The Auger spectra of CF4 in the light of foreign imaging
F. O. Gottfried and L. S. Cederbaum
Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, 69120 Heidelberg,
Germany
F. Tarantelli
Dipartimento di Chimica, Università di Perugia, I-06123 Perugia, Italy
~Received 29 December 1995; accepted 21 February 1996!
The fluorine and carbon Auger spectra of CF4 are investigated by computing very many dicationic
states in the valence region up to 120 eV with the Green’s function method. An analysis of the
double hole density in the correlated states of CF11
proves that pronounced hole localization
4
phenomena at the fluorine atoms take place in almost all the final states of the Auger decay. We
discuss how these phenomena are at the origin of the observed fluorine and carbon Auger spectral
profiles and, in particular, how they provide a complete and conclusive interpretation of the spectra.
The intra-atomic nature of the Auger process allows us, by a simple convolution of appropriate
~localized! one-site components of the computed two-hole density distribution, to obtain line shapes
which are in close agreement with experiment. To show the general validity of the presented
arguments we also compare the results for CF4 to the Auger spectra of BF3 . The central atom
spectrum of these molecules can be understood in the light of the recently introduced foreign
imaging picture of Auger spectroscopy.
© 1996 American Institute of Physics.
@S0021-9606~96!01420-5#
I. INTRODUCTION
Electron spectroscopists have been interested in the Auger spectra of the CF4 molecule for more than two
decades.1–5 Since the first electron-impact carbon and fluorine spectra of CF4 were published in 1969 by Siegbahn
et al.,1 and in 1982 by Rye and Houston2 it became a textbook example for a molecule exhibiting complex Auger
spectra.3 Although the spectra of Siegbahn were recorded at
higher resolution compared to those of Rye and Houston,
they could possibly contain peaks resulting from autoionization or resonance Auger processes as a consequence of using
electron impact excitation methods. To overcome these deficiencies the C 1s and F 1s Auger spectra were recently
recorded at a very high resolution by Griffiths et al.4 using
monochromatized Ala x-rays at 1487 eV. In these experiments peaks appearing from autoionization or resonance Auger processes are absent.
There is a strong imbalance between the highly resolved
experimental data available for CF4 and the accuracy of the
associated theoretical work. Rye and Houston carried out
calculations based on a semiempirical model.2 In order to get
the carbon spectrum aligned with their experimentally observed peaks they shifted the theoretical lines with two different values of the hole–hole interaction energy U. This led
them to the interpretation that the carbon spectrum is due to
two components, a localized component where the two holes
are located on the same C-F bond and a delocalized component with the two holes located on two different bonds. The
two different values of U correspond to the localized and
delocalized states, respectively. Larkins6 investigated the
CF4 molecule in a delocalized orbital model. He was able to
reproduce the main features of the fluorine spectrum in accordance with Rye and Houston who, in that case, had only
9754
J. Chem. Phys. 104 (24), 22 June 1996
to take into account one value of U. But the delocalized
orbital model, inherently unable to describe localization effects, failed to predict even the coarse structure of the C
spectrum. Recently Griffiths et al.4 undertook Green’s function calculations to simulate their experimental data. The
agreement between theory and experiment was not very satisfactory as there were still open points in the interpretation
and understanding of the spectra. Especially for the fluorine
KLL, the onset of the experimentally observed spectrum,
which lies almost 10 eV above the theoretical double ionization potential ~DIP! threshold, remains unsatisfactorily
explained.4 A detailed reproduction of the spectral profiles is
still missing as much as a deeper understanding of the general nature of the spectra. This is the aim of the present
paper. We have carried out extensive ab initio Green’s function calculations and analyzed the correlated wave functions
in the light of the recently introduced foreign imaging
phenomenon.7 The foreign imaging picture is founded on the
occurrence of pronounced hole localization and has proved
to be a suitable tool for the analysis and understanding of
complex Auger spectra.8 To show the generality of the foreign imaging phenomenon in Auger spectroscopy of ionic
systems we also give an analysis of the results of the previously published Auger spectra of BF3 .9
II. COMPUTATIONAL DETAILS
The theoretical framework, which was used to compute
the dicationic states of CF4 in the outer valence part of the
double ionization spectrum, is based on the second order
approximation scheme for the two-particle Green’s function,
known as the algebraic diagrammatic construction ~ADC~2!!.
It has already been discussed extensively in the
0021-9606/96/104(24)/9754/14/$10.00
© 1996 American Institute of Physics
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Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
literature.10–15 For a general overview of the theory and its
application to Auger spectroscopy see Ref. 16.
We briefly recall here that the approximation scheme
used for the pp-propagator leads to a symmetric eigenvalue
problem in the space of dicationic configurations of the system under consideration, in which the eigenvalues and eigenvectors can be directly related to the double ionization energies and to the residue amplitudes of the propagator,
respectively. This approximation takes into account the
2p – 2h ground state correlations as well as the 4h – 2 p contributions for the 2h main states, while the explicit configuration space only comprises the ionic 2h and 3h – 1 p configurations ~defined in the basis of the neutral ground state
Hartree–Fock orbitals!. The resulting eigenvalues give sizeconsistent ionization energies which are correct beyond second order for main states ~i.e., states perturbatively derived
from 2h space! and beyond first order for satellite states
~derived from 3h – 1p configurations!.
Using ADC~2! we computed double ionization potentials
~DIPs! and the pole strength distribution of the valence dicationic states of CF4 and BF3 in the energy range extending
up to 120 eV. It is expected that in the outer–outer region
where both holes are removed from the outer valence orbitals
~‘‘outer’’ means F 2 p carrying orbitals! the independentparticle picture is largely valid ~once hole localization effects
are accounted for! and therefore a relatively small number of
states with a large 2h hole weight are contributing to the
spectra. In the inner–outer ~‘‘inner’’ would refer to F 2s
carrying orbitals! and inner–inner region breakdown effects
may occur leading to a large number of states with a small
2h hole weight resulting in areas with a high density of
states. To extract roots from a dense inner part of the spectrum of a large eigenvalue equation is a very difficult problem per se. On the other hand, exactly because of the high
density of relevant states, rather than individual eigenvectors
we are interested in computing with enough accuracy the
envelope of the dense pole strength distribution which, as
will be discussed, can be related to the Auger spectrum. This
task can be accomplished very effectively by employing a
block-Lanczos procedure using as seed the 2h configuration
space ~main space!. This technique can be shown8,17 to provide a convergence rate on the ‘‘spectrum’’ of main space
components which is exponential in the width of the lines
making up the spectrum. In the present case, with an assumed width of ;1.5 eV, full convergence was obtained
after 100 block-Lanczos iterations. The states in the outer–
outer region of the spectra were also individually converged.
The calculations for the CF4 and BF3 molecules have
been carried out in a triple-zeta basis set18 consisting of
5s,3p Cartesian Gaussians on the first row elements enlarged by polarization functions on each atom with an exponent of 0.72 for C, 0.5 for B, and 1.62 for the F atom.19 The
experimental bond lengths for C-F and B-F of 1.32 Å and
1.295 Å, respectively, have been used.20 The active molecular orbital space in T d symmetry for CF4 in the ADC calculations comprises 90 valence-type Hartree–Fock orbitals.
Consequently the ADC matrices range in size from 25192 to
35552, depending on space/spin symmetry ~in the D2 sub-
TABLE I. Mulliken population analysis of the molecular orbitals of the
valence shell ~divided by the blank line into inner and outer valence shell! of
CF4.
HF energy ~eV!
State
C
CF
F
FF
249.3252
245.8723
1a 1
1t 2
0.0618
0.0278
0.2016
0.1335
0.6869
0.8522
0.0498
20.0136
227.6512
224.3521
220.9088
219.4960
218.7330
2a 1
2t 2
1e
3t 2
1t 1
0.1749
0.1645
0.0037
0.0207
0.0000
0.1266
0.1268
0.0380
0.0731
0.0000
0.6982
0.6978
0.9263
0.9454
1.0894
0.0003
0.0109
0.0320
20.0392
20.0894
group!. For BF3 72 valence-type Hartree–Fock orbitals have
been included in the ADC calculations leading to matrix
sizes of 8131 to 12398 ~in the C 2 v subgroup!. The results of
the calculation of the dicationic states of BF3 have already
been given elsewhere.9 Therefore, we concentrate in the next
section on the results of CF4 and come back to BF3 for
comparison while discussing localization effects and the
simulation of the Auger spectra.
III. DICATIONIC STATES AND DOUBLE IONIZATION
ENERGIES OF CF4
The ground state electronic Hartree–Fock configuration
of the CF4 molecule ~in T d symmetry! is given by
~ core!~ 1a 1 ! 2 ~ 1t 2 ! 6 ~ 2a 1 ! 2 ~ 2t 2 ! 6 ~ 1e ! 4 ~ 3t 2 ! 6 ~ 1t 1 ! 6 .
~1!
The core comprises the K-shells (1s) of the fluorine and
carbon atom. An interpretation of the orbitals can be obtained from a Mulliken population analysis ~see Table I!.
The inner valence part, namely the orbitals 1a 1 and 1t 2 are
essentially the fluorine 2s orbitals. The bonding sp 3 hybrid
orbitals are expressed by the 2a 1 and the 2t 2 orbitals. The
remaining eight orbitals are of fluorine lone pair character.
The double ionization energy and 2h composition of the
states of the outer-outer region of the spectrum are reported
in the appendix in Table IV. According to the large energy
gap between the fluorine lone pair orbitals and the orbitals
constituting the s -bonds and the character of the orbitals,
one expects well separated regions in the spectrum with the
following distribution of the outer valence two-hole states
with
increasing
energy:
(fluorine lone pair) 22 ,
21
21
(fluorine lone pair) ( s -bond) and ( s -bond) 22 .
At the onset of the spectrum in the energy range up to 42
eV one finds the states with ~fluorine lone pair!22 character,
followed by the ~fluorine lone pair!21 ( s –bond!21 states in
the interval between 43 eV and 49 eV. The singlet–triplet
splitting within this part of the spectrum is of the order of
several 1021 eV. But for higher energies the situation becomes less obvious. Together with the expected
( s -bond) 22 states one finds states with again
(fluorine lone pair) 21 ( s -bond) 21 character. As will be described in the next section the latter states refer to ‘‘one-site’’
states where the two holes are localized on the same fluorine
atom. The remaining states of the outer-outer part of the
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
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Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
spectrum are all of ‘‘one-site’’ character. By analyzing these
states at the high energy end, one finds singlet/triplet pairs of
states with the expected (fluorine lone pair!21 ( s -bond) 21
and ( s -bond) 22 composition. The singlet–triplet splitting
for these pairs is of the order of few eV with the corresponding singlet states lying higher in energy. This large singlet–
triplet splitting and the large energy gap between the states
2a 1 and 2t 2 exercising the s-bonds leads to the fact that the
various groups of states are not well separated in energy but
do overlap. At the high energy end of the outer–outer states
one finds a singlet ( s -bond!22 state where both holes are in
a 2a 1 orbital. For this state there is of course no triplet counterpart. In the following section these arguments will be
made more quantitative.
Another important result emerging from these data is
that already at low double ionization energies a very strong
two-hole configuration mixing in the composition of the
most states arises. These observations point very clearly to
an intractability of the CF4 double ionization spectrum
within an independent-particle framework and are consistent
with the requirements dictated by atomic localization of
positive charges. This was illustrated in detail for the case of
BF3 .9
IV. ATOMIC LOCALIZATION OF THE TWO HOLES IN
THE DICATIONIC STATES
A. Two-hole population analysis
To analyze the hundreds or even thousands of dicationic
states contributing to the observed Auger spectra of already
medium sized molecules is a challenging task. Because of
the large number of states the tool must be simple, but still
powerful enough to allow for the characterization and interpretation of the dicationic manifold. To answer the question
if and to what extent localization of the two valence holes in
the final states of the Auger process takes place, we use a
two-hole population analysis of the dicationic states.9 By this
analysis, the residue amplitudes of the two-particle propagator given by the total 2h part of the ADC eigenvectors are
decomposed into their localized atomic contributions. The
sum of the contributions of the atomic orbital hole pairs
p,q to the total pole strength, where both p and q refer to
basis functions centered on a given atom A, is the ‘‘onesite’’ pole strength of that atom, and measures the extent to
which both holes in the dicationic state are localized on atom
A. Similarly, the ‘‘two-site’’ character of a state is measured
by the sum of terms p and q which refer to basis functions
centered on two different atoms A and B. Thus the predominance of one of these contributions for a given state indicates
that the two vacancies are strongly localized in space ~either
at the same or each at another atomic center!, whereas states
for which more than one component is significantly present
are characterized as having correspondingly delocalized
holes.
In the case of CF4 we can thus separate the total 2h pole
strength of the ADC states in contributions which we denote
as C22 ~two holes on the carbon atom!, F22 ~two holes on
21
~two holes on different
the same fluorine atom!, F21
1 F2
21 21
fluorine atoms!, and C F ~one hole on the carbon and
one on a fluorine atom!. The relevant results for the outer–
outer valence dicationic states are reported in the appendix in
Table V.
By analyzing these states it becomes obvious that the
TABLE II. Average two-hole population and variance in percent of the total 2h pole strength of the eight groups of dicationic states of BF3 as well as the
21
average s 22 , s 21 p 21 and p 22 contribution ~in percent! to the main component, F22 or F21
1 F2 , respectively. The energy ranges of the groups do not overlap.
The labeling of the peaks refer to the figures. The underlined numbers represent the largest and hence most relevant numbers.
21
F21
1 F2
F22
Peak
B22
A
0.54060.518
B1
B2
C
s 22
s 21 p 21
p 22
1.58761.405
0.71860.375
79.64566.565
3.103
96.655
1.776
60.94567.116
7.084
91.290
1.04360.948
21
s 21
1 s2
21
s 21
1 p2
21
p 21
1 p2
0.123
83.93767.050
7.762
92.114
13.93665.829
0.259
2.04760.201
B21 F21
2.14062.546
16.36264.591
3.27562.857
26.54461.795
10.46467.993
17.79963.961
4.605
D1
D2
E
0.78160.586
0.028
59.938619.168
90.242
9.534
7.479
78.64665.876
89.960
2.608
1.08160.810
1.41860.329
3.32863.823
18.23164.785
21.050619.471
14.63161.830
5.64265.238
20.52561.720
91.111
F
1.27260.701
76.089
44.232627.477
17.335
79.01865.464
91.258
17.84863.949
6.523
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
74.72963.531
6.803
36.649629.939
4.123
2.119
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Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
TABLE III. Average two-hole population and variance in percent of the total 2h pole strength of the eight groups of dicationic states of CF4 as well as the
21
average s 22 , s 21 p 21 and p 22 contribution ~in percent! to the main component, F22 or F21
1 F2 , respectively. The energy ranges of the groups do overlap.
The labeling of the peaks refer to the figures. The underlined numbers represent the largest and hence most relevant numbers.
21
F21
1 F2
F22
Peak
C22
A
0.48260.739
B1
B2
C
s 22
s 21 p 21
p 22
4.01565.459
1.20561.031
64.395615.214
4.879
94.798
0.167
46.81763.612
11.936
87.777
1.20061.550
21
s 21
1 s2
21
s 21
1 p2
21
p 21
1 p2
0.143
81.425613.351
6.122
93.734
14.07869.116
0.325
4.65060.332
C21 F21
8.45368.871
20.21767.780
14.18468.594
36.01660.8818
12.51764.718
19.42066.494
11.786
D1
D2
E
0.83760.903
0.595
54.705613.746
82.228
17.270
12.785
67.875610.230
82.815
4.367
2.08660.9423
1.66260.998
9.62267.648
23.92067.767
20.53860.832
17.22963.715
12.81166.888
21.58463.802
78.504
F
2.37560.701
70.296
56.94369.927
22.109
70.927613.298
68.897
22.10463.663
67.13268.862
8.721
19.306
12.792
18.57869.469
7.027
states can be separated into two groups with a small overlap
region in between. At the lower energy end of the spectrum,
up to 48.0 eV we find ‘‘two-site’’ states which are dominated
21
by their F21
component. The one-site components C22
1 F2
22
and F are systematically one or two orders of magnitude
smaller. The coefficient C21 F21 , in the beginning of the
spectrum also an order of magnitude smaller, increases towards the end of the first group because of the more bonding
character of the 2a 1 and 2t 2 orbitals. In an overlap region
ranging from 48.4 eV to 51.0 eV we find states with a domi21
22
nating F21
component. The second group starts at
1 F2 or F
52.0 eV and ranges up to 61.5 eV. States in this group are
now dominated by the F22 component and can be characterized as ‘‘one-site’’ states. Therefore, in almost all the states
in the dicationic manifold the two valence holes are localized
on different fluorine atoms or on the same fluorine atom.
Only few states in the second group as well as in the overlap
region have components of comparable magnitude and are
delocalized. The remaining states at higher energies which
are not reported in Table V can be grouped in the same spirit.
B. Similarities in the dicationic states of BF3 and CF4
To show the generality of the grouping of states we compare in the present section the results for BF3 and CF4,
which are summarized in Tables II and III. The results of the
population analysis for BF3 and CF4 are illustrated in Fig. 1,
where we have reported separately contributions of the F22
21
~full area! and F21
1 F2 ~shaded area! components to the total
2h pole strength. The total pole strengths are convoluted
with a FWHM ~full width at half maximum! of 1.5 eV. Figure 1 illustrates clearly that the states cluster in six groups,
where the groups B and D actually split into two separate
groups, namely, singlet and triplet states. It also makes evi22
21
character
dent the alternating pattern of the F21
1 F2 and F
as well as the complete dominance of one or the other component. The difference between the sum of the fluorine contributions and the total 2h curve is essentially due to the
B21 F21 or C21 F21 component since the B22 and C22 terms
are systematically orders of magnitude smaller.
With the aid of Tables II and III and Fig. 1 we can now
analyze the various groups of states and their origin in more
detail. The onset of the spectrum for BF3 and CF4 takes place
at almost the same energy. The first group ~labeled A in the
21
tables and in the figures! is dominated by the F21
1 F2 population. These states are clearly characterized by their fluorine
21
p 21
1 p 2 character, and have holes localized on two different
fluorine atoms. The charge separation minimizes the hole–
hole repulsion so that this group is found at the low energy
side of the ionization spectrum ~high kinetic energy of the
Auger electrons!. In group A, due to the relatively large distance between the two localized holes, singlet/triplet pairs of
states lie close in energy, separated by only a few tenths of
eV. Despite their common characterization, the peaks A in
Fig. 1 of BF3 and CF4 exhibit obvious differences in their
complexity. The comprehensive description of the states of
CF4 in Tables III and IV and similar results of BF3 listed in
Ref. 9 allows us to analyze these differences in more detail.
According to the arrangement of the outer valence molecular
orbitals, as already stated in the previous section, one would
generally expect to find a 3-peak substructure underlying
group A, namely, ~i! ~fluorine lone pair!22 , ~ii!
~bonding orbital!21
and
~iii!
~fluorine lone pair!21
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
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Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
FIG. 1. The full curves represent the Gaussian convolution ~FWHM 1.5 eV!
21
of the total 2h pole strength, the shaded areas visualize the F21
1 F2 compo22
nents and the full areas reflect the F
component for ~a! BF3 and ~b!
CF4, respectively.
~bonding orbital!22 . But already for the less complex peak A
of BF3 this simple 3-peak structure is not obviously realized.
At first glance peak A looks like it is composed of four
peaks, where the fourth peak is found as a shoulder on the
higher energy side. The analysis of the data renders it obvious that the group of states characterized by ~ii! split into two
parts according to the localization of one hole either in the
bonding 2e 8 or 2a 18 orbital. Also peak A of CF4 fits into the
expected 3-peak structure after one realizes that in this case
the groups of states characterized by ~i! and ~ii! each split
into two peaks. The splitting of the group ~i! is a consequence of the gap of 1.4 eV between the outermost valence
orbitals and the 1e orbital. Correspondingly, one finds a
group of states with the two holes residing in the outermost
valence orbitals and another group of states where one hole
results out of a 1e orbital. The splitting in the second group
~ii! is again due to a gap between the molecular orbitals. The
bonding 2a 1 and the 2t 2 orbitals are separated in energy by
3.3 eV. This leads to two distinct groups, where the first
group always has one hole in the 2t 2 orbital and the second
always has one hole in the 2a 1 orbital. The groups ~i! and ~ii!
themselves are separated in energy by 1.4 eV. The bonding
character of one hole in group ~ii! is also reflected by an
increase of the C21 F21 population compared to the states of
group ~i! by about a factor of two. We would like to point
out that this detailed characterization of the groups holds for
the vast majority of states within each group and therefore
reflects the dominating character of the group. It cannot
strictly be applied to each individual state.
Group A is followed by its one-site counterpart group B,
comprising p 22 states with two holes confined on the same
fluorine atom, according to the large F22 component. While
for BF3 the highest lying states of group A and the lowest
lying states of group B are clearly separated in energy by a
gap of almost 1.5 eV, in the case of CF4 there is a small
overlap between the groups. This is due to the fact that for
CF4 group A covers a much broader energy range, because
of the larger energy difference between the outer valence
orbitals. The centroids of the peaks A and B are in both cases
separated in energy by more than 10 eV. The data show that
the states of group B actually split in two distinct subgroups,
B1 and B2 . Peak B1 comprises singlet and triplet states,
while the higher lying B2 is made up only of singlet states.
The singlet–triplet splitting within this group is of the order
of few eV and can easily be understood in view of the strong
localization of both positive charges on the same atom. Due
to this large singlet–triplet splitting, the states of group B
cannot be arranged according to the 3-peak substructure of
group A as already mentioned in the previous section. By
examining the groups B in Fig. 1, one finds that they exhibit
a 4-peak pattern with large similarities between BF3 and
CF4. The first three peaks, starting at about 50 eV, are clustered into B1 , whereas the fourth peak reflects B2 .
The next group of states ~labeled C! are characterized as
having the first hole in the outer valence shell of one fluorine
atom and the second in the inner valence shell of another
fluorine atom. The energy gap between the lowest lying
states of group C and the highest lying states of B2 is ;4.2
eV and ;0.9 eV for BF3 and CF4, respectively. These twosite states, again characterized by the dominance of the
21
F21
population, are followed by their one-site counter1 F2
parts D1 and D2 . Due to the presence of the inner valence
hole the singlet–triplet splitting is very large and therefore
triplets (D1 ) and singlets (D2 ) are widely separated in energy by about 10 eV. Again the triplets are lying lower in
energy than the singlets. Due to the broader energy interval
of the orbitals in the outer valence region of CF4 and the
therefore larger energy range covered by group C, the groups
C and D1 are not separated but D1 falls into the energy range
of C. The next group E comprises doubly ionized states of
the fluorine inner valence shell with the lowest lying states
clearly separated from the highest lying states of group D2
by almost 4.6 eV or 3.0 eV for BF3 and CF4, respectively.
Here again the two holes are localized on different atoms.
The remaining group of states, F, comprises essentially singlet states with two holes in the inner valence shell of the
same fluorine atom, lying in energy around 100 eV. Because
of the strong correlation effects in the energy range of groups
C to F, the intensity of the various peaks spreads over many
states with only very few states having a total 2h pole
strength larger than 0.1.
C. A simple localized picture for peaks A and B
The number and character of the main states underlying
peaks A and B can be understood within a simple model,
assuming that the outer valence orbitals are localized. This
localization is achieved by transforming the delocalized mo-
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
lecular orbitals by an appropriate symmetry transformation
to localized orbitals. In the case of BF3 the delocalized 2a 18
and 2e 8 will transform to three equivalent s p 2 localized
bonding orbitals, the 1a 92 and 1e 9 to three equivalent out-ofplane lone pair orbitals and the 3e 8 and 1a 82 to three equivalent in-plane lone pair orbitals. Therefore, to each fluorine
atom belongs one binding s p 2 orbital and two lone pairs.
The fluorine 2s lone pair is energetically well separated from
the outer valence region and need not be considered for our
purposes. Counting the number of states in which the two
holes could localize on two distinct fluorine atoms ~peak A!
yields 9 different states: 4 states where the two holes reside
in the lone pair orbitals, 4 states where one lone pair orbital
and one bonding orbital is involved and 1 state where both
holes are localized in the bonds. Consequently, taking into
F
2 e y 8 2 e y 8 1 ~ V y 8 y 8 y 8 y 8 6V y 8 y 8 y 8 y 8 !
1
2
1 2 1 2
1 2 1 2
1
2
1 2 1 2
The upper sign holds for singlet states and the lower holds
for triplet states, respectively. The 2-electron integrals are
defined as
1
u F ~ 1 ! F l~ 2 ! & .
u r12u k
Using localized orbitals obtained by a unitary transformation
of the occupied Hartree–Fock orbitals we are able to calculate the eigenvectors of the above simple model. The nonnormalized eigenvectors of the CI matrix are found to be of
the order of ~20.002,1! and ~1, 0002! for singlet states as
well as for triplet states. Hence, the two configurations practically decouple. The integral V y 8 y 8 y 9 y 9 is by a factor of 100
1 2 1 2
smaller than the integral V y 8 y 8 y 8 y 8 on the diagonal. The ex1 2 1 2
change integral V y 8 y 8 y 9 y 9 is even one order of magnitude
1 2 2 1
smaller.
These arguments no longer hold for the one-site states of
peak B. In this case the singlet CI matrix reads
F
22 e y 8 1V y 8 y 8 y 8 y 8
h.c.
1 2 2 1
2 e y 9 2 e y 9 1 ~ V y 9 y 9 y 9 y 9 6V y 9 y 9 y 9 y 9 !
h c.
V i jkl 5 ^ F i ~ 1 ! F j ~ 2 ! u
account the possibility of creating singlet and triplet states as
well as the number n of F-F pairs in the given molecule we
obtain 2*9*( n2 ) as the number of two-site states. But even
some details in the dicationic manifold of BF3 can be studied
within this simple localization picture. Analyzing the results
of the calculations for BF3 listed in Table I in Ref. 9 one
notices the absence of a mixing between the states where
both holes are in the in-plane orbitals and the states where
both holes are in the out-of-plane orbitals. This phenomenon
can be understood in the localization model by investigating
a CI matrix. To analyze this coupling the simple model has
to include only two 2h configurations, namely u v 81 v 82 & and
u v 91 v 92 & , where the subscript on the orbitals refers to the site
and the prime ~double prime! denotes in-plane ~out-of-plane!
symmetry. Thus, the singlet and triplet CI matrix for the
two-site states reads
1 ~ V y 8 y 8 y 9 y 9 6V y 8 y 8 y 9 y 9 !
1 2 2 1
1V y 8 y 8 y 9 y 9
22 e y 9 1V y 9 y 9 y 9 y 9
G
9759
1 2 2 1
G
.
the localization of the two holes on different lone pairs orbitals, with the triplet states lying considerably lower in energy than the singlet states. Having both holes localized in
the same lone pair accounts for 2 singlet states. These two
type of states are characterized as having the lowest
B21 F21 population of the one-site states. One hole localized
in the bond and the other localized on a fluorine lone pair
gives another 2 singlet and 2 triplet states. The triplet states
are again lower in energy than the singlet states. For these
type of states the B21 F21 population is increased by almost
a factor of two. After a gap in energy follows the last singlet
state, where both holes are confined to the bonding orbital.
This state is again characterized by a substantial increase of
the B21 F21 population. Based on these straightforward considerations and with the help of the appropriate tables given
in Refs. 8,9 it is possible to characterize the individual outerouter valence states in the dicationic manifold of BF3 and
SiF4 and they serve as a guideline for the less clear-cut example of CF4.
.
Here, the non-diagonal matrix element is given by the onecenter integral V y 8 y 8 y 9 y 9 . This integral is of considerable
size, leading to eigenvectors of the order ~20.970, 0.242!
and ~20.242, 20.970! and, consequently, to the observed
mixing of the configurations under consideration. The number and character of one-site states can also easily be derived
from our simple localization model. Counting the states,
where the two holes can localize on the same fluorine atom,
one finds 6 singlet and 3 triplet states. The total number of
one-site states is given by (9 * n), where n is the number of
bonds in the given molecule. Energetically most favorable is
V. AUGER SPECTRA
A. General remarks
Already medium sized polyatomic molecules exhibit
very complex Auger spectra. At the origin of these spectra
are very many dicationic states contributing to the observed
spectral profile. Therefore it seems practically impossible to
calculate exact transition rates for these hundreds or thousands of states. But the analysis of the correlated wave function in terms of localization of the two holes in the final
dicationic states ~as outlined in the previous section! provides us with the key not only for the simulation of the
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
9760
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
experimentally observed line shapes but also for the interpretation and understanding of these complex spectra.
Since the Auger decay is essentially an intra-atomic process the Auger intensities reflect the local two-hole density in
the dicationic states around the atomic site where the core
vacancy has been created. It is therefore clear that only states
with a significant relative component of the two-hole density,
which is located at the given atom, can have an appreciable
rate of decay proceeding from the corresponding core hole
state, i.e., the Auger process is site selective. We can further
expect to observe a qualitative correspondence between the
energy distribution of a given X22 component of the pole
strength and the experimental energy distribution of the intensity in the Auger spectrum originating from core ionization of atom X. This correspondence has indeed been shown
to hold with remarkable precision,14–16 especially when
many dicationic states contribute to an observed Auger band.
These arguments appear particularly appropriate for
highly ionic molecules like SiF4 7,8 and BF3 9 because of the
clear-cut localized character of the states.
For the fluorine spectrum of these and similar molecules
one expects a simple 3 regions Auger spectrum, according to
the outer–outer, outer–inner and inner–inner grouping of the
states. The outer–outer, and more pronouncedly the outer–
inner part, split in a lower lying triplet part and a higher lying
singlet part as it has been shown in Sec. IV. These groups are
labeled B1/2 , D1/2 and F. This simple atomic-like appearance
of the ligand spectrum has already been defined as the ‘‘selfimaging’’ picture in the Auger process.
The situation is different for the central atom spectrum,
where the two-hole density at the central atom is small and
very uniformly distributed over the entire spectrum of doubly ionized states. Consequently, all the atomic information
is lost and the spectrum reflects in every detail the full set of
dicationic states, whose energy distribution is exclusively determined by the surrounding molecular environment where
the electron vacancies are produced. This prevents the occurrence of any a priori strong selection rules similar to those
found in the fluorine spectrum and so all groups should be
visible in the central atom spectrum. Therefore each group of
the one-site states in the fluorine spectrum is preceded by its
two-site counterpart, which is lower in energy because of the
minimization of the hole–hole repulsion energy. Hence the
central atom spectrum consists in general of six groups labeled A, B1/2 , C, D1/2 , E and F. This characteristic of Auger
spectra has recently been defined as ‘‘foreign-imaging.’’7
In the following we show the validity of the arguments
for the highly ionic molecule BF3 , but also show that they
are qualitatively still correct for the less ionic molecule
CF4 and that they provide a general guideline for the analysis
of complex Auger spectra.
FIG. 2. Experimental ~upper! and theoretical ~lower! F Auger spectrum of
BF3 . The theoretical spectrum is obtained by Gaussian convolution
~FWHM 1.5 eV! of the F22 two-hole populations resulting from ADC~2!
calculations. Peaks indicated by s are satellites not belonging to the normal
Auger spectrum ~Ref. 22!.
spectively, under the experimental profiles taken from Ref.
21. As in Fig. 1, a FWHM of 1.5 eV and a singlet/triplet ratio
of 3:1 has been used for the convolution.
Since all of the six groups of states in the dicationic
manifold of BF3 are either fully one-site or fully two-site
states, they either appear in the fluorine spectrum ~one-site!
or have almost zero intensity ~two-site!. As a result the states
in groups A, C and E carry little intensity and these groups
have virtually disappeared from the fluorine spectrum of Fig.
2, leaving only groups B, D and F. The group B1 matches
exactly the experimental line shape at '52 eV. At the resolution we have used, the intense peak exhibits two components, as is also experimentally evident. At the high DIP side
B. BF3
In Fig. 2 and Fig. 3 we illustrate the results for the theoretical fluorine and boron KLL Auger spectra obtained from
the appropriate one-site 2h populations, F22 and B22 , re-
FIG. 3. Experimental ~upper! and theoretical ~lower! B Auger spectrum of
BF3 . The theoretical spectrum is obtained by Gaussian convolution
~FWHM 1.5 eV! of the B22 two-hole populations resulting from ADC~2!
calculations.
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
FIG. 4. Experimental ~upper! and theoretical ~lower! F Auger spectrum of
CF4. The theoretical spectrum is obtained by Gaussian convolution ~FWHM
1.5 eV! of the F22 two-hole populations resulting from ADC~2! calculations.
of the peak around 57 eV a further feature B2 is extremely
well reproduced. At the low DIP side of the main peak there
is a small structure which can be found as a shoulder in the
experimentally observed line shape. The next visible group
of states in the spectrum, which is located around 72 eV, is
labeled D1 and comprises only inner–outer triplet states, followed by their corresponding singlet states D2 at '80 eV.
The large singlet–triplet splitting of almost 10 eV is due to
the presence of the inner-valence hole. The obvious deviation of the relative intensity of peak D2 from the experiment
can be explained by an overestimation of the singlet states
due to the singlet/triplet ratio we applied. Finally the group
of states labeled F appears at around 100 eV. The states in
that group are of inner–inner character. The relative position
of that peak is somewhat underestimated in the calculation
due to the strong relaxation effects which are not completely
accounted for in our second-order approximation scheme
~ADC~2!!. The peak at around 65 eV and the broad structure
at around 90 eV in the experimental spectrum can be attributed to shake-up and shake-off satellites and do not belong to
the normal Auger spectrum.22 Therefore no states with a substantial F22 intensity are calculated in these energy ranges.
The boron Auger spectrum is much more complex. As
stated above the Auger process is here probing the two-hole
density at the central atom which is small but uniformly
distributed over the entire spectrum. As one can see from
Fig. 3 and a recently published experimental measurement23
all the peaks appear in the boron spectrum of BF3 . All the
one-site peaks B, D and F are preceded by their two-site
counterparts A, C and E, respectively. The singlet–triplet
9761
FIG. 5. Experimental ~upper! and theoretical ~lower! C 2 p Auger spectrum
of CF4. The theoretical spectrum is obtained by Gaussian convolution
~FWHM see text! of the C22 two-hole populations resulting from ADC~2!
calculations.
splitting for the two-site states is small and therefore singlet
and triplet states are contributing to these peaks. The position
of the peak F around 100 eV is in agreement with the last
peak of the experimental spectrum reported in Ref. 23 which
covers a broader energy range than the experimental profile
we used in Fig. 3 but is of much lower resolution.
C. CF4
Figure 4 and Fig. 5 report about the experimental and
theoretical fluorine and carbon KLL Auger spectra of CF4.
The theoretical fluorine spectrum is obtained by Gaussian
convolution of the F22 population. For the carbon spectrum,
because of the partly enhanced C21 F21 coefficient, we have
used a Gaussian convolution of only the diagonal C22 contributions to the population overlap matrix ~i.e., the sum of
overlap terms where both the left and right hole pairs are
strictly localized at the carbon atom, see Ref. 9!. We find the
theoretical spectrum in this case in slightly better agreement
with the experimental data than by using the full C22 population term. The experimental profiles and numbering of the
peaks are taken from Ref. 4.
In the fluorine spectrum a FWHM of 1.5 eV has been
used as well as the usual singlet/triplet ratio of 3:1. One can
identify the expected three regions structure of the spectrum,
and an assignment of the various peaks can be given. An
exception is the lower energy end of the spectrum where the
first two peaks ~labeled 1 and 2! are not reproduced correctly. This discrepancy at the lower energy end of the fluorine spectrum, not so pronounced but also present in the case
of BF3 , is probably due to strong vibrational effects for
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
9762
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
which we do not account. This may also be the reason for the
split in the peaks 4 and 5 which is not reproduced completely. However, an analysis of the underlying states of
peak 4/5 ~see Table V! allows for the characterization of the
two peaks. The states in the energy range between 51.9 eV
and 53.6 eV can be separated into two groups with a gap of
almost 0.6 eV in between. The first group of states from 51.9
eV up to 52.5 eV can therefore be attributed to peak 4 and
the states of the second group ranging from 53.0 eV to 53.5
eV belongs to peak 5. Vibronic effects may shift these
groups slightly against each other making the split in the
experimental spectrum visible.
A convincing assignment can be given for peaks 3, 6, 7,
8 and 9. The peaks are reproduced at the correct energy with
an exception of peak 9 in the outer–inner part which comes
out at a slightly too high energy. This is probably again due
to the strong relaxation effects which are not fully included
in our second-order approximation scheme. The same arguments may also apply for peak 10.
To account for the complicated vibrational effects in the
carbon spectrum we used for the outer–outer two-site states
a FWHM of 1.5 eV, but since a strong source of vibrational
broadening can be associated with the strong hole–hole repulsion in the one-site F22 states, we applied for the remaining states a FWHM of 2.5 eV. Furthermore to illustrate
peaks 1–3 the FWHM in that part of the spectrum has been
slightly reduced. We would like to stress that this choice of
the FWHM does not affect the following discussion of the
nature of the spectrum, and is only made to show that our
data permit the reproduction of many fine details of the experimental profiles once correct band broadenings are incorporated. As already mentioned, the carbon spectrum is an
example of a very complex Auger spectrum and, in contrast
to the fluorine spectrum, the Auger intensity is spread over
the whole energy range. Here an assignment of all peaks can
be given. Peaks 1–4 refer to the two-site outer–outer states.
Peaks 5 and 6 are of one-site outer–outer character with peak
5 having contributions from the overlapping two-site tail of
peak 4. Also at the energy range of peaks 7 and 8 two areas
are overlapping, namely the one-site triplets states referring
to the one-site singlet peak 9 fall into the same energy range
than peak 8. Peak 10 represents the outer–outer two-site
states and its higher energy counterpart ~peak 11! comprises
the outer–outer one-site singlet states.
The similarities between the central atom spectrum of
BF3 and CF4 are depicted in Fig. 6, where the shaded area
visualizes the intensity resulting from the two-site states
21
22
(F21
1 F2 . F ) and the full area reflects the states classified
21
22
as one-site (F22 . F21
1 F2 ). The full curve shows the B
22
and C
contribution to the total pole strength. Figure 6
visualizes the appearance of all the groups in the spectrum as
well as the alternating and complete dominance of one or the
other class of states and, therefore, clearly illustrates the
‘‘foreign-imaging’’ character of the central atom spectrum
for both molecules. The difference between the full curve
and the marked areas is always due to an overlap of the
one-site and two-site regions. The groups in the spectrum of
BF3 have no ~or very little! overlap but for CF4 they partly
FIG. 6. The full curves represent the central atom Auger spectra, the shaded
areas visualize the two-site components and the full areas reflect the one-site
components for ~a! BF3 ~FWHM 1.5 eV! and ~b! CF4 ~FWHM 2.0 eV!,
respectively.
do overlap. Thus the missing part of peak B1 of the CF4
spectrum is due to a small peak in the tail of peak A.
VI. SUMMARY AND CONCLUSIONS
The aim of the present paper was twofold. First, we
wished to show that by performing Green’s function calculations which are beyond second order perturbation theory it
is possible to reproduce the highly resolved experimental
Auger spectra available for CF4 and, in fact, that at least this
level of accuracy is required. Second, that the recently introduced foreign imaging scenario provides the key to a deeper
understanding of the nature of the central atom spectrum. To
show the wide range of applicability of the foreign imaging
picture, we illustrated the similarities of the central atom
spectrum of different molecules like BF3 and CF4.
The hundreds of correlated dicationic states in the
double ionization spectrum of CF4 and BF3 , obtained by the
use of a block-Lanczos method, were analyzed by studying
their two-hole density distribution. It is found that for the
ionic molecule BF3 all the states are dominated either by the
21
fluorine one-site (F22 @ F21
1 F2 ) or by the fluorine two-site
21 21
~F1 F2 @ F22 ) character. This is also true for the case of
SiF4 .8 A clustering in energy of the states occurs due to their
one-site or two-site localization and the outer or inner valence character of the ionized electrons. The two-hole density on the central atom is orders of magnitude smaller but
uniformly distributed over the entire energy space. These
observations hold also for the vast majority of states in the
less ionic CF4, where only very few states are delocalized
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
21
and therefore have F22 ' F21
1 F2 . As a result, also for CF4
almost all the states have the two holes strongly localized in
space, i.e., for a given state they are either localized on the
same fluorine atom or each on another fluorine atom.
The intra-atomic nature of the Auger decay allows us to
select the appropriate one-site contributions to the total pole
strength for a simulation of the fluorine and central atom
spectrum, respectively. The theoretical spectra obtained by
Gaussian convolution of these one-site components are
found to be in good agreement with experiment.
For the fluorine spectrum very strict selection rules are
imposed due to the clustering of the fluorine one-site and
two-site states, i.e., either a peak ~group of states! appears in
the spectrum due to the one-site character of its states or has
essentially zero intensity. Therefore the ligand spectrum is
strictly atomic-like and contains almost no information about
the molecular system.
The central atom spectrum is in general much more
complex. The fact that the two-hole density at the central
atom is small and distributed over the whole energy range
prevents the occurrence of similar selection rules. Because of
the hole localization on the ligands the central atom loses all
atomic information and yields instead a complete and detailed image of the surrounding molecular environment. This
foreign imaging picture is expected to play an important role
in the Auger spectra of all molecular systems in which hole
localization effects take place.
9763
In closing, we would like to comment on the general
influence of molecular dynamics on Auger spectra. For the
Auger decay, one can now distinguish between the cases
where the vibronic coupling is already important for the intermediate states or only for the final states. For instance in
the central atom Auger spectrum of BF3 and CF4, there is
only one B 1s and one C 1s orbital accessible as an intermediate state. Consequently, there is no vibronic coupling in
the core vacancy level and the Jahn–Teller modes play only
a minor role. On the other hand, the Auger decay of these
vacancies populate final dicationic degenerate states of E
symmetry for BF3 and in CF4 also of T symmetry, where
Jahn–Teller coupling prevails. In the situation of the fluorine
Auger spectrum of BF3 and CF4, where the F 1s electron is
ionized, there are several equivalent core levels and therefore
vibronic coupling takes place already for the intermediate
states as well as after the decay also for the final states.24
This may serve as an explanation of the reason why the
purely electronic calculations on the central atom spectra are
in better agreement with experiment than those on the fluorine spectra.
ACKNOWLEDGMENTS
Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. The authors also wish to
thank the ‘‘Vigoni’’ programme between C.R.U.I., Italy and
D.A.A.D., Germany for traveling funds.
APPENDIX
This appendix contains Tables IV and V, giving the detailed description of the dicationic states, their DIPs, composition
and localization properties.
TABLE IV. Computed double ionization potential ~DIP! and composition of the outer valence dicationic states of CF4 for the first group of one-side and
two-side states. The composition reported is given by the square of the 2h components of the ADC eigenvectors with a pole strength ~PS! larger than 0.01.
The 2h configurations are indicated by the occupied orbitals of CF4 from which the two electrons are removed. DIPs with a PS component larger than 0.1 are
boldfaced.
State
DIP ~eV!
PS
2h composition
0.6740(1t 1 )0.1285(3t 2 1t 1 )
0.0315(3t 2 )
0.0089(2t 2 1t 1 )
0.0075(1e) 0.0016(2t 2 3t 2 )
0.6992(1t 1 )
0.0721(3t 2 1t 1 )
0.0499(1e1t 1 )
0.0174(3t 2 )
0.0076(1e3t 2 ) 0.0028(2t 2 1t 1 ) 0.0015(2t 2 3t 2 ) 0.0011(2t 2 1e)
0.5660(1t 1 )
0.1130(3t 2 1t 1 )
0.0809(1e3t 2 )
0.0673(1e1t 1 )
0.0087(3t 2 ) 0.0079(2t 2 1t 1 ) 0.0030(2t 2 3t 2 ) 0.0026(2t 2 1e)
0.8482(3t 2 1t 1 ) 0.0056(2t 2 1t 1 )
0.5559(1t 1 ) 0.1985(3t 2 ) 0.0592(1e) 0.0349(2t 2 3t 2 )
0.6492(3t 2 1t 1 )
0.1261(1e1t 1 )
0.0400(1e3t 2 )
0.0241(1t 1 )
0.0051(3t 2 ) 0.0050(2t 2 1t 1 )
0.7464(3t 2 1t 1 ) 0.0515(1e3t 2 ) 0.0355(2t 2 1t 1 ) 0.0085(1e1t 1 )
0.0040(2t 2 3t 2 ) 0.0017(2a 1 3t 2 ) 0.0016(2t 2 1e)
0.7834(3t 2 1t 1 ) 0.0444(2t 2 1t 1 ) 0.0181(2t 2 3t 2 ) 0.0028(2a 1 1e)
0.4421(1e1t 1 ) 0.3952(3t 2 1t 1 ) 0.0041(2t 2 3t 2 ) 0.0033(2t 2 1t 1 )
0.0022(2t 2 1e) 0.0010(2a 1 1t 1 )
0.2787(3t 2 )
0.2388(1e1t 1 )
0.2036(3t 2 1t 1 )
0.0471(1e3t 2 )
0.0463(1t 1 ) 0.0186(2t 2 3t 2 ) 0.0080(2t 2 1e) 0.0043(2t 2 1t 1 )
0.0025(2a 1 3t 2 )
0.8370(1e1t 1 ) 0.0073(3t 2 1t 1 ) 0.0014(1e3t 2 )
0.3592(1e1t 1 )
0.2287(3t 2 )
0.2080(1e3t 2 )
0.0282(3t 2 1t 1 )
0.0107(2t 2 3t 2 ) 0.0077(1t 1 ) 0.0015(2t 2 1t 1 ) 0.0011(2a 1 1t 1 )
1
E
38.4111
0.8522
3
T1
38.4211
0.8518
1
T1
38.9210
0.8500
A2
A1
3
T1
39.1921
39.4192
39.5308
0.8539
0.8492
0.8515
3
T2
39.7069
0.8499
3
E
T1
39.7912
39.8175
0.8492
0.8489
1
T2
40.1802
0.8487
3
T2
T1
40.2004
40.5374
0.8463
0.8472
1
1
1
3
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
9764
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
TABLE IV. ~Continued.!
DIP ~eV!
PS
2h composition
A2
E
40.7637
40.9922
0.8449
0.8435
1
T1
41.3172
0.8437
1
A1
E
41.5085
41.6737
0.8420
0.8486
3
T2
41.6849
0.8431
1
T2
41.7333
0.8453
3
T1
41.8540
0.8435
1
T1
43.3254
0.8427
3
A2
T2
43.3264
43.7850
0.8401
0.8403
3
T1
43.8569
0.8356
1
E
44.1703
0.8387
1
A2
E
44.2281
44.7092
0.8396
0.8394
3
T1
44.8395
0.8361
1
T2
44.8653
0.8359
3
T2
44.8926
0.8377
3
A1
T1
45.5135
45.6185
0.8389
0.8342
3
T2
45.8494
0.8345
1
A1
46.1372
0.8379
3
T1
46.2710
0.8331
1
E
46.4049
0.8321
1
T1
46.6238
0.8371
3
E
T2
47.3602
47.4319
0.8248
0.8307
1
T2
48.0054
0.8291
3
T1
48.4902
0.8150
3
A2
E
49.3153
49.6460
0.7749
0.8217
0.5573(3t 2 1t 1 ) 0.2681(1e) 0.0181(2t 2 1t 1 ) 0.0013(1t 2 1t 1 )
0.3665(1e)
0.2846(3t 2 1t 1 )
0.1290(3t 2 )
0.0567(1t 1 )
0.0025(2t 2 3t 2 ) 0.0016(2t 2 )
0.5041(1e3t 2 ) 0.1380(3t 2 1t 1 ) 0.1300(1e1t 1 ) 0.0348(2t 2 1e)
0.0293(2t 2 3t 2 ) 0.0054(2a 1 1t 1 )
0.4324(1e) 0.3489(3t 2 ) 0.0299(1t 1 ) 0.0258(2t 2 3t 2 ) 0.0031(2t 2 )
0.5258(3t 2 ) 0.1402(2t 2 1t 1 ) 0.1222(3t 2 1t 1 ) 0.0299(2t 2 3t 2 )
0.0106(1e) 0.0092(2a 1 1e) 0.0081(2t 2 ) 0.0013(1t 2 3t 2 )
0.7350(1e3t 2 ) 0.0384(2t 2 1t 1 ) 0.0278(2t 2 1e) 0.0270(3t 2 1t 1 )
0.0106(2t 2 3t 2 ) 0.0025(2a 1 3t 2 )
0.3011(3t 2 ) 0.1835(3t 2 1t 1 ) 0.1522(1e3t 2 ) 0.0848(2t 2 1t 1 )
0.0581(2t 2 1e)
0.0394(2t 2 3t 2 )
0.0150(1t 1 )
0.0042(1e1t 1 )
0.0035(2a 1 2t 2 ) 0.0012(2a 1 3t 2 )
0.4641(3t 2 )
0.1723(1e3t 2 )
0.0635(1e1t 1 )
0.0557(2t 2 1t 1 )
0.0484(2t 2 1e) 0.0149(2a 1 1t 1 ) 0.0091(1t 1 ) 0.0082(2t 2 3t 2 )
0.0037(2t 2 ) 0.0019(3t 2 1t 1 ) 0.0010(1t 2 3t 2 )
0.4506(2t 2 1t 1 ) 0.1412(1e3t 2 ) 0.1245(3t 2 1t 1 ) 0.0926(1e1t 1 )
0.0259(2t 2 1e) 0.0039(2t 2 3t 2 ) 0.0021(2a 1 1t 1 ) 0.0013(1t 2 1t 1 )
0.4123(2t 2 1t 1 ) 0.3412(1e) 0.0859(3t 2 1t 1 )
0.2966(2t 2 1t 1 ) 0.1490(1e3t 2 ) 0.1468(3t 2 1t 1 ) 0.1348(2t 2 1e)
0.0571(1e1t 1 ) 0.0229(1t 1 ) 0.0198(2t 2 3t 2 ) 0.0071(2a 1 3t 2 )
0.0026(2t 2 )
0.2454(1e3t 2 ) 0.1958(2t 2 3t 2 ) 0.1624(2t 2 1t 1 ) 0.0950(1e1t 1 )
0.0526(3t 2 1t 1 ) 0.0301(1t 1 ) 0.0222(2t 2 ) 0.0139(3t 2 )
0.0135(2a 1 1t 1 ) 0.0025(2t 2 1e) 0.0011(1t 2 1t 1 )
0.2984(2t 2 3t 2 ) 0.2219(1e) 0.1412(2t 2 1t 1 ) 0.1148(3t 2 1t 1 )
0.0273(2t 2 ) 0.0201(1t 1 ) 0.0079(2a 1 1e) 0.0053(3t 2 )
0.0013(1t 2 1t 1 )
0.8302(2t 2 1t 1 ) 0.0061(3t 2 1t 1 ) 0.0034(1t 2 1t 1 )
0.5927(2t 2 1t 1 ) 0.2258(2t 2 3t 2 ) 0.0119(3t 2 1t 1 ) 0.0058(2a 1 1e)
0.0015(1t 2 1t 1 ) 0.0015(1t 2 3t 2 )
0.5132(2t 2 1t 1 ) 0.1198(2t 2 1e) 0.0992(2t 2 3t 2 ) 0.0321(3t 2 )
0.0232(2a 1 1t 1 ) 0.0203(1e3t 2 ) 0.0164(1e1t 1 ) 0.0073(1t 1 )
0.0014(1t 2 1t 1 ) 0.0014(2t 2 )
0.2316(2t 2 3t 2 )
0.1304(1e1t 1 )
0.1244(3t 2 )
0.1199(1e3t 2 )
0.0707(2t 2 1e)
0.0594(2t 2 )
0.0408(2a 1 3t 2 )
0.0405(1t 1 )
0.0086(3t 2 1t 1 ) 0.0046(2t 2 1t 1 ) 0.0022(2a 1 2t 2 ) 0.0014(1t 2 3t 2 )
0.5516(2t 2 1t 1 ) 0.2006(2t 2 3t 2 ) 0.0469(2t 2 1e) 0.0174(2a 1 3t 2 )
0.0154(3t 2 1t 1 ) 0.0016(1t 2 1t 1 ) 0.0016(1e3t 2 )
0.8363(2t 2 3t 2 ) 0.0018(1t 2 2t 2 )
0.3677(2t 2 3t 2 ) 0.3486(2t 2 1e) 0.0725(2t 2 1t 1 ) 0.0187(1e3t 2 )
0.0124(1e1t 1 ) 0.0102(3t 2 1t 1 ) 0.0016(2a 1 1t 1 ) 0.0014(1t 2 3t 2 )
0.4409(2t 2 3t 2 ) 0.3402(2t 2 1e) 0.0422(2t 2 1t 1 ) 0.0038(3t 2 1t 1 )
0.0037(2a 1 3t 2 ) 0.0013(2a 1 2t 2 )
0.2493(2t 2 3t 2 ) 0.2029(2t 2 ) 0.1848(3t 2 ) 0.1324(1e) 0.0550(1t 1 )
0.0084(2a 1 ) 0.0028(1t 2 3t 2 )
0.2678(2t 2 1e) 0.2560(2t 2 3t 2 ) 0.2475(2a 1 1t 1 ) 0.0364(1e3t 2 )
0.0095(1e1t 1 ) 0.0047(3t 2 1t 1 ) 0.0042(3t 2 ) 0.0040(1t 1 )
0.3207(2t 2 3t 2 ) 0.2286(2t 2 1t 1 ) 0.1754(2a 1 1e) 0.0519(3t 2 )
0.0371(3t 2 1t 1 )
0.0131(2t 2 )
0.0020(1e)
0.0013(1t 2 2t 2 )
0.0011(1t 2 3t 2 )
0.5326(2a 1 1t 1 ) 0.1475(2t 2 3t 2 ) 0.1038(2t 2 1e) 0.0273(2t 2 1t 1 )
0.0192(1e3t 2 ) 0.0023(1e1t 1 ) 0.0019(1t 2 3t 2 ) 0.0012(1t 2 1e)
0.3882(2t 2 3t 2 ) 0.3230(2a 1 1e) 0.0940(2t 2 1t 1 ) 0.0172(3t 2 1t 1 )
0.3672(2a 1 3t 2 ) 0.2684(2t 2 1e) 0.1038(2t 2 3t 2 ) 0.0430(2t 2 1t 1 )
0.0190(2a 1 2t 2 ) 0.0172(1e3t 2 ) 0.0087(3t 2 1t 1 )
0.3276(2a 1 3t 2 ) 0.1173(2t 2 1e) 0.1157(2t 2 1t 1 ) 0.0803(2a 1 2t 2 )
0.0633(2t 2 )
0.0450(3t 2 )
0.0366(2t 2 3t 2 )
0.0239(1e3t 2 )
0.0147(1e1t 1 ) 0.0026(3t 2 1t 1 )
0.2626(2a 1 1t 1 ) 0.2318(2t 2 3t 2 ) 0.1506(2t 2 ) 0.0523(1e3t 2 )
0.0432(1e1t 1 )
0.0275(2t 2 1e)
0.0212(1t 1 )
0.0179(3t 2 1t 1 )
0.0033(2t 2 1t 1 ) 0.0026(1t 2 2t 2 )
0.3868(2t 2 1t 1 ) 0.2124(1e) 0.1733(3t 2 1t 1 ) 0.0024(1t 2 1t 1 )
0.3545(2t 2 )
0.2794(2a 1 1e)
0.0987(2t 2 3t 2 )
0.0359(1e)
0.0216(3t 2 1t 1 ) 0.0128(2t 2 1t 1 ) 0.0125(1t 1 ) 0.0035(1t 2 2t 2 )
0.0014(3t 2 ) 0.0010(1t 2 1t 1 )
State
3
1
1
1
3
1
3
1
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
TABLE IV. ~Continued.!
DIP ~eV!
PS
2h composition
T2
49.8157
0.8209
3
T1
50.2856
0.7875
3
T2
51.0330
0.8045
1
T1
51.9871
0.7753
1
E
52.2376
0.7789
1
A1
52.3150
0.8069
1
T2
52.4903
0.7815
3
E
53.0695
0.7710
3
T1
53.3792
0.7731
1
T2
53.5121
0.7741
1
A1
54.4169
0.7899
3
T2
54.5457
0.7845
1
T1
55.0974
0.7621
1
E
55.8744
0.7643
1
T2
56.0099
0.7551
1
T2
56.1335
0.0104
1
T2
60.0234
0.1181
1
T2
60.1005
0.5532
T2
T2
1
T2
1
A1
60.3523
60.4146
60.7715
61.5160
0.0307
0.0370
0.0147
0.7455
0.2291(2t 2 3t 2 ) 0.2030(2t 2 ) 0.1217(2a 1 3t 2 ) 0.0915(2a 1 2t 2 )
0.0517(1e3t 2 ) 0.0459(2t 2 1t 1 ) 0.0295(1e1t 1 ) 0.0168(3t 2 1t 1 )
0.0151~1t1!
0.0094(2t 2 1e)
0.0027(3t 2 )
0.0019(1t 2 2t 2 )
0.0010(1t 2 2a 1 )
0.2781(2t 2 1e) 0.1765(2t 2 ) 0.0816(2a 1 1t 1 ) 0.0633(1e1t 1 )
0.0631(2t 2 1t 1 )
0.0425(3t 2 )
0.0332(1t 1 )
0.0227(2t 2 3t 2 )
0.0168(1e3t 2 ) 0.0062(3t 2 1t 1 ) 0.0017(1t 2 2t 2 )
0.3945(2a 1 3t 2 ) 0.1530(2a 1 2t 2 ) 0.0962(2t 2 1e) 0.0740(2t 2 1t 1 )
0.0426(2t 2 3t 2 ) 0.0201(1e3t 2 ) 0.0199(3t 2 1t 1 ) 0.0017(1a 1 3t 2 )
0.0017(1t 2 2a 1 )
0.2203(2t 2 1t 1 ) 0.1579(2t 2 1e) 0.1473(1e1t 1 ) 0.1388(3t 2 1t 1 )
0.1046(1e3t 2 ) 0.0023(1t 2 1t 1 ) 0.0018(2a 1 1t 1 ) 0.0015(2t 2 3t 2 )
0.1881(2t 2 1t 1 )
0.1760(1e)
0.1193(2t 2 )
0.0724(1t 1 )
0.0707(3t 2 1t 1 ) 0.0683(2t 2 3t 2 ) 0.0490(3t 2 ) 0.0314(2a 1 1e)
0.0014(1t 2 3t 2 ) 0.0012(1t 2 2t 2 ) 0.0011(1t 2 1t 1 )
0.3908(2t 2 3t 2 ) 0.1433(2t 2 ) 0.1422(2a 1 ) 0.0696(1e) 0.0579(1t 1 )
0.0014(1t 2 2t 2 )
0.1795(2t 2 1e)
0.1586(2t 2 )
0.1197(1e1t 1 )
0.1087(2t 2 3t 2 )
0.0521(1e3t 2 )
0.0442(3t 2 )
0.0195(2a 1 3t 2 )
0.0853(1t 1 )
0.0058(2t 2 1t 1 ) 0.0026(2a 1 2t 2 ) 0.0017(1t 2 3t 2 ) 0.0015(1t 2 2t 2 )
0.0012(1t 2 1e)
0.4724(2a 1 1e) 0.1777(2t 2 3t 2 ) 0.0887(2t 2 1t 1 ) 0.0280(3t 2 1t 1 )
0.0028(1a 1 1e)
0.4452(2t 2 ) 0.1658(2a 1 1t 1 ) 0.0661(2t 2 1e) 0.0287(1e3t 2 )
0.0268(3t 2 ) 0.0200(2t 2 1t 1 ) 0.0104(3t 2 1t 1 ) 0.0039(2t 2 3t 2 )
0.0020(1a 1 1t 1 ) 0.0015(1t 2 2t 2 ) 0.0012(1t 2 3t 2 )
0.1499(2t 2 1t 1 ) 0.1345(1e1t 1 ) 0.1345(2t 2 1e) 0.1064(1e3t 2 )
0.0321(1t 1 )
0.0318(2t 2 )
0.0274(2a 1 2t 2 )
0.1022(3t 2 1t 1 )
0.0219(2t 2 3t 2 )
0.0120(3t 2 )
0.0047(1t 2 )
0.0036(1a 1 1t 2 )
0.0034(1t 2 3t 2 ) 0.0028(1t 2 2a 1 ) 0.0022(1t 2 1t 1 ) 0.0019(1t 2 2t 2 )
0.0017(1t 2 1e)
0.2598(2t 2 ) 0.1717(2a 1 ) 0.1178(1e) 0.1080(1t 1 ) 0.0618(3t 2 )
0.0588(2t 2 3t 2 ) 0.0069(1t 2 ) 0.0035(1a 1 2a 1 ) 0.0013(1a 1 )
0.6251(2a 1 2t 2 ) 0.0374(2t 2 1e) 0.0348(2t 2 1t 1 ) 0.0329(2a 1 3t 2 )
0.0248(2t 2 3t 2 ) 0.0119(3t 2 1t 1 ) 0.0111(1e3t 2 ) 0.0023(1t 2 2a 1 )
0.0023(1a 1 2t 2 )
0.2571(2a 1 1t 1 ) 0.2564(2t 2 3t 2 ) 0.1359(2t 2 1e) 0.0425(1e3t 2 )
0.0415(2t 2 1t 1 ) 0.0197(3t 2 1t 1 ) 0.0046(1t 2 1t 1 ) 0.0018(1t 2 1e)
0.0016(1t 2 3t 2 )
0.2847(2a 1 1e) 0.2704(2t 2 ) 0.0855(2t 2 1t 1 ) 0.0448(3t 2 1t 1 )
0.0447(3t 2 ) 0.0123(1t 2 1t 1 ) 0.0093(1t 2 3t 2 ) 0.0073(2t 2 3t 2 )
0.0025(1a 1 1e) 0.0022(1t 2 2t 2 )
0.1865(2a 1 3t 2 ) 0.1847(2t 2 ) 0.1300(2a 1 2t 2 ) 0.0833(2t 2 1t 1 )
0.0766(2t 2 1e) 0.0353(3t 2 1t 1 ) 0.0316(1e3t 2 ) 0.0122(1t 2 1t 1 )
0.0050(3t 2 ) 0.0025(1a 1 2t 2 ) 0.0020(2t 2 3t 2 ) 0.0017(1t 2 1e)
0.0014(1t 2 2a 1 )
0.0028(2a 1 3t 2 ) 0.0019(2t 2 ) 0.0014(2a 1 2t 2 ) 0.0014(2t 2 1e)
0.0011(2t 2 1t 1 )
0.0616(2a 1 2t 2 ) 0.0162(2a 1 3t 2 ) 0.0132(2t 2 3t 2 ) 0.0101(2t 2 )
0.0045(1t 2 3t 2 ) 0.0032(3t 2 ) 0.0018(2t 2 1e) 0.0015(1t 2 2t 2 )
0.0011(1a 1 3t 2 ) 0.0010(1e1t 1 ) 0.0010(1t 2 )
0.2950(2a 1 2t 2 ) 0.0746(2t 2 3t 2 ) 0.0610(2t 2 ) 0.0593(2a 1 3t 2 )
0.0144(1t 2 3t 2 ) 0.0108(3t 2 ) 0.0062(1t 2 2t 2 ) 0.0049(1t 2 )
0.0049(3t 2 1t 1 ) 0.0042(2t 2 1e) 0.0031(1e1t 1 ) 0.0028(1a 1 2t 2 )
0.0026(1a 1 1t 2 ) 0.0019(1t 2 1t 1 ) 0.0017(2t 2 1t 1 ) 0.0017(1e3t 2 )
0.0013(1t 2 1e)
0.0154(2a 1 2t 2 ) 0.0046(2a 1 3t 2 ) 0.0039(2t 2 ) 0.0039(2t 2 3t 2 )
0.0179(2a 1 2t 2 ) 0.0054(2t 2 3t 2 ) 0.0052(2a 1 3t 2 ) 0.0045(2t 2 )
0.0053(2a 1 2t 2 ) 0.0035(2t 2 ) 0.0024(2a 1 3t 2 ) 0.0014(2t 2 3t 2 )
0.4052(2a 1 )
0.1612(2t 2 )
0.0506(1t 2 3t 2 )
0.0490(2t 2 3t 2 )
0.0389(3t 2 ) 0.0129(1t 2 ) 0.0099(1t 1 ) 0.0071(1e) 0.0066(1a 1 2a 1 )
0.0023(1t 2 2t 2 ) 0.0018(1a 1 )
State
1
1
1
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
9765
9766
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
TABLE V. DIPs and two-hole atomic population analysis of the Green’s function 2h pole strengths for the
outer valence dicationic states of CF4 ~first group of one-side and two-side states!. DIPs with a total pole
strength larger than 0.1 are bold faced.
Population
State
1
E
T1
1
T1
1
A2
1
A1
3
T1
3
T2
3
E
1
T1
1
T2
3
T2
3
T1
3
A2
1
E
1
T1
1
A1
1
E
3
T2
1
T2
3
T1
1
T1
3
A2
1
T2
3
T1
1
A2
1
E
3
E
3
T1
1
T2
3
T2
3
A1
1
T1
3
T2
1
A1
3
T1
1
E
1
T1
3
E
3
T2
1
T2
3
T1
3
A2
1
E
1
T2
3
T1
3
T2
1
T1
1
E
1
A1
1
T2
3
E
3
T1
1
T2
1
A1
3
T2
1
T1
1
E
1
T2
3
DIP ~eV!
38.4111
38.4211
38.9210
39.1921
39.4192
39.5308
39.7069
39.7912
39.8175
40.1802
40.2004
40.5374
40.7637
40.9922
41.3172
41.5085
41.6737
41.6849
41.7334
41.8540
43.3254
43.3264
43.7850
43.8569
44.2281
44.1703
44.7092
44.8395
44.8653
44.8926
45.5135
45.6185
45.8494
46.1372
46.2710
46.4049
46.6238
47.3602
47.4319
48.0054
48.4901
49.3153
49.6460
49.8157
50.2856
51.0330
51.9871
52.2376
52.3150
52.4903
53.0695
53.3792
53.5121
54.4169
54.5457
55.0974
55.8744
56.0099
C
22
0.0001
0.0000
0.0001
0.0000
0.0005
0.0001
0.0002
0.0002
0.0001
0.0007
0.0000
0.0007
0.0001
0.0007
0.0008
0.0012
0.0014
0.0009
0.0010
0.0016
0.0005
0.0002
0.0014
0.0026
0.0000
0.0033
0.0021
0.0022
0.0053
0.0023
0.0083
0.0054
0.0061
0.0080
0.0058
0.0086
0.0019
0.0054
0.0061
0.0075
0.0147
0.0001
0.0234
0.0263
0.0076
0.0177
0.0006
0.0038
0.0278
0.0057
0.0041
0.0191
0.0029
0.0157
0.0258
0.0040
0.0109
0.0133
F
22
0.0020
0.0053
0.0054
-0.0009
0.0082
0.0045
0.0129
0.0185
0.0030
0.0059
0.0002
0.0049
0.0313
0.0151
0.0120
0.0079
0.0260
0.0145
0.0259
0.0441
0.0262
0.0256
0.0324
0.0836
-0.0022
0.0338
0.0027
0.0260
0.0742
0.0080
-0.0006
0.0094
0.0063
0.0662
0.0303
0.0406
0.0111
0.0714
0.0405
0.0601
0.2059
0.6590
0.0685
0.1298
0.4465
0.2307
0.6715
0.6129
0.2981
0.5827
0.4763
0.4336
0.6570
0.4452
0.2599
0.5282
0.4476
0.4308
C21 F21
21
F21
1 F2
Total
0.0090
0.0065
0.0131
0.0418
0.0216
0.0386
0.0403
0.0406
0.0311
0.0442
0.0196
0.0446
0.0549
0.0497
0.0523
0.0521
0.0778
0.0589
0.0640
0.0750
0.1517
0.1279
0.1551
0.1300
0.1803
0.1412
0.1902
0.1809
0.1499
0.1899
0.2110
0.1930
0.1999
0.1615
0.1967
0.2107
0.1856
0.1965
0.2001
0.1981
0.1846
0.0720
0.2257
0.2364
0.1452
0.2248
0.0684
0.0903
0.2149
0.1051
0.1827
0.2204
0.0662
0.1552
0.2415
0.1998
0.2079
0.2141
0.8411
0.8399
0.8314
0.8131
0.8188
0.8084
0.7965
0.7900
0.8148
0.7979
0.8264
0.7970
0.7586
0.7779
0.7785
0.7807
0.7433
0.7687
0.7544
0.7228
0.6644
0.6864
0.6514
0.6194
0.6616
0.6604
0.6444
0.6270
0.6065
0.6375
0.6203
0.6265
0.6222
0.6023
0.6002
0.5722
0.6386
0.5516
0.5840
0.5634
0.4098
0.0438
0.5041
0.4284
0.1882
0.3312
0.0348
0.0719
0.2661
0.0880
0.1080
0.1000
0.0480
0.1738
0.2573
0.0301
0.0979
0.0969
0.8522
0.8518
0.8500
0.8539
0.8492
0.8515
0.8499
0.8492
0.8489
0.8487
0.8463
0.8472
0.8449
0.8435
0.8437
0.8420
0.8486
0.8431
0.8453
0.8435
0.8427
0.8401
0.8403
0.8356
0.8396
0.8388
0.8394
0.8361
0.8359
0.8377
0.8389
0.8342
0.8345
0.8379
0.8331
0.8321
0.8371
0.8248
0.8307
0.8291
0.8150
0.7749
0.8217
0.8209
0.7875
0.8045
0.7753
0.7789
0.8069
0.7815
0.7710
0.7731
0.7741
0.7899
0.7845
0.7621
0.7643
0.7551
J. Chem. Phys., Vol. 104, No. 24, 22 June 1996
9767
Gottfried, Cederbaum, and Tarantelli: Auger spectra of CF4
TABLE V. ~Continued.!
Population
State
1
T2
T2
1
T2
1
T2
1
T2
1
T2
1
A1
1
1
DIP ~eV!
56.1335
60.0234
60.1005
60.3523
60.4146
60.7715
61.5160
C
22
0.0002
0.0054
0.0264
0.0016
0.0018
0.0007
0.0303
F
22
0.0063
0.0550
0.2610
0.0155
0.0182
0.0071
0.2928
K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P. E. Heden, K.
Hamrin, U. Gelius, T. Bergmark, L. O. Werme, R. Manne, and Y. Beer,
ESCA Applied to Free Molecules ~North Holland, Amsterdam, 1971!.
2
R. R. Rye and J. E. Houston, J. Chem. Phys. 78, 4321 ~1983!.
3
M. Thompson, M. D. Baker, A. Christie, and J. F. Tyson, Auger Electron
Spectroscopy ~Wiley, New York, 1985!.
4
W. J. Griffiths, S. Svensson, A. Naves de Brito, N. Correia, C. J. Reid, M.
L. Langford, F. M. Harris, C. M. Liegener, and H. Ågren, Chem. Phys.
173, 109 ~1993!.
5
R. I. Hall, L. Avaldi, G. Dawber, A. G. McConkey, M. A. MacDonald,
and G. C. King, Chem. Phys. 187, 125 ~1994!.
6
F. P. Larkins, J. Electron Spectrosc. Relat. Phenom. 51, 115 ~1990!.
7
F. Tarantelli and L. S. Cederbaum, Phys. Rev. Lett. 71, 649 ~1993!.
8
F. O. Gottfried, F. Tarantelli, and L. S. Cederbaum, Phys. Rev. A 53, 2118
~1996!.
9
F. Tarantelli, A. Sgamellotti, and L. S. Cederbaum, J. Chem. Phys. 94,
523 ~1991!.
10
J. Schirmer and B. Barth, Z. Phys. A 317, 267 ~1984!.
11
A. Tarantelli and L. S. Cederbaum, Phys. Rev. A 39, 1639 ~1989!.
12
E. M.-L. Ohrendorf, H. Köppel, L. S. Cederbaum, F. Tarantelli, and A.
Sgamellotti, J. Chem. Phys. 91, 1734 ~1989!.
C21 F21
21
F21
1 F2
Total
0.0028
0.0419
0.2015
0.0114
0.0137
0.0052
0.2577
0.0012
0.0158
0.0643
0.0022
0.0033
0.0017
0.1646
0.0104
0.1181
0.5532
0.0307
0.0370
0.0147
0.7455
13
E. M.-L. Ohrendorf, F. Tarantelli, and L. S. Cederbaum, J. Chem. Phys.
92, 2984 ~1990!.
14
D. Minelli, F. Tarantelli, A. Sgamellotti, and L. S. Cederbaum, J. Chem.
Phys. 99, 6688 ~1993!.
15
D. Minelli, F. Tarantelli, A. Sgamellotti, and L. S. Cederbaum, J. Electron
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