PHYSICAL REVIEW A, VOLUME 64, 012719
High-resolution excitation-energy-dependent study
of the Auger decay of the O 1s-1 ␲ g core-excited state in oxygen
S. L. Sorensen,1,* R. Fink,2,† R. Feifel,2 M. N. Piancastelli,2,‡ M. Bässler,2,§ C. Miron,2 H. Wang,2,§ I. Hjelte,2
O. Björneholm,2 and S. Svensson,2
1
Department of Synchrotron Radiation Research, Institute of Physics, University of Lund, Box 118, S-221 00 Lund, Sweden
2
Department of Physics, Uppsala University, Box 530, S-751 21 Uppsala, Sweden
共Received 16 January 2001; published 15 June 2001兲
We report new measurements of the decay of the core-excited O 1s –1 ␲ g 3 ⌸ u state in molecular oxygen
under Auger resonant Raman conditions. The spectral features are interpreted with the aid of ab initio calculations using standard lifetime-vibrational interference, and in some cases, employing a model in which the
bond-length dependence of the Auger transition rates is taken into account. By analyzing a series of Augerdecay spectra using different excitation energies, several instances of fixed kinetic energy are pointed out in the
spectra; the nondispersive behavior arises from the decay between two potentials that are essentially parallel.
DOI: 10.1103/PhysRevA.64.012719
PACS number共s兲: 33.70.Ca, 33.80.Eh, 34.50.Gb
I. INTRODUCTION
It has recently become possible to study the dynamics of
molecular valence-electron excited states on an ultrashort
time scale using femtosecond laser pulses to excite and ionize molecules. For core electronic states, it is also possible to
monitor femtosecond dynamics under certain conditions. By
working under Auger resonant Raman conditions and exploiting the duration time concept and the short lifetime of
core-hole states, femtosecond dynamics of the wave functions may be studied in a precise way 关1兴. One consequence
of ultrashort electronic state lifetimes in molecules is interference when overlapping vibrational sublevels are coherently excited leading to lifetime-vibrational interference
共LVI兲. LVI is well understood and final states of many coreexcited molecules are predictable using standard potentials
共see, for example, Refs. 关2–8兴 and references therein兲. Measurements using narrow-bandwidth light to excite through a
core-electron resonance state make it possible to reduce the
inherent broadening of core-level states to that of the photon
bandwidth. This has been previously discussed extensively
共see Åberg and Crasemann 关8兴 and Armen et al. 关9兴兲 and the
advantages of measurement of electron or x-ray emission
spectra under Auger resonant Raman conditions 共ARR兲 are
well documented. The combination of very high-resolution
measurements with accurate calculation has illuminated
subtle aspects of molecular decay that may be of general
importance 关5兴.
In molecular oxygen, the short lifetime of the O 1s –1 ␲ g
core-excited state (⬃3 fs corresponding to the 0.15 eV lifetime width兲 coupled with the relatively small vibrational en*Corresponding author. FAX: ⫹46-46 222 42 21; email address:
[email protected]
†
Also at Theoretical Chemistry, University of Lund, Box 124,
S-221 00 Lund, Sweden.
‡
Permanent address: Department of Chemical Sciences and Technologies, University ‘‘Tor Vergata,’’ 00133 Rome, Italy.
§
Also at MAX-Lab, University of Lund, S-221 00 Lund, Sweden.
1050-2947/2001/64共1兲/012719共9兲/$20.00
ergy 共0.14 eV兲 creates a situation where the excitation and
decay are not separable. In short, interference arising from
coherently excited vibrational levels in the intermediate state
may be the dominant effect in a measured spectrum. For the
case of molecular oxygen, the coherence of the nuclear wave
function is high, as the vibrational energy spacing is close to
the broadening from the natural lifetime width of the excited
state. The comparable time scales for these processes lead
not only to interference, but also to dramatic changes in vibrational intensity envelopes.
We report a study on the decay of bound core-excited
states in molecular oxygen. This system has been a showcase
of lifetime-vibrational interference 共LVI兲 since using coincidence techniques with electron excitation 关10,11兴 and later
using synchrotron radiation 关12,13兴. In the latter study the
LVI picture was used to study the details of the outer valence
2
O⫹
2 X ⌸ g final state that exhibits striking changes in the vibrational intensity distribution. Despite the limited resolution
of these studies, a deeper understanding of the interplay between nuclear motion and electronic decay arose thanks to
the large number of simulations by theorists 关1,4,7,8,14,15兴.
More recently, angle-resolved studies were performed using
higher resolution synchrotron radiation and significantly better statistics were obtained 关16兴.
Although the latter study is of substantially better spectroscopic quality than previous studies, some state assignments
remain ambiguous in the high kinetic energy region of the
decay spectrum. There are a number of overlapping electronic states and because of the change in selection rules
upon resonant excitation, several states that are dipole forbidden in direct photoionization are rather intense in the
resonantly excited decay spectrum.
Raman-Stokes dispersion of a resolved vibrational peak in
truly molecular decay states has provided a signature for fast
electronic decay in dissociative intermediate states. Electronic states arising from the decay of atomic fragments will
not disperse with photon energy, although the intensity of
fragment peaks is sensitive to the energy detuning from a
dissociative state 关1兴. Thus, peaks that do not disperse are
assigned to fragment peaks in systems where the dissociation
64 012719-1
©2001 The American Physical Society
S. L. SORENSEN et al.
PHYSICAL REVIEW A 64 012719
time is comparable to the core-hole lifetime 关2,17–19兴. However, apparently nondispersive peaks have been reported that
are related to truly molecular decay states and can arise in
general if the potential energy curves 共PEC’s兲 of the intermediate and final states are essentially parallel. The spectral
features may thus easily be interpreted as being due to electronic emission from fragments. We have made a thorough
investigation of the high kinetic energy states in the decay of
the 3 ⌸ u state in oxygen and we find that all peaks arise from
molecular decay. Theoretical spectra from ab initio calculations provide a basis for identifying all of the peaks in the
spectrum and some aspects of the decay are brought out. For
all cases of apparent ‘‘nondispersive’’ behavior we find that
the final state potential is nearly parallel with the bound coreexcited 3 ⌸ u state.
II. EXPERIMENT
The measurements were performed at the undulator beam
line I 411 关20兴 at the 1.5 GeV MAX II electron storage ring
at the Swedish National Synchrotron Radiation Facility
MAX-Lab in Lund, Sweden. The beam line is based upon an
88-pole 2.65-m-long hybrid undulator 共58.85-mm-period兲
providing photons in the 50–1200 eV energy range. It is
equipped with a modified high-resolution SX-700 monochromator and a SES-200 rotatable hemispherical electronenergy analyzer. The main axis of the spectrometer lens was
fixed at 54.7° with respect to the plane of polarization of the
undulator radiation for all measurements.
The calibration of the spectra was made in the following
way: Electron-yield spectra were measured before and after
each decay spectrum. The absorption spectrum was fit and
the maximum energy was used as a benchmark photon energy. The electronic decay spectra were then measured directly in kinetic energy, which was calibrated using known
binding energy values for both the valence levels in oxygen,
and for nitrogen gas. Some spectra were measured using the
second diffraction order of the plane grating. The same procedure was used for these spectra. The photon bandwidth
was 65 meV in the second-order measurements and 130 meV
in first-order diffraction. The resolution of the electron spectrometer was set at 80 meV for all measurements.
III. THEORY
The theoretical calculations were performed according to
the guidelines described in Refs. 关21–24兴. Briefly, the molecular orbitals were set up using the cc-p v TZ basis set of
Dunning 关25兴. All core orbitals (1 ␴ g and 1 ␴ u ) as well as the
valence orbitals 2 ␴ g , 2 ␴ u , 3 ␴ g , 1 ␲ u , and 1 ␲ g were obtained from the Hartree-Fock ground-state wave function.
The missing 3 ␴ u valence orbital was generated in the modified improved virtual orbital procedure as described in Ref.
关21兴. The wave functions of the ground, core-excited, and
final electronic states were set up by the configuration interaction method where all possible excitations in the valence
orbitals were included. For the intermediate state, one electron was excited from the symmetric or antisymmetric linear
combinations of the O 1s orbitals 共1␴ u and 1␴ g , respec-
FIG. 1. 共a兲 PEC’s of the intermediate state 共b兲 the PEC’s of the
ground state of O2 共solid line兲 and the O⫹
2 X 共solid line兲,
A 2 ⌸ u ( . . . . . . ), C 2 ⌽ u (——), D 2 ⌬ g (— • •), and 3 2 ⌸ g ( – •)
O⫹
2 final states. 共c兲 the Auger decay rates ⌫ f as a function of the
O—O bond distance for the states shown in 共b兲.
tively兲 关21,26兴. The resulting PEC’s for the initial, intermediate, and some final states of particular interest are shown in
Figs. 1共a兲 and 1共b兲. These will be discussed further in the
text that follows.
The electronic part of the Auger transition rate depends
on the bond distance R as given by the Fermi-Wenzel formula
⌫ f 共 R 兲⫽
⫽
兩 A lm 兩 2
兺
lm
2
2 ␲ 兩 具 ⌽ i 共 R 兲 兩 Ĥ⫺E 兩 ⌽ lm
兺
f 共 R 兲典兩 .
lm
共1兲
共2兲
Here, ⌽ i (R) and ⌽ lm
f (R) represent the electronic wave functions of the intermediate and the f th electronic final state, lm
designates the channels of the Auger continuum electron,
and A lm is the Auger transition amplitude.
Transition amplitudes were calculated with the one-center
approximation as described in Refs. 关24,27–30兴. The energies and slopes for relevant final-state potentials at the
Franck-Condon point, as well as the transition rates at this
point are presented in Table I. The bond-distance dependence of some of these transition rates is shown in Fig. 1共c兲.
Thus, far more LVI calculations use the Franck-Condon
approximation; i.e., the dipole excitation (D) and the Auger
de-excitation amplitudes (A lm ) are assumed to be independent of the molecular geometry. The behavior of transitions
2
to the O⫹
2 X ⌸ g ground state in Fig. 1共c兲 shows that this is
not necessarily a valid assumption. The R dependence for
this transition may be calculated explicitly in the formula for
the cross section, ␴ ( ⑀ ) where ⑀ is the electron kinetic energy. In Ref. 关31兴 this was derived as
012719-2
HIGH-RESOLUTION EXCITATION-ENERGY-DEPENDENT . . .
PHYSICAL REVIEW A 64 012719
TABLE I. Assignment of the final states contributing to the RAES of O2 according to the theoretical
moment theory results. Available term symbols are taken from the literature. The other terms are numbered
in order to give them unambiguous names. Configurations are given with respect to the ground-state configuration of O2⫹
(1 ␴ 2g 1 ␴ 2u 2 ␴ 2g 2 ␴ 2u 3 ␴ 2g 1 ␲ 4u ). The partial transition rates ⌫ f are given in ␮ a.u. ⌬E is the
2
energy difference between the final and intermediate states at the Franck Condon point in eV. 具 E 典 and
FWHM correspond to the center and the full width at half maximum of the band in eV as obtained from the
moment method. dE/dR is the derivative of the potential energy curve at the Franck-Condon point in eV/Å.
3
The corresponding derivative for the 3 ⌸ (1 ␴ ⫺1
u 1 ␲ g ) core excited state is ⫺11.68 eV/Å.
Term and configuration
X2 ⌸g a
a 4⌸ u
A 2⌸ u a
b 4⌺ ⫺
g
C 2⌽ u a
D 2⌬ g a
2 2⌸ u a
a
B 2⌺ ⫺
g
2 ⫹ a
1 ⌺g
1 2⌺ ⫹
u
3 2⌸ u a
1 4⌬ u
1 4⌺ ⫹
u
a
1 2⌺ ⫺
u
4 ⫺ a
c ⌺u
2 2⌸ g a
1 2⌬ u a
a
2 2⌺ ⫹
u
a
2
1 ⌽g
3 2⌸ g a
2 2⌬ u
3 2⌺ ⫹
u
2 4⌺ ⫺
u
4 2⌸ g
89%(1 ␲ 1g )
2
96%(1 ␲ ⫺1
u 1 ␲ g)
⫺1
94%(1 ␲ u 1 ␲ 2g )
2
92%(3 ␴ ⫺1
g 1 ␲ g)
⫺1
96%(1 ␲ u 1 ␲ 2g )
2
90%(3 ␴ ⫺1
g 1 ␲ g)
⫺1
94%(1 ␲ u 1 ␲ 2g )
2
90%(3 ␴ ⫺1
g 1 ␲ g)
⫺1
88%(3 ␴ g 1 ␲ 2g )
83%(3 ␴ 1u )
2
64%(1 ␲ ⫺1
u 1 ␲ g)
⫺1
⫺1
96%(3 ␴ g 1 ␲ u 1 ␲ 3g )
⫺1
3
96%(3 ␴ ⫺1
g 1 ␲ u 1 ␲ g)
⫺1
⫺1
92%(3 ␴ g 1 ␲ u 1 ␲ 3g )
2
77%(2 ␴ ⫺1
u 1 ␲ g)
⫺2
89%(1 ␲ u 1 ␲ 3g )
⫺1
3
73%(3 ␴ ⫺1
g 1 ␲ u 1 ␲ g)
⫺1
⫺1
79%(3 ␴ g 1 ␲ u 1 ␲ 3g )
3
90%(1 ␲ ⫺2
u 1 ␲ g)
⫺2
83%(1 ␲ u 1 ␲ 3g )
2
51%(2 ␴ ⫺1
u 1 ␲ g)
2
57%(2 ␴ ⫺1
u 1 ␲ g)
⫺1
3
61%(3 ␴ g 1 ␲ ⫺1
u 1 ␲ g)
⫺2
3
85%(3 ␴ g 1 ␲ g )
⌫f
⌬E
具E典
FWHM
480
0
232
8
770
203
524
302
102
1
369
18
9
62
35
186
280
158
440
398
266
124
12
316
518.80
514.26
513.17
512.51
511.76
510.83
510.72
510.33
510.31
508.64
506.86
506.66
506.54
506.45
506.06
505.37
505.28
505.00
504.23
503.30
502.57
501.79
501.75
501.56
517.05
514.27
513.24
511.91
511.79
510.38
510.81
509.77
509.93
510.15
507.93
508.00
507.90
507.61
504.59
507.57
505.92
505.96
506.49
505.52
502.14
501.23
503.54
502.33
4.51
0.15
0.22
1.54
0.17
1.19
0.28
1.45
0.98
3.90
2.77
3.45
3.52
2.98
3.81
5.66
1.67
2.48
5.84
5.73
1.12
1.47
4.63
1.98
dE
dR
5.63
-11.83
-12.31
-5.78
-11.98
-7.14
-12.57
-6.14
-7.96
-26.63
-22.32
-24.93
-25.19
-23.11
2.93
-33.41
-18.06
-21.18
-34.10
-33.66
-7.43
-6.08
-29.44
-19.26
a
State was explicitly included in the LVI calculations.
␴共 ⑀ 兲⬀
冋冏
兺k 兺
兺n
lm
冏
具 ␹ k 兩 A lm 兩 ␹ n 典具 ␹ n 兩 D兩 ␹ 0 典
i
⑀ ⫹E k ⫺E n ⫹ ⌫
2
册
⫻ P 共 ⑀ ⫹E f ⫺E 0 兲 ,
2
共3兲
where the indices 0, n, and k designate the ground, intermediate, and final state vibrational quantum numbers and the
␹ ’s and E’s are the corresponding wave functions and energies, respectively. P(h ␯ ) is a Gaussian distribution representing the photon-energy distribution and ⌫ is the full width
at half maximum 共FWHM兲 of the natural linewidth of the
intermediate state. This width is determined by the sum of
the partial rates given in Eq. 共1兲 assuming that fluorescence
decay is negligible.
Theoretical spectra were obtained on three different levels
of accuracy. On the highest level, the transition to the X̃ 2 ⌸ g
ground state of O⫹
2 was treated explicitly with inclusion of
the R dependence 共RD兲 in the Auger transition rates of the
LVI formula. We designate this as RDLVI. Because of the
relatively large extent of vibrational mapping, the RD is well
motivated. The form of the potential energy curves 共PEC’s兲
has a great influence on these decay spectra, thus wellestablished experimental data for the PEC’s of the ground,
final 关32兴, and intermediate 关33兴 states are used. The existence of such significant geometry dependences is of general
interest in resonant Auger electron spectroscopy and a more
detailed investigation of this effect will be published elsewhere 关34兴.
The next level of theory is a distance independent
共Franck-Condon like兲 LVI theory. We denote this level
FCLVI. FCLVI was applied to the 15 low-lying final states
indicated in Table I. These states were selected as they give
rise to the most prominent part of the spectrum. The higherenergy states only produce broad, featureless structures in
the decay spectrum. The calculated decay spectra are very
sensitive to the accuracy of the spectroscopic parameters
012719-3
S. L. SORENSEN et al.
PHYSICAL REVIEW A 64 012719
FIG. 2. The O 1s electron yield spectrum measured at beam line
I411, MAX-Lab. The bandwidth 共140 meV兲 is indicated in the figure and the energies employed in the study are labeled. The spectroscopic parameters used in the calculation were taken from
Coreno et al. 关33兴.
used for the core-excited state, as well as for the final-ionic
states. The intermediate state PEC was shifted by an energy
of ⫺8.35 eV to match the experimental energies. For both
the RDLVI and FCLVI spectra, the lifetime energy width of
the core-excited state was taken as 0.1495 eV 关33兴. Further,
the photon energy distribution and the electron spectrometer
function were assumed to be of Gaussian form with FWHM
representative of the experimental conditions.
A treatment of a larger number of electronic states in the
FCLVI framework was not performed, as this is not justified
on the basis of the high density of electronic states with a
multitude of avoided crossings. Therefore, the less demanding moment theory of Cederbaum and Tarantelli 关4,21,23兴
was used to describe the complete resonant Auger electron
spectroscopy 共RAES兲 after broad band excitation of the O2
3
molecule in the 3 ⌸ (1 ␴ ⫺1
u 1 ␲ g ) resonance. In this theory,
the LVI formula was simplified such that only positions and
widths are obtained for the vibrational bands in the spectra.
For this, only the transition rates, the vertical energy, and its
derivative with respect to the internuclear bond distance are
required at the Franck-Condon point. Its main advantage is
that complete potential energy curves are not needed 共see
Refs. 关4,21,23兴 for further details兲. There, input data are
given in Table I together with the average positions and half
widths of the bands that result from the working equations of
the moment theory 共See, e.g., Eqs. 共8兲–共10兲 in Ref. 关23兴兲. We
designate these spectra ‘‘moment theory’’ spectra.
IV. RESULTS AND DISCUSSION
In Fig. 2, the measured electron yield spectrum of the
3
oxygen 3 ⌺ ⫺
g ⫺ ⌸ u core-excitation region is shown. The features in the electron yield spectrum are essentially equivalent
to those in the absorption spectrum. A simulation of the ab-
FIG. 3. 共a兲 The decay from the 3 ⌸ u core-excited state after
excitation to the resonance maximum 共530.80 eV兲. 共b兲 The direct
valence photoelectron spectrum measured with 70 eV photons with
a bandwidth of 15 meV. The total resolution is 55 meV. The configurations of the final states are indicated in the figure.
sorption spectrum including bars to represent the centroid
energy of each vibrational sublevel and a total simulated
absorption curve is included in the lower frame. The vibrational energy in the intermediate excited state is 0.140 eV
and a natural linewidth of 0.150 eV was used to represent the
full width half maximum 共FWHM兲 of the Lorentzian profile
of the vibrational sublevels. These values were taken from
the absorption spectra measured by Coreno et al. 关33兴. The
maximum intensity of the measured absorption spectrum is
found at 530.75 eV and was used as a reference energy for
all other excitation energies. The photon energies and bandwidth used in the study are indicated in the figure.
The decay spectrum after excitation to the resonance
maximum 共530.83 eV兲 is shown in Fig. 3共a兲. The features in
the spectrum are generally in accord with previous measurements 关10,12,13,16兴 but our spectrum is of higher spectral
resolution. A high-resolution direct photoelectron spectrum
is shown in Fig. 3共b兲. Although this spectrum is plotted on a
binding-energy scale, it is well established that the spectral
features may be compared directly to those in the resonantly
excited spectrum in Fig. 3共a兲 共see for example Ref. 关37兴兲.
The valence photoelectron spectrum is vibrationally resolved, and the state identification is taken from previous
studies 关38–40兴. There is not, however, a one-to-one correspondence between the spectral features visible in Figs. 3共a兲
and 3共b兲.
012719-4
HIGH-RESOLUTION EXCITATION-ENERGY-DEPENDENT . . .
At 513.5 eV, there is a feature with unresolved vibrational
structure that at first glance resembles the A 2 ⌸ u and a 4 ⌸ u
states at 16–18 eV binding energy. A comparison with the
binding energy shows that these peaks correspond to the
A 2 ⌸ u state but the vibrational progression is less extended
in the resonant decay spectrum. The B 2 ⌺ ⫺
g state at 20.2 eV
binding energy in the direct valence spectrum also appears in
the resonant decay spectrum but with a rather different vibrational intensity distribution. These changes will be discussed in the section on LVI. In the resonant spectrum the
dissociative 3 2 ⌸ u state is seen as an unresolved shoulder
extending from 506 eV to 510 eV. Even in the valence electron spectrum, the 3 2 ⌸ u state exhibits no apparent structure,
whereas the A 2 ⌸ u state appears with an extended unresolved vibrational progression. According to the information
obtained from the photoelectron spectrum, the strong peak
2
near 510.4 eV is clearly the B 2 ⌺ ⫺
g state. The 2 ⌸ u and D
states are also visible in the resonant spectrum. In order to
identify the origin of the majority of the peaks in the 505–
515 eV region, we need information from a range of photon
energies through the resonance.
When resonant excitation is made to the 3 ⌸ state, transitions to many dipole-allowed direct photoionized states are
unfavored. We will show that, as a general rule, all quartet
final-state configurations seen in the direct valence spectrum
are missing in the resonant excitation spectrum as was noted
earlier by Carroll and Thomas 关11兴. Another important point
is that several final states are much more intense in the resonantly excited spectrum. According to the independent particle model, the photoionization of the O2 X 3 ⌺ ⫺
g ground
states with occupied orbitals of ␴ and ␲ symmetry leads
exclusively to ⌸ and ⌺ ⫺ final states. However, in Table I,
some of the states with ⌬ and ⌽ symmetry show appreciable
transitions rates for the resonant decay. Even when the correspondence between states is clear, there are dramatic differences between the resonant and nonresonant decay spectra
in the extent of vibrational excitation within the electronic
state. Although LVI is needed for a quantitatevely correct
form of the resonant decay spectra part of these differences
can also be explained by the much simpler vibrational mapping phenomenon.
The following is the vibrational mapping concept: The
resonant excitation takes place within the Franck-Condon region centered about the ground-state equilibrium distance.
Because the ground and core-excited states are not well
aligned, the region of the intermediate state probed by the
excitation is highly dependent upon the photon energy. Excitation to low vibrational sublevels leads to the decay to a
wide range of final vibrational sublevels, but at higher energies, the picture changes. The higher vibrational sublevels
are populated mainly at the inner ‘‘turning point,’’ and because of the relatively short electronic state life-time, the
wave function does not have time to propagate and the decay
pattern will reflect the ‘‘width’’ of the vibrational state wave
function at the turning point. The concept of a wave packet
propagating on a bound-state potential surface while undergoing electronic decay has been discussed in some detail by
Pahl et al. 关15兴 and by Gortel and Menzel 关41兴. The strong
influence of the closeness of the core-excited state lifetime
PHYSICAL REVIEW A 64 012719
FIG. 4. The high-resolution decay spectra for photon energies at
the onset, the maximum, and on the high-energy flank of the resonance. The photon bandwidth is 60 meV for these spectra.
and the excitation time has been demonstrated and numerous
cases have been simulated for the core-excited oxygen state.
Because of the inherent experimental difficulties in achieving
sufficiently high resolution at the oxygen K edge, the theoretical simulations have, until now, presented a deeper understanding of these phenomena compared to measured spectra. For much the same reasons, the molecular oxygen
resonant Auger spectrum displays relatively dramatic
changes as the photon energy is tuned through the resonance
vibrational manifold. The final X 2 ⌸ g state can be considered
as a textbook example of these effects. Starting with the high
kinetic-energy states, we see an extended vibrational progression associated with the X state 共peak 1兲, especially when
compared to the direct valence spectrum in Fig. 3共b兲. This
high-resolution spectrum is measured with a photon bandwidth of 60 meV. Here, the ARR conditions are of great
value since we have a narrow linewidth through the decay
process. The X-state spectrum clearly shows the vibrational
mapping concept, and will be compared with calculated
spectra from different theoretical models in a later section of
the paper.
In Fig. 4, a series of high-resolution decay spectra for
photon energies corresponding roughly to excitation at the
␯ ⬘ ⫽0, ␯ ⬘ ⫽4, and ␯ ⬘ ⫽7 sublevels of the core-excited state
are shown. We observed the effects of vibrational mapping
manifested in features that appear to be fixed in kinetic energy associated with the four prominent peaks at 513.1,
511.8, 510.6, and 510.1 eV kinetic energy. These are indicated in the figure, and will be discussed in greater detail in
the context of the calculations.
First we compare spectra that are free from the effects of
narrow-band excitation to calculated spectra using the moment theory mentioned above. In Fig. 5, the measured RAES
spectrum using a bandwidth of approximately 280 meV is
compared with the calculated spectrum where FCLVI theory
is employed. The good agreement of these spectra support
the adequacy of the theoretical approach. The calculated parameters given in Table I appear to reproduce the main features in the decay spectrum. The calculations support most of
the assignments of this spectrum given above and in prior
012719-5
S. L. SORENSEN et al.
PHYSICAL REVIEW A 64 012719
FIG. 5. Broadband RAES of O2 from 共a兲 experimental spectrum, 共b兲 calculated FCLVI, and 共c兲 calculated ‘‘moment theory’’
spectra for a photon energy of 530.85 eV and a bandwidth of 230
meV. The bar spectrum in 共c兲 indicates the intensities and the average positions of the bands.
work 关11,16兴. The present assignments provide clear experimentally founded assignments through the resolution and the
dispersion studies. The transition rate to the a 4 ⌸ u state is
calculated to be zero as the transition to quartet final states
from the 3 ⌸ u intermediate state requires that the two electrons participating in the Auger process be triplet coupled. In
the transition to the a 4 ⌸ u state, these electrons stem exclusively from ␲ orbitals that are, in the one-center approach
used in this work, represented by 2p ␲ atomic orbitals.
KVV-type Auger transitions of triplet coupled electrons from
2 p orbitals are, however, parity forbidden like the
Ne⫹ 2 S e (1s ⫺1 )→Ne2⫹ 3 P e (2 p ⫺2 ) transition. However, in
the nonspherical electrostatic potential of the O2 molecule,
the ␲ orbitals contain a small contribution of d ␲ type atomic
orbitals, which could cause a nonzero transition rate to this
state. Despite this, the experimental observations do not indicate the a 4 ⌸ u state at its expected kinetic energy 共514.3
eV兲. Due to similar reasons the transition rates to quartet
states are small in general 共See Table I兲.
For the most intense peak in the experimental resonant
decay spectrum, the calculations give a different assignment
than what was proposed previously 关11,35兴. Most of the intensity of the peak at 510.8 eV is due to the 2 2 ⌸ u final state.
This peak has so far been assigned to the D 2 ⌬ final state, but
although it appears in the spectrum at the same position its
intensity is only half as big as that of the 2 2 ⌸ u state. In the
calculated moment theory spectrum 关Fig. 3共c兲兴 and even
more in the FCLVI spectrum 关Fig. 3共b兲兴 this part of the spectrum does not seem to agree well with the experimental one.
A comparison with experimental term energies shows that
due to the different size of dynamic electron correlation the
2
energies of electronic states with a (1 ␲ ⫺1
u 1 ␲ g ) configuration
4
2
2
2
共i.e. the a ⌸ u , A ⌸ u , C ⌽ u , 2 ⌸ u , and 3 2 ⌸ u states兲 are
systematically 0.3 eV too low compared to states with (1 ␲ 1g )
2
and (3 ␴ ⫺1
g 1 ␲ g ) configuration. This causes an artificial shift
of the rather sharp peak that is due to the 2 2 ⌸ u state to
higher kinetic energy. This peak in turn does not overlap
2
with the somewhat broader structures from the (3 ␴ ⫺1
g 1 ␲ g)
2
2 ⫺
2 ⫹
configuration states D ⌬ g , B ⌺ g , and 1 ⌺ g . The latter
two states are mainly responsible for the peak at 510.2 eV
that is strongly overlapping, with the most intense peak at
510.8 eV.
Furthermore, the calculations also allow to assign the
broad peak at 508 eV to the 3 2 ⌸ u and 2 2 ⌸ g states and that
at 506 eV to the 1 2 ⌽ g , 3 2 ⌸ g , and 1 2 ⌸ u states. FCLVI
calculations show that the structure around 509.5 eV kinetic
energy is not due to another distinct final electronic state, but
the result of extensive vibrational side bands of the strongly
repulsive 3 2 ⌸ u , 1 2 ⌬ u , and 1 2 ⌽ g states.
The two peaks at 502 and 503 eV can be assigned to the
2 2 ⌬ u and 4 2 ⌸ g states. At lower kinetic energies, no further
distinct structures can be seen in the moment theory spectrum. Therefore, we have not included their results in Table
I or in Fig. 3.
In prior works it was attempted to assign the resonant
decay spectrum on the basis of electronic states that are
known from photoelectron spectroscopy. In fact, the detailed
consideration of Carroll and Thomas 关11兴 is in remarkably
good agreement with the present calculations. However, it
has to be taken into account, that such an assignment can
only be tentative as a considerable part of the resonant decay
leads to states that are not visible in photoelectron spectra.
Thus, the five most intense final states, which are known
from photoelectron spectroscopy of the X 3 ⌺ ⫺
g ground state
and the a 1 ⌬ g first excited state of O2 , 共i.e. the C 2 ⌽ u ,
X 2 ⌸ g , 3 2 ⌸ u , and A 2 ⌸ u states兲 contribute only about as
much to the decay spectrum as the five most intense states
that are not observable with this technique (2 2 ⌸ u , 1 2 ⌽ g ,
3 2 ⌸ g , and 4 2 ⌸ u 兲. We also want to stress that even a
knowledge of all electronic states contributing to the resonant decay and the photoelectron spectrum is not giving a
complete picture of the multitude of electronic states. E.g.,
some of the quartet states discussed above are not detectable
with either of the two methods. The same also applies to the
1 2⌺ ⫹
u state, which has, however, been observed before by
electron transfer in alkali metal – O2⫹
2 collisions 关36兴.
Kivilompolo et al. 关42兴 have shown the close agreement
between moment theory calculations and the more demanding FCLVI calculations. However, only the latter theory is
012719-6
HIGH-RESOLUTION EXCITATION-ENERGY-DEPENDENT . . .
FIG. 6. The decay spectra for a series of photon energies
through the resonance. 共a兲 The experimental spectra measured with
a bandwidth of 140 meV. 共b兲 The theoretical FCLVI spectra.
able to describe the changes in the band forms for selective
narrow band excitation. In the next section, we investigate
the dispersion of the resonant peaks as the photon energy is
swept through the resonance.
A. LVI in the most prominent structures
In Fig. 6, we compare experimental spectra 关Fig. 6共a兲兴
measured with a series of photon energies with theoretical
FCLVI 关Fig. 6共b兲兴 for the states in the binding energy range
up to 25 eV. The photon energies include 529.85 eV 共0.5 eV
below the ␯ ⬘ ⫽0 peak兲 and 532.22 eV 共the ␯ ⬘ ⫽13 energy兲.
In both of these spectra, the contribution from direct photoionization is comparatively large, as seen by the relatively
strong c 4 ⌺ ⫺
u peak at ⬃508 eV kinetic energy in the
highest-energy spectrum. This peak is the only quartet state
seen in the spectrum, but is not populated through the resonant excitation. The intensity of the c state is mainly an
indication of the ratio of the cross sections for direct photoionization contra resonant photoionization. The remainder of
the features are from Auger decay.
The feature at 513 eV corresponds to the A 2 ⌸ u state.
This peak does not appear to disperse with photon energy.
Such a dispersive behavior is expected for fast transitions
between two states with nearly parallel potential curves. The
curves in Fig. 1 indicate that this is the origin of the ‘‘fixed
kinetic energy’’ of this peak. There is a sharper feature indicated with an arrow in Fig. 6 that also belongs to this decay.
This interpretation is strengthened by the fact that the behavior of this state is reproduced very well in the calculation in
Fig. 6共b兲.
We identify the states in descending kinetic energy as the
c 2 ⌽ u at 511.64 eV, the D 2 ⌬ g at 510.33 eV, and the B 2 ⌺ ⫺
g
(1 ␲ 3u 1 ␲ 2g ) at 509.72 eV kinetic energy for excitation at the
resonance maximum. This is in accord with previous studies
关11,43,16兴. It should be noted that the D 2 ⌬ g and the 2 2 ⌸ u
states are very weak or invisible in the direct valence electron spectrum in Fig. 3共b兲.
Another feature related to LVI is the sharp structure at
approximately 511 eV kinetic energy. The higher-resolution
PHYSICAL REVIEW A 64 012719
2
FIG. 7. 共a兲 The decay to the O⫹
2 X ⌸ g final state as the photon
energy is tuned through the resonance region. Note that the energy
scale is in binding energy. 共b兲 The simulated spectra using the
RDLVI and FCLVI level of theory, full and dotted lines, respectively.
spectra 共Fig. 4兲 show this feature more clearly. This feature
arises from the decay at the intermediate state turning point,
this time to the C 2 ⌽ u state (1 ␲ 3u 1 ␲ 2g hole state兲. Another
case in hand is the 2 2 ⌸ u at 510.76 eV, a many-electron state
arising from the (1 ␲ u ) 3 (1 ␲ g ) 2 configuration with the two
electrons in the 1 ␲ g orbitals being preferentially singlet
coupled. From the PEC’s in Fig. 1 and the derivatives in
Table I we see that these potentials are nearly parallel to that
of the 3 ⌸ intermediate state with a minimum at a distance
close to 1.5 Å. The same explanation given for the A 2 ⌸ u
state applies in this case as well. Transitions between aligned
potentials with the same curvature will not disperse with
photon energy. The other three states follow the dispersion
expected under ARR conditions and the general features of
all of these spectra are well reproduced in the calculated
spectra.
B. Bond-length dependent RDLVI for the X final state
One particularly noteworthy difference between the
FCLVI and the experimental spectra in Fig. 6 is seen for the
higher vibrational levels. The general description of the overall intensity distribution among the vibrational states has
been put forth in some detail in previous works 关15兴 and
calculations using a time-dependent theory for the decay of
the core-excited state are available 关41兴. Although previous
measurements have been qualitatively in agreement with
simulated spectra, the resolution of these spectra is not sufficient to show such deficiencies. Further calculations have
thus been made in which the bond-length dependence of the
Auger transition rate to the the X 2 ⌸ g final state is taken into
account. In the case of decay to large bond distances where
curve crossings may affect the potentials, this has been found
to be of great importance 关31兴. We will show that the bondlength dependence of this transition is of importance, although not as much as the tremendous change of the transition rates shown in Fig. 1 may indicate.
012719-7
S. L. SORENSEN et al.
PHYSICAL REVIEW A 64 012719
We present a series of experimental spectra together with
the calculated RDLVI and FCLVI spectra for the decay to
the X 2 ⌸ g (1 ␲ 1g ) final state in Figs. 7共a兲 and 7共b兲. The lowest
and highest photon-energy spectra in the calculation are
scaled in order to make the features clearer. For the lowest
photon energy, the weak features in the spectrum resemble
strongly the direct photoionization spectrum. This is in accord with the duration time concept for detuning below the
onset of the resonance cross section. As the photon energy is
tuned to the ␯ ⬘ ⫽0 level of the intermediate state, an excitation of higher vibrational levels becomes stronger as the
wave packet propagates on the intermediate state potential.
The fast decay rapidly depopulates the state, hence, the tapering off of intensity. As the excitation energy is tuned to
higher vibrational sublevels, the decay spectra span a wider
kinetic energy range. This trend continues, albeit modulated
by the resonance cross section, except that the asymmetric
intensity distribution in the images of the turning points becomes more pronounced. This is a direct consequence of the
emptying out of the intermediate state that is described by
the exponential population decrease of the excited state. The
calculated spectra show this trend very clearly.
A comparison of the experimental and theoretical spectra
in Fig. 7 shows that the FCLVI theory clearly overestimates
the intensity of the low kinetic-energy part of the X 2 ⌸ g
band. This effect is already clear for a spectrum taken at the
maximum of the absorption curve where this structure decreases by approximately a factor of 2. It becomes even more
prominent for higher-excitation energies. However, for these,
the nonnegligible amount of direct photoionization causes
additional intensity in the high kinetic-energy part of the
spectrum that has not been included in the calculations.
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V. CONCLUSIONS
In this paper, the details of the decay of the 3 ⌸ intermediate state in oxygen have been investigated with ab initio
calculation. Furthermore, electron spectra of unprecidented
resolution are presented and analyzed.
An unambiguous identification of the final states and their
dispersion behavior has been made. We have been able to
identify the configurations of these states by a combination
of valence spectra, photon-energy-dependent studies, and advanced calculations. We identify all final states as belonging
to the molecular electronic structure, and all states are found
to disperse according to the Raman-Stokes dispersion law.
The details of the vibrational intensity distribution in the
lower valence states of O⫹
2 have been reproduced to a satisfactory degree. For that purpose, we used bond-length independent Auger transition rates. A significant R dependence
of the transition rate was found for the decay to the X final
state by comparing the experimental measurement with the
results of standard and with bond-length dependent Auger
transition rates.
ACKNOWLEDGMENTS
The authors are grateful to the staff of the MAX laboratory for assistance beyond the call of duty. This work was
supported by the Swedish Research Council for the Natural
Sciences 共NFR兲, the Foundation for Strategic Research
共SSF兲, and the Swedish Council for Technical Development
共TFR兲. One of the authors 共M.N.P.兲 wishes to thank the NFR
for a Guest Chair in Atomic and Molecular Physics. The
Wennergren Foundation is acknowledged for financial support of one of us 共R.F.F.兲.
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