Improved RCI techniques for atomic 4fn excitation
energies: application to Sm I 4f66s2 5DJ levels
Donald R. Beck and Steven M. O’Malley†
Physics Department, Michigan Technological University, Houghton, MI 49931-1295
U.S.A.
DRAFT SUBMITTED TO J. Phys. B
9/17/2010
Abstract.
We complete the development of a relativistic energy dependent efficient method
by which important pair correlation effects associated with open subshells can be
incorporated into the Relativistic Configuration Interaction (RCI) methodology. We
apply this to predict the positions of the 4f6 6s2 5 DJ levels of Sm I. Relative to 5 D1 ,
we predict 5 D0 lies at -1613 cm−1 and 5 D4 at 6589 cm−1 . For 5 D2 and 5 D3 , we are
22 cm−1 and 123 cm−1 below the observed difference, respectively. We also calculate
magnetic dipole transition rates among these levels and the ground state 7 FJ levels,
which may be of interest to future parity nonconservation (PNC) studies.
PACS numbers: 31.15.am, 31.15.vj, 32.70.Cs
E-mail: donald@mtu.edu
† Present address: AER, Inc. 131 Hartwell Avenue Lexington, MA 02421. e-mail: somalley@aer.com
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels2
1. Introduction
Obtaining accurate wavefunctions for open f-subshell atoms is difficult when the fsubshell occupation/term changes (as in transitions) because that means the paircorrelation effects associated with the f-subshell must be included. An example is given
for the 4f7 6 G – 8 S transition in Gd IV in table 1 of reference [1] for which the fully
expanded basis set for the important correlation contributions contains ∼36K vectors,
which we managed [1] to reduce to ∼8.5K vectors. For this transition, there was no
change in 4f occupation. In the case of 4f7 - 4f6 5d transitions the problem is more severe
[1]. A further illustration of how the complexity grows with occupation (n) is given
for dn occupations in ref. [2]. Note that since dn and d10−n are “equivalent”, that the
closed d10 subshell has the same core complexity as d4 as regards bi-virtual pair d2 pair
correlation; i.e. d8 vs d2 .
In our recent calculations on Fe II f -values [3], we did careful Relativistic
Configuration Interaction (RCI) calculations at the singles and doubles level, and then
shifted by small amounts diagonal energy matrix elements to agree with the observed
spectrum [4, 5]. This leads to competitively accurate wavefunctions. In the case of
rare earths, however, the spectrum is largely unknown especially for the positive ions,
because these have been determined in the condensed phase where there are substantial
ligand field effects for the 5d electrons.
The computationally most expensive pair correlation arises from the highest angular
(l) contributions, e.g. 4f2 →vh2 +vi2 · · ·, yet their contributions, while necessary, are
somewhat modest [1, 6]. So, if they could be computed “outside” the original problem
(e.g. Gd IV) for a much simpler case, such as all closed subshells (e.g. Yb I) and then
inserted into the complicated problem as shifts of diagonal matrix elements, a great
improvement of efficiency could be obtained with little loss of accuracy. This approach,
described below, has been used on Gd IV [7] with a reduction of the energy difference
error by 33% and an efficiency improvement of 67%.
In late 1960s Oksuz and Sinanoglu calculated wavefunctions of ground and excited
states of second row (Li-Ne) atoms and ions [8], without including bi-virtual pair
correlation effects. Given the development of the theory and computational resources at
the time (e.g. obtaining excited state energy eigenvalues for matrices of order larger than
350 was rare), the computational absence of these bi-virtual effects (e.g. 2p2 →vd2 +vf2
· · ·) was unsurprising. What they did do was calculate the net effect of the absence by
subtracting their computed result from experiment, yielding Evv0 . Theoretically, they
were able to show that Evv0 was a linear combination of a product of KNOWN angular
factors (βs) and radial pair energies (²r ), as shown in eqn. (1)
Evv0 (C2 ; LS) =
X
βC2 ;LS (na la , nb lb ; ls)²r (na la , nb lb )
(1)
ls
where C2 = (na la )qa (nb lb )qb , i.e. the a, b subshell portion of the reference configuration.
An explicit LS coupling based formula for the βs is given in the Appendix of
reference [7]. Some explicit β values and relationships between electron and hole
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels3
configurations are given in table 1 of that work [7]. A computer program, COFE,
was written to evaluate βs for any LS-coupled single configuration [9]. Within a closed
subshell, nl, β(nl2 ; S, L) = (2S + 1)(2L + 1).
The striking thing was that a least squares fit of the few (9, with 1s2 remaining
closed) radial pair energies to 113 Evv0 gave an accurate rendering of Evv0 for the
individual levels [8]. The implication was that the ²r depended only weakly on Z and
N.
If this transferability is valid for open f subshell states (e.g. in Gd IV), then radial
pair energies could be extracted from a much simpler closed f subshell system, e.g. Yb
I 4f14 which has been treated by Jankowski et al [6]. In fact, our recent work comparing
²r ’s directly computed for Gd IV with those of Yb I exhibit satisfactory agreement,
except for the largest “low-l” bi-virtual pair 4f2 →vf2 which can be efficiently retained
in the RCI Gd IV calculation. Other than the nearby 4f2 →5d2 pair correlation, the
method also worked well for the complete Pr III J=9/2 spectrum [7].
2. Methodology
2.1. Overview
So how do we use this LS based (non-relativistic) methodology in a intermediate coupled
(relativistic) situation? We generate approximate LS eigenstates from the relativistic
reference space by setting the minor component of the radial function to zero, and
assume the major component is independent of j. The L2 and S 2 matrices are then
simultaneously diagonalized in this basis. Since all possible “LS” functions are kept,
we are still spanning the reference space. The LS βs are then evaluated (Appendix A
of ref [7]) and the diagonal matrix element of each “LS” matrix element (for, e.g. Gd
IV) is shifted by the appropriate linear combination of products of β and ²r .
So what remains to be done? In this work, we are interested in the position of
the lowest 4f6 6s2 5 D levels in Sm I relative to the 4f6 6s2 7 F0 ground state. Since the
5
DJ levels are “LS” degenerate (3-fold), the βs should be calculated using energetically
determined reference function coefficients. Previously, we have been using average βs
(e.g. an average over the three 5 D terms), because these could be obtained without
using energy determined coefficients, but in fact there can be a substantial deviation
from the average for the individual terms. For example, for the three 5 D terms, the β for
the 3 H pair energy varies from 5.46 to 6.77 (average is 5.98), and that for 1 I from .045 to
1.504 . It is interesting to note that for at least Sm I 4f6 6s2 5 D, the relativistic and nonrelativistic energy dependent βs are very similar. So the COFE program [9] was modified
to accept energy determined reference coefficients, as an alternative to generating them
individually arbitrarily from the set of LS eigenvectors (three 5 D here).
However, this program [9] is entirely LS based; its elements are Slater determinants
(SD) formed from spin-orbitals |nlml ms i, whereas RCI is spinor based , viz |nljmj i (l
labels the major component). In order to use the RCI energy coefficients in COFE [9] ,
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels4
one would need to map the spinor SD to the spin-orbital SD, which would require too
much hands on work by user. Instead, we have created in this work a relativistic version
of COFE called RCOFE [10]. Below and in the Appendix we give the details necessary
to do this.
We start with a reference relativistic spinor based wave-function, viz:
Φ=
X
Ck ∆k (· · · κa · · · κb · · ·)
(2)
k
where κa etc are spinors. We then formally evaluate the reference energy using SlaterCondon rules (the basis is orthonormal). Only the Coulomb operator is used, because
the one-particle Dirac operator doesn’t give rise to pair correlation, and the two-particle
Breit operator is considered negligible here. The N -electron SD matrix elements are
reduced to two-electron “He” SD matrix elements formed from one particle spinors, viz:
|nsljmi.
We need to recouple the two electron SD in the bra and ket to attain the desired
result, viz |(n1 s1 l1 j1 m1 )(n2 s2 l2 j2 m2 )SLJM i.
While this appears to be a standard recoupling problem, there seems to be a general
absence in the literature of a detailed derivation, other than references back to the
original work of Condon and Shortley [11]. We will supply details here based on this
work [11] which we had earlier employed in our work on three open-subshell electrons
[12].
We have to take care in the case of equivalent electrons. Specifically, (nl)2 “He”
terms survive only if S + L = even and (nj)2 J “He” levels survive only if J is even.
For situations where SDs are unique, as for us, it is essential that these conditions be
imposed. Cowan [13] provides specific detail as to what is needed for the recoupling
process we are doing. The remaining details can be found in the Appendix.
2.2. Usage
The single configuration reference function Φ, an energy determined linear combination
of SD whose elements are spinors is input to RCOFE. RCOFE then computes the βs
(coefficients of the irreducible pair energies, ²r ) for that function. For open 4f subshells,
one can use the Jankowski et al pair energies, ²r , which were obtained from the closed
subshell Yb I ground state. Note that for fixed S, L all contributions have the same β.
For example, h4f 6 5D|H|4f 4 (vf 2 1S) 5Di and h4f 6 5D|H|4f 4 (vg 2 1S) 5Di have the same β.
In practice, in comparing an actual case (e.g. Gd IV) to Yb I, a few of the largest ²r ’s
(e.g. 4f2 → vf2 ) may differ substantially, and so these should be computed directly (and
removed from the Jankowski et al ²r total)
Returning to the actual case, one then uses an “LS” basis (all vectors kept) for the
reference function, and then shifts the diagonal energy of each LS reference function by
the appropriate β∗²r sum.
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels5
3. Location of Sm I 4f6 (5 D) 6s2 J=0, 4 Levels and Magnetic Dipole Rates
among 7 F and 5 D levels
Recently we had an inquiry from Dr. Budker [14] as to whether we could locate the Sm
I 4f6 (5 D) 6s2 J=0, 4 levels. The 5 D1,2,3 levels have already been observed [4]. The 4f6 6s2
5
D levels can be quite close to some of the 4f6 6s6p levels [4], and there has been past
interest [15] in using them for non-conservation of parity studies, but more recent work
involving the 5 D J=1, 2, 3 levels (the only ones known) was not particularly encouraging
[16].
Our methodology is that of Relativistic Configuration Interaction (RCI) and for
properties like energy differences in complicated atoms, we prefer to minimize the
amount of correlation we have to include. In the present instance, we have used the
known [4] 5 D1 state as a reference and computed the 5 D0 and 5 D4 positions relative
to that. Essentially, this means only including single excitations from the valence (4f,
6s) and shallow core subshells (n=4, 5), the large complex preserving double excitations
(i.e. 4d2 →4f2 ), 6s2 bi-virtual pair correlation, and the single configurational Dirac-Fock
[17] energy differences, including level dependent magnetic Breit effects.
Our results, shown in table 1, place the 4f6 6s2 J=0 level 1613 cm−1 below the
5
D1 level, and the 5 D4 level 6589 cm−1 above the 5 D1 level. Our results for the relative
positions of the 5 D2 and 5 D3 level (relative to 5D1) are 1928 cm−1 and 4158 cm−1 .
Comparing to the experimental positions [4] our 5 D2 level is 22 cm−1 too low and are
5
D3 level is 123 cm−1 too low.
Our calculations included the following configurations (in terms of excitations from
0 0
6 2
4f 6s ), based on our experience with the Gd IV 4f7 ground state: 6s2 →vlv l , 4f→vf+vh;
5p→4f+vp+vf; 5s→vd; 4d→vg; 4p→4f; 4d2 →4f2 . These calculations have been done
separately (DF+ 1 correlation configuration) in interests of efficiency. A comprehensive
calculation would include all these correlations in a single matrix, and the larger pair-pair
(single-pair) interactions would have to be “partially balanced” through the inclusion
of selected triple and quadruple excitations. For example, the 4d2 →4f2 and 4d→vg
contributions are absolutely (but not relatively) so large, that they would (mainly)
falsely influence the smaller contributions [1].
Due to near degeneracy of 4f6 (5 D1 )6s2 and 4f6 (7 F)5d6s (7 F0 , 7 P3 ) [4] it may seem
that the relative position of these levels needs to be accurately determined. However, we
find the matrix elements between these configurations to be small, such that the 5d 6s
correlation energy is ∼0.002 eV. This holds, even when we “manually” shift the diagonal
matrix elements of the two configurations to make them coincide with observation [4] .
It may be noted that the most nearly degenerate levels occur for different J’s, so then
the interaction is identically zero.
Another point of interest is that the 6s2 bi-virtual pair correlation is a small
contributor to the 5 DJ – 5 D1 energy difference. This is consistent with the fact that
the < r > for the 6s radial depends only weakly on J for the 5 D values, and the 7 F0
level is similar. Effectively this means (see below) the 4f, 6s pair correlation will not be
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels6
contributing significantly to what is of interest here.
But it is also useful to extend our interest to see how accurately we can position the
4f6 (5 D0 )6s2 level relative to the ground 4f6 (7 F0 )6s2 level. Here we would need to include
4f2 pair correlation, which gives us a good opportunity to use the new relativistic β
results of the first section.
We saw that bi-virtual pair correlation effects involving an open subshell a
(occupation qa ) and a closed subshell b is given by the expression:
(qa /[2(2la + 1)]
X
(2s + 1)(2l + 1)²r (a, b; sl)
(3)
s,l
where the sum is over all combinations of s, l which can arise from the configuration
na la nb lb [18]. The validity of this expression rests on the same criteria as used in section
1 of this work.
Computationally, what this means is that because the subshell occupation is
NOT changing for the 7 F and 5 D levels above, we need not include bi-virtual pair
correlation associated with closed [1s· · ·5p, 6s] and open [4f] subshells. Since correlation
contributions from 4f6 5d6s are small (see above) we also need not evaluate differential
effects due to 6s and 5d interchange.
To the excitations used to compute 5 DJ – 5 D1 energy differences (see table
1), we need to add the bi-virtual contributions from 4f2 pair excitations. These
slowly convergent contributions are computationally expensive to obtain. However,
we compute them using our relativistic β methodology. We use the actual energy
determined Dirac-Fock coefficients for the lowest 5 D0 to obtain the βs. From tables
I and II in the Jankowski et al [6] work we obtain the Yb I radial pair energies. We
directly compare the largest excitations (4f2 →vf2 + vg2 ) for Sm I (obtained via RCI)
and Yb I and find these radial pair energies satisfactorily similar.
The net result is that the 5 D0 level has -0.4578 eV more energy than the 7 F0 from
these excitations. This is used to shift the 5 D0 matrix elements down by this amount,
relative to the 7 F0 matrix element. Rediagonalizing the Dirac-Fock matrix with this
shift results in a net 5 D0 lowering relative to 7 F0 of -0.260 eV. With the inclusion of the
few remaining excitations used for Gd IV [1], we have a further lowering of 408 cm−1 .
We then predict the 5 D1 – 7 F0 energy difference to be 18756-2787-408+1613 = 17, 174
cm−1 . The last number is the (computed) difference between 5 D0 and 5 D1 . The observed
5
D1 – 7 F0 energy difference [4] is 15915 cm−1 , so our result is 1259 cm−1 too high. The
order of magnitude (1000 cm−1 ) seems typical for both transition metal and rare earth
atoms at this stage of computational development [1].
4. Magnetic Dipole Transition Rates
We give magnetic dipole transition rates involving 4f6 (7 F)6s2 and 4f6 (5 D)6s2 levels in
Sm I in Table 2. These were computed at the shifted Dirac-Fock level using relativistic
formulae [19]. The diagonal 7 FJ energy matrix elements were shifted to yield the
observed [4, 5] or calculated (5 D0 ,5 D4 ; this work) 7 FJ – 5 DJ energy differences. Since
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels7
the 7 FJ – 5 DJ 0 transitions are only non-zero due to the mixing of 5 DJ into 7 FJ (7 FJ 0
into 5 DJ 0 ), it is important to have the mixings determined as accurately as possible. For
5
D0 , for example, the unshifted mixing was about 3.0%, whereas the shifted mixing was
about 5.0%. This increased the 7 F0 – 5 D1 f -value about 50%. The amount of mixing
decreases with increasing J.
For comparison, we have available magnetic dipole matrix elements [16, 20] for
7
the FJ – 5 DJ 0 transitions. These are [20] Ofelt’s [21] mixing coefficients for Sm III
obtained semi-empirically from crystal spectra. We have converted these to f -values
using the formula in Table III of an NBS volume [22], and assuming that squaring the
matrix elements [20] produces the required [22] line strength, with no additional factors
needed. These results are displayed in Table 2 as well.
Except for the 7 FJ – 5 DJ transitions, there is fair to good agreement between the
two results. The J – J transitions are much smaller (two orders of magnitude or more)
because [20] the matrix element is directly proportional to the common Landé g-value,
and the overlap integral between the 7 FJ and 5 DJ states [20]. A detailed derivation of
this result can be found in Griffiths [23]. For these f -values, the differences between
the two sets of results are considerably larger, which is not unexpected given their small
size.
The 7 FJ – 7 FJ 0 and 5 DJ – 5 DJ 0 results appear to be entirely new.
5. Conclusion
We have completed the development of a relativistic energy dependent efficient method
by which important pair correlation effects associated with open-f subshells can be
included in RCI. Small to moderate sized, but expensive to compute, correlation effects
can be included as shifts of diagonal energy matrix elements. These shifts are linear
combinations of products of angular factors and radial pair energies. Explicit formula
have been given which enables the user to determine the angular factors. The few
(seven) 4f2 radial pair energies seem to be significantly independent of Z and N , the
number of electrons, based on computational experience to date. This enables us to
compute them for a much simpler system such as ones with a closed 4f subshell, like
Yb I [6]. Using these shifts, we have decreased the error in the 7 F0 – 5 D0 splitting ∼2100
cm−1 , leaving our result for 7 F0 – 5 D1 1259 cm−1 too high. This order of magnitude
error (1000 cm−1 ) seems to be typical of what can be done in both transition metal and
rare earth atoms at this stage of computational development,in positioning the excited
states relative to the ground state level.
Finally, we have located the missing 5 D0 and 5 D4 levels of Sm I, by using the
observed 5 D1 state [4, 5] as a reference. This enables us to avoid including a significant
amount of correlation effects (e.g. 4f2 pair energies) common to the 5 DJ levels. As a
check, we compared our result for the 5 D2,3 – 5 D1 energy differences. We are only 22
cm−1 and 123 cm−1 below the observed difference for these two levels, respectively.
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels8
Acknowledgments
The authors thank the National Science Foundation, Grant # 968205 for support of
this work.
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels9
Table 1. Relative energy contributions to Sm I 4f6 6s2 5 DJ and 7 F0 levels (in cm−1 )
5
J=0
-1184
a
J=1
0
7
DJ
J=2
J=3
1876
DF+Maga
1196
6830
F0
Excitation
-18756
N/A
-5139
-354
-12703
-167
-495
-834
-735
-849
-24919
2097c
408
²(6s2 )
4f→vf+vh
4d2 →4f2
5p→4f
4p→4f
5p→vf
5s→vd
5p→vp
4d→vg
²(4f2 )
Miscd
J=4
-5131
-509
-14235
-184
-590
-718
-659
-444
-23231
-5132
-423
-14046
-174
-559
-753
-677
-469
-23039
-5135
-422
-14008
-177
-560
-707
-658
-483
-23070
Correlationb
-5135
-5120
-424
-428
-14015 -14026
-218
-258
-551
-642
-742
-822
-663
-684
-482
-464
-23080 -23069
-1613
17174f
15915f
1928
1949
Final Results
4158
6589
4281
RCIe
Obse
relative to the 5 D1 DF+Magnetic value.
relative to the 5 DJ (7 F0 ) DF+Magnetic value. 5 DJ radials are used which enhances
the 5p→vp contribution for 7 F0 .
c
Using relativistic βs + Jankowski et al [6] ²r to shift the diagonal energy matrix
element of 5 D0 . The net result is to decrease the 7 F0 – 5 D0 splitting by 2097 cm−1
(see text).
d
Contribution from (4p2 +5p2 )→4f2 , 4p→vp, 4s4d→4f2 .
e
From ref. [4]. Relative to 5 D1 except as noted.
f
Relative to 7 F0 .
b
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels10
Table 2. Sm I Magnetic dipole transition ratesa
J →J
0
(b)
Aki (s−1 )(b)
fik
7
0→1
1→0
1→1
1→2
2→1
2→2
2→3
3→2
3→3
3→4
4→3
4→4
FJ → 5 DJ 0
0.116×10−7
0.173×10−7
0.493×10−11
0.176×10−8
0.242×10−7
0.108×10−10
0.417×10−9
0.215×10−7
0.947×10−11
0.897×10−10
0.159×10−7
0.209×10−10
0.655
7.206
0.802×10−3
0.217
6.137
0.210×10−2
0.746×10−1
5.388
0.221×10−2
0.217×10−1
4.392
0.811×10−2
7
0→1
1→2
2→3
3→4
0.139×10−6
0.155×10−6
0.162×10−6
0.148×10−6
a
a
b
0.280×10−6
0.271×10−6
0.225×10−6
0.151×10−6
0.165×10−7
0.205×10−7
0.165×10−9
0.242×10−8
0.266×10−7
0.923×10−10
0.491×10−9
0.222×10−7
0.622×10−10
0.263×10−9
0.166×10−7
0.303×10−9
FJ →7 FJ 0
0.266×10−2
0.167×10−1
0.355×10−1
0.470×10−1
5
0→1
1→2
2→3
3→4
(c)
fik
DJ →5 DJ 0
0.886×10−1
0.412
0.583
0.651
Except for the 5 D0 and 5 D4 levels, experimental values [4, 5] are used for energies.
This work.
Ref. [20] (see text).
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels11
Appendix A.
Reduction of the N -electron matrix element
hΦ |
X
1/rij | Φi
i<j
where Φ is given by eqn. (2), to two-electron (“He”) matrix elements, using SlaterCondon rules.
Appendix A.1. The five 3j Symbol Result for “He” Matrix Elements
We reverse the process, then invert the result.
Step 1 : Express |SLJM i states as a linear combination of |SMS LML i states using
standard coupling theory.
Step 2:
s
| s1 l1 s2 l2 SLML MS i = (−1)l1 +l2
Ã
×
s1
s2
S
ms1 ms2 −MS
!Ã
l1
l2
L
ml1 ml2 −ML
!
(2S + 1)(2L + 1) X X X X
N
ms1 ms2 ml1 ml2
∆(s1 l1 ms1 ml1 ; s2 l2 ms2 ml2 )
(A.1)
where N =2 for equivalent subshells (=1, otherwise) and the ( ) are Wigner 3j symbols.
The expression is a modern version [24] of Condon and Shortley’s result [11].
Step 3: Express the one particle |sms lml i functions in the SD of Step 2 in terms of
one particle |sljmi using standard coupling theory.
Step 4: Invert the combined result, using the 3j orthonormality expressions as in,
for example, Edmonds [25].
The result is shown below. Here N =2 if n1 l1 = n2 l2 (=1, otherwise), and the ( )
are Wigner 3j symbols.
q
∆(n1 s1 l1 j1 m1 ; n2 s2 l2 j2 m2 ) =
(2j1 + 1)(2j2 + 1)
Ã
q
S−L
× (2S + 1)(2L + 1)(2J + 1)N (−1)
Ã
!Ã
s1
l1
j1
l1
l2
L
×
ms1 ml1 −m1
ml1 ml2 −ML
× | (n1 s1 l1 j1 )(n2 s2 l2 j2 )SLJM i
X X X XXX X
JM SMS LML ml1 ml2 ms1 ms2
S
L
J
MS ML −M
!Ã
!Ã
s1
s2
S
ms1 ms2 −MS
s2
l2
j2
ms2 ml2 −m2
!
!
(A.2)
Several sums can be performed immediately from the conditions: M = m1 + m2 ,
MS = ms1 + ms2 , ML = ml1 + ml2 , m1 = ms1 + ml1 , m2 = ms2 + ml2 .
With these results, the two particle “He” spinor matrix elements can be evaluated.
Since 1/r12 commutes with S 2 , L2 , J 2 , Jz operators both bra and ket must have the same
values of S, L, J, M . Coefficients for each S, L combination appearing are collected
into the β for that S, L pair energy.
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels12
Appendix A.2. An alternative: the 3j-9j result for “He” matrix elements
Step 1: Express the SD |j1 m1 j2 m2 i as a linear combination of |j1 j2 JM i states using
standard coupling theory. For equivalent electrons, be careful to have J = even and
include a normalizing factor.
Step 2: Use the result in Shore and Menzel [26] which converts | j1 j2 JM i to a sum
over |l1 l2 SLJM i states. For equivalent electrons, the sum over the 9j symbol is to be
restricted such that L + S = even; if j1 = j2 , J=even, and if j1 < j2 , we multiply by
√
2, following Cowan [13]. The result is:
q
∆(n1 s1 l1 j1 m1 ; n2 s2 l2 j2 m2 ) =
×
X√
JM
Ã
2J + 1
j1 j2
J
m1 m2 −M
(2j1 + 1)(2j2 + 1)N2 N3 (−1)j1 −j2 +M
!
SL



 l1

s1 j1 


(2S + 1)(2L + 1) l2 s2 j2



 L S J 

Xq
× | (n1 s1 l1 j1 )(n2 s2 l2 j2 )SLJM i
(A.3)
where N2 = 2 if n1 j1 = n2 j2 and =1, otherwise. N3 = 2 if n1 l1 = n2 l2 AND j1 6= j2 , and
=1, otherwise [13].
Since s1 = s2 = 21 this means S = 0, 1 only. In this case the 9j symbol can be reduced
to 1 (S=0) or 3 or 4 (S=1) 6j symbols [25]. Evaluation of eqn. (A.3) is computationally
more efficient than eqn. (A.2), but neither requires significant computational resources.
Appendix A.3. Equivalence of Equations (A.2) and (A.3)
Two formula are needed to show this: (1) the recoupling formula for the product of two
3j symbols given as problem (3-1) in Judd [27] and equation (6.2.8) in Edmonds [25] for
summing the product of three 3j symbols. Note that one of the l2 ’s in the phase factor
should be l1 [25].
P
We start from eqn. (A.2), recoupling the sum over ml2 , replacing it with l3 ,u3
over a product of two 3j symbols and a 6j symbol. Then we perform the sum of ml1 ,
ms1 and ms2 using (6.2.8) which yields a product of a 3j and 6j symbol (phase factors
appear throughout and must be carefully treated).
Then we use (6.2.8) again to perform the sum of ML , MS and µ3 , which yields a
product of a 3j symbol (the one appearing in eqn. (A.3)) and a 6j symbol. Finally, the
l3 sum over the product of three 6j symbols and (2l3 +1) is replaced by a 9j symbol using
(6.4.3) of Edmonds [25]. Using the 9j symbol permutation symmetry [25] we finally get
eqn. (A.3).
Improved RCI techniques for atomic 4fn excitation energies: application to Sm I 4f6 6s2 5 DJ levels13
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