A theoretical study of the C- 4So_3/2 and 2Do_{3/2,5/2} bound states and C ground configuration: fine and hyperfine structures, isotope shifts and transition probabilities

0.51 MB·English

by  T. Carette

A theoretical study of the C− 4So and 2Do bound states and C ground
3/2 3/2,5/2
configuration: fine and hyperfine structures, isotope shifts and transition probabilities
T. Carette∗ and M. R. Godefroid†
Chimie quantique et photophysique, CP160/09, Universit´e Libre de Bruxelles, B 1050 Brussels, Belgium
(Dated: January 28, 2011)
This work is an ab initio study of the 2p3 4So , and 2Do states of C− and 2p2 3P ,
3/2 3/2,5/2 0,1,2
1D , and 1S states of neutral carbon. We use the multi-configuration Hartree-Fock approach,
2 0
focusing on the accuracy of the wave function itself. We obtain all C− detachment thresholds,
including correlation effects to about 0.5%. Isotope shifts and hyperfine structures are calculated.
The achieved accuracy of the latter is of the order of 0.1 MHz. Intra-configuration transition
1 probabilities are also estimated.
1
0 PACSnumbers: 32.10.Hq,31.15.aj,31.15.ac,32.10Fn,31.15.ag
2

n I. INTRODUCTION 1s2 2s2 2p2 1S
a J=2
J
Negative ions have always attracted broad attention J=1
7 J=0
2 fromthescientificcommunity[1,2]. Theychallengeboth
the experimentalist and theoreticians. The first because

h] theyareweaklybound,andthereforefragile,andbecause 1s2 2s2 2p2 1D
they do not possess a lot of features allowing measure-
p
- ments. The latter because the binding of an extra elec-
m tron is granted only by a arrangement of the electrons in C
o a highly correlated system [3]. Moreover, the fact that
x 100
t theelectronsinnegativeionsareboundbyashortrange 1s2 2s2 2p2 3P
a
. potential confer them unique properties. 1s2 2s2 2p3 2Do
s C− is the lightest negative ion to have two bound
c
i terms: its ground state 4So and the 2Do excited state C-
s which both arise from the 2p3 configuration. The level
y
h diagram of the states studied in this work is given in 1s2 2s2 2p3 4So J=5/2
p Figure 1. J=3/2
[ Carbon is among the most abundant components in

1 the universe and a key element in life chemistry. The FIG. 1. Level diagram of the C and C−. The fine structure
v carbon negative ion is attracting in astrophysics and ofthe3P stateoftheneutralcarbonandthe2Do stateofC−
8 atmosphere physics since nitrogen-like 2p3 4So − 2Do are magnified (×100).
1 forbidden lines are recognized as useful transitions for
3 abundances determination [4–6]. It has also recently
5 been suggested by Le Padellec et al. [7] that C− nega- oreticalside,veryaccuratecarbonelectronaffinitieswere
.
1 tive ion could intervene in astrophysical reactions. The obtained with coupled-cluster-based methods [15–17].
0 photodetachment cross-sections of the C− have been re- The structure of the C− has not been studied thor-
1 peatedly studied, both theoretically [8] and experimen- oughlyandespeciallylittleisknownaboutthe2Do mul-
1
tally[9],forphotonenergiesaddressingvalenceelectrons tiplet. Inlaboratoryplasmas,lifetimesoftheorderofthe
:
v and core electrons [10, 11]. Recently, an isotope separa- ms were measured for the C−(2Do), the electron detach-
i tion method was tested by Andersson et al. [12], based ment being principally caused by the black-body radia-
X
on the isotopic dependence of the Doppler shift of the tionand,toalesserextent,tocollisions[18]. Significantly
ar C− detachment thresholds in an accelerator. longer lifetimes could be reached in the cold and diluted
interstellar media where molecular anions have already
A binding energy of 1.262 119(20) eV for the C−(4So)
beendetected[19]. However, theC−(2Do)is, forvarious
has been measured by Scheer et al. [13] who could not
reasons, very difficult to study experimentally. In this
improve the old value of 33(1) meV for the C−(2Do) de-
context, a firm theoretical knowledge of this system is
tachment threshold, measured by Feldmann [14]. The
particularly precious.
fine-structureofthe2Do stateisnotknown. Onthethe-
Elementsfromborontofluorinearethenexttargetsaf-
ter beryllium in the working line of “exact” calculations.
High accuracy can be achieved for systems with up to
∗ [email protected] fourelectronsusingwavefunctionsexpandedinexplicitly
† [email protected] correlated gaussian or in Hylleraas coordinates [20–22].
2
Although the precision that can be achieved for atoms tions (CSF) [27] for describing a given term
with more electrons is limited by the complexity of the
(cid:88)
electroncorrelationmathematicaltreatment, theground Ψ(γLSMLMS)= ciΦ(γiLSMLMS). (1)
states of the second period p−block atoms from B to F i
aresatisfactorilydescribedbyanon-relativisticapproach
on top of which relativistic corrections are added.
B. Hyperfine interaction
A critical benchmark quantity for highly correlated
models is the isotope shift (IS) on the electron affin-
The level hyperfine structure is caused by the interac-
ity (eA) that is doubly sensitive to correlation effects:
tionoftheangularmomentumoftheelectrons(J)andof
through the negative ion structure and through the
thenucleus(I),formingthetotalatomicangularmomen-
specific mass shift parameter. The multi-configuration
tum F=I+J. The theory underlying the computation
Hartree-Fock method has been successfully used for cal-
ofhyperfinestructureusingMCHFwavefunctionscanbe
culating the IS on the eA of O [23], S [24] and Cl [25].
found in references [29–31]. It is possible to express the
Thepurposeofthepresentworkisdouble. Ourprinci-
non relativistic hyperfine interaction in terms of the J-
pal objective is to obtain the crucial informations about
independent orbital (a ), spin-dipole (a ), contact (a )
the C− electronic structure for stimulating experimen- l sd c
and electric quadrupole (b ) electronic hyperfine param-
tal research on the 2Do state. Therefore, we focus on q
eters defined as [29]
quantities that are especially difficult for experimental-
ists to measure: isotope shifts, hyperfine structures and N
absolute transition probabilities. As for the energy sep- al ≡(cid:104)ΓLSMLMS|(cid:88)l0(1)(i)ri−3|ΓLSMLMS(cid:105), (2)
arations themselves, we do not try to compete nor with i=1
the observation, nor with the previous coupled-cluster N
calculations. Weinsteadusethesereliablereferencedata asd ≡(cid:104)ΓLSMLMS|(cid:88)2C0(2)(i)s(01)(i)ri−3|ΓLSMLMS(cid:105),
forassessingthequalityofourcomputationalprocedure. i=1
(3)
Oursecondobjectiveistoobtainnon-relativistic(NR)
wavefunctionsasaccurateaspossibleusingthestandard N
tools of the ATSP2K package [26]. For getting the best ac ≡(cid:104)ΓLSMLMS|(cid:88)2s(01)(i)ri−2δ(ri)|ΓLSMLMS(cid:105),
estimationoftheaccuracy,wechoosetousethesamesys- i=1
tematicalconstructionofourCandC− models,avoiding (4)
any arbitrary compensation of the “additional” electron N
correlation of the negative ion compared to the neutral b ≡(cid:104)ΓLSM M |(cid:88)2C(2)(i)r−3|ΓLSM M (cid:105), (5)
q L S 0 i L S
atom. i=1
andcalculatedforthemagneticcomponentM =Land
L
In Section II, we present large scale numerical multi- M =S [32]. The diagonal hyperfine interaction energy
S
configuration Hartree-Fock (MCHF) calculations (Sec- correction is usually expressed in terms of the hyperfine
tion IIE), relativistic calculations estimated using the magnetic dipole (A ) and electric quadrupole (B ) con-
J J
Breit-Pauli approach (BPCI, Section IIF) [27] and rel- stants as follows
ativistic configuration interaction based on the Dirac
C
equation (RCI, Section IIG) [28]. In Section III, we W(J,J)=A
J 2
present accurate results for hyperfine structures (Sec-
3C(C+1)−4I(I+1)J(J +1)
tion IIIA), total energies including the fine structure +B . (6)
(Section IIIB), mass polarization shift parameters (Sec- J 8I(2I−1)J(2J −1)
tion IIIC) of C 2p2 3P, 1D, 1S and C− 2p3 4So, 2Do. The first three parameters (2), (3), and (4) contribute
In Section IIID, we present the M1 and E2 transition to the magnetic dipole hyperfine interaction constant
probabilities within the 2p2 and 2p3 configurations of C through
and C−.
A =Al +Asd+Ac , (7)
J J J J
with [33]
II. COMPUTATIONAL METHOD Al =G µI a (cid:104)LLL·JJJ(cid:105) , (8)
J µ I l LJ(J +1)
1 µ
A. The MCHF expansion Asd = G g I a
J 2 µ s I sd
3(cid:104)LLL·SSS(cid:105)(cid:104)LLL·JJJ(cid:105) − L(L+1)(cid:104)SSS·JJJ(cid:105)
The multiconfiguration Hartree-Fock (MCHF) varia- × , (9)
SL(2L−1)J(J +1)
tional approach consists in optimizing the one-electron
functions spanning a configuration space and the mixing 1 µ (cid:104)SSS·JJJ(cid:105)
Ac = G g Ia , (10)
coefficients of the interacting configuration state func- J 6 µ s I c SJ(J +1)
3
while the last one (b ) constitutes the electronic contri- D. Transition probabilities
q
bution to the electric quadrupole hyperfine interaction
The Einstein A coefficient of spontaneous emission
if
B =−G Qb
J q q is defined as the total probability per unit of time for an
6(cid:104)LLL·JJJ(cid:105)2 − 3(cid:104)LLL·JJJ(cid:105) − 2L(L+1)J(J +1) atominagivenenergylevelitomakeatransitiontoany
× (.11)
L(2L−1)(J +1)(2J +3) state of the energy level f [36].
A transition between levels of same parity is forbid-
Expressing the electronic parameters a , a and a in denintheelectricdipoleapproximation,beingingeneral
l sd c
atomic units (a−3) and µ in nuclear magnetons (µ ), many orders of magnitude lower than an allowed transi-
0 I N
the magnetic dipole hyperfine structure constants A tion. Two interactions of the same order of magnitude
J
are calculated in units of frequency (MHz) by using cancontributetotheappearanceofsuchtransitions: the
G = 95.41067. Similarly, the electric quadrupole hy- dipole magnetic and the electric quadrupole radiation-
µ
perfine structure constants B are expressed in MHz matterinteractions. Atthenon-relativisticlevel,adipole
J
whenadoptingatomicunits(a−3)forb ,barnsforQand magnetic transition (M1) is governed by the electronic
0 q
G = 234.96475. The expectation values of the angular magnetic dipole operator that is
q
momenta scalar products are given by
AM1 ∝(E −E )3|(cid:104)Γ J ||LLL+g SSS||Γ J (cid:105)|2. (18)
i f f f s i i
(cid:104)LLL·JJJ(cid:105)=[J(J +1)+L(L+1)−S(S+1)]/2, (12)
(cid:104)SSS·JJJ(cid:105)=[J(J +1)−L(L+1)+S(S+1)]/2, (13) Inthemono-configurationapproximation,theabovema-
trix element is non-zero only between states of the same
(cid:104)SSS·LLL(cid:105)=[J(J +1)−L(L+1)−S(S+1)]/2. (14)
configuration and LS. This selection rule is relaxed by
configurationandLS mixings, theremainingconstraints
when calculated with non-relativistic LSJ wave func-
being that J =J ,J ±1, and that Ψ and Ψ have the
tions. The expression for the off-diagonal hyperfine in- f i i i f
same parity. For its part, an electric quadrupole (E2)
teraction, depending on the hyperfine constants A ,
J,J−1 transition rate is proportional to the electric quadrupole
B and B , are developed in reference [31]. Hib-
J,J−1 J,J−2 moment matrix element
bert [32] gives the expressions of A in terms of the
J,J−1
hyperfine parameters (2–5). (cid:12)(cid:42) (cid:12)(cid:12) (cid:12)(cid:12) (cid:43)(cid:12)2
(cid:12) (cid:12)(cid:12)(cid:88) (cid:12)(cid:12) (cid:12)
AE2 ∝(E −E )5(cid:12) Γ J (cid:12)(cid:12) r2CCC(2)(k)(cid:12)(cid:12)Γ J (cid:12) ,
i f (cid:12) f f(cid:12)(cid:12) k (cid:12)(cid:12) i i (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)
k
(19)
C. The isotope shift
the sum running on all spatial electron coordinates k.
Neglecting the LS term mixing, a necessary condition
The first order isotope shift on an energy level is de-
forA tobenon-zeroisthatS −S =0,|L −L |≤2,
E2 f i f i
composed in a field shift (or volume shift) and a mass
|L +L |≥2 and that the atomic parity is conserved.
f i
shift [34]. The first is proportional to the change in nu-
cleusrmsradiusandchangeofthemodifiedelectronden-
sity at the origin. It is negligible in our context.
E. Non-relativistic calculations
The energy corrected for the first order mass shift, on
the other hand, can be estimated using [35]
We first select a zero-order set of CSFs, the multi-
M Mm ¯h2 reference (MR). For all studied states it is the set of sin-
EM = m+ME∞+ (M +m)2 mSsms (15) gle and double excitations of the main configuration to
then=2,3shells. AlltheCSFsinteractingtofirstorder
withtheMRareselectedandwechoosethereverseorder
wheremistheelectronmass,M isthebarenucleusmass,
for the subshell coupling [24]. The orbital active set is
E the infinite mass nucleus and
∞ defined as the set of all orbitals characterized by quan-
(cid:12) (cid:12) tum numbers n ≤ n and l ≤ l , and is denoted
(cid:42) (cid:12) (cid:12) (cid:43) max max
Ssms =− Ψ∞(cid:12)(cid:12)(cid:12)(cid:88)∇i·∇j(cid:12)(cid:12)(cid:12)Ψ∞ . (16) (cid:100)thnemsapxalmceasx(cid:101)(cid:100).4fW(cid:101)etofi(cid:100)rs1t2kp(cid:101)e,rfdoernmoetdedcMalcRu-lIa(cid:100)tniomnasxldmeafixn(cid:101)e.d in
(cid:12)i<j (cid:12)
Foreachactivespace(cid:100)10k(cid:101),(cid:100)11k(cid:101)and(cid:100)12k(cid:101),weorder
the configurations according to their weight [37]. We
The first term contains the normal mass shift
then construct several new MRs following this hierarchy,
m independently for each state and active set, by selecting
NMS=− E (17)
m+M ∞ the minimum group of configurations that add up to a
certain percentage p of the total wave function. Those
and the second one is the specific mass shift (SMS). The multi-references are denoted MR for each given MR-
p
masspolarizationparameter, S , hasthedimensionof I wave function. Unsurprisingly, the MR sets are not
sms p
an inverse square length. sensitive to the used active set.
4
Configuration-interaction calculations are performed andinTableIIforC−. Itshouldbestressedthatthedif-
on each MR -I CSF sets, p being limited to 99.8% for ferences between the results obtained with p=99.9 and
p
C−(4So) and to 99.3% for C−(2Do). An example of the p = 99.95 are close to the expected numerical accuracy.
convergenceofthecalculationswiththenumberofcorre- We estimate that the dominant error in the neutral car-
lationlayers(n−2)andpisgiveninFigure2. Itpresents bon calculations is due to the nl-truncation of the active
results on the 2p2 3P state of neutral carbon. The black set. ThisisnotthecaseintheC− calculationsforwhich
squares show h¯2δS with the p-truncation is the most limiting.
m sms
From Tables I and II, we observe that the differences
δS =|S (MR-I(cid:100)nl(cid:101))−S (MR-I(cid:100)12k(cid:101))|,(20)
sms sms sms between the MR -I(cid:100)11k(cid:101) and MR -I(cid:100)12k(cid:101) results do not
p p
versus depend strongly on p. In fact, the error made by re-
porting the impact of the 12th shell on the calculation
δE =|E(MR-I(cid:100)nl(cid:101))−E(MR-I(cid:100)12k(cid:101))| (21) with p = 99, on the results of the largest MR -I(cid:100)11k(cid:101) is
p
smallerthan410−7E ontheenergy,than210−6a−2on
for n = 6 − 11. Similarly, the white squares com- S and than 9 10−5ha−3 on the hyperfine parame0ters.
pare the E and S convergences of the MR -I(cid:100)12k(cid:101), sms 0
sms p
p=99.0−99.9,resultstowardtheMR -I(cid:100)12k(cid:101)model. Usingthisobservation,weaddacorrectionforthe13th
99.95
Theenergyalwaysdecreasesalongasequenceofincreas- correlation layer and for the l=8 orbitals. First MCHF
inglylargecalculations,accordingtothevariationalprin- calculations are performed on the MR-I(cid:100)13k(cid:101) and MR-
ciple,whiletheS ofMR -I(cid:100)nl(cid:101)calculationsdecreases I(cid:100)12l(cid:101)CSFspaces,fixingallone-electronradialfunctions
sms p
with n and increases with p. δS and δE show a close attheMR-I(cid:100)12k(cid:101)levelandvaryingonlytheneworbitals.
sms
to linear correlation, i.e. the angular coefficients in the We use the so optimized orbitals in MR99-I(cid:100)13l(cid:101) CI cal-
log-log figure is ∼ 1. The slope of this correlation is culations, omitting the 13l subshell in the active set and
slightly smaller than one for the convergence in n and is using the multi-reference obtained with the (cid:100)12k(cid:101) active
about 10−20 for the convergence in p, as can be seen set. The two contributions (higher n and l) are of same
from the offsets in the log-log figure. Similar behaviors order of magnitude, as far as the energy is concerned.
wherefoundintheopen-coreCIcalculationsofS/S− [24] Still,theadditionalcorrelationlayertendstodominatein
and Cl/Cl− [25]. In general, we can make the following C− whiletheadditionalangularflexibilityhasthelargest
observations [38]: the CSFs that are important for the impact in the neutral carbon calculations.
energy are accordingly important for the Ssms, and the Aswealreadymentioned, theconvergencesineithern
Ssms value is more sensitive to the choice of the CSF or p, of a state energy and Ssms are monotone and cor-
space than to the orbital basis set. related (see e.g. Figure 2). This fact could help strongly
for extrapolating the energy and S value. However,
sms
evenforatwo-electronsystem,theprecisebehaviorofthe
energy convergence with the principal quantum number
1e-04 in the high n-limit is unknown [39, 40]. Froese-Fischer
used the following extrapolation function for studying
four electrons systems [41]
]h
E
[s
m
Sds 1e-03 ∆En =a4/(n−δn)4+a5/(n−δn)5+a6/(n−δn)6 (22)
m
2/
-h
the a , a and a parameters, and δn being chosen such
4 5 6
thata <0and|a |∼|a |∼|a |. Fittingthen=10−12
4 4 5 6
1e-02 results of Table I to equation (22) for extrapolating to
n → ∞, we obtain −37.84465 E for the energy of the
h
1e-02 1e-03 1e-04 1e-05 C(3P)state. Thisproceduredoesnotextrapolatetol→
d E [E ]
h ∞. The error on the extrapolation is of the order of
10−5 E and the truncation in l of about 10−4 E . We
FIG. 2. Bi-logarithmic plot of the convergence of the C(3P) h h
are in fair agreement with the non-relativistic energy of
masspolarizationexpectationvalueversusthecorresponding
−37.8450E estimatedbyChakravortyetal.[42]. Toour
energy,inHartrees(Eh). Thecoordinatesoftheblacksquares h
are the differences (in absolute value) between the results of knowledge, the values of Table I are the best ab initio
the MR-I(cid:100)nl(cid:101) and MR-I(cid:100)12k(cid:101) calculations, n = 6−11 from estimated energies, even without the n = 13 and l =
left to right. Similarly, the coordinates of the white squares 8 corrections. Finally, let us mention that Sarsa et al.
showtheconvergenceofMRp-I(cid:100)12k(cid:101)towardMR99.95-I(cid:100)12k(cid:101), [43]calculatedtheSsms expectationvalueforthecarbon
p=(99−99.9)% from left to right. 3P state using the Monte Carlo (MC) approach with an
explicitly correlated wave function and obtained S =
sms
The energy, S and hyperfineparameters calculated −0.38(2) a−2. Our final estimated value (−0.40314 a−2)
sms 0 0
in this work are presented in Table I for neutral carbon falls a bit outside the statistical MC error bars.
5
hyperfinedditively Ssms −1.29745−0.38101−0.38198−0.35936−0.36255−0.36275−0.36358−0.36384−0.36417−0.36436 −0.36693−0.36179−0.35869−0.35805−0.35766−0.35750 −0.36712−0.36357−0.36198−0.35871−0.35821−0.35782−0.35766 −0.36736 −0.35790
a
−2aandthe0shavebeen E MCHF −37.549610−37.720089−37.733658−37.740170−37.742742−37.743993−37.744656−37.745021−37.745234−37.745361 CI −37.744893−37.745357−37.745523−37.745561−37.745588−37.745600 −37.745020−37.745388−37.745483−37.745655−37.745689−37.745716−37.745728 −37.745181 −37.745889
intal
msbi
Ssor 68553279783173721164 100476818683 74776742505856 49 31
e8 05496201140701060603 060605050505 05050505050505 05 05
,thhl= 1Sbq −1.3−1.1−1.1−1.2−1.2−1.2−1.2−1.2−1.2−1.2 −1.2−1.2−1.2−1.2−1.2−1.2 −1.2−1.2−1.2−1.2−1.2−1.2−1.2 −1.2 −1.2
Ed 2
n p
Eareinshella 221s2s2al 3.264203.238623.274253.276613.277343.276943.276733.276693.276653.27155 3.276843.276773.276513.276533.276653.27661 3.276753.276923.276683.276443.276493.276613.27657 3.27690 3.27672
h
est3
gi1 F
enerthe MCH 557151677238697842938991029051 CI 136819500429388374 156938838523452407394 182 420
Thectof Ssms −1.35−0.40−0.37−0.36−0.36−0.36−0.36−0.36−0.37−0.37 −0.37−0.36−0.36−0.36−0.36−0.36 −0.37−0.36−0.36−0.36−0.36−0.36−0.36 −0.37 −0.36
a
s.p
mm
ri 1853054166 980400 0690412 0 1
tee D 33724959062888234355 305674798384 43596886919596 58 11
h 1 1362566777 777777 7777777 7 8
1andSwhicht 2222s2pE −37.63−37.77−37.78−37.79−37.79−37.79−37.79−37.79−37.79−37.79 −37.79−37.79−37.79−37.79−37.79−37.79 −37.79−37.79−37.79−37.79−37.79−37.79−37.79 −37.79 −37.79
D s
n 1
1 o
3P,ons 7672017710233127356631103025319531743133 319231863181318531833182 3150 31453140314531433142 3153 3144
2bon2palculati bq 0.60.60.60.60.60.60.60.60.60.6 0.60.60.60.60.60.6 0.6 0.60.60.60.60.6 0.6 0.6
cark(cid:101)c 753931199598763183815197401 079529821312521549 288 731023486670698 353 699
the(cid:100)12 ac 0.00.260.590.460.420.440.480.460.470.47 0.470.470.460.450.450.45 0.47 0.470.470.450.450.45 0.47 0.45
forger
dr
mela 36381543873507223637 414339505151 41 4340505252 37 48
perforofthe asd 0.3380.3750.3650.3590.3600.3620.3610.3610.3610.361 0.3610.3610.3610.3610.3610.361 0.361 0.3610.3610.3610.3610.361 0.361 0.361
alculationstheresults 232pPal MCHF 1.691811.685771.702161.704501.704801.704631.704601.704601.704611.70462 CI 1.705001.704951.704741.705101.705081.70507 1.70501 1.704951.704741.705101.705091.70507 1.70505 1.70511
ce 2s
CIar 22
s
ndues 1 418410791926361499598637673695 241643455344282274 263 664476392303295 282 314
CHFaalval Ssms −1.39−0.43−0.40−0.39−0.40−0.40−0.40−0.40−0.40−0.40 −0.41−0.40−0.40−0.40−0.40−0.40 −0.41 −0.40−0.40−0.40−0.40−0.40 −0.41 −0.40
n
Mfi
ee 8483957550 099511 4 34666 3 5
thTh 861309417979200307360388406417 378406417425429430 388 417428434439440 400 452
tsof−3.0 E 37.6837.8237.8337.8337.8437.8437.8437.8437.8437.84 37.8437.8437.8437.8437.8437.84 37.84 37.8437.8437.8437.8437.84 37.84 37.84
ula −−−−−−−−−− −−−−−− − −−−−− − −
Resin
TABLEI.parameterstransferred. modelnlp HF45678k9k10k11k12k 11k99.099.599.799.899.999.95 12k99.099.399.599.799.899.999.95 13l99.0 Final
6
TABLEII.ResultsoftheMCHFandCIcalculationsperformedfortheC− 2p3 4So and2Do terms. TheenergiesE areinE ,
h
theS ina−2 andthehyperfineparametersina−3. Thefinalvaluesaretheresultsofthelarger(cid:100)12k(cid:101)calculationsonwhich
sms 0 0
the impact of the 13th shell and l=8 orbitals have been additively transferred.
model 1s22s22p3 4So 1s22s22p3 2Do
nl p E Ssms ac E Ssms al asd ac bq
MCHF MCHF
HF −37.708844 −1.60530 0.0 −37.642589 −1.54597 2.35963 0.47193 0.0 0.0
4 −37.862042 −0.56521 0.33257 −37.810185 −0.51517 2.27217 0.50749 0.23109 0.10808
5 −37.876688 −0.56168 0.18050 −37.827492 −0.51410 2.28209 0.54068 0.23971 0.12912
6 −37.884109 −0.54166 0.55357 −37.836040 −0.49077 2.25909 0.52516 0.35738 0.15402
7 −37.887227 −0.54675 0.43564 −37.839993 −0.49006 2.23484 0.52138 0.31616 0.18659
8k −37.888691 −0.54785 0.41389 −37.841966 −0.49306 2.22493 0.52273 0.30328 0.19303
9k −37.889449 −0.54912 0.46136 −37.842933 −0.49444 2.21796 0.52136 0.32378 0.19962
10k −37.889853 −0.54975 0.45976 −37.843464 −0.49534 2.21563 0.51911 0.32252 0.20107
11k −37.890085 −0.55017 0.45517 −37.843751 −0.49583 2.21476 0.51910 0.32312 0.20291
12k −37.890213 −0.55042 0.45714 −37.843927 −0.49614 2.21462 0.51922 0.32144 0.20238
CI CI
11k 99.0 −37.890143 −0.54957 0.46140 −37.844634 −0.48786 2.20369 0.51222 0.30714 0.21266
99.3 −37.890301 −0.54774 0.45872 −37.844971 −0.48375 2.20081 0.51100 0.30452 0.21527
99.5 −37.890474 −0.54414 0.44761
99.7 −37.890640 −0.54204 0.43942
99.8 −37.890692 −0.54132 0.43823
12k 99.0 −37.890271 −0.54982 0.46325 −37.844826 −0.48798 2.20269 0.51165 0.30450 0.21262
99.3 −37.890440 −0.54797 0.46098 −37.845165 −0.48386 2.19983 0.51040 0.30176 0.21519
99.5 −37.890603 −0.54438 0.44919
99.7 −37.890769 −0.54227 0.44080
99.8 −37.890822 −0.54151 0.43945
13l 99.0 −37.890429 −0.55003 0.46636 −37.845003 −0.48828 2.20286 0.51161 0.30547 0.21300
Final −37.890980 −0.54172 0.44257 −37.845343 −0.48415 2.20000 0.51036 0.30273 0.21557
TABLE III. Relativistic corrections (µE ) to the total energies evaluated by BPCI calculations (see text).
h
Model C 1s22s22p2 C− 1s22s22p3
nl p 3P 3P 3P 1D 1S 4So 2Do 2Do
0 1 2 2 0 3/2 3/2 5/2
Main spectroscopic terms only (see text)
10k 99 −14 437.25 −14 362.42 −14 239.95 −14 288.85 −14 256.62 −14 200.89 −14 192.92
11k 99 −14 409.01 −14 334.21 −14 211.76 −14 330.16 −14 296.56
11k 99.5 −14 455.75 −14 380.91 −14 258.35 −14 331.96 −14 299.28
With additional spectroscopic terms (see text)
10k 99 −14 437.25 −14 362.63 −14 240.02 −14 288.99 −14 256.62 −14 129.53 −14 201.14 −14 193.15
11k 99 −14 409.01 −14 334.42 −14 211.82 −14 330.30 −14 296.56
7
TABLEIV.RelativisticcorrectionstotheS specificmassshiftparameterofeachconsideredstateevaluatedbyMCHF-RCI
sms
MR-SD calculations. Results are presented in µa−2.
0
model C 2p2 C− 2p3
3P 3P 3P 1D 1S 4So 2Do 2Do
0 1 2 2 0 3/2 3/2 5/2
mono-configurational
RCI −47 −47 −47 −49 −53 −58 −61 −61
DF −876 −560 65 −243 −233 −91 −57 −40
multi-configurational, mono-reference
3 62 190 375 316 343 449 439 442
4 4555 4769 5143 4933 4859 5080 4993 5011
5 3996 4228 4636 4388 4413 4595 4678 4700
6 3837 4067 4473 4259 4376 4230 4131 4154
7 3918 4157 4579 4338 4324 4305 4180 4210
8 3939 4175 4593 4337 4338 4329 4272 4304
multi-configurational, multi-reference
3 98 225 412 351 390 441 442 449
4 4750 4967 5349 5128 5134 5201 5106 5128
5 4167 4402 4817 4559 4633 4713 4794 4820
6 3999 4233 4647 4426 4570 4347 4241 4268
7 4081 4323 4752 4506 4509
final 4102 4341 4766 4506 4523 4446 4382 4418
F. Breit-Pauli calculations Forthereasonsexpressedabove,wealsoperformBPCI
calculations that focus on term mixing. For each LSJ
and active set (cid:100)nl(cid:101) (nl = 4f −12k for C and 4f −8k
A first way to include relativistic effects is to use the
for C−), the MR -I list is merged with the MR-I set
Breit-Pauli Hamiltonian that includes the 1/c2 relativis- 98
obtained using the reference containing all allowed LS
tic correction operators to the non-relativistic atomic
couplings of the 2s → 3d, 2p → 3p and 2s2 → 2p2 exci-
Hamiltonian [27].
tations from the main configuration.
Since the radiative transitions we consider are essen-
tially authorized by L and S mixing, we need to have
a good description of the term mixing. On the other G. Relativistic configuration interaction
hand, itisthecalculationofthescalarrelativisticeffects calculations
thatisneededforestimatingtherelativisticeffectsonthe
electron affinity since the fine structures of the involved We use essentially the same method as in [33]. First,
species are usually known experimentally. We therefore we perform reference MCHF calculations with all single
choose two distinct Breit-Pauli models. The first BPCI and double configuration excitations (SD) of the ground
CSF lists are used for the term separation and detach- stateinactivesetsrangingfrom(cid:100)3d(cid:101)to(cid:100)8k(cid:101). Theresult-
mentthresholdscorrectionswhilethesecondapproachis ing non-relativistic radial orbitals P (r) are then con-
nl
used for the transition probabilities calculations. verted to Dirac spinors using the Pauli approximation
Focusingonthecorrelation,wemergetheMR -I(cid:100)10k(cid:101)
99 P (r)=P (r) (23)
lists of the studied terms for both C− and C. Then we nκ nl
(cid:18) (cid:19)
α d κ
extendthismodelbyaddingtheCSFsinteractingtofirst Q (r)= + P (r) (24)
order with the CSFs 2p3p LS, LS = 3D, 3S, 1P for C nκ 2 dr r nl
and the CSFs 2p2 3p LS, LS = 2Po, 2Fo, 4Do, 4Po where α is the fine structure constant and κ is defined
for C−. For the neutral carbon we test the impact of
(cid:26)
−l−1 when j =l+1/2
additional correlation on the relativistic corrections by κ= (25)
l when j =l−1/2
using the MR -I(cid:100)11k(cid:101) and MR -I(cid:100)11k(cid:101) spaces. We
99 99.5
finally diagonalize the Breit-Pauli Hamiltonian in those
We finally perform the corresponding RCI calculations
CSFspacesusingthecorrespondingactivesetsoptimized
usingthesetofSDexcitationsofthemainconfiguration.
in the non-relativistic calculations. The relativistic cor-
Larger configuration sets are explored by means of non-
rections to the energy are summarized in Table III. We
relativistic CI and RCI calculations using the references
see that the effect of the additional LS mixing on the
energy levels is so small that only the corrections on the MR(C−)={1s22s22p3,1s22s12p33d1}, (26)
fine structures are meaningful. MR(C) ={1s22s22p2,1s22s12p23d1,1s22p4}. (27)
8
TABLE V. Relativistic corrections on A I [kHz/µ ] and B/Q [kHz/barn] of each considered state evaluated by MCHF-RCI
µI N
MR-SD calculations.
model 2p2 3P 2p2 1D 2p3 4So 2p3 2Do
(n ) A B A B A B A B A B A B
max 1 1 2 2 2 2 3/2 3/2 3/2 3/2 5/2 5/2
mono-configurational
RCI 93 2 118 90 −141 −82 2 0.1 69 855 −24 0
DF −158 −10 164 −177 −26 150 −122 −0.2 −464 550 371 0
multi-configurational, mono-reference
3 −221 −51 −52 38 −166 −130 −404 −0.4 −62 1317 −113 26
4 −226 −31 157 −111 −23 61 −301 −0.1 −18 1276 50 24
5 −256 −34 152 −121 −13 70 −322 0.0 −16 1280 68 23
6 −254 −35 163 −123 −3 64 −319 0.2 −19 1356 73 27
7 −279 −39 158 −135 6 71 −357 −0.1 −19 1393 77 36
8 −285 −35 160 −146 11 85 −368 −0.1 −16 1392 81 36
multi-configurational, multi-reference
3 −221 −52 −52 38 −168 −132 −414 −0.4 −66 1267 −117 25
4 −217 −34 166 −106 −24 52 −289 −0.1 −26 1210 52 25
5 −246 −37 162 −115 −14 59 −306 0.0 −25 1218 71 24
6 −242 −38 174 −116 −4 52 −298 0.1 −30 1291 78 28
7 −266 −42 170 −128 6 57
final −272 −38 172 −138 10 71 −347 −0.1 −27 1328 86 38
Therelativisticeffectsareestimatedfromthedifferences
between the non-relativistic CI and corresponding RCI TABLE VI. AµII [kHz/µN] and B/Q [kHz/barn] theoretical
results. values for carbon 3P, 1D and C− 4So, 2Do.
InTableIV,wecomparetherelativisticcorrectionson C C−
Ssms obtained with our calculations to the ones deduced state A I B/Q state A I B/Q
fromsingleconfigurationcalculations(RCI–HFandDF– µI µI
Non-relativistic
HF). It seems that correlation plays an important role
in the estimation of these corrections, as could be ex- 2p2 3P 2 296 74 184 2p3 4So 9 394 0
1 3/2
pected from an operator that measures the correlation 2p2 3P 105 883 −148 367 2p3 2Do 53 836 −35 455
2 3/2
between the momenta of the electrons. The convergence 2p2 1D 156 317 283 206 2p3 2Do 107 317 −50 650
2 5/2
ofthemono-referenceapproachwithnissufficient. How- + relativistic corrections
ever,weseealargechangebetweenthemono-andmulti-
reference approaches. In Table V, we present the correc- 2p2 3P 2 024 74 145 2p3 4So 9 048 0
1 3/2
tionsforA I andB/Qthatarebothindependentofthe 2p2 3P 106 055 −148 505 2p3 2Do 53 809 −34 128
nuclear spinµII and multipole moments (µI,Q). 2p2 1D22 156 327 283 276 2p3 2D35o//22 107 403 −50 613
Similarly to the non-relativistic calculations, we note
that for neutral carbon, the impact of the 7th and 8th
shellsisnotmuchaffectedbythechoiceofreference. We perfine structures of the carbon ground state of 11C and
then estimate the final value as in the non-relativistic 13C,respectively. Inthelatterarticle,amagneticdipole-
case. moment of 11C of −0.964(1) µ was deduced from the
N
then available µ(13C) value. We update this estimation
byusingthemodernµ(13C)value[47]combinedwiththe
III. RESULTS AND COMPARISON TO two measured A(3P2) constants:
EXPERIMENT
(cid:18)A (11C) I µ(13C)(cid:19)
µ(11C)= J 11 =−0.9642(2) µ .
A. Hyperfine Structures A (13C) I N
J 13 exp
(28)
In this work, we focus on the isotopes 13 and 11 of Theerroronthisvalueisnowdominatedbytheaccuracy
carbon, respectively of nucleus spin 1/2 and 3/2. The of the A(3P2) hyperfine constants measurements.
11C nucleus decay into 11B by e+-emission with a half- As mentioned in Section IIE, it is difficult to have a
lifetime of 20.4 minutes. Haberstroh et al. [46] and Wol- rigorous estimation of the uncertainty on the hyperfine
ber et al. [44] performed experimental studies of the hy- parameters. Wehoweveradvancealearnedguessoftheir
9
TABLE VII. Comparison of our calculated hyperfine constants of 13C to other works. The experimental values are adjusted
according to our analysis of the off-diagonal JJ(cid:48) interaction. All values are in MHz.
13C 13C−
A (3P) A (3P) A (1D) A (4So) A (2Do) A (2Do)
1 2 2 3/2 3/2 5/2
Original exp.a 2.838(17) 149.055(10)
This work 2.84 148.99 219.61 12.71 75.59 150.88
Prev. workb 2.28 148.1
reliability. First, we see in Table I that the integrals a , are involved, one must diagonalize the matrix
l
a andb ofCchangelessthan0.05%aftertheaddition
ofsdthe (cid:100)13ql(cid:101) correction. The a parameter of the 3P is (cid:18) 0 W(JJ(cid:48);F) (cid:19)
c (29)
only slightly more affected (∼ 0.1%). These effects are W(JJ(cid:48);F) ∆JJ(cid:48)E(LS F)
representative of the accuracy of our results for neutral
carbon. In the case of C− we face two additional limita- where ∆JJ(cid:48)E(LS F) = E(LSJ(cid:48)F)−E(LSJF) is dom-
tions: the structure of the negative ion converges more inated by the fine structure splitting and W(JJ(cid:48);F) is
slowly and we are limited in our expansions. Moreover, governed by the off-diagonal hyperfine constants – here
only the most troublesome contact term is responsible AJ,J−1, BJ,J−1 and BJ,J−2 (see Section IIB). The off-
for the non-relativistic HFS of 2p3 4So. For these rea- diagonalelectricquadrupoleinteractionisnegligibleand,
sons, and comparing the values of Table II with results at the non-relativistic level, we obtain for C(3P)
obtainedwiththeactiveset(cid:100)10k(cid:101),wemustallowforrel-
IA /µ =50.47 MHz/µ , (30)
ative uncertainties on the HFS parameters roughly ten 1,0 I N
times larger for C− than for C. IA2,1/µI =62.71 MHz/µN, (31)
(32)
In Table VI, we present the non-relativistic A I and
B/Qresultscalculatedusingthefinalvaluesofa ,µaI ,a while for C−(2Do) we have
l sd c
and b of Tables I and II. In the same table, we add
q
the relativistic corrections of Table V to those values. IA5/2,3/2/µI =34.20 MHz/µN. (33)
TheA(3P )constantistheplaceofseverecompensations
1
The hyperfine interaction between states belonging to
between the orbital (a ) and spin-dipole (a ) contribu-
l sd
different terms is negligible.
tions so that the uncertainties on those sum up to an
error of the order of 102 kHz/µ . The other nuclear- Wolber et al. [44] measured two hyperfine splittings in
N the 13C 3P multiplet, allowing the determination of the
parameters-independent hyperfine constants of neutral J
A andA diagonalconstantsbutnotoftheoff-diagonal
carbon suffer of a non-relativistic uncertainty of about 1 2
10−102 kHz/µ . Thesearelargerthanthefluctuations constants so that they had to deduce the contribution of
observed in TabNle V. As far as C− is concerned, on the the JJ(cid:48)–interaction theoretically. The level shifts that
onehandthe4So hyperfinestructureisessentiallydueto they obtained from their computations are significantly
higherthanours. However, theA constantsthatrepro-
the contact term, itself arising only from correlation ef- J
fects,andontheotherhand,the2Do hyperfineconstants duce the experimental hyperfine splittings when using
our results for the JJ(cid:48)-interaction, A =2.829(17) MHz
aresmallbuttheachievedconvergenceofthecalculations 1
and A = 149.052(10) MHz, do not differ largely from
islessgood. Thereforetherelativenon-relativisticuncer- 2
taintiesonC−hyperfinestructuresarelargerastheysum the experimental constants presented in Table VII.
Haberstroh et al. [46] measured three hyperfine split-
up to about 50−100 kHz/µ (kHz/barn). We conclude
N tings for 11C 3P , which is insufficient for determining
that the reliability of all normalized hyperfine constants J
is of the order of 102 kHz/µ (kHz/barn) with the ex- allfourAJ andBJ ofthisterm. Hencetheydeducedthe
N
ception of the B(4So) that is certainly negligible. value of B1 from the relation B2/B1 =−2 which is only
valid in the Hartree-Fock model. From Table VI, we see
Our results are compared with observations in Ta- that this formula holds very well at the non-relativistic
bles VII and VIII for C and C− respectively. We ob- level but that, including relativistic corrections, we have
serveagoodagreementwithexperiment,betterthanex-
pected from the above discussion. This represents a sig- B2/B1 =−2.0029. (34)
nificant improvement compared to the theoretical study
TheeffectoftherefinedB /B ratioandJJ(cid:48)-interactions
of J¨onsson et al. [45]. 2 1
canceleachotherintheestimationofthediagonalhyper-
Theobservedhyperfinesplittingsarisefromthediago- fine constants so that the resulting A constants do no
J
nalhyperfineinteraction,parametrizedbytheA andB differ significantly from the experimental ones quoted in
J J
constants, and, to higher order, from the non-diagonal Table VIII. For the electric quadrupole interaction, the
(JJ(cid:48)) interaction of states of same F. If only two levels accuracy of our results is such that we can safely update
10
TABLE VIII. Comparison of our calculated hyperfine constants of 11C to other works. The experimental values are adjusted
according to our analysis of the off-diagonal JJ(cid:48) interaction and of B(3P )/B(3P ). All values are in MHz.
1 2
11C
A (3P) B (3P) A (3P) B (3P) A (1D) B (1D)
1 1 2 2 2 2
Original exp.a −1.308(24) 2.475(14) −68.203(7) −4.949(28)
This work −1.30 2.474 −68.17 −4.955 −100.49 9.450
11C−
A (4So) B (4So) A (2Do) B (2Do) A (2Do) B (2Do)
3/2 3/2 3/2 3/2 5/2 5/2
This work 5.82 ≈0 34.59 −1.139 69.04 −1.688
the electric quadrupole moment of the 11C nucleus with Our systematic procedure is not particularly efficient
the formula for predicting the negative ion binding energy. In par-
ticular, for the 4So detachment threshold, the coupled-
(cid:0)B (11C)(cid:1)
Q(11C)= 2 exp. (35) cluster approaches are much more impressive [15, 16].
(B /Q) The recent value of Klopper et al. [16] indeed achieves a
2 th
sub-meV (< 8 cm−1) agreement with the experimental
UsingtheB constantofHaberstrohetal.[46],weobtain electron affinities for all first- and second-period atoms
2
a value of +0.03333(19) (2) barns but if we use our (H-Ne). A similar accuracy had already been achieved
exp th
theoreticalparametersintheanalysisoftheobservations, more than 10 years before by de Oliveira et al. [15] for
we obtain the second and third period p−block atoms.
By trying various extrapolation schemes on our C−
Q(11C)=+0.03336(19)exp(2)th barns. (36) calculations, we explain up to ∼ 20 cm−1 of the differ-
ence between our calculation of the 4So binding energy
This value is used for estimating the theoretical BJ con- and the experimental value (about 5 cm−1 for each n
stants of this work presented in Table VIII. The differ- and l extrapolations, and about another 10 cm−1 for
encebetweentheoryandexperimentforthe B constant
2 the extrapolation to a complete active set). Turning
followsdirectlyfromthefactthat(36)includestherefine-
to the relativistic effects calculations, we see that the
ments of the theoretical parameters needed in the anal-
scalarcontributionscalculatedwiththeCCmethodsgive
ysis of the observed hyperfine splittings. −21.54 cm−1 [16] and −22.83 cm−1 [15] while we obtain
Let us mention the previous calculations of the bq pa- −37.95 cm−1. The extrapolation being reliable to about
rameter (we get bq = 0.6314 a−03): bq = 0.6325 a−03 [48], a couple of tenth of percents and since the additional
b = 0.6319 a−3 [31]. Using the experimental constant expected contributions are of the order of the cm−1, we
q 0
B quoted in Table VIII, Sundholm and Olsen [48] pro- conclude that our BPCI relativistic corrections are still
2
posed Q(11C) = +0.03327(24) barns which would only unbalanced. The problem of our relativistic corrections
tenuously agree with our estimation if the (B (11C)) on the detachment thresholds is confirmed by the fact
2 exp
value was to be improved. that, looking to Table III, they are not well converged.
In the case of C−, the small 2Do fine structure Aside a possible unbalance in the relativistic effects
(1.75cm−1,seebelow),leadstoJJ(cid:48)-interactionshiftson estimation, our error is roughly proportional to the cor-
the energy levels that are roughly 10 times larger than relation contribution. We see that the differences be-
in the neutral atom ground term, i.e. of the order of tween the HF and experimental energy separations (see
0.1 MHz. TableIX)arereproducedto∼0.1−0.7%,whichisabout
thepercentageoftheC(3P)correlationenergyweget. It
meansthatpandnaregoodindicatorsofthepercentage
B. Energy differences of the correlation effects included in a model. However,
theuncertaintyonourrelativisticcorrectionsandonthe
TableIXpresentsseveralcalculatedenergyseparations 2Do missing correlation is too large for an extrapolation
and compares them to other works. based on this observation to be useful, e.g. for improv-
Our C− term splitting is in very good agreement with ingtheexperimentaldeterminationofthepositionofthe
experiment but, as will be seen below, this is partially 2Do levels. Indeed,a0.5%uncertaintyonourcalculated
J
accidental. correlation energies, which is no overestimation, reflects
Our results on the neutral atom ground configuration in corrections ranging from ∼60 cm−1 in the case of the
level spacings are systematically better than the ones of largest HF-experiment discrepancy, to about 7.5 cm−1
Froese-FischerandTachiev[49]. Itindicatesthat,inthis forthe2Do−1S threshold,i.e.ofthesameorderofmag-
context, our relativistic corrections are reliable. For the nitude than the experimental uncertainty.
3P finestructure, weobtainas accurateresultsas recent Wewouldliketostressanotheradvantageofusingthe
fully relativistic calculations [50]. number of correlation layers n and p as parameters for