Towards the electron EDM search: Theoretical study of HfF+

0.13 MB·English

by  A. N. Petrov

Towards the electron EDM search: Theoretical study of HfF+
A.N. Petrov,1,∗ N.S. Mosyagin,1 T.A. Isaev,1 and A.V. Titov1,†
1
Petersburg Nuclear Physics Institute, Gatchina, Leningrad district 188300, Russia
We report first ab initio relativistic correlation calculations of potential curves for ten low-lying
electronic states, effectiveelectric field on theelectron and hyperfineconstantsfor the3∆1 state of
cation of aheavytransition metalfluoride,HfF+, thatissuggested tobeused astheworkingstate
7 in experiments to search for the electric dipole moment of the electron. It is shown that HfF+ has
0 deeply bound 1Σ+ ground state, its dissociation energy is De =6.4 eV. The 3∆1 state is obtained
0
tobetherelatively long-lived firstexcited statelyingabout 0.2 eV higher. The calculated effective
2 electric field Eeff =Wd|Ω| acting on an electron in this state is 5.84×1024Hz/(e·cm).

n
a
J Introduction. The search for the electric dipole mo- of required spectroscopic properties, E and hyperfine
eff
9 ment(EDM)oftheelectron,d oreEDMbelow,remains structure of HfF+ as a candidate for the eEDM experi-
e
1 one of the most fundamental problems in physics. Up ment. Suchacompleteanalysisisthefirstoneperformed

to now only upper limits were obtained for d . The for the given kind of molecules.
] | e|
h tightest bound on d was obtained in the experiment on E , hyperfine structure and working state in HfF+.
e eff
p the Tl atom [1], which established an upper bound of One of the most important features of such experiments
m- de <1.6 10−27 ecm (e is the charge of the electron). is that the knowledge of the effective electric field, Eeff,
| | × ·
It seems unlikely that experiments on atoms can give seen by an unpaired electron is required for extracting
o
tighter bounds. Molecular systems, on the other hand, d from the measurements. E can not be obtained in
t e eff
a provideawayformuchenhancedsensitivity,sincetheef- an experiment, rather, electronic structure calculations
.
s fective intramolecularelectric fieldacting onelectronsin are required for its evaluation. It is presented by the
c
polar molecules can be five or more orders of magnitude expressionE =W Ω, whereW isaparameterofthe
i eff d| | d
s higherthanthemaximalfieldavailableinthe laboratory P,T-odd molecular Hamiltonian that is given in Refs.[4,
y
[2, 3, 4]. 14, 15]
h
p ThenewgenerationoftheeEDMexperiments,employ-
[ ingpolarheavy-atommolecules,isexpectedtoreachsen-
1
v2 rsietsiuvlittsyaorfe1ex0−pe30c−te1d0t−o2d8er·acmma/t√icdaallyyi(nefl.gu.,ensceeea[5ll])t.heTphoepir- Wd = ΩdehΨ|Xi Hd(i)|Ψi , (1)
4 ular extensions of the standard model, in particular su-
where Ψ is the wavefunction for the working 3∆ state,
5 persymmetry,evenifboundsontheP,T-oddeffectscom- 1
and Ω = ΨJ nΨ = 1, J is the total electronic
12 patible with zero are obtained (see [6, 7] and references momentumh,n| is·the| uinitv±ectoralongthemolecularaxis
therein). These studies include the beam experiments
1 directed from Hf to F,
carriedoutonYbFmolecularradicalsbyE.Hindsgroup
6
0 atImperialCollege,London[5],andthevaporcellexper-
/ iment on the metastable a(1) state of PbO prepared by 0 0
s H =2d , (2)
c the group of DeMille at Yale University (see [8] and ref- d e(cid:18)0 σE (cid:19)
i erencestherein). NewwaysofsearchingforeEDM,using
s
y trappedcoldmolecularcations,wereinvestigatedduring Eistheinnermolecularelectricfield,σarethePaulima-
h trices. Recently,theE valueforthe3∆ statewasesti-
last two years at JILA by E. Cornell group. The first eff 1
p candidate was HI+ [9], but subsequent estimate [10] and mated in the scalar-relativistic approximation by Meyer
:
v accurate calculations [11] have showed that the cation et al. [16] and the method used for calculation of Eeff is
i close in essence to that developed by us earlier [17] and
X hasrathersmallEeff (seebelow). Thensomemoremoti-
vated candidates were considered, HfH+ [12], HfF+ [13] appliedtofirsttwo-stepcalculationsofthePbFmolecule
r
a etc., having, possibly, working 3∆ as the ground or at [14,17]. Inthepresentwork,amorereliablevalueofEeff
1
is calculated using the advanced two-step techniques de-
least long-lived first excited state. Even experimental
veloped by us later [18, 19, 20, 21]. The hyperfine con-
study ofspectroscopicpropertiesforthis cationis a very
stants for 177Hf and 19F nuclei (A [Hf] and A [F]) for
difficultproblemandtherearenosuchdatameasuredup k k
the 3∆ state of HfF+ are also calculated. When the
todatethoughtheyarerequiredalreadytoanalyzebasic 1
experimental value A [Hf] is measured, it will provide
stagesoftheeEDMexperiment[13]. Inturn,themodern k
an accuracy check for the calculated E value. Both
relativistic computational methods can now give reliable eff
A [Hf] and A [F] values are useful for identifying HfF+
answers on almost all the questions of interest even for k k
by his spectrum from another species in the experiment.
compounds of heavy transition metals as is in our case.
Moreover, preparation of the HfF+ cations in the work-
So, the goal of the present paper is a theoretical study
ingstatefortheeEDMexperiment,registrationofeEDM
2
signalsetc.requirestheknowledgeofspectroscopicprop- orbital of Hf with admixture of 6p orbital of Hf, δ and
z
erties. Inparticular,anopenquestionistoclarify,which π are mainly the 5d orbitals of Hf.
state is the ground one. If 3∆ is not the ground state, Toobtainspectroscopicparameters,sixpointsbetween
1
itsradiativelifetimeisalsorequired. Inthepresentwork 3.1and4.0a.u.andpointat100a.u.ofthe HfF+ poten-
these data are obtained, though precise studying of the tial curves were calculated for ten lowest-lying states in
compoundswithtransitionelementsisadifficultproblem 10-electroncalculationsandforfourstatesin20-electron
for modern molecular theory. The needed spectroscopic ones. The 20-electron calculation is substantially more
information cannot be obtained now from other sources. time-consuming and for the rest six states were calcu-
Methods and calculations. The 12-electron general- lated only for one point, 3.4 a.u., in the present study.
ized relativistic effective core potential (GRECP) [22] Comparingthelattercalculationswithcorresponding10-
simulating interaction with the explicitly excluded 1s to electron ones the core (5s2,5p6 shells of Hf and 1s2 shell
4f electrons of Hf is used in 20-electron calculations of ofF)relaxationandcorrelationcorrectionstotheT val-
e
HfF+. In10-electroncalculationswhicharesubstantially ues, called “core corrections”below and in Table I, were
less time consuming, 5s and 5p spinors of hafnium and estimated. The core properties A and E were calcu-
k eff
1s orbital of fluorine are frozen from the states averaged lated only for the working 3∆ state at point 3.4 a.u.,
1
over the nonrelativistic configurations 5d26s0.66p0.4 for which is close to the equilibrium distance (see Table I).
Hf+ and 2s22p5 for F and not treated explicitly. The Beforecalculatingcorepropertiesthe shapesofthe four-
generalized correlation atomic basis set (GCBS) [23, 24] component molecular spinors are restored in the inner
(12s16p16d10f10g)/[6s5p5d3f1g] is constructed for Hf. coreregionafterthetwo-componentGRECPcalculation
The ANO-L (14s9p4d3f)/[4s3p2d1f] atomic basis set ofthemolecule. Forthispurposethenonvariationalone-
listed in the molcas 4.1 library [25] was used for flu- centerrestoration(NOCR)method[17,18,20,21,27,33]
orine. The molecular orbitals are obtained by the com- is applied.
pleteactivespaceself-consistentfield(CASSCF)method Resultsanddiscussion. Theresultsofcalculationsfor
[25, 26] with the spin-averagedpart of the GRECP [27], HfF+ spectroscopic parametersare presented in Table I.
i.e. only scalar-relativistic effects are taken into account Thefirstpointtonoteisthatthecationisdeeplybound.
at this stage. In the CASSCF method, orbitals are sub- Secondimportantresultisthatthe1Σ+ stateappearsto
divided into three groups: inactive, active and virtual. bethegroundoneandtheworkingstate,3∆ ,isthefirst
1
Inactive orbitals are doubly occupied in all the config- excited one. Besides, the excitation energy from 1Σ+ to
urations, all possible occupations are allowed for active 3∆ is increased from 866 cm−1 to 1633 cm−1 after in-
1
orbitals, whereas virtual orbitals are not occupied. So, cluding the 5s and 5p shells of Hf and 1s shell of F to
wavefunctionisconstructedasafullconfigurationinter- the relativistic correlation calculation. Excitation ener-
actionexpansioninthe space ofactive orbitals andboth gies from 1Σ+ to other calculated low-lying states are
active andinactive orbitalsare optimized for subsequent also increased. Note also that the values obtained in 10-
correlation calculations of HfF+. Using the C point electron calculations for lowest four states when just the
2v
group classification scheme, five orbitals in A , four in corecorrectiondescribedaboveis takeninto accountare
1
B and B and two in A irreps are included into the in a good agreement with the purely 20-electron calcu-
1 2 2
active space. In 10-electron calculations, one orbital in lations. (Note that accounting for correlation/relaxation
theA irreps(whichismainly2sorbitalofF)belongsto of the 4f-shell would be also desirable but it is too con-
1
the inactive space. In 20-electron CASSCF calculations, suming and we expect it will result in smaller change of
the5sand5porbitalsofHfand1sorbitalofFareadded spectroscopicpropertiesthanthatfortheabovecorecor-
to the space of inactive orbitals. rection.) The SO splittings of the 3∆ and 3Π states are
Next, the spin-orbit direct configuration interaction mainly due to the SO splitting of the 5d shell of Hf:
(SODCI) approach [28, 29, 30] modified by our group
to account for spin-orbit interaction in configuration se- Hslso =a·(l5d·s5d) , (3)
lection procedure [31] with the selected single and dou-
The atomic Dirac-Fock calculation of Hf+ gives
ble excitations from some multiconfigurational reference
ε ε = 3173 cm−1 a = 1269 cm−1, where ε
states (“mains”) is employed on the sets of different ΛS 5/2− 3/2 ⇒ 5/2
and ε are orbital energies of 5d and 5d states of
many-electron spin- and space-symmetry adapted basis 3/2 5/2 3/2
Hf+. The SO interaction(3)averagedoverthe 3∆ or 3Π
functions (SAFs). Details on features of constructing
states is reduced to
the reference space and selection procedure are given in
Refs. [21, 31, 32]. Hso = (LS), (4)
Ten lowest states with the leading configura- LS A· ·
tions [...]σ12σ22 (1Σ+), [...]σ12σ21δ1 (3∆1,2,3; 1∆), where = 1269/2= 635 cm−1, L and S are the orbital
[...]σ12σ21π1 (3Π0−,0+,1,2; 1Π) were calculated. Here andspiAnmomentaofHfF+. TheSOinteraction(4)leads
σ1 orbital is mainly formed by 2pz orbital of F with tosplittingbetweencomponentsofthe3∆and3Πstates
admixture of 6pz and 6s orbitals of Hf, σ2 is mainly 6s on 1269 and 635 cm−1, correspondingly. It is in a good
3
TABLE I:Calculated spectroscopic parameters for HfF+ TABLE II: Calculated parameters Ak[Hf] and Ak[F] (in
MHz) and Eeff (in 1024Hz/(e · cm)) for the 3∆1 state of
177Hf19F+ at internucleardistance of 3.4 a.u.
State Re ˚A Te cm−1 Te with core we cm−1 De cm−1
a
correction
10-electron calculation Ak[Hf] Ak[F] Eeff
1Σ+ 1.784 0 0 751 51107 10-electron calculation
3∆1 1.810 866 1599 718
3∆2 1.809 1821 2807 719 -1250 -33.9 5.89
3∆3 1.807 3201 4324 721 20-electron calculation
1∆2 1.814 9246 11519 696
3Π0− 1.856 9466 11910 689 -1239 -58.1 5.84
3Π0+ 1.854 9753 12196 699
3Π1 1.860 10190 12686 687
3Π2 1.856 11898 14438 703
1Π1 1.870 12642 14784 679
20-electron calculation
1Σ+ 1.781 0 790 51685 ∗Electronic address: [email protected]; Also at St.-
33∆∆21 11..880056 21862383 774486 †PAelstoersabtuSrgt.-SPteatteersUbnuirvgerSsittayt,eSUt.-nPiveetersristbyu,rSgt,.R-Puestseirasburg,
3∆3 1.804 4273 749 Russia
[1]B. C. Regan, E. D. Commins, C. J. Schmidt,and D. De-
Mille, Phys.Rev.Lett. 88, 071805 (2002).
a See paragraph “Methods and calculations” for details.
[2]O. P. Sushkov and V. V. Flambaum, Sov. Phys.–JETP
48, 608 (1978).
[3]V.G. Gorshkow, L. N. Labzovsky,and A.N. Moskalyov,
Sov.Phys.–JETP 49, 209 (1979).
[4]A. V. Titov, N. S. Mosyagin, A. N. Petrov, T. A. Isaev,
andD.P.DeMille,Progr.Theor.Chem.Phys.B 15,253
agreement with the splitting of the 3∆ state calculated (2006).
in Table I but not with the 3Π one because of the SO [5]B.E.Sauer,H.T.Ashworth,J.J.Hudson,M.R.Tarbutt,
interaction with closely lying 1∆ and 1Π states. and E. A.Hinds, AtomicPhysics 20, 44 (2006).
The calculated A and E for the 3∆ state are pre- [6]J. S. M. Ginges and V. V. Flambaum, Phys. Rep. 397,
k eff 1 63 (2004).
sented in Table II. Inopposite to A [F], A [Hf] andE
k k eff [7]J.ErlerandM.J.Ramsey-Musolf,Prog.Part.Nucl.Phys
are not seriously changed when the outer core electrons
54, 351 (2005).
are included into calculation. Such behavior of Ak[F] is [8]D.Kawall,F.Bay,S.Bickman,Y.Jiang,andD.DeMille,
explainedbythefactthatfluorinecanbeconsideredwith Phys. Rev.Lett. 92, 133007 (2004).
good accuracy as a closed shell subsystem (thus having [9]R. Stutz and E. Cornell, Bull. Amer. Phys. Soc. 49, 76
A [F] 0) in the 3∆ state, i.e. large compensation of (2004).
k
≈ [10]B.Ravaine,S.G.Porsev,andA.Derevianko,Phys.Rev.
contributions from orbitals with different projections of
Lett. 94, 013001 (2005).
totalelectronicmomentumfor A [F]takesplace. There-
k [11]T. A. Isaev, N. S. Mosyagin, A. N. Petrov, and A. V.
fore, even small perturbation can seriously change the
Titov, Phys. Rev.Lett. 95, 163004 (2005).
Ak[F] value. Calculated Eeff is large and comparable [12]L. Sinclair, J. Bohn, A. Leanhardt, P. Maletinsky,
with the corresponding value for the a(1) state of the E. M. R. Stutz, and E. Cornell, in 36th Meeting of
PbO molecule [21]. If one does not pay attention to the the Division of Atomic, Molecular and Optical Physics
difference of signs our value is in 1.34 times larger than (DAMOP) (APS, Lincoln, Nebraska, 2005), poster
M6.00119.
value obtainedin the paper [16] in scalar-relativisticcal-
[13]E.Cornell andA.Leanhardt (2006), privatecommunica-
culations. Our rather crude estimate of the 3∆ lifetime
1 tion.
forthe3∆ 1Σ+transitionisabout1/2sec. Thisvalue
1 [14]M. G. Kozlov, V. I. Fomichev, Yu. Yu. Dmitriev, L. N.
→
is difficult for accurate calculations because of small ab- Labzovsky,and A. V.Titov, J. Phys. B 20, 4939 (1987).
solutevaluesofbothtransitionenergiesand,particularly, [15]M.KozlovandL.Labzowsky,J.Phys.B28,1933(1995).
transition dipole moments between those states whereas [16]E.R.Meyer,J.L.Bohn,andM.P.Deskevich,Phys.Rev.
absolute errors are similar to those for other transitions. A 73, 062108 (2006).
[17]A. V. Titov, Ph.D. thesis, St.-Petersburg (Leningrad)
Acknowledgments. This work is supported by the
State University,Russia (1985).
RFBR grant 06–03–33060. T.I. and A.P. are also sup-
[18]A. V.Titov, Int. J. QuantumChem. 57, 453 (1996).
portedbygrantsofRussianScienceSupportFoundation.
[19]N.S.Mosyagin, M.G.Kozlov, andA.V.Titov, J.Phys.
A.P. is grateful for the grant of Governor of Leningrad B 31, L763 (1998).
district. [20]A. N. Petrov, N. S. Mosyagin, T. A. Isaev, A. V. Titov,
4
V. F. Ezhov, E. Eliav, and U. Kaldor, Phys. Rev. Lett. [28]R.J.BuenkerandS.D.Peyerimhoff,Theor. Chim.Acta
88, 073001 (2002). 35, 33 (1974).
[21]A. N. Petrov, A. V. Titov, T. A. Isaev, N. S. Mosyagin, [29]R. J. Buenker and S. Krebs, in Recent Advances in Mul-
and D.P. DeMille, Phys. Rev.A 72, 022505 (2005). tireference Methods,editedbyK.Hirao(WorldScientific,
[22]A. V. Titov and N. S. Mosyagin, Russ. J. Phys. Chem. Singapore, 1999), pp.1–29.
74, Suppl.2, S376 (2000), [arXiv: physics/0008160]. [30]A.B.Alekseyev,H.-P.Liebermann,andR.J.Buenker,in
[23]N. S. Mosyagin, E. Eliav, A.V. Titov, and U. Kaldor, J. Recent Advances in Relativistic Molecular Theory, edited
Phys. B 33, 667 (2000). byK.HiraoandY.Ishikawa(WorldScientific,Singapore,
[24]T. A. Isaev, N. S. Mosyagin, M. G. Kozlov, A. V. Titov, 2004), pp. 65–105.
E. Eliav, and U. Kaldor, J. Phys.B 33, 5139 (2000). [31]A. V. Titov, N. S. Mosyagin, A. B. Alekseyev, and R. J.
[25]K. Andersson, M. R. A. Blomberg, M. P. Fu¨lscher, Buenker,Int.J. QuantumChem. 81, 409 (2001).
G. Karlstr¨om, R. Lindh, P.-A. Malmqvist, P. Neogr´ady, [32]N.S.Mosyagin,A.V.Titov,R.J.Buenker,H.-P.Lieber-
J.Olsen,B.O.Roos,A.J.Sadlej,etal.(1999),quantum- mann, and A. B. Alekseyev, Int. J. Quantum Chem. 88,
chemical program package “molcas”, Version 4.1. 681 (2002).
[26]J.OlsenandB.O.Roos,J.Chem.Phys.89,2185(1988). [33]A.V.Titov,N.S.Mosyagin,andV.F.Ezhov,Phys.Rev.
[27]A.V.Titov andN.S.Mosyagin, Int.J.QuantumChem. Lett. 77, 5346 (1996).
71, 359 (1999).