIume 101, number 4,5
CHEMICAL PHYSICS LET-I-ERS
f, = (ZY?)-~{2nvl[exp(27r+ll] X @co -kJ4
have been adopted, whereby EA =Eg at variable azimuthaI angIe a, with the two analyzers kept at equal and opposite scattering angles (@A 7 6, = 44Y) with respect to the incident beam. Data taken under these kinematic conditions have been shown to be less sensitive to possible distortion effects occurring in the ioniz&on even [7,S]_ Under these conditions,
i- I$) -k,?
- Ike - kA1-21k0 -k&2
v= IkA
28October 1983
q = (k;+ 4k;
-k&t
~0~~0~-
4k,k,
cos OA cm @)‘I’,
so that at Q, = 0” totally
Approximation periments.
(9) is used for the analysis of the ex-
3. Results and discussion The experimental apparatus has already been described [3]. The energy spectra reported here have been recorded by varying the incident energy E. at fved EA and En and are therefore directly drawn in terms of the binding (or separation) energy eh. The measurements have been carried out at an energy of = 1600 eV, corresponding to an energy resolution Z=X 1.5 eV_ Non-coplanar symmetric kinematics ~fi”hm
1”
Fig. 1. Ionization energy spectrum ofCI+CHCI on the basis of the many-body results.
20
symmetric states are evidenced, whereas in the @ # 0” condition, corresponding to higher momenta, states of different symmetry are prominent. In the present work, we discuss the theoretical and experimental results for the C2H313r moIecuIe together with preliminary data on C2H3CI. The following contracted gaussian basis sets have been used in the calculations: (9s,Sp) @s,2p] on carbon and (4s)[2s] on hydrogen 191; (12s,9p)[6s,4p] [ 101 augmented by a d term (or = 0.68) on chlorine; (15s, lZ?.p,2d)[Ss,Eip,Zd] on bromine [l I]. The Green’s function calculations have been performed using the programs originally provided by W_ von Niessen, modifled to allow for CPU-time monitoring and restart facili-
30
&~W)
4”
at 0 = 0” (q = O&t>_ Top: experimental data; bottom: the spectrum as computed
479
-
Vofume 101. number 3.5
CHEMICAL PHYSICS LETT.ERS
28 October 1983
ties and interfaced to the ATMOL SCF and four-index tmnsformation package f 12]_ C,H,CI and C2H3Br are 18 valence-electron closedshell systems having Cs symmetry. The Hartree-Fock orbitals show the following order. starting from outervalence inward. for both molecules: 2s” I Oa’ I a*’ Pa’ ga’ 7a’ 6a’ 5a’ 4a’.. This order is confirmed by the many-body calculations. It is useful to analyze the main SCF components of some of the innermost valence orbit3ls through the C?H,S (X = F, Cl, Br) series. In CzH,F [5], 6a’ is a 2ac antibonding orbital. 5a’ is a 20, bonding and 4a’ the outermost s non-bonding orbital on the fluorine. As the atomic number of the substituent increases. the binding ener,r of the last atomic orbital decreases so that it can mix heavily with the carbon 2s. In the case of the C2H3Br molecule. the 4a’ becomes a crC_8r bonding orbital. the 53’ a oC_nr antibonding and the 6a’ a 0=-c sIightly antibonding orbital. The spherically averaged Fourier transforms of these wavefunctions show a maximum for q =#=0 in the case of 6a’ and 5a’, whereas the F(q) behaviour is monotonicaliy decreasing with 4 for 4a’. The experimental (e.2e) energy spectra at ctr = 0” are reported in Bg_ I a for C2H;Br and in fig. 2a for C2H3CI. The analogous spectrum at 9 = 3“ for C,H,Br is shown in fig. _ k It should be noted that the states whose Fourier transfomi has a nia..inumi for q = 0 mainly contribute to the spectra at cft = 0” (corresponding indeed to 4 = 0). thus allowing an easier assignment of the various b&ids. as will be shown below for a particular case. A deconvolution of the experimental data by means of gaussians with full width at half maximum of I .5 eV has been performed in order to resolve as many peaks as possible. The experimental IPs, as deduced from the spectra, are shown in tables 1 and :! for those bdnds for which an assignment is directly possibie. namely the outer-vnlence ionizations 2a” to 6a’ for both molecules. The accuracy of these data is deemed to be slightly inferior to that of the UPS measurements (also shown in the tables), since they have been obtained by a deconvolution procedure on the energy curves. For the inner-valence ionizations (a11 of a’ symmetry), the intensity appears to be spread over a large number of final ionic stares. This difference in the behaviour of the outer- and inner-valence states is quite common; it has been found in particular by Dixon et al. [ 131 for 460
Fig. 2. Ionization energy spectra of CH2=CHBr. (a) Experimental spectnm~ at Q, = 0” (q = 0 a;'>:(b) theoretical spectrum at Q = OJ; (c) esperimentaf spectrum at @ = 3O (4 = OS a<*); (d) theoretical spectrum at 9 = 3O.
the 2b,, and ‘a8 orbitals of ethylene (which directly correspond to 6a’ and 5a’ of its monohalo derivatives) and for the C2H,F molecule [S]. In the present case, apart from a satellite line close to the 6a’ main ionization of C,H,Br, the spectra of figs. la, ‘a and 3e exhibit diffuse bands extending up to =40 eV. As a general rule, the above features are accounted for by the many-body computations_ These have been perfomred within two different approximations, the first of which [I] (referred to 3s OVGF) is particularly suited to the outer-valence ionizations, where the quasiparticle picture of the ionization process is valid (absence or negligible presence of satellite lines). it is correct up to third order in the electron-electron interaction, higher-order contributions being taken into account by a renormalization procedure f I] _For the OVGF calcuIations, the virtual-orbital space has been completely exhausted, except for the core states and their virtual
5
19.82 (0.79) 21.53 (0‘03) 24002 (0602) 47,35 (0804) 16.44 (0.87) 22.03 (0.02) 24.83 (OaO2) 46.58 (0.03) 15.35 (0.89) 22.03 (0.01) 46.80 (0.03) 13,21 (0,92) 45,96 (0.02) 12,80 (0,91) 16.30 (0.02) 23,55 (0.01) 11.24 (0.94 44815 (0.01) 9.48 (0,93)
?a’
8~1’
9u’
la”
100’
2a”
IP
6a’
state
b, Rof, [ 181,
23879(0,06) 24,29 (0‘05) 24.34 fO+Ol) 24,83 (0,04) 24,97 (0.37) 25‘22 (0.16) 26.79 (0.02) 26,89 (0,02) 29J7 (0401) 29.33 (0*03) 29.76 (0.01) 31.80 (0,02) 51.72 (0.06)
a) This work.
Sa’
24.97 (0,Ol) 26e79(0.04) 28,07 (OJ2) 28014(0,Ol) 28.53 (0.03) 28.60 (0406) 29.76 (0,02) 30.06 (0.09) 30.26 (0.12) 3034 (0,04) 30.57 (0,08) 30.64 (O,Ol) 30,90 (0,02) 32861(0402) 32.65 (0.02)
48’
53.07 (0.08)
IP
state
2ph TDA approximation
9.72
2a”
12.88
la” 11,27
13,24
9a’
10a’
15,35
IP
8a’
state
0,92
0,93
0‘92
0.93
0.91
OVGF approximation a)
2B0
101’
la”
9a’
8a’
state
9,83
11.40
13.02
13.40
15.31
IP
0‘92
0.93
0,92
0.92
0.91
OVGF approximation b,
13.6 13.2 11.7 10.2
la” 10a’ 2a”
15,4
16.3
1940
23.0
9a’
&I
7a’
6a’
Sa’
2a”
10a’
la”
9a’
8a’
7a’
6a’
10,2
1260
13s
1447
16J
17,7
19,9
IP
state
state
IP
Uxp. (e,2c)
Exp, UPS
reported, OVGF results from ret [ 181 MCalsoshown for comparison. Experimental values from UPS (raf, ( 181)and from (c,20) experiments (this work)
Tabic i Verticalionization potentials (II%)in eV and pole strengths (in parentheses) for C!Ii~=CliC1obtained by Green’s function methods, Only pole strengths 20.01 arc
Volume
CHEMICAL
101, number 4.5
PHYSICS
28 October
LETTERS
1983
Table 2 Vertical ionization potentials in eV and pole strengths (in parentheses) for CHz=CHBr obtained by Green’s function methods. Only pole strengths >O.Ol are reported. Experimental values from UPS (ref. [ 181) and from (e.2e) experiments (this work) 2ph TDA
approximation
state
IP
4a’
25.47 (0.02)
OVGF state
5a’
26.44 (0.01) 27.42 (0.07) (0.04)
27.85 (0.04) ‘8.12
(0.03)
state
25.92 (0.02)
26.06 26.44 27.16 27.42 28.39 29.87
26.06 (0.02)
2757
IP
(0.04) (0.01) (0.02) (0.02) (0.06) (0.03)
appoximation IP
Exp. UPS state
Exp. (e,2e) IP
state
IP
6a’
195 1 (0.86)
4a’
29.00
76
16.46 (0.89)
5a’
23.9
8a’
14.98 (0.91)
66
18.9
6a’
19.0
9a’
13.00 (0.93)
7a’
16.0
7a’
16.5
la”
12.28 (0.92)
8a’
15.0
8a’
15.4
1Oa’
1056 (0.94)
9a’
13.0
9a’
13.4
26’
9.44 (0.93)
1a”
12.3
1a”
12.3
1Oa’
10.9
1Oa’
10.9
Za”
9.9
2a”
9.9
28.39 (0.04) 29.05 (0.32)
6a’
29.23 (0.05) 295 1 (0.03) 29.56 (0.01)
19.71 (0.73) 20.91 (0.08) 21.86 (0.01) 22.70 (0.03)
27.79 (0.01) 29.87 (0.01)
30.76 (0.05) 30.85 (0.02) 30.99 (0.01) 3253
(0.02)
7a’
16.71 (0.85)
32.70 (0.02)
2 1.45 (0.03)
35.11 (0.04)
‘2.70
(0.02)
37-70 (0.02)
Sa’
23.51 23.92 24.01 24.61 24.93 25-47
(0.01) (0.01) (0.04) (0.03) (0.51) (0.05)
8a’
15.36 (0.92)
9a’
12.99
1d"
12.40 (0.90) 15.81 (0.04)
1 On’
10.60 (0.96)
Za”
9.50
(0.95)
(0.94)
counterparts. For the inner-valence part of the spectra, where a strong satellite structure is present, the 2ph TDA approsimstion [ 141 has been employed, which is known to yield a qualitative agreement with the experimental results, being correct up to second order in the
electron interaction. It gives account of the breakdown of the quasi-particle picture of the ionization and of the satellite structure accompanying the main lines in the inner-valence part of the spectrum. This approximation has been recently improved. within the completely general “algebraic diagrammatic construction” (ADC) 482
scheme for the Green-s function [ 151, with the introduction of the so-called “extended 2ph TDA” method (see, for example, ref. [ 161) which is correct up to third order. However, since both methods give only a qualitative description of the satellite structure of the inner-valence bands, and given the present resolution of the (e,2e) experiments, the 2ph TDA results of this work, together with the OVGF ones, can be used equally well the purpose of interpretation. The method is capable, in principle, of using a large virtual space, but the cost of the computation increases
Volume 101. number 45
CHEMICAL PHYSICS LETTERS
enormously with increasing number of active orbitals. For this reason, and due to the large dimension of the matrices involved, the 2ph.TDA orbital space has been truncated to comprise 19 and 21 orb&k out of the full number of 50 and 64 for C2H3Cl and C,H,Br, respectively_ Since no configuration could be excluded in principle, due to the large number of states contributing to the spectrum, the excluded configurations have been accounted for in mean by including them as the centroids of their energy [ 171. It will be worthwhile in the future to perform largerscale calculations on the same molecules_ Although. in general, the 2ph TDA results for the outer-valence energies are not as accurate as the OVGF ones, in the present case the figures are on the same level of accuracy; the same situation was found, for instance, in an invkigation of CHF, and CF4 [6] _The reconstructed spectra, based on the OVGF and 2ph TDA IPs and pole strengths, are reported in fig. 1b for C,H,Cl and in figs. 2b and 2d for CZH3Br. It is immediately seen that the agreement with experiment is quite good in the low-energy part of the spectra (corresponding to the OVGF case), being only roughly
28 October 1983
for both molecules. Due to the absence of a satellite structure (or, conversely, to the presence of a sharp main line), the assignment of these bands is straightforward- On the other hand, the assignment of the inner 5a’ and 4a’ bands is not a clear-cut one, since these bands are greatly overlapped and their intensity is spread over many states, as discussed before. According to the discussion above, the spectrum can be analyzed with the aid of the electron momentum distribution (EMD) curves. These are sections of the surfaces at constant energy, representing the cross sections as functions of Ed and q_ We refer in what follows to the C2H3Br molecule. The seven EMDs in fig. 3 refer to the outer-valence ionizations (the 6a’ orbital is also included, though it shows, both experimentally and theoretically, a non-negligible satellite structure). The method used to obtain the theoretical EMD curves C2H3Br 10-Z
I
a)
b1
qualitative on the right side of the spectra. The intensity of the reconstructed bands between 25 and 40 eV
has been reduced in order to fit into the drawings; the arbitrariness of the normalization stems from the fact that no absolute meaning can be attributed to the pole strengths relative to different ionizations (even at high incident energy [ 11). As to the position of the bands on the energy axis, it should be noted that the 2ph TDA aproximation generally yields inner-valence ionization bands that are shifted to higher energy by 2-4 eV_ This inaccuracy can clearly be ascribed to the truncation of the basis set and to the neglect of the 3h2p and higher configurations (which are likely to be of importance in the inner-shell region), and also to the omission of the constant part of the self-energy [I] in our computations. Apart from this shift, the method permits one in the present case to give account of the noticeable overlap of the 5a’ and 4a’ band, and of the very complicated nature of the configuration interaction. The results of the many-body calculations are reported in tables 1 and 2 for C,H,Cl and C,H,Br, respectively, together with the UPS experimental values from ref. [18] and the (e,2e) II’s as deduced from the spectra in figs. 1b, 2b and 2d. The good numerical agreement of the computed IPs is evident for the outer ionizations, the maximum discrepancy being x0.5 eV
Fig. 3. Electron momentum distributions of the outer-valence shell of CH2=CHBr measured at different Q. The solid lines are the calculated curves, obtained by including contributions from all the states convoluted by the energy resolution (see ref. [5]):
483
28 October 1983
CHEMICAL PHYSICS LETTERS
Volume 101. number 4.5
C2H3Br
and corresponds to satellite lines of the 5a’ and 4a’ bands, the former being responsible for the shoulder appearing at 23 eV in the spectrum at @ = 3”. Owing to the different shape of the Fourier transforms of these orbitals, each of the satellite lines can be unam-
bigously
assigned in this way.
Acknowledgement
“,
The financial support fully acknowledged.
10-l
0
0”
of the CNR (Rome) is grate-
2.
References 1.
[ 1] L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 t Fig. 4. Electron momentum distributions in the inner-valence region of CI12=C~iBr. Top: experimentd distributions At tv.o particulx scpxatiun energies; bottom: theoretical curves obt.dnrd from the computed 5.1’ and 4a’ orbitals.
(based on the 2ph TDA results) has been reported elsewhere [5 ] _The qualitative agreement with the experimental curves of figs. 3a-3g allows one to analyze the contribution of each single SCF eigenvector to the final cross secUon, thus confirming the assignment of the bands. It is clear, in this case. that the computed pole strengths lead to the correct qualitative behaviour of the reconstructed distributions. The situation is not as simple for the two EMDs at 24.0 and 29.1 eV_ In this inner-valence region, the theoretical reconstruction fails to reproduce the correct shape of the EMD curves due to the computed shift of the bands discussed previously together with rhe pronounced satellite broadening and the strong overlap of the 5a’ and 4a’ bands. From fig. 4. it is clear tllat the main contribution to the experimental curves comes from the 4a’ ionization, whose pure theoretical distribution has the correct behaviour at srn~llq values, whereas the full theoretical reconstruction overemphasizes the 53 contribution. On the other hand. the band between 22 and 27 eV in the spectruni at cf, = O” (4 = 0) modifies in sha e when the spectrum is recorded at @ = 3” (q = 0.5 ~0 P) (cf. figs. 2a and 2~) in such a way so as to show that it is mainly due to the superposition of two different ionizations 484
(1977) ‘05. [Z] E. Wcigold and LE. McCarthy, Advan. At. Mol. Phys. 14 (1978) 127. 131 A. GiardiniCuidoni. R. Fantoni and R. Camilloni. Comm. At. ?.lol. Phys. 10 (1981) 107. [4] R. Cambi, G. CiuoUo, A. Sgamellotti, F. Tarantelli, R. Fantoni, A. GiardiniGuidoni and A. Sergio, Chem. Phys. Letters 60 (1981) 295. [5] R. Fantoni, A. GiardiniCuidoni, R. Tiibelli, R. Cambi, G. CiuBo, A. Sgamellotti and F. Tarantelli, Mol. Phys. 45 (1982) S39. [6] R. Cambi, G. Ciullo. A. Sgamellotti, F. Tarantelli, R. Fantoni. A. GiardiniCuidoni, R. Tiribelli and hl. Rosi, Chem. Phys. Letters 90 (1982) 445. [ 7] LE. hlcCarthy and E. Weigold, Phys. Rept. C27 (1976) 27.5. [S] R. Camilloni. G. Stefani, R. Fantoni and A. GiardiniGuidoni, J. Electron Spectry. 17 (1979) 209. [9] T-H. Dunning, Jr.. J. Chem. Phys. 53 (1970) 2823. [ 101 T-H. Dunning Jr., Chem. Phys. Letters 7 (1970) 423. [ 1 l] P.G. Mezey, M.H. Lien, K. Yates and G. Csbmadia, Theoret. Chim. Acta 40 (1975) 75. [ 121 V-R. Saunders and M-F. Guest, “AWlOL3”, Daresbury Laboratory SERC, Wanin~ton WA4 4AD. [ 131 A-l. Dixon, S.T. Hood, E. Weigold and G.RJ. Wiams. J. Electron Spectry. 14 (1978) 267. [ 141 J. Schirmer and L.S. Cederbaum, J. Phys. Bll (1978) 1889. [ 151 J. Schirmer, Phys. Rev. A26 (1982) 2395; J. Schiimer, LX Cederbaum and 0. Walter, Phys. Rev_ A, to be published. [ 161 0. Walter and J. Schiier. J. Phys. B14 (1981) 3805; J. Scbirmer and 0. Walter, to be published. [ 171 W. von Niessen and L-S. Cederbaum, Mol. Phys. 43 (1981) 897. [ 181 W. van Niessen, L. Asbrink and G. Bieri, J. Electron Spectry. 26 (1982) 173.
CHEMICAL
PHYSICS LETTERS
AN (e, 2e) SPECTROSCOPIC INVESTIGATION OF THE IONIZATION
-28 October
1983
AND A GREEN’S FUNCTION STUDY
OF CHLORO- AND BROMO-ETHYLENE
R. CAMBI, G. CIULLO, A. SGAMELLOTTI, Diparrimento di chimica. Uniwrsit~ di Pen@.
F. TARAIWELLI
06100 Perugin. Ifaly
and
R. FANTONI, A. GIARDINI-GUIDONI,
I.E. MCCARTHY * and V. DI MARTIN0 **
ENEA. Centro Ricerche Energin Frascati, 00044 Ftaswti. Rome. Italy Received 1 August 1983
A Green-s function study and an (e,2e) spectroscopic investigation have been carried out on the CzH-&l and CzHSBr molecules. The complete valence ionization spectra of both molecules are discussed_ An analysis of the experimental and theoretical electron momentum distributions is also presented, allowing an unambiguous assignment of the spectral features.
1. Introduction
The application of the one-electron Green’s function to photoelectron spectroscopy has been discussed in detail by Cederbaum and Domcke 1I] _The position of the poles of the Green’s function give the electronic levels of the ion, while the pole strengths provide the ratios of the cross sections for ionizations leading to the corresponding energy levels in the high-energy limit. The kinematics of dipole (e,Ze) spectroscopy is arranged to simulate the photoelectron reaction. The scattering process is characterized by large impact energies and low momentum transfer, the scattering angle being = zero. Under these conditions, the ionization involves distant collisions and the cross section can be taken as proportional to the optical oscillator strength, as in the case of photoabsorption. On the other hand, binary (e,2e) spectroscopy (used in the present work) is a large momentum-transfer coincidence technique, in which the (e,2e) cross section is a function of the momentum (q) of the ejected electron before ionization. It is now well known that binary * On leave from The Flinders University of South Australia. ** Guest. 0 009-2614/83/0000-0000/$03.00
0 1983 North-Holland
(e,2e) spectroscopy gives the low-momentum range for each ion state at an arbitrary total energy. This means that we can choose an energy where the highener=v limit is valid, i.e. where the plane-wave approximation for the continuum orbitals holds, and measure both the distribution of the ion states and their corresponding low-momentum profiles; these are usually well described by the spherically averaged square of the appropriate molecular orbital multiplied by a spectroscopic factor [2,33. The high-energy limit, where the experimentally determined structural information is independent of the total energy, is reached at least at 1000 eV. The analysis of (e,2e) experiments by the Green’s function method has been successful in several recent applications to small molecules [4-61. Here we report an investigation of the considerably larger systems C,H,Cl and C,H,Br. In section 2 we point out the direct connection of the Green’s function to the (e,2e) cross section_ In section 3 the theoretical results obtained for CZH3Cl and C2H3Br are compared with the experimental data; a complete investigation of the electronic structure of C,H,Br performed by means of binary (e,2e) spectroscopy is presented, including the momentum distribution curves, and preliminary results on C,H,Cl are also reported. 477
CHESIICAL PHYSICS LETTERS
Volume 101, number 4.5 2. Relationship of the one-particle the (e,2e) cross section
Green’s function
to
Cederbaum and Domcke [I] have considered in great detail the application of the one-particle Green‘s function to ionization processes_ The total cross section P(w) for ionization is expressed in terms of the target Hartree-Fock orbitals 4 and one-particle photoionization amplitudes:
where 4 is a continuum orbital for the ejected electron. The spectroscopic application of the binary (e,2e) reaction is optimal in non-coplanar symmetric kinematics. Here we cannot distinguish the ionized electron from the scattered one_ We label them B and A respectively. and treat B in the formalism in such a way as to bring out the analogy with photoionization. Antisymmetrication with respect to A and B is implicit when necessary. Apart from initial- and final-state degeneraties, we have for the transition probability P(w) in the (o.2e) reaction p(w) =
x
(3r)4(k,k,/ko)
cF
I(FI(~,I~I~~,I~~)I~~(~
=
(k&jItlQ/iO)’
(4)
r^ being the electron-electron I matrix. introducing the notation Z, for the appropriate sum and average over initial- and final-state degeneracies, we have = (3+4(x-
P(o)
k /srx- )
AB
0
X CT* 7 Im[GI,(m ant Bn Bt
-w.
- i$]S(w
-EA
(5) -EB)
where
Expressions (5) and (6) are identical (apart from the kinematic constant) to eqs. (Z-4) of ref. [I]. The Green’s furlction (6) is, in general, not diagonal_ It has been shown [I] that, if we assume it is ahnost diagonal, we may write the approximation P(w) = (27r)4(~,kB/XO)
-15,
-Q.
(3
where the labelling of electron energies and momenta is as in refs. [2,3] _The N-electron molecular ground state is I*$> and the observed ior]-electron B state is IF>. Instead of a dipole operator, we connect these states through an operator (X-,ITIX-,) for particular I;lomenta of electrons 0 (incident) and A. The operator T contains all the complexity necessary to describe the reaction fully. A great advantage of the (e.2e) reaction is that we can perform the whole spectroscopic application in the energy region where it is valid ro approximate IF) by the product of a plane wave and the final ion I*r-l>. In non-coplanar symmetric kinematics there is a large body of experimental information [2.3] confirming the validity of the plane-wave impulse approximation where. for a particular momentum k,, the secondquantized foml of*the (e,2e) amplitude is
(3
478
TBI
28 October 1983
X oG I-r,,l’Pt(x)s(w
-E,
- EB),
(7)
where Pt(h) is the pole strength for ionization from orbital & to the final state h. We note that (4) factorizes into the product of a half-off-shell t-matrix element and the momentum-space orbital (q I+$ for
q =k, -k,
-k,.
(8)
Performing the sum and average over the electron spin states. the spherical average for rotational degeneracy and averaging over vibrations by calculating 4 at the average internuclear spacing. we obtain the following expression for the (e,2e) cross section leading to a particular ion state X: aA(kA ,kB) = (2~)~ (kAkB/ko) x &e&=&h
Ibld#,
(9)
where by JdQ we indicate integration over the angular dependence of q_fee is the half-off-shell Mott scattering cross section,
VoIume 101, number 4,5
CHEMICAL PHYSICS LET-I-ERS
f, = (ZY?)-~{2nvl[exp(27r+ll] X @co -kJ4
have been adopted, whereby EA =Eg at variable azimuthaI angIe a, with the two analyzers kept at equal and opposite scattering angles (@A 7 6, = 44Y) with respect to the incident beam. Data taken under these kinematic conditions have been shown to be less sensitive to possible distortion effects occurring in the ioniz&on even [7,S]_ Under these conditions,
i- I$) -k,?
- Ike - kA1-21k0 -k&2
v= IkA
28October 1983
q = (k;+ 4k;
-k&t
~0~~0~-
4k,k,
cos OA cm @)‘I’,
so that at Q, = 0” totally
Approximation periments.
(9) is used for the analysis of the ex-
3. Results and discussion The experimental apparatus has already been described [3]. The energy spectra reported here have been recorded by varying the incident energy E. at fved EA and En and are therefore directly drawn in terms of the binding (or separation) energy eh. The measurements have been carried out at an energy of = 1600 eV, corresponding to an energy resolution Z=X 1.5 eV_ Non-coplanar symmetric kinematics ~fi”hm
1”
Fig. 1. Ionization energy spectrum ofCI+CHCI on the basis of the many-body results.
20
symmetric states are evidenced, whereas in the @ # 0” condition, corresponding to higher momenta, states of different symmetry are prominent. In the present work, we discuss the theoretical and experimental results for the C2H313r moIecuIe together with preliminary data on C2H3CI. The following contracted gaussian basis sets have been used in the calculations: (9s,Sp) @s,2p] on carbon and (4s)[2s] on hydrogen 191; (12s,9p)[6s,4p] [ 101 augmented by a d term (or = 0.68) on chlorine; (15s, lZ?.p,2d)[Ss,Eip,Zd] on bromine [l I]. The Green’s function calculations have been performed using the programs originally provided by W_ von Niessen, modifled to allow for CPU-time monitoring and restart facili-
30
&~W)
4”
at 0 = 0” (q = O&t>_ Top: experimental data; bottom: the spectrum as computed
479
-
Vofume 101. number 3.5
CHEMICAL PHYSICS LETT.ERS
28 October 1983
ties and interfaced to the ATMOL SCF and four-index tmnsformation package f 12]_ C,H,CI and C2H3Br are 18 valence-electron closedshell systems having Cs symmetry. The Hartree-Fock orbitals show the following order. starting from outervalence inward. for both molecules: 2s” I Oa’ I a*’ Pa’ ga’ 7a’ 6a’ 5a’ 4a’.. This order is confirmed by the many-body calculations. It is useful to analyze the main SCF components of some of the innermost valence orbit3ls through the C?H,S (X = F, Cl, Br) series. In CzH,F [5], 6a’ is a 2ac antibonding orbital. 5a’ is a 20, bonding and 4a’ the outermost s non-bonding orbital on the fluorine. As the atomic number of the substituent increases. the binding ener,r of the last atomic orbital decreases so that it can mix heavily with the carbon 2s. In the case of the C2H3Br molecule. the 4a’ becomes a crC_8r bonding orbital. the 53’ a oC_nr antibonding and the 6a’ a 0=-c sIightly antibonding orbital. The spherically averaged Fourier transforms of these wavefunctions show a maximum for q =#=0 in the case of 6a’ and 5a’, whereas the F(q) behaviour is monotonicaliy decreasing with 4 for 4a’. The experimental (e.2e) energy spectra at ctr = 0” are reported in Bg_ I a for C2H;Br and in fig. 2a for C2H3CI. The analogous spectrum at 9 = 3“ for C,H,Br is shown in fig. _ k It should be noted that the states whose Fourier transfomi has a nia..inumi for q = 0 mainly contribute to the spectra at cft = 0” (corresponding indeed to 4 = 0). thus allowing an easier assignment of the various b&ids. as will be shown below for a particular case. A deconvolution of the experimental data by means of gaussians with full width at half maximum of I .5 eV has been performed in order to resolve as many peaks as possible. The experimental IPs, as deduced from the spectra, are shown in tables 1 and :! for those bdnds for which an assignment is directly possibie. namely the outer-vnlence ionizations 2a” to 6a’ for both molecules. The accuracy of these data is deemed to be slightly inferior to that of the UPS measurements (also shown in the tables), since they have been obtained by a deconvolution procedure on the energy curves. For the inner-valence ionizations (a11 of a’ symmetry), the intensity appears to be spread over a large number of final ionic stares. This difference in the behaviour of the outer- and inner-valence states is quite common; it has been found in particular by Dixon et al. [ 131 for 460
Fig. 2. Ionization energy spectra of CH2=CHBr. (a) Experimental spectnm~ at Q, = 0” (q = 0 a;'>:(b) theoretical spectrum at Q = OJ; (c) esperimentaf spectrum at @ = 3O (4 = OS a<*); (d) theoretical spectrum at 9 = 3O.
the 2b,, and ‘a8 orbitals of ethylene (which directly correspond to 6a’ and 5a’ of its monohalo derivatives) and for the C2H,F molecule [S]. In the present case, apart from a satellite line close to the 6a’ main ionization of C,H,Br, the spectra of figs. la, ‘a and 3e exhibit diffuse bands extending up to =40 eV. As a general rule, the above features are accounted for by the many-body computations_ These have been perfomred within two different approximations, the first of which [I] (referred to 3s OVGF) is particularly suited to the outer-valence ionizations, where the quasiparticle picture of the ionization process is valid (absence or negligible presence of satellite lines). it is correct up to third order in the electron-electron interaction, higher-order contributions being taken into account by a renormalization procedure f I] _For the OVGF calcuIations, the virtual-orbital space has been completely exhausted, except for the core states and their virtual
5
19.82 (0.79) 21.53 (0‘03) 24002 (0602) 47,35 (0804) 16.44 (0.87) 22.03 (0.02) 24.83 (OaO2) 46.58 (0.03) 15.35 (0.89) 22.03 (0.01) 46.80 (0.03) 13,21 (0,92) 45,96 (0.02) 12,80 (0,91) 16.30 (0.02) 23,55 (0.01) 11.24 (0.94 44815 (0.01) 9.48 (0,93)
?a’
8~1’
9u’
la”
100’
2a”
IP
6a’
state
b, Rof, [ 181,
23879(0,06) 24,29 (0‘05) 24.34 fO+Ol) 24,83 (0,04) 24,97 (0.37) 25‘22 (0.16) 26.79 (0.02) 26,89 (0,02) 29J7 (0401) 29.33 (0*03) 29.76 (0.01) 31.80 (0,02) 51.72 (0.06)
a) This work.
Sa’
24.97 (0,Ol) 26e79(0.04) 28,07 (OJ2) 28014(0,Ol) 28.53 (0.03) 28.60 (0406) 29.76 (0,02) 30.06 (0.09) 30.26 (0.12) 3034 (0,04) 30.57 (0,08) 30.64 (O,Ol) 30,90 (0,02) 32861(0402) 32.65 (0.02)
48’
53.07 (0.08)
IP
state
2ph TDA approximation
9.72
2a”
12.88
la” 11,27
13,24
9a’
10a’
15,35
IP
8a’
state
0,92
0,93
0‘92
0.93
0.91
OVGF approximation a)
2B0
101’
la”
9a’
8a’
state
9,83
11.40
13.02
13.40
15.31
IP
0‘92
0.93
0,92
0.92
0.91
OVGF approximation b,
13.6 13.2 11.7 10.2
la” 10a’ 2a”
15,4
16.3
1940
23.0
9a’
&I
7a’
6a’
Sa’
2a”
10a’
la”
9a’
8a’
7a’
6a’
10,2
1260
13s
1447
16J
17,7
19,9
IP
state
state
IP
Uxp. (e,2c)
Exp, UPS
reported, OVGF results from ret [ 181 MCalsoshown for comparison. Experimental values from UPS (raf, ( 181)and from (c,20) experiments (this work)
Tabic i Verticalionization potentials (II%)in eV and pole strengths (in parentheses) for C!Ii~=CliC1obtained by Green’s function methods, Only pole strengths 20.01 arc
Volume
CHEMICAL
101, number 4.5
PHYSICS
28 October
LETTERS
1983
Table 2 Vertical ionization potentials in eV and pole strengths (in parentheses) for CHz=CHBr obtained by Green’s function methods. Only pole strengths >O.Ol are reported. Experimental values from UPS (ref. [ 181) and from (e.2e) experiments (this work) 2ph TDA
approximation
state
IP
4a’
25.47 (0.02)
OVGF state
5a’
26.44 (0.01) 27.42 (0.07) (0.04)
27.85 (0.04) ‘8.12
(0.03)
state
25.92 (0.02)
26.06 26.44 27.16 27.42 28.39 29.87
26.06 (0.02)
2757
IP
(0.04) (0.01) (0.02) (0.02) (0.06) (0.03)
appoximation IP
Exp. UPS state
Exp. (e,2e) IP
state
IP
6a’
195 1 (0.86)
4a’
29.00
76
16.46 (0.89)
5a’
23.9
8a’
14.98 (0.91)
66
18.9
6a’
19.0
9a’
13.00 (0.93)
7a’
16.0
7a’
16.5
la”
12.28 (0.92)
8a’
15.0
8a’
15.4
1Oa’
1056 (0.94)
9a’
13.0
9a’
13.4
26’
9.44 (0.93)
1a”
12.3
1a”
12.3
1Oa’
10.9
1Oa’
10.9
Za”
9.9
2a”
9.9
28.39 (0.04) 29.05 (0.32)
6a’
29.23 (0.05) 295 1 (0.03) 29.56 (0.01)
19.71 (0.73) 20.91 (0.08) 21.86 (0.01) 22.70 (0.03)
27.79 (0.01) 29.87 (0.01)
30.76 (0.05) 30.85 (0.02) 30.99 (0.01) 3253
(0.02)
7a’
16.71 (0.85)
32.70 (0.02)
2 1.45 (0.03)
35.11 (0.04)
‘2.70
(0.02)
37-70 (0.02)
Sa’
23.51 23.92 24.01 24.61 24.93 25-47
(0.01) (0.01) (0.04) (0.03) (0.51) (0.05)
8a’
15.36 (0.92)
9a’
12.99
1d"
12.40 (0.90) 15.81 (0.04)
1 On’
10.60 (0.96)
Za”
9.50
(0.95)
(0.94)
counterparts. For the inner-valence part of the spectra, where a strong satellite structure is present, the 2ph TDA approsimstion [ 141 has been employed, which is known to yield a qualitative agreement with the experimental results, being correct up to second order in the
electron interaction. It gives account of the breakdown of the quasi-particle picture of the ionization and of the satellite structure accompanying the main lines in the inner-valence part of the spectrum. This approximation has been recently improved. within the completely general “algebraic diagrammatic construction” (ADC) 482
scheme for the Green-s function [ 151, with the introduction of the so-called “extended 2ph TDA” method (see, for example, ref. [ 161) which is correct up to third order. However, since both methods give only a qualitative description of the satellite structure of the inner-valence bands, and given the present resolution of the (e,2e) experiments, the 2ph TDA results of this work, together with the OVGF ones, can be used equally well the purpose of interpretation. The method is capable, in principle, of using a large virtual space, but the cost of the computation increases
Volume 101. number 45
CHEMICAL PHYSICS LETTERS
enormously with increasing number of active orbitals. For this reason, and due to the large dimension of the matrices involved, the 2ph.TDA orbital space has been truncated to comprise 19 and 21 orb&k out of the full number of 50 and 64 for C2H3Cl and C,H,Br, respectively_ Since no configuration could be excluded in principle, due to the large number of states contributing to the spectrum, the excluded configurations have been accounted for in mean by including them as the centroids of their energy [ 171. It will be worthwhile in the future to perform largerscale calculations on the same molecules_ Although. in general, the 2ph TDA results for the outer-valence energies are not as accurate as the OVGF ones, in the present case the figures are on the same level of accuracy; the same situation was found, for instance, in an invkigation of CHF, and CF4 [6] _The reconstructed spectra, based on the OVGF and 2ph TDA IPs and pole strengths, are reported in fig. 1b for C,H,Cl and in figs. 2b and 2d for CZH3Br. It is immediately seen that the agreement with experiment is quite good in the low-energy part of the spectra (corresponding to the OVGF case), being only roughly
28 October 1983
for both molecules. Due to the absence of a satellite structure (or, conversely, to the presence of a sharp main line), the assignment of these bands is straightforward- On the other hand, the assignment of the inner 5a’ and 4a’ bands is not a clear-cut one, since these bands are greatly overlapped and their intensity is spread over many states, as discussed before. According to the discussion above, the spectrum can be analyzed with the aid of the electron momentum distribution (EMD) curves. These are sections of the surfaces at constant energy, representing the cross sections as functions of Ed and q_ We refer in what follows to the C2H3Br molecule. The seven EMDs in fig. 3 refer to the outer-valence ionizations (the 6a’ orbital is also included, though it shows, both experimentally and theoretically, a non-negligible satellite structure). The method used to obtain the theoretical EMD curves C2H3Br 10-Z
I
a)
b1
qualitative on the right side of the spectra. The intensity of the reconstructed bands between 25 and 40 eV
has been reduced in order to fit into the drawings; the arbitrariness of the normalization stems from the fact that no absolute meaning can be attributed to the pole strengths relative to different ionizations (even at high incident energy [ 11). As to the position of the bands on the energy axis, it should be noted that the 2ph TDA aproximation generally yields inner-valence ionization bands that are shifted to higher energy by 2-4 eV_ This inaccuracy can clearly be ascribed to the truncation of the basis set and to the neglect of the 3h2p and higher configurations (which are likely to be of importance in the inner-shell region), and also to the omission of the constant part of the self-energy [I] in our computations. Apart from this shift, the method permits one in the present case to give account of the noticeable overlap of the 5a’ and 4a’ band, and of the very complicated nature of the configuration interaction. The results of the many-body calculations are reported in tables 1 and 2 for C,H,Cl and C,H,Br, respectively, together with the UPS experimental values from ref. [18] and the (e,2e) II’s as deduced from the spectra in figs. 1b, 2b and 2d. The good numerical agreement of the computed IPs is evident for the outer ionizations, the maximum discrepancy being x0.5 eV
Fig. 3. Electron momentum distributions of the outer-valence shell of CH2=CHBr measured at different Q. The solid lines are the calculated curves, obtained by including contributions from all the states convoluted by the energy resolution (see ref. [5]):
483
28 October 1983
CHEMICAL PHYSICS LETTERS
Volume 101. number 4.5
C2H3Br
and corresponds to satellite lines of the 5a’ and 4a’ bands, the former being responsible for the shoulder appearing at 23 eV in the spectrum at @ = 3”. Owing to the different shape of the Fourier transforms of these orbitals, each of the satellite lines can be unam-
bigously
assigned in this way.
Acknowledgement
“,
The financial support fully acknowledged.
10-l
0
0”
of the CNR (Rome) is grate-
2.
References 1.
[ 1] L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 t Fig. 4. Electron momentum distributions in the inner-valence region of CI12=C~iBr. Top: experimentd distributions At tv.o particulx scpxatiun energies; bottom: theoretical curves obt.dnrd from the computed 5.1’ and 4a’ orbitals.
(based on the 2ph TDA results) has been reported elsewhere [5 ] _The qualitative agreement with the experimental curves of figs. 3a-3g allows one to analyze the contribution of each single SCF eigenvector to the final cross secUon, thus confirming the assignment of the bands. It is clear, in this case. that the computed pole strengths lead to the correct qualitative behaviour of the reconstructed distributions. The situation is not as simple for the two EMDs at 24.0 and 29.1 eV_ In this inner-valence region, the theoretical reconstruction fails to reproduce the correct shape of the EMD curves due to the computed shift of the bands discussed previously together with rhe pronounced satellite broadening and the strong overlap of the 5a’ and 4a’ bands. From fig. 4. it is clear tllat the main contribution to the experimental curves comes from the 4a’ ionization, whose pure theoretical distribution has the correct behaviour at srn~llq values, whereas the full theoretical reconstruction overemphasizes the 53 contribution. On the other hand. the band between 22 and 27 eV in the spectruni at cf, = O” (4 = 0) modifies in sha e when the spectrum is recorded at @ = 3” (q = 0.5 ~0 P) (cf. figs. 2a and 2~) in such a way so as to show that it is mainly due to the superposition of two different ionizations 484
(1977) ‘05. [Z] E. Wcigold and LE. McCarthy, Advan. At. Mol. Phys. 14 (1978) 127. 131 A. GiardiniCuidoni. R. Fantoni and R. Camilloni. Comm. At. ?.lol. Phys. 10 (1981) 107. [4] R. Cambi, G. CiuoUo, A. Sgamellotti, F. Tarantelli, R. Fantoni, A. GiardiniGuidoni and A. Sergio, Chem. Phys. Letters 60 (1981) 295. [5] R. Fantoni, A. GiardiniCuidoni, R. Tiibelli, R. Cambi, G. CiuBo, A. Sgamellotti and F. Tarantelli, Mol. Phys. 45 (1982) S39. [6] R. Cambi, G. Ciullo. A. Sgamellotti, F. Tarantelli, R. Fantoni. A. GiardiniCuidoni, R. Tiribelli and hl. Rosi, Chem. Phys. Letters 90 (1982) 445. [ 7] LE. hlcCarthy and E. Weigold, Phys. Rept. C27 (1976) 27.5. [S] R. Camilloni. G. Stefani, R. Fantoni and A. GiardiniGuidoni, J. Electron Spectry. 17 (1979) 209. [9] T-H. Dunning, Jr.. J. Chem. Phys. 53 (1970) 2823. [ 101 T-H. Dunning Jr., Chem. Phys. Letters 7 (1970) 423. [ 1 l] P.G. Mezey, M.H. Lien, K. Yates and G. Csbmadia, Theoret. Chim. Acta 40 (1975) 75. [ 121 V-R. Saunders and M-F. Guest, “AWlOL3”, Daresbury Laboratory SERC, Wanin~ton WA4 4AD. [ 131 A-l. Dixon, S.T. Hood, E. Weigold and G.RJ. Wiams. J. Electron Spectry. 14 (1978) 267. [ 141 J. Schirmer and L.S. Cederbaum, J. Phys. Bll (1978) 1889. [ 151 J. Schirmer, Phys. Rev. A26 (1982) 2395; J. Schiimer, LX Cederbaum and 0. Walter, Phys. Rev_ A, to be published. [ 161 0. Walter and J. Schiier. J. Phys. B14 (1981) 3805; J. Scbirmer and 0. Walter, to be published. [ 171 W. von Niessen and L-S. Cederbaum, Mol. Phys. 43 (1981) 897. [ 181 W. van Niessen, L. Asbrink and G. Bieri, J. Electron Spectry. 26 (1982) 173.