3s23p–3s3p2 transitions in S iv

3s23p–3s3p2 transitions in S IV

Alan Hibbert,1P Tomas Brage2 and Janine Fleming1

1Department of Applied Mathematics and Theoretical Physics, Queen’s University, Belfast BT7 1NN2Department of Physics, University of Lund, S-221 00, Lund, Sweden

Accepted 2002 February 27. Received 2002 February 25; in original form 2002 January 4

A B S T R A C T

We use both the CIV3 and MCHF codes to calculate oscillator strengths of allowed and

intercombination lines in the 3s23p–3s3p2 multiplets. Valence, core–valence and some

core–core correlation effects are included. The two approaches give results in excellent

agreement. Core effects are particularly important for the intercombination lines, though

relatively minor for allowed transitions.

We obtain a branching ratio of 1.42 with an estimated accuracy of 0.02 for the

3s23p 2PoJ –3s3p2 2S1=2 transitions, compared with an experimental value of 1:12 ^ 0:1. The

A-values of the 2PoJ – 4PJ 0 intercombination lines are substantially different from those of

previous calculations.

Key words: atomic data.

1 I N T R O D U C T I O N

We have undertaken a series of extensive calculations of oscillator

strengths of transitions among low-lying levels of S IV. Here we

address two particular problems related to solar observations.

(i) The 3s23p 2PoJ –3s3p2 2S1=2; J ¼ 3=2, 1/2 transitions.

Engstrom et al. (1989) undertook beam-foil measurements, and

reported that the experimental intensity ratio of these two lines

differs significantly from the algebraic value of 2.0 which would be

obtained by assuming LS coupling and by ignoring the difference

between the wavelengths of these two transitions. In a more recent

extended analysis, Engstrom et al (1995) gave a revised value of

1:12 ^ 0:1 for this ratio. Previous calculations, using the

multiconfigurational Dirac–Fock (MCDF) method (Huang 1986)

or using the Relativistic Parametric Potential method (RELAC)

(Farrag, Luc-Koenig & Sinzelle 1982) obtain a value of 1.38 for

this ratio. We seek to resolve this discrepancy.

(ii) The 3s23p 2PoJ –3s3p2 4PJ0 transitions.

These intercombination lines are observed in the solar spectra as

well as in the star RRTel, recently observed from the Hubble Space

Telescope. Some earlier calculations, using the configuration inter-

action method with the code CIV3 (Hibbert 1975), were undertaken

by Dufton et al (1982). Only valence–shell correlation was

considered. We wish to investigate whether the results of those

calculations, relatively simple by today’s standards, are in need of

revision.

Allowed E1 transitions in S IV have formed the subject of other

theoretical studies. Glass (1979) used CIV3 to study transitions

within the n ¼ 3 complex, but included no correlation configur-

ations except those obtainable from the n ¼ 3 orbitals. Gupta &

Msezane (1999), again with CIV3, have considerably extended this

work, both in the degree of electron correlation included and also

in the range of ionic states considered. Froese Fischer (1976, 1981)

used the MCHF method to study transitions among low-lying 2Po,2D, 2Fo states of Al-like ions, including S IV. Aashamar, Luke &

Talman (1984) used the multiconfigurational optimized potential

method to study a wide range of transitions in Al-like ions. In this

method, a set of effective potentials is calculated variationally for

each l-value of the orbital functions, and the potentials are

incorporated into the code SUPERSTRUCTURE (Eissner, Jones &

Nussbaumer 1974) to determine the orbitals and thence the

energies and oscillator strengths. All of these calculations were

undertaken in LS coupling, and so would not address the two issues

detailed above.

The only recent calculation which incorporates intermediate

coupling was undertaken by Tayal (1999). He used CIV3 with the

major Breit–Pauli terms included in the Hamiltonian, again to

study a wider range of transitions than we discuss here. However,

he included only valence–shell correlation. With only three

electrons outside closed shells, and with the 3s-subshell open in the

upper states, we considered it appropriate for the present work to

consider also the effect of core polarization.

2 M E T H O D O F C A L C U L AT I O N

2.1 General considerations

We will use two independent approaches, though both of

configuration interaction character. In intermediate coupling, the

wave functions take the form

CðJpÞ ¼i

XaiFiðLiSiJpÞ; ð1Þ

PE-mail: [email protected]

Mon. Not. R. Astron. Soc. 333, 885–893 (2002)

q 2002 RAS

which allows for configuration state functions (CSFs) Fi with

different L and S but a common J to be included in the expansion; p

denotes the parity of the wave function.

First, we will extend the work of Dufton et al. (1982) using the

code CIV3, which is based on the superposition of configurations

(SOC) method. The CSFs are systematically added to the wave

function so that an assessment of the convergence of the oscillator

strengths can be made with respect to the number of orbital

functions used, and also to the type of correlation (valence, core

polarization etc) included. The radial functions of the one-electron

orbitals used in these configurations are determined variationally.

We will express them in analytic form as the sum of Slater-type

orbitals (STOs)

PnlðrÞ ¼Xk

j¼1

cjnlxjnlðrÞ; ð2Þ

where the STOs are of the form

xjnlðrÞ ¼ð2jjnlÞ

2Ijnlþ1

ð2IjnlÞ!

� �1=2

r Ijnl expð2jjnlrÞ: ð3Þ

The parameters jjnl, and any cjnl which are not determined from the

orthonormality requirements amongst the orbitals, are determined

variationally by minimizing one or other of the energy eigenvalues

of the Hamiltonian matrix formed using the expansion (1) for the

trial wave function. This latter feature allows us to represent all

relevant atomic states with comparable accuracy. In the CIV3 code,

it is customary to determine the orbital functions in a sequential

manner: once a radial function has been optimized, it is not re-

optimized when further orbitals are added.

Secondly, we will use the multiconfigurational Hartree–Fock

(MCHF) method and associated programs (Froese Fischer, Brage &

Jonsson 1997). In this approach, the variational principle is used to

set up coupled integro-differential equations for the radial func-

tions of the orbitals. These are solved self-consistently.

Convergence studies are also undertaken in this approach, but

the calculations differ from those of the SOC method in that, as the

orbital set is extended, functions which were included in earlier

stages of the process are re-optimized in the later calculations. This

of course allows for greater flexibility. It does, however, mean that,

unlike the first approach, it is not so easy to see which orbitals are

being used to take a particular correlation effect into account.

In both approaches, we will extend the calculations of Dufton

et al. (1982), who incorporated correlation effects only in the outer

ðn ¼ 3Þ shell. This valence–shell correlation (denoted by ‘val’ in

our tables below) uses only configurations with a common

1s22s22p6 core. We will also take into account core polarization

through the inclusion of core–valence (‘c-v’) correlation, for

which we use configurations constructed by replacing at least one

core (1s,2s,2p) and at least one valence ðn ¼ 3Þ orbital by

‘correlation’ orbitals, optimized as discussed above, and also

core–core (‘c-c’) correlation, for which two or more of the core

orbitals are so replaced. We will represent the relativistic effects,

needed to separate the LS terms into levels and to allow the

calculation of intercombination lines, by means of the Breit–Pauli

approximation. For an ion such as S IV, and for transitions between

low-lying levels, this should be very adequate.

2.2 Optimization of the CIV3 radial functions

In determining the manner of optimization of the radial functions,

it is first necessary to decide which states will contribute to the

proposed calculations. The 2Po – 4P intercombination lines arise

principally because of the interactions between 3s3p2 4PJ and 2SJ,2PJ,

2DJ, and the doublets also interact with 3s24s 2S and 3s23d 2D.

A small contribution to the intercombination lines also occurs from

the interactions between 3s23p 2PoJ , 3p3 2Po

J and, for J ¼ 3=2, with

3p3 4So3=2. The allowed transitions highlighted in the Introduction

will also be influenced by these interactions.

The method of optimization of the orbitals is displayed in

Table 1. All the optimizations were done in LS coupling, that is,

without the inclusion of the Breit–Pauli effects. The 1s, 2s, 2p, 3s,

3p orbital functions are those given by Clementi & Roetti (1974)

for the Hartree–Fock representation of the ground state. The 4s and

4p functions were chosen to minimize the energies of the 3s24s 2S

and 3s24p 2Po states respectively. The optimal 3p function for the

3s3p2 states has a slightly different radial form from that of the

ground state. We optimized the 5p function to allow for this

difference, choosing the 2D state energy as the functional because

for that state, the contribution of the 5p was the most significant.

The 3d and 4f functions are used to incorporate the main ‘semi-

internal’ correlation effects (Oksuz & Sinanoglu 1969). This 3d

function is not optimal for the 3s23d 2D state, so the true 3d orbital

for this state is represented by a linear combination of our 3d and

4d functions. The 5d function is used to represent the most

important core–valence correlation effect (achieved by opening

the 2p6 subshell).

We found that with this set of orbitals, the correlation energy of

the 3p2 pair, particularly in the 2S state, was not well represented,

so we optimized a further orbital for each l # 4 to improve this

representation. We will refer to this set of orbitals as ‘basic

orbitals’. The parameters of the radial functions are displayed in

Table 2.

With these orbitals, we calculated the energies of the relevant

atomic states. The spin–other orbit operator was approximated as

described in Section 2.3 below, and the results were determined in

LSJ coupling, at each of the three different levels of approximation

(valence correlation only, valence and core–valence correlation,

and then with some core–core correlation added). The results of

these calculations are displayed in Table 3. The column headed

‘val’ gives results which incorporate valence–shell correlation

only; that is, all the configurations have a common core of

1s22s22p6. Specifically, we allow all single and double replace-

ments from the valence orbitals using all the radial functions

available, from the multireference sets consisting of 3s23p and 3p3

Table 1. Optimization processes for the radial functionparameters.

Orbital Optimized on

Basic orbitals:1s,2s,2p,3s,3p HF orbitals of 3s23p 2Po

4s 3s24s 2S4p 3s24p 2Po

5p 3s3p2 2D (using 3s3pnp)3d 3s ! 3d correlation in 3s23p 2Po

4d 3s23d 2D (using 3s23d þ 3s24dÞ5d ð2p63s23p þ 2p53s3p2ndÞ2Po; (c-v orbital)4f 3p3 2Do (using 3p3 þ 3p24fÞ5s,6p,6d,5f,5g 3s3p2 þ 3snln0l0 2S

Extra orbitals:6s c-v correlation in the ground state7s,7p,7d,6f,6g,6h 3s3p2 þ 3snln0l0 2S8s,8p,8d c-v correlation in the 3s3p2 2S state

886 A. Hibbert, T. Brage and J. Fleming

q 2002 RAS, MNRAS 333, 885–893

for odd parity states and 3s3p2, 3s23d, 3s24s, 3p23d and 3p24s for

even parity states. All of these basic configurations have substantial

CI coefficients ai in (1). The inclusion of core–valence (c-v)

correlation is effected by allowing single and double replacements

from the same reference sets, but with one of 2s or 2p being

replaced. Finally, core–core (c-c) correlation allows both the two

orbitals replaced to come from the 2s22p6 core. In the case of c-c

correlation, the number of possible configurations is very large –

several tens of thousands for each LSp symmetry. Many of these

configurations have very small CI coefficients, and contribute very

little to the energy of any of the states. When we come to consider

the intercombination lines, several LS symmetries must be coupled

together through the spin-dependent operators of the Breit–Pauli

Hamiltonian. It therefore seemed sensible to exclude from the c-c

(and also c-v) configurations those for which jaij is very small.

Specifically, we excluded c-v configurations with jaij , 0:00015

and c-c configurations with jaij , 0:001. We found a considerable

reduction in the number of configurations remaining, although the

energies changed by at most a few cm21. The results of Table 3

were obtained with this reduced set of configurations.

It may be seen from Table 3 that although the calculated energy

levels of the doublet states (particularly the 2S states, which form

one of our primary concerns) are quite accurately given by this set

of orbitals and configurations, the same is not true of the 4P state.

We undertook a series of calculations which revealed that one of

the causes of this inaccuracy was the still limited degree of valence

correlation included in our calculations to date, although some

further important core polarization was introduced with our 6s

orbital. Valence correlation has a more pronounced effect on

doublets than on quartets, because of the parallel nature of the spins

of the three valence electrons in the quartet state. We therefore

optimized a further set of orbitals, one for each l # 5 (see Table 1)

on the 3s3p2 2S energy. Finally, to provide more assurance of the

convergence of c-v correlation, we introduced the orbitals 8s, 8p

Table 2. Optimized radial functionsparameters.

nl cjnl Ijnl jjnl

4s 0.06686 1 12.6367820.25478 2 5.35068

0.81054 3 2.3374521.26057 4 1.43558

5s 0.28929 1 5.4115424.59212 2 2.4539110.14209 3 2.59194

26.99214 4 2.519461.43635 5 1.67893

6s 0.08865 1 5.4115421.48657 2 2.45391

3.58651 3 2.5919422.98695 4 2.51946

1.45748 5 1.6789321.29840 6 1.00000

7s 2.63970 1 2.07115225.20093 2 2.07115

98.40843 3 2.071152202.83163 4 2.07115

234.14248 5 2.071152143.75298 6 2.07115

36.66049 7 2.07115

8s 5.76062 1 2.77698249.81207 2 2.77698194.05609 3 2.77698

2434.19206 4 2.77698597.39871 5 2.77698

2501.75823 6 2.77698236.62720 7 2.77698

248.05127 8 2.77698

4p 0.15725 2 6.3024020.78069 3 1.94895

1.30887 4 1.28659

5p 1.11494 2 2.3732323.37935 3 2.13775

4.22183 4 1.7099322.29935 5 1.36950

6p 4.31809 2 2.5759828.76807 3 2.91632

6.99348 4 2.5454923.71951 5 1.98328

1.19372 6 1.46703

7p 10.75211 2 2.06247261.84055 3 2.06247155.43053 4 2.06247

2207.24142 5 2.06247144.10223 6 2.06247

241.34407 7 2.06247

8p 18.82705 2 2.647732113.00181 3 2.64773

315.66221 4 2.647732506.19760 5 2.64773

480.76953 6 2.647732252.71588 7 2.64773

56.86504 8 2.64773

3d 1.00000 3 2.09598

4d 3.47990 3 1.4240324.05244 4 1.58813

5d 29.57164 3 1.89681215.20678 4 3.16332216.46176 5 2.39540

6d 19.84595 3 2.45502220.64985 4 3.44289

Table 2 – continued

nl cjnl Ijnl jjnl

3.92815 5 3.4611224.52943 6 2.79175

7d 12.11851 3 2.87738255.08977 4 2.87738101.91707 5 2.87738

288.10661 6 2.8773829.49229 7 2.87738

8d 26.71469 3 3.351382131.80535 4 3.35138

289.18826 5 3.351382341.45715 6 3.35138

212.50543 7 3.35138254.97658 8 3.35138

4f 1.00000 4 2.27784

5f 6.10631 4 2.0227926.35438 5 2.26443

6f 7.39619 4 2.94807214.09534 5 2.94807

7.31687 6 2.94807

5g 1.00000 5 3.04511

6g 3.38765 5 3.2005523.45236 6 3.20055

6h 1.00000 6 3.61825

3s23p–3s3p2 transitions in S IV 887

q 2002 RAS, MNRAS 333, 885–893

and 8d, optimized on this effect in 3s3p2 2S. The orbital parameters

of the ‘extra’ orbitals also are shown in Table 2 and the energies

with this improved set of orbitals (and therefore extended

configuration set, truncated on the basis of the size of jaij as

above) are shown in Table 3.

2.3 Inclusion of relativistic effects

With such an extensive set of configurations, the use of the full

form of the two-electron operators of the Breit–Pauli Hamiltonian

would lead to prohibitively time-consuming calculations. For

elements with several closed shells, the two-electron spin-other-

orbit can be quite accurately represented by a modified form of the

spin–orbit operator (Hibbert & Bailie 1992), so that these two

operators are replaced by

H 0so ¼

XN

i¼1

Zzl

r3i

li·si; ð4Þ

where zl are parameters which depend only on the l-value of the

interacting electron. In the present CIV3 calculations, we have used

this operator plus the mass correction and Darwin terms to model

the full Breit–Pauli Hamiltonian. We found that the contribution of

the spin–spin operator was of the order of 1 or 2 cm21, so that the

time taken to calculate its matrix elements could not be justified.

The parameters zl may be chosen to give the best possible

representation of the fine structure of the states under investigation.

For the calculations with the ‘basic’ orbitals, we found that the

values z1 ¼ 0:91, z2 ¼ 0:33 and z3 ¼ 1:0 ¼ z4 (and z0 ¼ 0Þ gave

the fine structure of the ground state and the 3s23d 2D states quite

accurately.

When we added further configurations, arising from the ‘extra’

orbitals, and more particularly from the inclusion of c-v or c-c

correlation, we found that this choice of zl led to fine-structure

splittings which were larger than experiment. We could have

modified the zl to bring the calculated splittings back into

agreement with experiment, essentially recalculating the par-

ameters for each new set of configurations considered. Instead, we

preferred the alternative approach of choosing the zl so that the

matrix elements of (4) closely agreed with those of the full spin–

orbit plus spin-other orbit operators. In fact, we found that for the

matrix elements (both diagonal and off-diagonal) of these two

operators between the dominant configurations of the states which

involve 3p, the magnitude of the matrix element of the spin-other

orbit operator was consistently about 18–19 per cent of that of the

spin–orbit operator, and of opposite sign. A similar analysis was

undertaken for states involving 3d. Accordingly, for calculations

involving the extended set of orbitals, we used z0 ¼ 0:0,

z1 ¼ 0:814, z2 ¼ 0:30 and zl ¼ 1:0 for l . 2. This choice is then

independent of the configuration set used and gives us an indepen-

dent measure of how accurately the fine structure is calculated, and

indirectly the mixing between quartets and doublets which is

related to the accuracy of the intercombination lines.

2.4 MCHF calculations

The MCHF approach is based on the active set, ðN;N þ 1Þ method

(Froese Fischer 1994; Brage, Froese Fischer & Judge 1995;

Fleming et al. 1995). According to this, configuration state func-

tions (CSFs) are generated from an active set of orbitals. A primary

and a secondary atomic state is selected for each parity. In each

step, all orbitals with main quantum number n # N are then

optimized on the primary atomic state, while additional orbitals

with n ¼ N þ 1 are optimized on the secondary state. This gives a

balanced way of representing more than one term at the same time.

This is important when dealing with intercombination lines, since

we do optimize the orbitals in a non-relativistic approach, while the

transitions are induced through relativistic interaction. More than

one LS-symmetry will therefore be involved for each parity. As the

primary and secondary state for the even parity we select the 3s3p2

4P and 2S terms, respectively. This is motivated by the fact that the

intercombination line is induced primarily through spin-dependent

Table 3. Experimental and calculated energies (cm21).

CIV3 MCHF

Expt basic z1 ¼ 0:91 extended z1 ¼ 0:814Martin et al. (1990) val val þ c-v val þ c-v þ c-c val val þ c-v val þ c-v þ c-c val val þ c-v

3s23p 2Po1=2 0 0 0 0 0 0 0 0 0

2Po3=2 951 948 959 960 856 970 965 855 959

3s3p2 4P1/2 71184 69881 70482 70247 70031 70825 70758 70059 712054P3/2 71529 70342 71189 71120 70375 715774P5/2 72074 70837 71744 71672 70871 72113Dfs 890 806 919 914 812 908

2D3/2 94103 93472 94156 94264 943992D5/2 94150 93516 94222 94330 94459Dfs 47 44 66 66 60

2S1/2 123510 124062 123784 123943 123785 123544 123539 123702 123917

2P1/2 133620 135318 133604 134208 135172 133763 134618 1339812P3/2 134246 135743 134391 135245 134607Dfs 626 571 628 627 626

3s23d 2D3/2 152133 153145 153060 1539822D5/2 152147 153178 153074 153997Dfs 14 33 14 15

3s24s 2S1/2 181448 181338 181131 181575 181173 181689 181417


q 2002 RAS, MNRAS 333, 885–893

interaction between these two terms, and that we are specifically

interested in intensity abnormalities in transitions involving the

latter of the two. For the odd parity we choose the 3s23p 2Po as the

primary atomic state. Since the mixing between this and the 3p3 4So

also contributes to the intercombination rates, this will be a natural

choice as a secondary state.

To represent valence correlation, we include in our expansion

CSFs with the form . . .2p6{3; 4; . . .;N}3; where the notation

implies a distribution of three electrons among the shells with

n [ {3; 4; . . .;N}. To represent core–valence correlation, we add

to these CSFs of the form . . .2p5{2; 3}3{3; 4; . . .;N}1, with the

constraint on the azimuthal quantum number l # 2 for all orbitals.

For the primary level, our active set has N ¼ 8. The numbers of

orbitals for different azimuthal symmetries are limited to 6 for

l ¼ 0; 1; 2 (up to 8s,8p,8d), 3 for l ¼ 3 (from 4f to 6f), 2 for l ¼ 4

(5g and 6g) and 1 for l ¼ 5 (only 6h). For the secondary state only

orbitals with the same l-symmetry as for the primary are included,

but up to N ¼ 9. In a final CI-calculation, all these CSFs are

included. For the even levels we also include all possible 2P and 2D,

as generated from the active set with n # 9. Since no orbitals are

optimized on these levels, the accuracy is much less for transitions

from these than from 4P and 2S. This is apparent by looking at the

length and velocity forms of the oscillator strengths in Table 4. For

the 2Po – 2S transition, the agreement is excellent in the MCHF

calculations. For the other two transitions, there is less close

agreement between the two forms than in the CIV3 results. We have

retained the very systematic approach for the MCHF calculations,

which focus on only two terms, to allow this comparison with CIV3

which involves a more balanced treatment of states.

2.5 Use of experimental energies

The MCHF calculations are entirely ab initio, except where we

indicate (by ‘adjusted’ in the tables) that experimental transition

energies have been used. In the CIV3 approach, we have the

possibility of additional adjustments whereby we add to certain

diagonal elements of the Hamiltonian matrix small corrections

which have the effect of bringing the energy eigenvalue separations

into agreement with the experimental separations (for a discussion,

see Brage & Hibbert 1989 and Hibbert 1996). This adjusting

process was also used by Tayal (1999).

3 R E S U LT S A N D D I S C U S S I O N

The use of two different methods (CIV3 and MCHF) undertaken

independently assists in our goal of assessing the accuracy of our

calculations. Moreover, in the CIV3 calculations, we have

considered two sets of orbitals – the basic set, which is itself

fairly extensive, and an extended set in which the basic set is

augmented with further orbitals, collectively optimized on both

valence and core–valence correlation effects. As Section 2.4

shows, the MCHF basis set is in fact slightly more extensive still,

although the focus of the optimization is different from that of the

CIV3 process. Nevertheless, this sequence of basis sets allows us to

assess the degree of convergence with respect to orbital basis size.

3.1 Energies

The calculated energies of the relevant states are displayed in

Table 3, at various levels of approximation. Experimental energies

are also included, to allow comparison and to assess convergence

of the calculations.

The CIV3 calculations with the basic set of orbitals were

undertaken to provide a starting point from which to improve. Only

the J ¼ 1=2 levels of the even parity states were calculated, along

with the ground-state levels. This is sufficient to allow us to

consider the 2PoJ – 2S1=2 transitions and some of the intercombina-

tion lines. The 4P levels are almost 1000 cm21 too low relative to

the ground state, which is an indication that correlation effects are

inadequately represented in this approximation, as we remarked

earlier. However, the energy separations of the doublets are

relatively good, so that the limitations apply consistently across the

doublet states. One might therefore expect that the allowed

transitions will be already quite accurate. Clearly, the inclusion of

only valence–shell correlation is insufficient, but the separations

are much improved by the addition of configurations representing

core–valence (c-v) effects. While all possible configurations

representing valence correlation were included, only a limited

range of c-v and even more limited core–core (c-c) configurations

were added. The improvements to the even parity state energies are

significant, but the fine-structure splitting of the ground state is

hardly changed. The value of z1 ¼ 0:91 in equation (4) was

maintained for all the calculations with the basic orbitals. We shall

see that this constancy of the fine structure of the ground state is an

indication of the limited inclusion of the core effects.

The extended orbital set was optimized to improve the

convergence of valence and core–valence correlation effects.

The orbitals listed on the penultimate line of Table 1 were

optimized on the valence–shell correlation. Their inclusion results

in a significant lowering of the energies, although trial calculations

with yet more orbitals led to little change in the ‘val’ results. This

suggests to us that, with the extended orbital set, the calculations

are fairly well converged with respect to valence–shell correlation.

This is supported by the comparison with the corresponding MCHF

calculations, as shown in Table 3. These were carried out

independently with a (slightly) larger basis set. Moreover, the way

of introducing the Breit–Pauli operators was different. The MCHF

calculations used the full (i.e., normal) form of the operators. In the

CIV3 calculations, the spin–spin operator was omitted (its effect is

very small) and the spin–orbit and spin-other orbit operators are

replaced with equation (4), but now with zl chosen to reproduce as

closely as possible the main single configuration matrix elements

Table 4. Oscillator strengths in length and velocity form from LS calculations.

CIV3 –val CIV3 –val þ c-v MCHF –val þ c-vTransition fl fv fl fv fl fv

3s23p 2Po –3s3p2 2D 0.0513 0.0502 0.0522 0.0536 0.0522 0.06273s23p 2Po –3s3p2 2S 0.1020 0.1041 0.0954 0.0936 0.0967 0.09673s23p 2Po –3s3p2 2P 0.7543 0.7609 0.7159 0.7149 0.7297 0.78023s23p 2Po –3s23d 2D 1.1732 1.1916 1.1269 1.11103s23p 2Po –3s24s 2S 0.0891 0.0892 0.0925 0.0932

Note: The CIV3 results are calculated with the extended sest of orbitals.


q 2002 RAS, MNRAS 333, 885–893

of these operators, rather than the experimental fine-structure

splittings of various states. In this way, we can use these splittings

as an indicator of the degree of accuracy of our calculation. In these

circumstances, the close agreement between the CIV3 and MCHF

calculations with valence–shell correlation alone is pleasing.

It will be noticed from Table 3 that with the extended set of

orbitals, the calculated fine structure of the ground state with

valence correlation alone does not agree with experiment. It

requires c-v correlation, augmented by c-c correlation, to bring

theory into line with experiment, especially for the ground state.

The fact that it does so, to a good approximation, is an indicator

that our calculations of oscillator strengths with the same wave

functions should be quite accurate.

The MCHF energies in the ‘val þ c-v’ approximation are closer to

experiment than are those of CIV3 for the quartet levels, although

the situation is reversed for the doublets. We note that for the even

parity states, the MCHF work includes orbitals up to n ¼ 9, so that

these are more fully converged. We note also that the CIV3

calculations include the 3s23d 2D and 3s24s 2S states explicitly, CI-

expanded to the same degree of correlation as are the other states.

By contrast, the MCHF calculations do not (although these

configurations are included), so that the CIV3 results incorporate

the interactions with the 3s3p2 2D and 2S states more precisely.

The inclusion of core–core effects is rather limited, partly

because we did not optimize further orbitals on these effects, and

partly because the number of configurations necessary to produce

convergence is huge. The results are included here to give an

indication of the size of the effect. The deterioration of the

agreement between theory and experiment suggests imbalance in

the way in which the c-c correlation is included in the different

states, and the val þ c-v þ c-c results should be treated with some

caution.

3.2 Oscillator strengths from LS calculations

One indicator of the accuracy of oscillator strengths is the level of

agreement between length and velocity forms in LS calculations.

Some of our results are displayed in Table 4. Even with only

valence–shell correlation included, the length/velocity agreement

is close – to within 2 per cent. This same level of agreement is

achieved in the CIV3 calculations when core–valence effects are

added ðval þ c-vÞ. It will be noted, however, that the magnitudes of

most of the oscillator strengths have changed by much more than

this – up to 7 per cent. This demonstrates the general principle that

agreement to within n per cent of length and velocity forms does

not imply their accuracy to within n per cent. Instead, we must call

on the apparent convergence of our energy levels at the val þ c-v

level of approximation, together with the expected small effect

(especially on the length form) of core–core correlation, to argue

that our LS results are correct to within a few (perhaps 5) per cent.

There is good agreement between the CIV3 and MCHF

calculations of the length form for the 3s23p–3s3p2 transitions.

We have already remarked that the difference in the corresponding

velocity values arises because no MCHF orbitals were optimized on

the 2P or 2D states. We also note that the 3s24s and 3s23d states

were not included in the MCHF calculations.

3.3 Branching ratios of the 3s23p 2PJo –3s3p2 2S1/2 transitions

We present in Table 5 three comparisons between different sets of

calculations, which are chosen to show the extent of the

convergence of our results.

In section (a) of Table 5, we show the effect on the CIV3 results of

the extra orbitals in the extended set. We choose the val þ c-v

calculation in both cases, so that the variation is entirely due to the

Table 5. Branching ratios of 3s23p 2PoJ –3s3p2 2S1=2.

(a) Convergence as orbital set is increased (val þ c-v – CIV3 results)

basic set extended setab initio adjusted ab initio adjusted

J ¼ 0:5 fl 0.120 0.119 0.118 0.118fv 0.123 0.122 0.115 0.116

Al (ns21) 1.226 1.210 1.199 1.201

J ¼ 1:5 fl 0.085 0.085 0.085 0.085fv 0.087 0.087 0.083 0.083

Al (ns21) 1.715 1.709 1.710 1.707

Branching ratio 1.399 1.412 1.426 1.421

(b) Effect of core–valence and (partial) core–core correlation (CIV3 results – extended orbital set)

Tayal (1999) CIV3 – present workval val val þ c-v val þ c-v þ c-c

ab initio ab initio adjusted ab initio adjusted ab initio adjustedJ ¼ 0:5 fl 0.123 0.121 0.123 0.118 0.118 0.120 0.122

fv 0.121 0.123 0.125 0.115 0.116 0.109 0.111Al (ns21) 1.253 1.238 1.254 1.199 1.201 1.222 1.241

J ¼ 1:5 fl 0.092 0.093 0.092 0.085 0.085 0.089 0.089fv 0.089 0.094 0.093 0.083 0.083 0.080 0.079

Al (ns21) 1.852 1.882 1.845 1.710 1.707 1.793 1.773

Branching ratio 1.478 1.520 1.471 1.426 1.421 1.467 1.429

(c) Comparison between CIV3 and MCHF (val þ c-v – adjusted)

CIV3 MCHF

J ¼ 0:5 Al(ns21) 1.201 1.209J ¼ 1:5 Al(ns21) 1.707 1.718

Branching ratio 1.421 1.421


q 2002 RAS, MNRAS 333, 885–893

additional orbitals. For these transitions, the magnitudes of the

oscillator strengths change only a little as the extra orbitals are

included. This is especially true of the length form: the change in

the velocity form is a little more marked, as might be expected

since several of the orbitals were optimized on the c-v effect, which

influences the velocity form more than the length form. The other

feature to note is that the ‘adjusted’ calculations – undertaken as

described in Section 2.5 – change less than do the ab initio ones.

This indicates that much of the change in the ab initio calculations

brought about by the extra orbitals is due to improvements in the

calculated energies.

In section (b) of Table 5, we use just the extended set of orbitals,

but vary the type of correlation effects included. Our length values,

for valence–shell correlation only, agree closely with similar but

independently undertaken calculations of Tayal (1999). There are

significant changes to the oscillator strengths when c-v effects are

added. The inclusion of c-c effects changes the results again, and in

particular causes a deterioration in the length/velocity agreement.

As we remarked in Section 3.1, only a limited number of c-c

configurations were included for the calculation of these oscillator

strengths, with the calculated energies being slightly inferior to

those obtained in the val þ c-v calculations. We would expect the

velocity form to be the more sensitive to c-c effects, and not

converged in our calculation. We note that, in the length form, the

adjusted values result in a similar ratio of the two A-values (the

branching ratio) for val þ c-v and val þ c-v þ c-c.

The MCHF calculations were undertaken in the val or val þ c-v

approximations. Section (c) of Table 5 compares the two A-values

obtained with CIV3 and with MCHF in the val þ c-v case.

Experimental transition energies were used for these A-values,

although the adjusted values differ only marginally from the ab

initio values. The results from the two methods differ by less than 1

per cent, and the ratios are in complete agreement.

It would appear, then, that the branching ratio of the two

transitions is quite stable to changes in our orbital basis set, and is

reasonably well converged once c-v effects have been included.

The range of 1:42 ^ 0:02 for this ratio should provide a

conservative estimate of accuracy.

It is interesting to note how this ratio comes to differ from the LS

ratio of 2.0. First, we observe that the wavelengths of the two lines

differ by 0.78 per cent. This effect alone reduces the ratio to 1.95.

However, the principal cause of the reduction of the ratio to 1.42 is

the mixing of 3s3p2 2S1/2 and 2P1/2. Its effect is to decrease the

3=2–1=2 transition rate by about 10 per cent, and to increase the

1=2–1=2 rate by a little over 20 per cent.

3.4 The 3s23p 2PJo –3s3p2 4PJ0 intercombination lines

The A-values of intercombination lines are generally much more

sensitive to the degree of convergence of the calculations than are

those of allowed transitions. This sensitivity is related to the level

of accuracy of the calculated mixing between states which ‘drive’

the intercombination lines – in the present case, the mixing

between 4PJ0 and 2S1/2, 2PJ0,2DJ0, and also between 2Po

3=2 and 4So3=2.

Just how varied the results can be is shown in the final part of

Table 6. Here we compare our valence–shell correlation results

with similar calculations obtained by other authors. We note that

Dufton et al. (1982) and Tayal (1999) also used CIV3 and also the

same energy adjustment procedure (see Section 2.5) as that used in

the present work. In spite of this, there is considerable variation in

the sets of results.

The earlier part of Table 6 shows the changes brought about by

the inclusion of various types of correlation effect. For the CIV3

calculations with the basic set of orbitals, the zl in equation (4)

were chosen to fit the calculated fine-structure separations to the

corresponding experimental values. Hence we were able to obtain

Table 6. A-values (104 s21) of 3s23p 2PoJ –3s3p2 4PJ0 .

val val þ c-v val þ c-v þ c-cJ J0 ab initio adjusted ab initio adjusted ab initio adjusted

CIV3: basic orbitals (excluding 3p3 4So3=2Þ

0.5 0.5 6.06 6.94 6.35 6.77 6.37 6.971.5 0.5 4.19 4.72 4.23 4.55 4.31 4.73

CIV3: extended orbitals (excluding 3p3 4So3=2Þ

0.5 0.5 6.19 6.391.5 0.5 4.18 4.31

CIV3: extended orbitals (including 3p3 4So3=2Þ

0.5 0.5 6.19 6.39 6.29 6.581.5 0.5 4.59 4.72 4.78 4.920.5 1.5 0.107 0.105 0.105 0.0931.5 1.5 2.11 2.16 1.79 1.881.5 2.5 5.07 5.13 4.53 4.51

MCHF

0.5 0.5 6.371.5 0.5 4.650.5 1.5 0.1011.5 1.5 1.911.5 2.5 4.73

Comparison of calculations which include only valence–shell correlationJ J0 This work Tayal (1999) Dufton et al. (1982) Bhatia et al. (1980)0.5 0.5 6.94 4.84 5.50 2.701.5 0.5 4.72 3.84 3.39 1.680.5 1.5 0.072 0.014 0.0641.5 1.5 1.34 1.95 1.101.5 2.5 3.42 3.95 2.73


q 2002 RAS, MNRAS 333, 885–893

A-values for all three stages of the inclusion of correlation effects.

The process of energy adjustment is less significant for the val þ c-v

than for the valence correlation alone, but the ‘adjusted’ results give

quite similar values in all three stages.

An indication of the effect of the additional orbitals used in the

extended set is shown in Table 6 for the val þ c-v case only. The

changes are not large, suggesting that the orbital set is reasonably

sufficient.

In our earlier calculations in the present work, we represented

the ground state by the 2Po symmetry alone. For the J ¼ 3=2 level,

there is a small mixing of 2Po3=2 and 4So

3=2 which, since the upper

levels of these intercombination lines are mainly quartets, give rise

to a contribution to the A-values for the transitions involving 2Po3=2:

for the example shown, there is a consequential rise of about 10 per

cent in the A-value.

For the calculations with the extended orbitals, the zl were

chosen to reproduce specific matrix elements of the Breit–Pauli

Hamiltonian, and as we saw from Table 3, the fine-structure

splittings were poor when only valence–shell correlation was

included. For this reason, only the val þ c-v and val þ c-v þ c-c

results are presented in Table 6. Core–core correlation has some

influence on the A-values, changing the results by up to 15 per

cent.

It is interesting to note the comparison between the CIV3 and

MCHF results, for the val þ c-v calculations. The agreement

between them is much closer than that between these results and

earlier work. In view of this level of agreement, and of the

relatively minor changes across the CIV3 results of Table 6, we

would expect the correct A-values to be within 10 per cent of the

CIV3 results listed under val þ c-v (adjusted).

3.5 Other allowed transitions

We present in Table 7 the oscillator strengths of a number of other

‘allowed’ transitions, in both length and velocity forms, where we

have used the extended set of orbitals in the CIV3 calculation. The

calculated energies of the doublet states are correct to better than 1

per cent, so the effect of energy adjustment is minor for these

transitions, and we show it only for the val þ c-v case. The

length/velocity agreement is consistently close.

The ‘val’ results are compared with Tayal (1999), and the

agreement is good, apart from the very small oscillator strengths

where the length/velocity agreement of the present work suggests

that the present results are superior; however, the differences are

small. The inclusion of core–valence effects changes the oscillator

strengths by a few per cent.

3.6 Recommended oscillator strength values

We have seen that the inclusion of core–core correlation has only a

small effect on the oscillator strengths, particularly in the case of

the length form. Its inclusion does, however, tend to worsen

slightly the agreement between length and velocity forms, and this

suggests that larger numbers of configurations would be needed to

stabilize the velocity value. In view of this uncertainty, we prefer to

recommend our results which include only valence and core–

valence correlation effects. The differences between the CIV3 and

MCHF results are quite small at this level of approximation, and so

in compiling Table 8, which gives our recommended values for

oscillator strengths or transition probabilities, we have included

just the CIV3 results. We would assign uncertainties of within 5 per

cent for allowed transitions, and 10 per cent for the intercombina-

tion lines.

Table 7. Oscillator strengths of other transitions.

Transition CIV3 – val Tayal (1999) – val civ3–val þ cv civ3–val þ cv(ab initio ) (adjusted) (ab initio ) (adjusted)

fl fv fl fv fl fv fl fv

3s23p 2Po1=2 –3s3p2 2D3=2 0.052 0.050 0.043 0.038 0.053 0.053 0.050 0.048

3s3p2 2P1/2 0.487 0.486 0.477 0.490 0.459 0.456 0.459 0.4563s3p2 2P3/2 0.261 0.260 0.257 0.263 0.248 0.246 0.249 0.2473s23d 2D3/2 1.172 1.180 1.17 1.18 1.129 1.109 1.126 1.1183s24s 2S1/2 0.089 0.088 0.089 0.088 0.092 0.093 0.092 0.093

3s23p 2Po3=2 –3s3p2 2D3=2 0.0043 0.0040 0.0035 0.0028 0.0042 0.0042 0.0039 0.0037

3s3p2 2D5/2 0.045 0.043 0.037 0.033 0.045 0.046 0.042 0.0413s3p2 2P1/2 0.135 0.135 0.134 0.138 0.131 0.129 0.131 0.1303s3p2 2P3/2 0.628 0.627 0.619 0.633 0.598 0.594 0.597 0.5943s23d 2D3/2 0.123 0.124 0.123 0.124 0.118 0.116 0.118 0.1173s23d 2D5/2 1.060 1.072 1.06 1.07 1.019 1.004 1.018 1.0123s24s 2S1/2 0.090 0.090 0.090 0.089 0.094 0.094 0.094 0.094

Table 8. Recommended values for oscillator strengths and transitionprobabilities.

Transition l (in A) f A (in s21)

3s23p 2Po1=2 –3s3p2 4P1/2 1404.80 0.189 (24)* 6.386 (4)

3s3p2 4P3/2 1398.05 0.614 (26) 1.048 (3)3s3p2 2D3/2 1062.66 0.050 1.464 (8)3s3p2 2S1/2 809.66 0.118 1.201 (9)3s3p2 2P1/2 748.39 0.459 5.463 (9)3s3p2 2P3/2 744.90 0.249 1.494 (9)3s23d 2D3/2 657.32 1.126 8.692 (9)3s24s 2S1/2 551.12 0.092 2.021 (9)

3s23p 2Po3=2 –3s3p2 4P1/2 1423.83 0.718 (25) 4.721 (4)

3s3p2 4P3/2 1416.90 0.651 (25) 2.160 (4)3s3p2 4P5/2 1406.02 0.228 (24) 5.129 (4)3s3p2 2D3/2 1073.52 0.0039 2.251 (8)3s3p2 2D5/2 1072.97 0.042 1.626 (8)3s3p2 2S1/2 815.94 0.085 1.707 (9)3s3p2 2P1/2 753.76 0.131 3.063 (9)3s3p2 2P3/2 750.22 0.597 7.073 (9)3s23d 2D3/2 661.46 0.118 1.801 (9)3s23d 2D5/2 661.40 1.018 1.034 (10)3s24s 2S1/2 554.03 0.094 4.071 (9)

Note: *Power of 10 in parentheses.


q 2002 RAS, MNRAS 333, 885–893

4 C O N C L U S I O N S

We have described in this paper a series of very extensive

calculations, using two independent procedures (CIV3 and MCHF),

of oscillator strengths of transitions involving the ground state

levels of S IV. We have demonstrated the following.

(1) The branching ratio of the 3s23p 2PoJ –3s3p2 2S1=2 transitions

is around 1.42, to which we have assigned an uncertainly of ^0.02,

which we consider to be conservative. This largely confirms earlier

calculations using other methods, rather than the most recent

experimental ratio of 1:12 ^ 0:1. The deviation of this ratio from

the value 2.0 is partly a consequence of the small difference in the

wavelengths of the two transitions, but largely due to the mixing

between the 3s3p2 2S1/2 and 2P1/2 levels, which is itself an

intercombination effect.

(2) The A-values of the 3s23p 2PoJ –3s3p2 4PJ0 transitions were

indeed in need of revision. We estimate that our new results are

correct to better than 10 per cent.

We have also obtained improved oscillator strengths for a number

of other transitions from the ground state levels.

AC K N OW L E D G M E N T S

We thank PPARC, UK for support under Rolling Grant GR/L20276

and the associated Visiting Fellowship Programme.

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3s23p–3s3p2 transitions in S iv