S. S. Tayal
Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA
Received 29 June 2004 / Accepted 5 July 2004
Abstract
Non-orthogonal orbitals technique in the multiconfiguration Hartree-Fock approach
is used to calculate oscillator strengths and transition probabilities for allowed
and intercombination lines in Cl II. The relativistic corrections are included
through the Breit-Pauli Hamiltonian. The Cl II wave functions show strong term
dependence. The non-orthogonal orbitals are used to describe the term dependence
of radial functions. Large sets of spectroscopic and correlation functions are
chosen to describe adequately strong interactions in the
,
and
Rydberg series and to properly account for the important
correlation and relaxation effects. The length and velocity forms of oscillator strength
show good agreement for most transitions. The calculated radiative lifetime for
the
state is in good agreement with experiment.
Key words: atomic data - atomic processes - line: formation
Emission lines of Cl II are detected at optical and ultraviolet wavelengths in various astrophysical plasmas. The emission features of Cl II are observed in the spectra of the Io torus from both ground-based (Küppers & Schneider 2000) and Far-Ultraviolet Spectroscopic Explorer (FUSE) (Feldman et al. 2001) observations. The Cl II lines are also identified in the high-resolution spectrum of the symbiotic star RR Telescopii as well as in the spectra of the planetary nebulae NGC 6741 and IC 5117 (Keenan et al. 2003). The oscillator strengths and transition probabilities of allowed and intercombination lines in Cl II are needed for the determination of diagnostic line ratios that can subsequently be compared to astrophysical observations.
Previously Deb et al. (2003) reported oscillator strengths for some allowed
transitions between low-lying fine-structure levels. They used structure
code CIV3 (Hibbert 1975; Glass & Hibbert 1978)
in their configuration-interaction (CI) calculations. They utilized a set of orthogonal spectroscopic and
correlation orbitals to construct CI wavefunctions of the fine-structure
levels belonging to the
,
,
,
and
configurations. The R-matrix method (Berrington et al. 1995)
in the LS coupling scheme has been used by Berrington & Nakazaki (2002) to
calculate radiative atomic data such as excitation energies, oscillator
strengths and ground state photoionization cross sections for chlorine atom
and its ions. More recently, Tayal (2003) investigated the term dependence of
one-electron radial functions and strong interactions in the different
,
and
Rydberg series. The importance of various correlation
corrections was also examined and it was found that Cl II wavefunctions exhibit
significant core-valence correlation or core polarization for the outer electron.
The other earlier studies include the calculations of Fawcett (1986) and Biemont
& Hansen (1986) who used Slater integrals as variable parameters to minimize
the discrepancies between the computed and measured energy levels. Biemont &
Hansen (1986) and Mendoza & Zeippen (1983)
reported radiative transition probabilities
of forbidden transitions in the ground
configuration.
Fawcett (1986) reported energy
levels and oscillator strengths for the
-
,
-
and
-
transitions.
The measurement of lifetimes of the
levels has been reported by
Bashkin & Martinson (1971) and Lawrence (1969).
Bashkin & Martinson (1971) used
the beam-foil technique and Lawrence (1969) used a pulsed-electron technique to
measure the lifetimes of the
excited levels. Because of the need of
accurate oscillator strengths for allowed and intercombination lines of Cl II in a
reliable and sophisticated approximation, we extended the work of Tayal (2003) to
include relativistic terms such as mass correction, Darwin term, spin-orbit,
spin-other-orbit and spin-spin interactions in the Breit-Pauli Hamiltonian. It may
be noted that significant mixing between a number of Cl II levels is caused by
spin-orbit interaction in intermediate coupling. The diagonalization of the
Hamiltonian matrix gives eigenfunctions of the total angular momentum J and
parity
.
We used non-orthogonal orbitals to represent
different Cl II states. The non-orthogonal orbitals provide much greater
flexibility in the choice of wavefunctions than the orthogonal orbitals
and also allow to include correlation with a reasonable number of
configurations and correlated orbitals (Zatsarinny & Tayal 2001).
Our calculations are performed using the multiconfiguration Hartree-Fock
(MCHF) method (Froese Fischer 1991;
Zatsarinny & Froese Fischer 1999).
In the MCHF approach the atomic state is represented by an atomic state
function (Brage et al. 1993)
The Cl II wavefunctions exhibit large correlation corrections and
strong term dependence of the one-electron orbitals. We have constructed Cl II state
wavefunctions from non-orthogonal orbitals that are optimized
for each atomic state separately. The low-lying states in Cl II are defined
by configurations
,
,
,
,
,
and
and these states show different correlation patterns.
The details of wavefunctions can be found in Tayal (2003) and here we give only a
brief description. The term dependence of the valence orbitals in the
(l=0, 1, 2) states is due to both the intermediate term and the final total term
obtained by coupling the valence electron with the
core. For example,
the 3d orbital in the
,
and
states
is term-dependent with average radius in the range 2.20-5.12 au. Similarly,
the 4d orbital is very term dependent with an average radius in the range 6.41-11.04 au for the
,
and
states. The term dependent 5d, 6d and 7d orbitals are obtained in separate optimization for each state to
represent lower members of the different
,
and
Rydberg series. A set of 4s, 5s and 4p spectroscopic orbitals is obtained
to describe other states considered in our work.
Earlier calculations used the same set of orthogonal
orbitals to represent all states of Cl II and included specific correlated
orbitals to account for the term-dependence of one-electron orbitals.
Members of various Rydberg series
display strong interaction
with each other and also with the
states which can be considered as perturbers.
Accurate representation of these strong mixing among different Rydberg series
and perturbers is crucial
for the accurate determination of oscillator strengths and transition probabilities.
In addition to the spectroscopic orbitals, we obtained several correlation s, p, d
and f orbitals. A set of correlated s, p, d and f orbitals are optimized on the
state. A set of three
d correlation functions was determined to accurately
represent the strong interaction between the
and
series. Two f correlation orbitals were obtained to
represent core-valence electron correlation where the valence 3d electron plays a
spectator role and one of the 3p electron is excited to f orbitals. The polarization of
the ground state is represented by another set of s, p, d and f correlation
functions. Different sets of correlation functions were determined to describe interactions
between the members of the
Rydberg series and the
Rydberg series and
the
pertuber state. These wavefunctions are then used to calculate the
length (
)
and velocity (
)
forms of oscillator strengths and transition probabilities
for allowed and intercombination transitions among the fine-structure levels.
The spectroscopic and correlation functions are used to construct CI expansions
for different atomic states by allowing one-electron and two-electron excitations
from all the basic configurations
,
,
,
,
,
and
used in our calculation. Progressively
larger calculations are performed in a systematic manner to make sure that the
strong interactions between various Rydberg series and between Rydberg series and
perturber states are properly accounted for. The various correlation corrections
are well represented as the mean radii of correlation orbitals are comparable to
the spectroscopic orbitals. In the construction of CI expansions for fine-structure
levels with various J and
we used configurations generated in this excitation
scheme for the atomic LS states and with insignificant configurations with coefficients
less than 0.0075 omitted from the expansions. The excitation energies of
fine-structure levels relative to the ground level are given in Table 1 where
comparison is made with the experimental values from the National Institute of
Standard and Technology (NIST) (http://physics.nist.gov). We have also included
in Table 1 the difference between the present ab initio calculation and
the experiment. The accuracy of calculated excitation energies on
an average is about 0.005 au. The assignment and ordering of the excited levels
in our calculation agrees with the experiment except for the levels of the
and
terms. The energy
gap between the levels of various terms is very small.
There are strong interactions in the levels of the ,
and
Rydberg series and also between the
and
perturber states with the
and
Rydberg series, respectively. These interactions lead to low eigenvector purities
of the
,
,
,
,
,
,
,
,
,
and
states with the main
configuration contributing in the range 26-79% to the composition of states. On
the other hand the members of the
s
,
and
Rydberg series show weak interactions with the other
series of the same symmetry. However, the core
correlation of the type
represent large corrections for these series. In addition, one-electron virtual
excitation of one of the 3p core electrons to the correlated f orbitals also
represent significant correlation effect. Some of these levels show strong
interactions with each other because of their close proximity. The energy
difference for some levels is negative because these excited levels are slightly
over-correlated compared to the ground level in our calculation. Comparison with the
calculation of Deb et al. (2003) is not made because they reported only excitation
energies adjusted to experimental values after a fine-tuning to their calculated
values.
The length and velocity values of oscillator strengths and length form of
transition probabilities for dipole-allowed transitions between the fine-structure
levels of the ground
and excited
,
,
,
and
configurations are listed in Tables 2 and 3 where our
results are compared with the calculation of Deb et al. (2003) who reported
oscillator strengths for some of these transitions. We have also listed the
experimental wavelengths for the dipole-allowed transitions. The agreement between the
length and velocity forms of oscillator strengths may to some extent indicate
the accuracy of the wavefunctions and the convergence of the CI expansions.
However, it is not necessarily a sufficient condition
for the reliability of the results.
There is normally excellent agreement between the present length and velocity
forms of oscillator strengths for strong fine-structure transitions with
oscillator strengths larger than 0.1. The transitions with oscillator strengths
larger than 0.01 display good to excellent agreement between the length and
velocity forms. Some of the transitions are very weak with very small oscillator
strengths and do not show good agreement between the length and velocity values.
The small oscillator strengths arise due to delicate cancellations of the
dipole matrix elements in the summation of oscillator strengths. The strong
mixing among several fine-structure levels may have dramatic effects on
oscillator strengths. As investigated in details by Tayal (2003), the oscillator
strengths of many transitions in Cl II are very sensitive to the representation
of the strong interactions between various Rydberg series
of the same symmetry and the Rydberg series
and perturbers and to various correlation corrections. The dipole matrix elements
between the main configurations comprising the initial and final states of a transition
may cancel or add up resulting in small or large value of oscillator strength.
For example, the strong interaction between the
d
Rydberg series and
perturber state as well as between the
d
and
d
series have dramatic
effects on the oscillator strengths for the
-
,
-
and
-
transitions.
The cancellations of the dipole matrix elements for
the
-
and
-
transitions
give rise to small oscillator
strength for these transitions and addition of dipole matrix elements for the
-
transition
gives rise to larger oscillator
strength as is clear from Table 2. The correlation effect plays particularly
important role for weaker transitions.
There is reasonable
agreement between the present results and the calculation of Deb et al. (2003)
for the
3P1-
,
-
,
-
and
-
fine-structure transitions. Our results differ significantly from the calculation
of Deb et al. (2003) for other transitions. The main difference between our
calculation and the calculation of Deb et al. (2003) is in the choice of radial
functions and the set of configurations in the CI expansions of levels. We used
term-dependent s, p and d non-orthogonal spectroscopic orbitals and several sets
of s, p, d and f correlation orbitals to adequately account for various interactions
and core correlation corrections. Deb et al. (2003) used same set of orthogonal
spectroscopic 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p and correlation 4d, 4f, 5s, 5p, 5d
radial functions to represent several levels of the
,
,
,
and
configurations. The most significant
discrepancies between our results and the values of Deb et al. (2003) are for
fine-structure transitions involving the
levels. The discrepancies are
caused by the inadequate treatment of the strong mixing between the levels of
symmetry and the missing important core correlation corrections of the
type
in the calculation of Deb et al. (2003).
For the accurate representation of these interactions in Cl II, the basis set
of functions should not only describe the low-lying members of a Rydberg
series, but also should approximate the higher bound and continuum parts of
the series.
Our results for all the
-
transitions
are larger than the calculation of Deb et al. (2003). The members of the
series show weak mixing, but core correlation
is significant for this series. Our results
for the
-
and
-
transitions
are also larger than the calculation of Deb et al. (2003). The multiplet oscillator
strength for the
-
transition from the present work is
in excellent agreement with the R-matrix calculation of Berrington & Nakazaki
(2002). Our results are also supported by a good agreement with the measured
lifetime for the
excited state. The calculated multiplet
mean lifetime (8.48 ns) for the
excited state compares very well
with the experimental lifetime of Bashkin & Martinson (
ns) and
Lawrence (
ns). The largest discrepancy between our calculation
and Deb et al. (2003) occurs for the
-
transitions where our results are several orders smaller than the calculation
of Deb et al. (2003). The R-matrix calculation of Berrington & Nakazaki
(2002)
also predicted very small multiplet oscillator strength for the
-
transition which is in reasonable agreement with
our results. The composition of the
state in our
calculation is 62.4%
+ 27.6%
+ 1.7%
+ 1.6%
.
The leading percentage for the
same state from the calculation of Deb et al. (2003) is about 80%. Clearly
our calculation predicts this state to be much less pure than Deb et al.
(2003) due to strong mixing and it explains to a major part the discrepancy
with their calculation.
The length and velocity values of oscillator strengths and
length value of transition probabilities for intercombination lines
of significant strengths with oscillator strengths larger than 0.0001
are listed in Table 4. The
values of oscillator strengths for intercombination lines are usually
several orders of magnitude smaller than those for the allowed transitions.
However, some of the intercombination lines have comparable oscillator
strengths to dipole-allowed transitions. For example, some fine-structure
transitions involving higher excited
,
and
terms have
oscillator strengths larger than 0.01. The oscillator strengths for the
-
and
-
transitions are small and have values of
2.59(-5) and 5.19(-5) respectively.
The intercombination lines are induced by spin-orbit interaction by causing
mixing between differnt LS symmetries with the same set of quantum numbers J and
.
The velocity values of oscillator strengths are listed to
indicate the comparison between length and velocity values to somewhat assess the
quality of wave functions. The length values should be preferred over
velocity values because these remain stable with respect to the addition
of more configurations. We have also shown the experimental wavelengths
of intercombination lines in this table.
In conclusion, we have presented accurate oscillator strengths and transition
probabilities of allowed and intercombination lines among the levels of the
,
,
,
,
and
configurations.
We used term-dependent non-orthogonal orbitals for the construction
of CI wavefunctions for these levels. Different sets of spectroscopic and
correlation orbitals are used to adequately describe the interactions between
various Rydberg series and Rydberg series and perturber states. The wavefunctions
display large correlation corrections. The calculated energies and lifetime for
excited states show good agreement with the measured values.
There is good agreement
between the length and velocity values of oscillator strengths for most
lines. Significant discrepancies with available oscillator strengths are noted
for some transitions.
Our oscillator strengths should be useful to model astrophysical plasmas
and to interpret the recent ground-based and FUSE observations.
Acknowledgements
This research work was supported by NASA grant NAG5-13340 from the Planetary Atmospheres Program.