ARTICLE IN PRESS
Radiation Physics and Chemistry 76 (2007) 404–411
www.elsevier.com/locate/radphyschem
Relativistic calculations for highly correlated atomic and
highly charged ionic systems
Fumihiro Koikea,, Stephan Fritzscheb
a
Physics Department, School of Medicine, Kitasato University 1-15-1 Kitasato, 228-8555, Japan
b
Universitaet Kassel, Heinrich-Plett-Strasse, 40, 34132, Kassel, Germany
Received 31 August 2005; accepted 2 October 2005
Abstract
The effect of electron correlations in many electron atoms or many electron atomic ions are discussed extensively
based on the MultiConfiguration Dirac Fock (MCDF) calculations. In a precision atomic physics the relativistic
treatment of the system is indispensable. The correlation effects and the relativistic effects are no more additive if one
want to treat the many electron systems accurately. In the excited states, the single electron orbitals are modified in
accordance with the vacancies near the atomic center. The electron correlations may be evaluated from the nonorthogonality of the single electron orbitals. Several examples have been given. The electronic configurations with the
same total parities may interact each other even in the cases the constituent single electron orbitals have opposite
parities. Such the configuration interactions may provide us with characteristic interference structures in the optical
emission or absorption spectra. The anormally of the extreme ultra-violet optical emission spectra of highly charged tin
ions has been illustrated as an example.
r 2006 Elsevier Ltd. All rights reserved.
PACS: 32.80.t; 32.80.Fb
Keywords: Atomic structure; Atomic transition; Optical emission; EUV; Relativistic theory; MCDF; Configuration interaction;
Electron correlation
1. Introduction
In recent years, the spectroscopic measurement in
atomic physics experiments gives us quite a sophisticated
set of data that requires an accurate theoretical
treatment. Also in the field of plasma physics, requirement for detailed and sophisticated description of the
plasma is becoming noticeable in the study of confinement nuclear fusion as well as the other plasma
applications. Because a plasma model is based on the
Corresponding author.
E-mail address: [email protected] (F. Koike).
atomic models that may facilitate appropriately the
corresponding plasma simulations, a precision plasma
model requires precision atomic data. For plasmas
containing heavy elements, we must take into account
the effects arising from such complex species. In Fig. 1,
we illustrate the situation schematically.
To obtain theoretical values with reasonable precision, we find that a fully relativistic treatment for the
problem is indispensable. In fact, the value b ¼ v=c,
which is the ratio of the orbital electron velocity v and
the speed of light c, is about 14=1370:1 in silicon 1 s
orbitals, for example. The relativistic correction could be
a few percent of the total atomic state energy and also of
0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.radphyschem.2005.10.044
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F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
405
Multi-Configuration
and
Non-Local Treatment
Dirac Equation
and more
Electron Correlation
Schroedinger Equation
Precise Atomic
Structure Calculation
Relativistic Treatment
Local State-Independent Potentials
Fig. 1. A schematic illustration for the steps towards precision atomic structure calculations. Because the effects of electron
correlations and of the relativistic behavior of atomic electrons are not additive, both the effects must be evaluated simultaneously.
the electronic transition energy if one considers the deep
inner-shell excitations; the relativistic effects cannot be
canceled out by subtraction of the lower state energies
from the upper state ones. Large spin–orbit term
splittings may also appear. The electron correlation
energy is in general a few percent of the total atomic
electron energy. Consideration of the electron correlations between various electronic configurations is also
indispensable if we want to determine the atomic
transition energies within the accuracy of several
electron volts or less. And, furthermore, an important
point is that we cannot, in general, discriminate between
the relativistic effects and the correlation effects out of
the actual atomic transition energies or excitation
strengths, and sometimes the relativistic shift of the
orbital energies could even enhance or suppress the
electron correlation effects. In this sense, we should
point out that both the relativistic and correlation effects
must be treated simultaneously.
In this report, we, firstly, in Section 2, discuss the
characteristic properties of Dirac Hartree Fock or
multiconfigutration Dirac Hartree Fock methods, which
are a group of methods based on variational principles
for the set of single electron orbitals. It is pointed out
that the concept of a single electron orbital would have
to be modified if one would optimize the atomic states
individually. We briefly review extensive efforts to carry
out the atomic structure calculations by a number of
authors. Secondly, the electron correlations in highly
excited atomic or ionic states including the deep inner
shell excitations and/or multiple excitations will be
reviewed. We illustrate a couple of numerical examples
with the corresponding experimental data. The electron
correlation effects is important even in highly ionized
atoms. In heavy elements, a pair of electrons may come
close to provide us with large correlation energies. The
energy of correlations of atomic electrons can be
considered as typically of the order of a few electron
volts. We must take into account the electron correlation
effects, when the required accuracy of the atomic
transition energies falls in this range. The orbital
contractions in the atomic excitation may play a key
role, which will be shown in Section 3. Thirdly, in
Section 4, we discuss the electron correlations due to the
inter-(sub)shell mixing. In general, the electronic configurations with the same total parity may interact each
other even in the cases the constituent single electron
orbitals have opposite parities; a configuration state
function recovers its total parity after a two-electron
replacement. As an example of such configuration
interactions, we illustrate an anomaly in the extreme
ultra-violet optical emissions of highly charged tin ions
Sn12þ . Finally, in Section 5, we give concluding remarks.
2. Characteristics of the Dirac Hartree Fock methods
As for the description of the Hartree Fock method, we
may refer to the formalism of Lipkin (1973). We
consider an N-electron wavefunction under the independent particle model. We take ayki as a creation
operator of the single electron orbital. The N-electron
wave function may be represented by
jUi ¼
N
Y
i¼1
ayki j0i,
(1)
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F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
406
where j0i represents the vacuum and the anti-symmetrization of the electron orbitals will be made if
necessary. In the Hartree Fock method we apply a
variational condition
dhUjHjUi ¼ hUjHjdUi ¼ 0
(2)
under an appropriate ortho-normality constraint for
wavefunctions. We take dyi as a small variation of ayki .
Then, the variation jdUi may be represented as
jdUi jU þ dUi jUi ¼
N
X
ð1ÞP dyj
j¼1
N
Y
ayki j0i,
i¼1ðiajÞ
(3)
where P in ð1ÞP is the number of permutations that is
required to place the operator dyj on top of the product.
Substituting Eqs. (1) and (3) into Eq. (2), we obtain
hUjHjdUi ¼
N
X
ð1ÞP h0jakj jHjdyj j0i ¼ 0.
(4)
j¼1
From this equation, we see that the (D)HF method
diagonalizes H with respect to the single electron
displacements on the basis of independent particle view
point. We also find that the atomic states that
diagonalize the multi-electron displacements cannot be
represented by one unique configuration state function
(CSF) jUi. And, further on, we may recognize that an
atomic state jUi may be optimized for each atomic state
independently of the other states. The basis sets fayki g
may differ between different atomic states. And,
furthermore, we may note that we can obtain excited
states by choosing the basis sets fayki g as appropriate;
there is no restriction for the atomic state jUi to remain
in the ground state in the framework of the (D)HF
method.
In the multiconfiguration Dirac Hartree Fock method, we define the atomic state function (ASF) in terms of
configuration state functions (CSFs). We define an ASF
jWm i by CSFs jUn i as
jWm i X
jUn icnm
with jUn i n
N
Y
i¼1
ay ðnÞ j0i.
ki
(5)
We optimize jWm i applying the variational condition
that
*
+
X
d
Un cnm jHjWm
n
*
¼
X
+
dfUn g cnm jHjWm
n
*
þ
X
+
Un dfcnm gjHjWm
¼0
ð6Þ
n
under an appropriate orthonormality constraint for the
orbital basis sets, CSFs and ASF. We note here that the
ASF jWm i may be optimized individually with respect to
the index m, and, therefore, orbital basis sets are not
necessarily orthogonal between the sets for different m.
For example, a 1s orbital in the first set jð1sÞ1 i may be a
linear combination of the second set jðnsÞ2 i; we may here
write as
X
X
jð1sÞ1 i ¼
jðnsÞ2 ihðnsÞ2 jð1sÞ1 i ¼
jðnsÞ2 isn .
(7)
n
n
The states described by the other set {ai ð2Þ} may be
represented by a state with multiple excitations on the
basis of fai ð1Þg; the effect of electron correlations may be
evaluated by introducing a separate optimization for
individual states. However, in the calculation of atomic
states, we should note that the symmetry quantum
numbers are not the subject of the optimization and
therefore the basis sets are always orthonormal with
respect to the symmetry quantum numbers.
In the relativistic calculations, the electronic Hamiltonian of the system is chosen as in the following:
X
X 1
H¼
hDC ðiÞ þ
r
i
ioj ij
!
X 1
ðai ri Þðaj rj Þ
þ
ai aj þ
,
ð8Þ
2rij
r2ij
ioj
where hDC ðiÞ is the single electron Dirac Hamiltonian,
and the last term is the Breit correction term. The
symmetries of the ASFs are specified by their parity,
total angular momentum, and its magnetic component.
And they are expanded in terms of the configuration
state functions (CSFs) that represent virtual single, or
multiple excitations from occupied to unoccupied (sub)shells.
In these couple of decades, extensive efforts have been
made for computer codes to calculate the atomic
structures on the basis of multiconfiguration relativistic
treatment. Grant and coworkers (Dyall et al., 1989;
Parpia et al., 1996) have developed a set of programs
called GRASP (General Purpose Relativistic Atomic
Structure Program). As an extension of GRASP,
Fritzsche and his coworkers have developed a set of
codes (Fritzsche et al., 2002a,b; Fritzsche, 2001; Fritzsche
et al., 2000; Fritzsche and Anton, 2000; Fritzsche et al.,
1999; Fritzsche and Grant, 1997; Fritzsche and Froese
Fischer, 1997; Fritzsche and Grant, 1995) that calculate
the atomic transition and other properties, which is called
RATIP (Relativistic Atomic Transitions and Ionization
Properties). Kagawa and his coworkers have developed a
set of relativistic configuration interaction codes (Kagawa, 1975; Kagawa et al., 1991). Another set of multiconfiguration Dirac Fock programs has been developed
by Kim and his coworkers (Mohr and Kim, 1992). And,
Klapisch and his coworkers have developed a set of
programs that is called HULLAC with local approximation of the exchange integrals (Klapisch et al., 1997).
ARTICLE IN PRESS
F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
processes. They observed a pair of 1s1 2s1 ð3 SÞ 2p6 3s1 3p1 ,
and 1s1 2s1 ð1 S Þ2p6 3s1 3p1 double electron excitations at
924.78 eV and at about 935 eV of photon energies. Those
two peaks are about 10 eV apart from each other. This
value is somewhat large compared to the usual understanding of the singlet-doublet separation in neon 1s1 2s1
configurations. To find out the origin of this largeness,
we have calculated the electron density distribution of
the Ne2þ 1s1 2s1 2p6 configuration using both the MCDF
wavefunctions of neutral neon and the doubly charged
neon atomic ion. Fig. 3 shows the charge density plots
from those two sets of wavefunctions. We see that in the
actual Ne2þ 1s1 2s1 2p6 ions, the electronic charges are
relaxed to the atomic center. It may be concluded that
this contraction of the electronic wavefunctions due to
the 1s2s double hole creation is the cause of the
enlargement of the singlet–triplet separation in Ne
1s1 2s1 ð3;1 S Þ2p6 3s1 3p1 states.
In the rest of the present report, we introduce a couple of
calculations on the remarkable phenomena relating to the
electronic correlations of atoms using GRASP family codes.
3. Importance of the state dependent atomic orbital
wavefunctions
3.1. The 4d–4f photoexcitation of Xeþ
It is well known that there is a strong concentration of the
oscillator strength in the 4d photoexcitations of xenon
atoms in the photon energies around 100 eV, which is called
the giant resonance. Sano et al. (1996) made a measurement
on the photoionization of 4d-electrons in singly charged Xe
ions. And a number of subsequent measurements have been
carried out by Koizumi et al. (1997, 1996). They found a
double hump structure in the photoion spectra of Xeþ and
analyzed them by means of the MCDF calculation. They
found that the 4f orbital suffers a strong collapse when the
system has a simultaneous 5p to 6p shake up excitation. We
illustrate the usual and collapsed wavefunctions in Fig. 2.
Due to this 4f orbital collapse, the 4f orbital could have a
strong interaction with 4d orbital which realize the double
hump structure in the observed spectra. And we also note
here that this type of collapsed orbital may not be expanded
in terms of the nf orbital wavefunctions without collapse;
the optimization of the collapsed orbital is compulsory if we
want to gain an insight into the Xeþ 4d 9 4f 1 5s2 5p4 6p1 shake
up excitations.
3.3. Li 3s orbital contraction triggered by K, L electron
escape by photoabsorption
In the last decade, there have been an extensive study
of the lithium hollow atomic state excitations (Azuma
et al., 1995). Azuma et al. (1997) have observed for the
first time the K, L double shell hollow atomic state in
lithium atoms. Fig. 4 shows their experimental spectrum
with the results of an elaborate MCDF calculation. We
can see that the oscillaor strength of the three electron
excitations is still large to make the states experimentally
observable. The largeness of the oscillator strength
originates from the 3s and 3p orbital contraction in
the hollow atomic states. In Fig. 5, we illustrate the
change of the charge distributions from the lithium
ground state to the lithium 333 resonance states. We can
3.2. Ne 1s and 2s resonant double photoexcitation
Recently, Oura et al. (2004) have made an experiment
on neon 1s and 2s resonant double photoexcitation
Xe + 4 d 94 f 1 5s2 5 p5 6 p0
6p
0.5
5f
6f
7f
0.0
Radial Wavefunction
4f
4d – 4 f
+
-0.5
4d
6p
4f
0.5
5p – 6 p
Collapse
-1.0
Xe + 4 d 94 f 1 5s2 5 p4 6 p1
0.0
7p
-0.5
5p
-1.0
4d
0
5
407
10
Radial Distance (a0 )
15
20
Fig. 2. The collapse of the 4f orbital wavefunction due to the 5p–6p shake up excitation in Xeþ 4d photoionizations.
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WithOUT Relaxation
After the Inner-Shell Hole
Creation
Charge Density (1 /a0)
8
6
With Relaxation
After the InnerShell Hole Creation
4
Contraction
of electron
orbitals
2
0
0
0.5
1
1.5
2
Radial Distance (a0 )
2.5
3
Fig. 3. The charge density plots from two sets of wavefunctions. We see that in the actual Ne2þ 1s1 2s1 2p6 ions, the electronic charges
are relaxed to the atomic center.
× 10- 4
2 s 2 2p
3s 23p + 3p3
2
2s2p3s+2 s 23p
Cross Section (kb)
Oscillator Strength
3
1
15.5
15.0
14.5
14.0
13.5
13.0
175.3
12.5
174.5 175.0 175.5 176.0
Photon Energy (eV)
eV
233
175.6 eV
0
140
150
160
170
180
Photon Energy (eV)
Fig. 4. Calculated oscillator strength distribution for Li hollow atomic excitations. The inset is the experimental photoion spectra
around 175 eV of photon energy. The theoretical resonance peaks are convoluted with small artificial natural width and instrumental
width.
see that the range of the charge density in 333 resonance
is rather short compared with the ones in 223 or 233
resonance; this may give a fairly large overlap of the 333
state configuration with the ground state configuration.
4. Electron correlations due to the inter-(sub)-shell
mixing
In these decades, the 13.5 nm range extreme ultraviolet (EUV) light emissions of many electron atomic
ions have become of a strong interest in relation to the
short wavelength light source of the semiconductor
lithography technologies. One of the best candidates for
such an EUV light source is considered to be of the intra
N shell (n ¼ 4 shell) transitions or inter N–O shell
transitions of tin (Sn) or xenon (Xe) multi-charged
atomic ions. Extensive efforts to produce an accurate
atomic data for the use of radiation transport calculations have been carried out by a number of authors
(Sasaki et al., 2004; Koike et al., 2005; Churilov, 2004;
O’Sullivan and Faukner, 1994; Svendsen and O’Sullivan, 1994). Because the range of the wavelength required
by the side of the lithography optics instrumentation is
very narrow, the required accuracy of the atomic
structure calculation is also high accordingly. We have
to evaluate, on one hand, the intra N shell electron
correlations precisely for excited states as well as the
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F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
409
3
Charge Density (1 /a0)
Ground State
222 Resonance
2
223 Resonance
233 Resonance
1
333 Resonance
0
0
2
4
6
8
10
12
Radial Distance (a0)
Fig. 5. Charge density distributions of Li hollow atomic states.
Configuration Mixing
4f 0
4f1
81 eV
4 d3
4 d1
72 eV
4 p5
4 p6
47 eV
4 s2
4s2
E = 81 – 72 = 9 eV ~ 4% of 4 d binding energy 253 eV
Fig. 6. The scheme of configuration interactions in Sn12þ 4d–4f excited states. An excited state odd parity electronic configuration
4s2 4p6 4d 1 4f 1 can be generated by replacing two even parity 4d electron orbitals by two odd parity electron orbitals 4p and 4f from a
configuration 4s2 4p5 4d 3 4f 0 . If the unperturbed energies of these two configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 are not very
different in values, strong interactions between these two configurations may be expected.
ground states. On the other hand, we have to notice
that, in this type of ions, especially of the Sn ions, a
peculiar behavior in the emission spectra has been
observed by O’Sullivan and Faukner (1994). They have
pointed out that the narrowing and the shift take place
in the 4f –4d EUV light emission spectra. They have
explained these phenomena as due to the interactions
between 4p6 4d w1 4f 1 and 4p5 4d wþ1 4f 0 configurations,
where w is an integer that runs from 1 to 9. This is
another type of the electron correlations which are
different from the ones discussed in the previous section,
and we may note that those correlations may also be
interesting from the view point of basic atomic physics.
To gain a further insight into the effects, we have carried
out careful MCDF calculations for 4d w (w ¼ 1–9)
atomic ions with atomic number Z ¼ 48–56, using
GRASP and RATIP family computer codes. At first,
we chose the system Sn12þ as a candidate for detailed
investigation of the electron correlation effects, since the
ground state of Sn12þ has 4d 2 partially filled open
subshell as its outermost subshell, and the EUV light
emission falls near to the range of the wavelength
around 13.5 nm. In Fig. 6, we illustrate the scheme of
configuration interactions that we took into consideration. An excited state odd parity electronic configuration
4s2 4p6 4d 1 4f 1 can be generated by replacing two even
parity 4d electron orbitals with two odd parity electron
orbitals 4p and 4f from a configuration 4s2 4p5 4d 3 4f 0 . If
the unperturbed energies of these two configurations
4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 are not very different in
values, we may expect strong interactions between these
two configurations. In the result of an elaborate MCDF
calculation, we found that in the case of Sn12þ , the
energy differences between the neighboring single
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electron orbitals are quite close as indicated in Fig. 6; the
difference of the energy separations between 4f and 4d,
and 4d and 4p has turned to be 9 eV, which is only 4% of
the 4d electron orbital energy. And, furthermore, we
have found that the major peak positions of the 4s, 4p,
4d, and 4f radial wavefunctions almost coincide. The
configurations 4s2 4p6 4d 1 4f 1 , and 4s2 4p5 4d 3 4f 0 have
good reasons to suffer strong mixing as of the effect of
configuration interactions. The optical 4p–4d and 4f –4d
transitions cannot be discriminated from the others, in
other words, the transitions take place coherently at the
same time, providing us with quite a peculiar EUV
emission spectrum. In Fig. 7, we have plotted the
oscillator strength distribution for 4d–4f band transitions of Sn12þ ions. Each spectrum gives a simple sum
over the oscillator strengths for transitions of all the
possible combinations of the members of multiplets in
the excited and ground states. The calculated spectrum
has been smoothed out by convoluting a natural width
function of the artificial width 0.05 nm. The dashed
curve represents the spectrum in which the coherent
transitions of 4d–4f and 4p–4d has been taken into
account. The dotted curve represents the distribution of
only the 4d–4f transitions, and the solid curve represents
the distribution of only the 4p–4d transitions. We can
see that a constructive interference between the 4d–4f
and 4p–4d transitions is taking place at a shorter
wavelength region, and also that a destructive interference is taking place at a longer wavelength region. We
can confirm the enhancement of the optical transition
due to the interference effects in a wavelength region at
13 eV. We have, further, explored if any similar effects
can be found in the atomic ions with different atomic
numbers and charge states. We finally realized that those
interference phenomena are quite common to the ions in
this range; a similar effect has been observed also in Xe
ions, and further on, in almost all the atomic ions with
Z ¼ 48–56.
5. Discussion
We have reported a couple of characteristic features
of the electron correlation effects based on the multiconfiguration Dirac Fock type calculations using the
atomic codes from GRASP and RATIP. In these
calculations, the atomic states are optimized individually
to the specific energy levels. The single electron
wavefunctions with the same principal quantum numbers may differ between the atomic energy levels; the
concept of the independent particle wavefunctions are
restricted in the framework of the MCDF theory.
However, if we trace the change in the nature of a
single electron orbital with the same set of symmetry
index in the atomic excitation and de-excitation
processes, the concept of the single particle model will
be recovered by introducing the primitive idea of the
contraction or stretching of the orbital. And we can
4 d - 4 f & 4 p - 4 d Transitions of Sn12+ Ions
4d - 4f + 4p - 4d
Intensities (arb. units)
Interference considered
4 d - 4 f only
4 p - 4 d only
Wavelength (nm)
10
12
14
Constructive
16
18
20
Destructive
Interference
Fig. 7. The oscillator strength distribution for 4d–4f band transitions of Sn12þ ions. Each spectrum gives a simple sum over the
oscillator strengths for transitions of all the possible combinations of the members of multiplets in the excited and ground states. The
calculated spectrum has been smoothed out by convoluting a natural width function of the artificial width 0.05 nm. The dashed curve
represents the spectrum in which the coherent transitions of 4d–4f and 4p–4d has been taken into account. The dotted curve represents
the distribution of only the 4d–4f transitions, and the solid curve represents the distribution of only the 4p–4d transitions.
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F. Koike, S. Fritzsche / Radiation Physics and Chemistry 76 (2007) 404–411
organize such a picture on good theoretical foundations.
The key to distinguishing the single electron quantum
numbers are the number of nodes in the orbital
wavefunction. The MCDF method may give a good
base for those pictures. The parity of the multiple
electron system may be recovered by the replacement of
the two single electron orbitals of opposite parities with
respect to the parity of the original occupied orbitals. In
the atomic ions with 4d partially filled shell, the orbitals
4s, 4p, 4d, and 4f may sometimes be almost equidistant
in the single electron orbital energy. In these cases, the
4p–4d excitation and 4d–4f excitation may have almost
the same energies as far as the difference of the single
electron orbital energy concerned. In the framework of
the MCDF calculations, these two excitations cannot be
discriminated between each other with respect to the
total parity and the total angular momentum of the ionic
systems; they are mixed through the configuration
interactions, in other words, they undergo serious
interference in the course of excitation or de-excitations.
We have introduced the modification of the photoemission spectra in tin atomic ions. These types of
phenomena induced by the interference between the
configurations with two opposite parities of the single
electron orbitals are found to be quite common to the
systems of highly charged ions with the atomic number
from 48 to 56.
Acknowledgments
This work is partly supported by the Leading Project
for advanced semiconductor technology of MEXT
Japan. This work is partly supported by Grant in Aid
of scientific Research from JSPS Japan. One of the
authors (F. Koike) would like to express his thanks to
Professor Nishihara of the Institute of Laser Engineering in Osaka University for his valuable suggestions and
his continuous encouragement.
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