Electron Electric Dipole Moment and Hyperfine Interaction Constants for ThO

0.15 MB·

by  Timo Fleig

4
1
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2 Electron Electric Dipole Moment and Hyperfine Interaction Constants
for ThO
r
a
M
Timo Fleig1 and Malaya K. Nayak2
8 1Laboratoire de Chimie et Physique Quantiques,
2 IRSAMC, Universit´e Paul Sabatier Toulouse III,
118 Route de Narbonne, F-31062 Toulouse, France
] 2Bhabha Atomic Research Centre, Trombay, Mumbai - 400085, India
h
(Dated: March 31, 2014)
p
- A recently implemented relativistic four-component configuration interaction approach to study
m
P- and T-odd interaction constants in atoms and molecules is employed to determine the electron
o electric dipole moment effective electric field in the Ω=1 first excited state of the ThO molecule.
at We obtain a value of Eeff = 75.6(cid:2)GcmV(cid:3) with an estimated error bar of 3% and 10% smaller than a
. previouslyreportedresult[arXiv:1308.0414 [physics.atom-ph]]. Usingthesamewavefunctionmodel
cs weobtainanexcitationenergyofTvΩ=1=5329[cm−1],inaccordwiththeexperimentalvaluewithin
2%. In addition, we report the implementation of the magnetic hyperfine interaction constant
i
s A as an expectation value, resulting in A = −1335 [MHz] for the Ω = 1 state in ThO. The
y || ||
smallereffectiveelectricfieldincreasesthepreviouslymeasuredupperboundtotheelectronelectric
h
dipolemomentinteractionconstant[arXiv:1310.7534v2[physics.atom-ph]]andthusmildlymitigates
p
constraints to possible extensions of theStandard Model of particle physics.
[

3
v INTRODUCTION transition energy and Eeff is the internal elec-
4 tric field at the position of the electron, giving
8
Polardiatomicmoleculesarepromisingcom- rise to a dipolar interaction with the electron’s
2
2 plex systems [1, 2] in search of the electric postulated EDM. Eeff cannot be measured ex-
. dipole moment (EDM) of the electron. The perimentally but has to be determined from
1
measurementof a non-zeropermanent molecu- atheoreticalelectronic-structurecalculationon
0
4 lar EDM and the establishment of its origin in the respective molecule in the respective quan-
1 fundamental charge (C) and spatial parity (P) tum state. On the one hand, large Eeff is a se-
: violating interactions [3], for example inducing lection criterion for systems with a large EDM
v
i anelectronEDM,wouldbe a signatureof New interaction and therefore holding promise for
X Physics beyond the Standard Model (SM) of the electron EDM to actually be found. Sec-
ar elementary particles [4]. ond, an accurate value of Eeff is required for
The thorium monoxide (ThO) molecule has reliably constraining the parameter ranges of
been found to be one of the most interesting New Physics models going beyond the SM [9].
candidates in this quest [5, 6], among other Greatcareistakeninassessingandminimizing
aspects due to its very large internal effec- errors in the determination of ∆Et. It is obvi-
tive electric field E on unpaired electrons ous that the same care should be taken in the
eff
[7]. Recently, the ACME collaboration re- theoretical assessment of the effective electric
ported an order of magnitude smaller upper field Eeff.
bound to the electron EDM interaction con- Previous calculations of E in the relevant
eff
stant, |d | < 8.7×10−29ecm, obtained from a “science” state Ω = 1, arising mainly from the
e
spin-precessionmeasurementonapulseofThO configuration7s16d1(Th2+O2−),havebeenre-
molecules [8]. This upper bound for d is prin- portedbyMeyeretal. [7]andSkripnikovetal.
e
cipally determined from Eq. (1), [10]. Intheformer[7]E =104 GV hasbeen
eff cm
obtained based on a semi-empirical model cal-
∆Et (cid:2) (cid:3)
d = (1) culation which in part employs non-relativistic
e
E
eff approximations and a very limited set of elec-
where ∆E is an upper bound to a measured tronic configurations. The latter study [10] de-
t
termines E = 84 GV by means of a two- position~r of an electron in an atom is [14]
eff cm
component relativistic single-reference coupled
(cid:2) (cid:3)
cluster (CC) approach. ~µ ×~r
A~ = K (3)
Theworkpresentedinthispaperis aimedat r3
an accurate determination of E in the Ω=1
eff
excited state of the ThO molecule, along with we can derive the parallel magnetic hyperfine
a clarification of which physical aspects play interaction constant A|| as the z projection of
the decisive role in obtaining this quantity re- theexpectationvalueofthecorrespondingper-
liably. In addition, we investigate the excita- turbative Hamiltonian in Dirac theory
tion energy of the Ω=1 state and the parallel
hyperfine coupling constant which is regarded µ n α~ ×~r
Th i i
A = (4)
as a measure of the quality of molecular wave- || IΩ r3
functions employedinthe determinationofrel- *Xi=1 (cid:18) i (cid:19)z+ψ
ativistic EDM enhancement factors.
where I is the nuclear spin quantum number,
α isa Hamiltonian-formDiracmatrixforpar-
k
ticlek,andnisthenumberofelectrons. Again,
THEORY
we evaluate Eq. (4) over the CI wavefunction
for the state ψ .
Ω=1
Electron EDM Hamiltonian
The potential energy due to the electron
APPLICATION TO THO
EDMinteractioninthe molecule is determined
as an expectation value [11] over the one-body
Hamiltonian Hˆ as follows Technical Details
edm
N N
General Setup
Hˆ (j) = −d γ0 Σ ·E
edm e j j
* + * +
Xj=1 ψ Xj=1 ψ For the determination of the nuclear hyper-
N
= 2ıcde γ0γ5 p~ 2 (2), fine coupling constant we use the thorium iso-
e~ j tope 229Thfor whichthe nuclearmagnetic mo-
* +
Xj=1 ψ ment has been determined to be µ = 0.45µ
N
[15]. Its nuclear spin quantum number is I =
where N is the number of electrons, γ are
5/2. In all calculations the speed of light was
the standard Dirac matrices, d is the electron
e ~σ 0 set to 137.0359998a.u.
EDM interaction constant, Σ= with
0 ~σ
~σ the vector of Pauli spin matric(cid:18)es, and(cid:19)E is
j
the electric field at the position of an electron Atomic basis sets
(j). The wavefunction ψ is determined from
relativistic4-componentConfigurationInterac-
Fully uncontracted atomic Gaussian basis
tion (CI) theory [12] for the ψ first elec-
Ω=1 sets ofdouble-ζ,triple-ζ andquadruple-ζ qual-
tronically excited state of the ThO molecule,
ity were used for the description of electronic
using the all-electron Dirac-Coulomb Hamilto-
shells. For thorium we used Dyall’s basis
nian. Details on the implementation ofEq. (2)
sets [16, 17] and for oxygen the Dunning
can be found in reference 13. The optimized
cc-pVNZ-DK sets [18] with N ∈ {2,3,4},
coefficients in the linear expansion of ψ in the
as well as the aug-cc-pVTZ-DK set [18].
basis of Slater determinants over 4-component
For thorium all 5d,5f,7s,6d correlating ex-
Dirac spinors contain the approximate effects
ponents were added to the basic n-tuple-
ofelectroncorrelationsamongtheelectronsex-
ζ sets, amounting to {26s,23p,17d,13f,1g}
plicitly treated in the CI expansion.
uncontracted functions in case of double-
ζ (in the following abbreviated as vDZ),
{33s,29p,20d,14f,4g,1h}in the case of triple-
Magnetic Hyperfine Interaction Constant ζ (vTZ) and {37s,34p,26d,17f,8g,4h,1i} in
the case of quadruple-ζ (vQZ), respectively.
Since the magneticvectorpotentialA~ due to Thelattersetinadditioncontainsall6s,6pcor-
themagneticmoment~µ ofanucleusK atthe relating exponents.
K
2
Molecular Wavefunctions
FIG. 1: Generalized Active Space models for ThO
CI wavefunctions. The parameters m,n,p and q
Molecular calculations were carried out with are defined in the text and determine the occupa-
amodifiedlocalversionoftheDirac11program tionconstraintsofthesubspacesofKramers-paired
package [19]. Optimized molecular spinors spinors. Themolecularspinorsaredenotedaccord-
ing to their principal atomic character. The space
have been obtained using the Dirac-Coulomb
with 183−K virtualKramerspairs (forvTZbasis
Hamiltonian and all-electron four-component
sets) is comprised by all canonical DCHF orbitals
Hartree-Fock calculations. The basic model
below an energy of 38 EH.
used for these open-shell calculations is based
on an average-of-configuration Fock operator
accumulated
for two electrons in the Th (7s,6dδ) Kramers # of Kramers pairs
# of electrons
pairs with all other (88) electrons restricted min. max.
to closed shells. This model, called (av.2in3),
Deleted (176)
denoting an averaging with 2 electrons in 3
Kramers pairs in the following, is appropriate
for the region close to the equilibrium bond
distance of the molecule where the dominant Virtual 183−K 36 36
configurations correspond to the system Th2+
O2− [20]. The open-shell averaging ensures a Th: 6d σ π ,7p, 8s
balanceddescriptionofthe low-lyingelectronic Th: 7s, 6dδ K 36−m 36
states ofinterestinthis study. In afew models
usingthesmallestbasisset(double-ζ)the6dπσ Th: 6s, 6p 8 34−n 34
and7pshellsofThwereincludedduetopartial O: 2s, 2p
mixingwiththeTh6dshell,defining(av.2in9). Th: 5d 5 18−p 18
For the larger basis sets we have restricted the
Th: 5s, 5p 4 8−q 8
open-shell averaging to (av.2in3).
We exploit a Generalized Active Space Frozen (31)
(GAS) concept for defining CI wavefunctions core
of varying quality. Figure 1 shows the parti-
tioning of the space of Kramers-paired spinors
into seven subspaces, five of which are active
for excitations. Based on this
and K =10 which adds energetically low-lying
partitioning and four parameters (m,n,p,q)
σ-type spinors to this space.
which define the accumulated occupation con-
The different wavefunctionmodels are inad-
straints of the subspaces, we choose four dif-
ditiondefinedbythenumberofcorrelatedelec-
ferent CI wavefunction models for our calcula-
trons in total(in parenthesis)and the included
tions:
excitation ranks, where “SDT” stands for Sin-
gle, Double, and Triple excitations, as an ex-
ample. The value of a parameter, e.g. p = 2,
parameter values correlationmodel label
denotes the maximum hole rank of the respec-
m=2,n=2,p=0,q=0 MR -CISD(18)
K tiveactivespace. Inthatparticularexampleall
m=3,n=2,p=0,q=0 MRK-CISDT(18) Slater determinants with zero up to two holes
m=2,n=2,p=2,q=0 MR -CISD(28) in the Th (5d) space would be included in the
K
m=2,n=2,p=2,q=2 MR -CISD(36) wavefunction expansion.
K
The parameter K (see Fig. 1) has been in-
troducedtodefinevariableactivevalencespinor Electronic-structure results
spaces. K = 3 includes only the Th (7s,6dδ)
spinorsinthefourthactivespaceandthuscom- We first establish molecular wavefunctions
prises a minimal model for a balanced descrip- whichaccuratelydescribetheexcitationenergy
tion of the ground Ω = 0 state and the ex- of the Ω = 1 electronic state. Table I displays
cited Ω ∈ {1,2,3} states which derive from vertical excitation energies as a function of ba-
3∆ in the Λ-S coupling picture. We have fur- sis set. The vTZ set leads to a large correc-
thermore used K = 5 which adds two π-type tion of −15%, whereas the vQZ set only yields
spinors to the fourth space, K = 7 adding an- another −4%, less than 200 cm−1 on the ab-
other two π-type spinors, and finally K = 9 solute. We therefore continue our investigation
3
with the set of vTZ quality.
TABLEI:Verticalexcitationenergy,effectiveelec-
The next criterion we consider is the elec-
tricfield,andhyperfineconstantataninternuclear
tronic shells included in the explicit treatment
distance of R=3.477 a0 for Ω=1 usingbasis sets
of dynamic electroncorrelation. In Table II we withincreasingcardinalnumberandthewavefunc-
compileresultsfromonly2correlatedelectrons tion model MR3-CISD(18)
(Th(7s,6d)shells)upto38correlatedelectrons
(Th(5s,5p,5d,6s,6p,6d,7s),O(2s,2p)shells).
Whereas correlations among the valence elec- Basis set/CI Model Tv [cm−1] Eeff(cid:2)GcmV(cid:3) A|| [MHz]
trons of both atoms are seen to be important, vDZ/MR3-CISD(18) 4535 80.8 −1283
core-valence and core-core correlations change vTZ/MR3-CISD(18) 3832 81.0 −1292
the excitation energy by only −3%, on the or- vQZ/MR3-CISD(18) 3643 80.7 −1298
der of −100 cm−1. Exp. (Te)a 5317
As a third criterion we take the size and
structure of the active spinor space into ac- aReference[20]
count. X givingthenumberofKramerspairsin
theactivespace,theexcitationenergiesforfour
exponent of the added function. In the work
different models are comparedin Table III.We
of Skripnikov et al. [10] diffuse functions are
observe that increasing the active space leads
used in the atomic basis set for the oxygen
to non-negligible corrections. In particular the
atom, since the dominant contribution to the
last step, from X =7 to X =10, where σ-type
electronic states in question arise from Th2+
spinorsareaddedtothe activespace,provesto
O2− configurations[20],andthereforesuchdif-
be important. It is only at this level that the
fuse functions may affect the present results.
vertical excitation energy becomes satisfacto-
ReplacingtheoxygenvTZbasissetbytheaug-
rily accurate as compared to the experimental
cc-pVTZ-DKsetleadstoachangeofE ofless
value of T = 5317 cm−1. Theoretical studies eff
e than0.01%andreducesT forΩ=1byonly32
haveshown[20]thatthedifferencebetweenthe v
cm−1. We therefore conclude that the vTZ ba-
equilibrium bond lengths in the Ω=0 and the
sis set on the oxygen atom without additional
here considered Ω = 1 excited state amounts
diffuse functions yield sufficiently accurate re-
to only 0.03 a.u. From this result we infer a
sults.
small non-parallelity correction on the order of
It has been argued that the EDM effective
−100 cm−1 for our vertical excitation energy
electric field is predominantly a core property
determined at the experimental minimum of
[21] and thus could be sensitive to the corre-
the ground state potential-energy curve.
lated movement of the inner-shell electrons of
the respective heavy atom. Table II suggests
that these correlationcontributions
Electron EDM and Hyperfine Interaction
Constants
TABLEII:Verticalexcitationenergy,effectiveelec-
tricfield,andhyperfineconstantataninternuclear
Having studied the quality of the wavefunc-
tionindescribingthe excitedstate ofrelevance distanceofR=3.477a0 forΩ=1correlatingonly
the atomic valence shells down to including core-
totheelectronEDMmeasurement,weturnour
valenceandcore-corecorrelationandusingthevTZ
attention to the determination of the effective basis sets
electric field and the hyperfine interaction con-
stants in this state. CI Model Tv [cm−1] Eeff(cid:2)GcmV(cid:3) A|| [MHz]
The results in Table I show that E is vir- MR-CISD(2) 5929 68.5 −1264
eff
tually insensitive MR3-CISD(18) 3832 81.0 −1292
to the size of employed atomic basis sets for MR3-CISD(28) 3752 80.0 −1297
ThO. The hyperfine interaction constant A||, MR3-CISD(36)a 3742 80.8 −1287
changes by hardly more than 1% in magnitude Exp. (Te)b 5317
when increasing the basis set cardinal number
from2to4. Sinceincreasingthe cardinalnum- aDue to extreme computational demand the virtual
berimprovesthesestandardbasissetspredom- cutoffis5a.u. here.
inantly in the outer core and valence atomic bReference[20]
regions,wehavealsotestedtheeffectofadding
steep functions to the thorium atomic core. It arenegligible,atleastforthe ThOmolecule.
is observed that E changes no more than by Whereas the inclusion of only the two valence
eff
0.03%, depending on the respective Gaussian electronsisinsufficientforanyoftheproperties
4
discussed here, E and the hyperfine coupling
eff
TABLE III: Vertical excitation energy, effective
constant prove to be sufficiently converged al-
electric field, and hyperfine constant at an inter-
ready at the valence level of 18 correlatedelec-
nuclear distance of R = 3.477 a0 for Ω = 1 using
trons. As we have previously discussed for the the vTZ basis set and varyingactive spinor spaces
HfF+ molecularion[13]thiscanberationalized
as follows: CI Model Tv [cm−1] Eeff(cid:2)GcmV(cid:3) A|| [MHz]
Inorbitalperturbationtheory,thefirst-order MR3-CISD(18) 3832 81.0 −1292
corrected expression for a given spinor ϕi is MR5-CISD(18) 4054 79.7 −1291
MR7-CISD(18) 4321 80.1 −1318
ϕ ≈ϕ(0)+ ϕ(k0)|Vˆ|ϕ(i0) ϕ(0) (5) MR10-CISD(18) 5329 75.6 −1335
i i Dε(0)−ε(0)E k Exp. (Te)a 5317
kX(6=i) i k
aReference[20]
where ϕ(0) is the mth unperturbed spinor, ε(0)
m m
the corresponding spinor energy and Vˆ is the
electron correlation fluctuation potential. In 75.6 GV . This reduction is accompaniedby a
thepresentcaseweareinterestedinthechange cm
striking improvement of the vertical excitation
of the valence spinor ϕ which is the domi- (cid:2) (cid:3)
7s energy of the Ω = 1 state to a value in ex-
nant contributor to the electron EDM expec-
cellentagreementwiththeexperimentalresult,
tation value, Eq. (2). If a spinor ϕ(k0) is a(n) even after applying the non-parallelity correc-
(outer)corespinor,the energydifferenceinthe tion discussed earlier. The hyperfine constant
denominator will become very large and thus exhibitsanincreaseofslightlymorethan3%in
the correlation contribution to the form of the magnitude when increasing the size of the ac-
valence spinor will be very small. As an ex- tivespinorspacefromMR toMR whichisa
3 10
ample, we consider ϕ(k0) = ϕ1s and ϕ(i0) = significantly larger change than those observed
ϕ7s. The fluctuation potential matrix element for different basis sets and varying number of
ϕ(0)|Vˆ|ϕ(0) is on the order of E , whereas correlated electrons.
k i H
Dthe energy dEenominator becomes ε(0) −ε(0) ≈ In view of the significant discrepancy of the
i k presentMR -CISD(18)resultforE fromthe
(−0.05+4058.5)E = 4058.45E . The per- 10 eff
H H valuereportedbySkripnikovetal. itisinstruc-
turbationcoefficientoftheϕ spinortotheϕ
1s 7s tive to discuss the different wavefunction mod-
spinor is therefore strongly suppressed. This
els thathaveledto theseresults. Thefactthat
analysis is clearly confirmed by our results in
an increase of the size ofthe active space has a
Table II. Both E and A are largely unaf-
eff || noticeable effect ona propertyof the molecule,
fected by including more than the 18 valence
hereE andA ,pointstothe importanceofa
electrons in the explicit treatment of electron eff ||
certainclassof higherexcitations inthe molec-
correlation. Evenwithouthavingtakenintoac-
ular wavefunctionexpansion,here suggesting a
counttheinnermostelectronicshellsofthetho-
multi-reference character of the Ω = 1 excited
riumatominthecorrelationtreatment,wecan
state. In order to gain more insight we have
conclude that such correlations, due to the in-
carried out additional studies including excita-
creasing magnitude of the energy denominator
tion ranks higher that Doubles into the virtual
in Eq. (5), will lead to negligble contributions
spinor space, see Table IV, which
to the wavefunction of valence spinors.
We now turn our attention to wavefunction
models built from active spinor spaces of vari-
TABLE IV: Vertical excitation energy, effective
able size, the results for which are given in Ta- electric field, and hyperfine constant at an inter-
ble III. The minimal active space (MR3-CISD) nuclear distance of R = 3.477 a0 for Ω = 1 using
yields a value of E = 81.0 GV which is thevDZbasissetandvaryingmaximumexcitation
eff cm
quiteclosetothemostelaborateresultofSkrip- rank
(cid:2) (cid:3)
nikov et al. [10] which is 84.0 GV . As ex-
pected, the augmentation of the camctive spinor CI Model Tv [cm−1] Eeff(cid:2)GcmV(cid:3) A|| [MHz]
space with spinors of different s(cid:2)ymm(cid:3)etry rep- MR3-CISD(18) 4535 80.8 −1283
resentation than ϕ does not significantly af- MR9-CISD(18) 5703 73.8 −1321
7s
fect E for Ω = 1. However, upon including MR3-CISDT(18) 5166 74.4 −1340
eff
the σ-type spinors in the active space, yielding Exp. (Te)a 5317
themodelMR -CISD(18),weobserveastrong
10
drop of the effective electric field to a value of aReference[20]
5
was computationally feasible for the small- CONCLUSION
est employed basis sets, vDZ. Again, the in-
clusion of higher excitations leads to the char-
By means of a careful study of effects due
acteristic drop in E , here by ≈ −6.5 GV
eff cm to basissets, dynamic electroncorrelation,and
(MR -CISDT(18) relative to MR -CISD(18)).
3 3 (cid:2) (cid:3) active spinor spaces within a rigorously rela-
Interestingly, nearly the same decrease is ob-
tivistic all-electron four-component formalism,
servedwhenconfigurationsincludingthreepar-
we have determined the EDM effective electric
ticlesinthevirtualspace,v3,areexcludedfrom
field for the Ω = 1 first electronically excited
the CI expansion, model MR -CISD(18), and
9 state at the experimental internuclear distance
thehigherexcitationsarerestrictedtoasubset
fortheThOmoleculargroundstate. Weobtain
involving additional σ-type spinors, i.e., con-
a value of E = 75.6 GV with an estimated
figurations of the type σ1v2. We note that eff cm
errorbarof3%and10%smallerthantheresult
the Ω = 1 excitation energy behaves in very (cid:2) (cid:3)
previously reported by others [10]. With the
similar way, with higher excitations involving
same wavefunction model we have obtained an
the enlarged active spinor space playing the
excitation energy for the Ω = 1 state of T =
major role and increasing the excitation en- v
5329cm−1,inexcellentagreementwiththe ex-
ergymarkedly(theovershootingvalueforMR -
9 perimental result which confirms the quality of
CISD(18) is due to basis set incompleteness,
the molecular wavefunctions employed in the
as the results in Table I confirm). Also for
present study of P and T violating effects in
A we observe a notable dependence on higher
|| ThO. The magnetic hyperfine interaction con-
excitations in the wavefunction, to a large de-
stant is obtained to A = −1335 [MHz] us-
gree coveredby the augmentationof the active ||
ing our most reliable wavefunctionmodel. Our
spinor space. Again, the most elaborate result
detailed analysis shows that the ThO Ω = 1
ofSkripnikovetal. (−1296[MHz]inourunits)
state hasastrongmulti-referencenaturewhich
is very close to our result using the insufficient
mustbeadequatelyaccountedforinelectronic-
active spinor space (−1292 [MHz]).
structure theoretical treatments.
Apparently, it is configurations involving Our result for Eeff has consequences for the
excitations into the augmenting active space recently determined upper bound to the elec-
spinors, combined with correlation excitations tron EDM interaction constant of |de| < 8.7×
into the virtual space that are of major impor- 10−29ecm [8]. This upper bound has been ob-
tance for obtaining an accurate value for Eeff, tainedbasedonEeff =84 GcmV [10]. DuetoEq.
and not the higher excitations such as v3 pro- (1) a 10% smaller value for Eeff requires a cor-
(cid:2) (cid:3)
vided by the Coupled Cluster (CC) model of responding adjustment of the upper bound for
Skripnikov et al. The CC model of the latter de toalargervalue. Consequently,theelectron
authors, in turn, is a standard single-reference EDMconstraintto models extending the Stan-
CC expansion which is applied to the property dardModelofelementaryparticlesissomewhat
of an electronically excited state which clearly attenuated.
exhibits a significant multi-reference charac- In ongoing work we are studying potential
ter, as our results demonstrate. In addition, energy curves for ThO along with dipole and
Skripnikov et al. use ground-state 1Σ0 spinors transition dipole moments for relevant molecu-
and the same state as the Fermi vacuum for lar electronic states. Furthermore, we are con-
the CC expansion which biases the wavefunc- tinuing with the implementation of further op-
tion towards this ground state. This is con- erators of importance in the search for P and
firmed by the too high excitation energy of T violating effects in the universe, in partic-
the Ω = 1 state obtained by those authors ular the scalar-pseudoscalarP,T-odd electron-
(5741 cm−1) as compared to the experimen- nucleon interaction.
tal value of 5317 cm−1. In contrast to this
result, our best model vTZ/MR -CISD(18)
10
which is in addition based on configuration-
averagedspinors yields an excitation energy in
much better agreement with the experimental [1] E.R.Meyer,J.L.Bohn,andM.P.Deskevich.
Phys. Rev. A, 73:062108, 2006.
value, and, furthermore, shows that the ade-
[2] J. J. Hudson, D. M. Kara, I. J. Smallman,
quatedescriptionofallrelevantphysicaleffects
B. E. Sauer, M. R. Tarbutt, and E. A. Hinds.
results in a value of E = 75.6 GV , 10%
eff cm Nature, 473:493, 2011.
smallerthantheearlierpredictionofSkripnikov [3] M.PospelovandA.Ritz.Ann.Phys.,318:119,
(cid:2) (cid:3)
et al. 2005.
6
[4] G.C.Branco,R.G.Felipe,andF.R.Joaquim. nenko, L. P. Yatsenko, A. V. Romanenko,
Rev. Mod. Phys., 84:515, 2012. M. Schreitl, G. Winkler, and T. Schumm.
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vich,N.R.Hutzler,M.Parsons,D.Patterson, [16] K. G. Dyall. Theoret. Chim. Acta, 90:491,
E.Petrik,B.Spaun,J.M.Doyle,G.Gabrielse, 2006.
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[6] A.C.Vutha,B.Spaun,Y.V.Gurevich,N.R. 2012.
Hutzler,E.Kirilov,J.M.Doyle,G.Gabrielse, [18] T. H. Jr. Dunning. J. Chem. Phys., 90:1007,
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