Table Of ContentEnhanced effect of CP-violating nuclear magnetic quadrupole moment in HfF+
molecule
L.V. Skripnikov,1,2,∗ A.V. Titov,1,2,† and V. V. Flambaum3
1National Research Centre “Kurchatov Institute” B.P. Konstantinov Petersburg
Nuclear Physics Institute, Gatchina, Leningrad District 188300, Russia
2Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia
3School of Physics, The University of New South Wales, Sydney NSW 2052, Australia
(Dated: 12.01.2017)
7
1 HfF+ cationisaverypromisingsystemtosearchfortheelectronelectricdipolemoment(EDM),
0 andcorresponding experiment is carried out by E.Cornell group. Herewe theoretically investigate
2 the cation to search for another T,P-odd effect – the nuclear magnetic quadrpole moment (MQM)
interactionwithelectrons. Wereportthefirstaccurateabinitiorelativisticelectronicstructurecal-
n
a culationsofthemolecularparameterWM=0.494 1e03c3mH2z thatisrequiredtointerprettheexperimental
J data in terms of theMQM of Hfnucleus. For thiswe have implemented and applied the combined
2 Dirac-Coulomb(-Gaunt)andrelativisticeffectivecorepotentialapproachestotreatelectroncorrela-
1 tioneffectsfromalloftheelectronsandtotakeintoaccounthigh-ordercorrelationeffectsusingthe
coupled cluster method with single, double, triple and noniterative quadruple cluster amplitudes,
] CCSDT(Q). We discuss interpretation of the MQM effect in terms of the strength constants of
h T,P-odd nuclear forces, proton and neutron EDM, QCD parameter θ and quark chromo-EDM.
p
-
m
INTRODUCTION THEORY
o
t
a Qualitatively the effect under consideration corre-
.
s sponds to the interaction of the nuclear magnetic
c HfF+ cation is a very promising system to search for
quadrupole moment with the gradient of the magnetic
si the electron electric dipole moment (eEDM) [1–8] (see field producedby electrons. This is the T,P-oddinterac-
y also [9–15]). At present E. Cornell’s group prepares the
tion which mixes states of opposite parity in atoms and
h
ion trap experiment on the cation [6, 9]. In contrast to
p molecules [16, 17]. Relativistic Hamiltonian of the inter-
the 232ThO molecule which was used to obtain the best
[ action is given by the following expression [16, 20, 21]:
current limit on eEDM [10] one can use available sta-
1 ble isotope of Hf, e.g. 177Hf to search for the magnetic M 3[α×r] r
8v quadrupole moment of the 177Hf nucleus in the 177HfF+ HMQM =−2I(2I−1)Tik2 r5 i k, (1)
2 cation [11]. This is because the 177Hf nucleus posses nu-
3 clear spin I>1/2 [16–18] while 232Th has I =0. where Einstein’s summation convention is implied, α =
3 0 σ
are the 4x4 Dirac matrices, r is the displace-
0 (cid:18)σ 0 (cid:19)
1. As was shown in [18] MQM can be strongly enhanced mentoftheelectronfromtheHfnucleus,I isthenuclear
0 due to the collective nuclear effect. Below we study this spin, M is the nuclear MQM,
7 effect for the case of Hf nucleus.
1 3M
: Mi,k = Ti,k (2)
v 2I(2I−1)
Xi ouEsllyecsttruodniiecdstinruRcteufrse. [o1f, H3–fF5]+focrattihoenehEaDsMbeepnropbrleemvis- Ti,k =IiIk+IkIi− 32δi,kI(I +1). (3)
r
a – calculation of the effective electric field (Eeff) which In the subspace of ±Ω states (Ω = hΨ|J ·n|Ψi, J is
is required to interpret the experimental energy shift in the total electronic momentum, Ψ is the electronic wave
terms of eEDM. In Refs. [3, 4] the two-step relativistic function for the considered 3∆ state of HfF+) expres-
1
effective core potential approach was used. In Ref. [5] a
sion (1) is reduced to the following effective molecular
direct approach within the Dirac-Coulomb Hamiltonian
Hamiltonian [16]:
wasapplied. Inthepresentpaperwefollowthenewcom-
bined Dirac-Coulomb(-Gaunt) and two-step relativistic W M
HMQM =− M S′Tˆn, (4)
pseudopotential scheme [19] to study the electronic part eff 2I(2I−1)
of the problem of calculation of the interaction between
theMQMofHfnucleusandelectronsofHfF+ inthefirst where n is the unit vector along the molecular axis ζ
excited3∆ stateofHfF+ cation. Thisschemeallowsone directedfromHftoF,S′istheeffectiveelectronspin[22]
1
to treat all of the important effects including correlation defined by the following equations: S′|Ω >= Ω|Ω >,
ζ
of the inner-core electrons. S′ |Ω = ±1 >= 0 [20, 23], S=|Ω|=1. W parameter is
± M
2
defined by the following equation: Inadeformednucleusthe MQMinthe“frozen”frame
(rotating together with a nucleus) may be estimated us-
3 αi×ri ing the following formula [18]:
W = hΨ| r |Ψi . (5)
M 2Ω (cid:18) r5 (cid:19) ζ
Xi i ζ
Mnucl = Msingle(I,I ,l)n(I,I ,l), (9)
zz zz z z
As was shown in Ref. [24] for a completely polarized X
molecule the energy shift due to MQM interaction is: where the sum goes over occupied orbitals,
Msingle(I,I ,l) is given by Eqs. (8) and (2),
zz z
δ (J,F)=(−1)I+FC(J,F)MW Ω , (6) T = 2I2 − 2I(I + 1), n(I,I ,l) are the orbital
M M zz z 3 z
occupation numbers, which may be found in Ref. [26].
J 2 J
The sum over a complete shell gives zero; therefore, for
(2J +1)(cid:18)−Ω 0 Ω (cid:19) J I F
C(J,F)= , (7) shells more than half-filled, it is convenient to use hole
2 I 2 I (cid:26) I J 2 (cid:27)
numbers in place of particle numbers, using the relation
(cid:18)−I 0 I (cid:19)
Msingle(hole)=−Msingle(particle).
zz zz
The nucleus 177Hf has the following occupa-
where(...)meanselementswith3j−symbolsand{...}are
tion numbers: 13 neutron holes in the orbitals
those with 6j−symbols [25], F is the total angular mo-
[¯l ,I ] = [f¯ ,−7/2], [¯i ,±13/2,±11/2,±9/2],
mentum and J is the number of rotational level. Note, I z 7/2 13/2
[h¯ ,±9/2,±7/2], [p¯ ,±3/2], and 8 proton holes
that δ depends on J and F quantum numbers. Be- 9/2 3/2
sides HMMQM has non-zero off-diagonal matrix elements [d¯3/2,±3/2], [d¯5/2,±5/2], [h¯11/2,±11/2,±9/2].
eff
The MQMinthe laboratoryframe, M ≡M ,canbe
on J quantum number (between different rotational lev- lab
expressed via MQM in the rotating frame (9):
els). This should be taken into account when mixing of
different rotational levels become significant. In Eq. (6)
thiseffectisneglected. For177HfF+ (I=7/2)andground Mlab = I(2I −1) Mnucl =
rotationallevelJ=1Eq.(6)givestheMQMenergyshifts, (I+1)(2I+3) zz
|δ(J,F)|, equal to 0.107W M, 0.143W M, 0.05W M (1.5η −1.1η )·10−33(e·cm2)
M M M p n
for F =5/2,7/2,9/2,correspondingly. −(4.0d +2.9d )·10−13cm, (10)
p n
where I =7/2 is the nuclear spin of 177Hf.
CALCULATION OF THE NUCLEAR MAGNETIC The T,P-odd nuclear forces are dominated by the π
0
QUADRUPOLE MOMENT
meson exchange [16]. Therefore, we may express the
strength constants via strong πNN coupling constant
The angularmomentum I of a sphericalnucleus is de- g = 13.6 and T,P-odd πNN coupling constants corre-
termined by a valence nucleon. In the single-valence- spondingtotheisospinchannelsT =0,1,2: η =−η =
n p
nucleonmodelthenuclearMQMisgivenbythefollowing 5·106g(g¯ +0.4g¯ −0.2g¯ ) (see detailes in [24]). As a re-
1 2 0
expression: sult, we obtain
M =[d−2·10−21η(µ−q)(e·cm)]λ (2I−1)t , (8) M(g)=−[g(g¯ +0.4g¯ −0.2g¯ )·1.0·10−26e·cm2.
p I 1 2 0
(11)
where t = 1 for I = l +1/2 and t = −I/(I +1) for
I I
I = l − 1/2, I and l are the total and orbital angu- Possible CP-violation in the strong interaction sector is
lar momenta of a valence nucleon, η is the dimension- described by the CP violation parameter θ˜. According
less strength constant of the T,P-odd nuclear potential to Ref. [27] gg¯0 =−0.37θ˜. This gives the following value
ηG/(23/2m )(σ · ∇ρ) acting on the valence nucleon, ρ of MQM for 177Hf:
p
is the total nucleon number density, the nucleon mag-
netic moments are µ = 2.79 for valence proton and M(θ)=−7·10−28θ˜·e·cm2. (12)
p
µ = −1.91 for valence neutron, q = 1 and q = 0,
n p n
λ = h¯/m c = 2.1·10−14 cm. The contribution of the Finally, we can express MQM in terms of the quark
p p
valence nucleon EDM d was calculated in Ref. [17] , the chromo-EDM d˜ and d˜ using the relations gg¯ =
u d 1
contributionoftheT,P-oddnuclearforceswascalculated 4.·1015(d˜ −d˜ )/cm, gg¯ =0.8·1015(d˜ +d˜ )/cm [28]:
u d 0 u n
in [16]. Using a natural assumption that in any model
of CP-violation the π meson exchange gives significant M(d˜)=−4·10−11(d˜ −d˜ )·e·cm. (13)
u d
contributions it was concluded in [16] that the contribu-
tionoftheT,P-oddnuclearforcestoanyT,P-oddnuclear The contributionsofd andd to MQMinEqs.(11-13)
p n
moment is 1-2 orders of magnitude larger than the con- arefromonetotwoordersofmagnitudesmallerthanthe
tribution of the nucleon EDM [16]. contributions of the nucleon T,P-odd interactions.
3
ELECTRONIC STRUCTURE CALCULATION contributions; (III) correction on inclusion of the Gaunt
DETAILS interaction; (IV) contribution of high-order correlation
effectsuptothecoupledclusterwithsingle,double,triple
It follows from Eq. (5) that W parameter is mainly and noniterative quadruple amplitudes for the valence
M
determined by the behavior of the electronic wave func- electronswithinthe2-component(withspin-orbiteffects
tion in the region close to the heavy atom nucleus. We included)two-stepapproach;(V)calculationofthe basis
call such parameters as the Atoms-In-Compounds char- set correction for 52 outer electrons of HfF+ within the
acteristics or properties [29–31]. Other examples are the scalar-relativistic two-step approach.
hyperfine structure interaction constants, effective elec- Forstep(I)weusedtheCVQZbasissetforHf[51,52]
tric field, chemical shifts, etc. To compute such param- and aug-ccpVQZ basis set [53, 54] with two removed
eters we have previously developed the two-step method g-type basis functions for F. The inner-core electrons
[29, 32, 33] which allows us to avoid direct 4-component (1s..3d of Hf) were excluded from the correlation treat-
relativistic treatment. In the first stage, one consid- ment. For the outer-core/valence correlation calculation
ers the valence (and outer-core) part of the molecular we set cutoff equal to 50 Hartree for the virtual spinors.
wavefunctionwithinthe generalizedrelativisticeffective Theinner-corecorrelationcontributionwascalculatedat
corepotential(GRECP)method[34–36]. Theinner-core the CCSDlevelasthe difference betweenthe WM values
electrons are excluded from the explicit treatment. The calculatedwithcorrelationofall80electronsofHfF+ in-
feature of this stage is that the valence wave functions cluded into correlation treatment and with 52 electrons
(spinors)aresmoothedinthespatialinnercoreregionof as in stage(I). For these calculations we used the CVDZ
aconsideredheavyatom. Thisleadstoconsiderablecom- [51, 52] basis set on Th and the cc-pVDZ [53, 54] basis
putationalsavings. Sometechnicaladvantageisthatone set on F. We set cutoff equal to 7000 Hartree for vir-
can also use very compact contracted basis sets [19, 37]. tual molecular spinors in these calculations to be sure
This is of crucialimportance to treat high-order correla- that the necessary correlation functions present in the
tioneffects. Besides,onecanexcludethespin-orbitterm one-electron spinor basis. Correction at the step (III)
of the GRECP operator and consider scalar-relativistic has been calculated at the Hartree-Fock level. In the
approximation with a good nonrelativistic symmetry. stage (IV) 20 electrons of HfF+ were correlated. Cor-
Due to the corresponding savings one can use very large rection was estimated as the difference in the calcu-
basis sets to consider basis set corrections and analyze lated values of WM within the CCSDT(Q) versus the
its saturation. At the second step, one uses the nonvari- CCSD(T) method. For Hf we use slightly reduced ver-
ational procedure developed in [29, 32, 33, 38] to restore sion[12,16,16,10,8]/(6,5,5,1,1)of the basisset whichwas
the correct 4-component behavior of the valence wave used in Refs. [3, 4, 7]. For F the ANO-I basis set [55]
function in the spatial core region of a heavy atom. The reduced to [14,9,4,3]/(4,3,1) was used. In the stage (V)
procedure is based on a proportionality of the valence weconsideredthe influenceofadditional7g−,6h−and
and low-lying virtual spinors in the inner-core regions of 5 i− basis functions on Hf (with respect to the basis
heavy atoms. The procedure has been recently extended functions of these types included in the CVQZ basis set,
to consider not only the atomic and molecular systems used at step (I)). For stages (IV) and (V) we used the
but also three-dimensional periodic structures (crystals) semilocalversionsof12-electronand44-electronGRECP
inRef. [39]. GRECPandthe restorationprocedurewere operators [3, 4, 7, 35, 36].
alsosuccessfullyusedforpreciseinvestigationofdifferent In all the calculations the Hf−F internuclear distance
diatomics[7,29,40–49]. Thetwo-stepmethodallowsone in the 3∆1 state was set to 3.41 Bohr [1].
to consider high-order correlation effects and large basis For the Hartree-Fock calculations and integral trans-
sets with rather modest requirements to computer re- formations we used the dirac12 code [56]. Relativistic
sourcesincomparisonto4-componentapproaches. How- correlationcalculationswereperformedwithinthemrcc
ever, some uncertainty remains due to the impossibility code [57]. For scalar-relativisticcalculations we used the
to consider the full version of the GRECP operator in cfour code [58–61]. The code to compute matrix ele-
the currently available codes and neglect of the inner- ments of the MQM Hamiltonian has been developed in
core correlationeffects. In Refs. [19, 50] we suggested to the present paper.
combinethetwo-stepapproachandthedirectrelativistic
Dirac-Coulomb(-Gaunt) approach to take advantages of
both approaches. RESULTS AND DISCUSSIONS
Computational scheme of the molecular W param-
M
eter (5) assumes evaluation of the following contribu- The final value of W is 0.494 1033Hz.
M e cm2
tions: (I) the main correlation contributions within the Inner-core contribution to the final value of W is
M
52-electron 4-component Dirac-Coulomb coupled cluster about 3%. Gaunt contribution is about -1.6%. High-
with single, double and noniterativetriple cluster ampli- order correlation effects give -0.3%. This means that
tudes, CCSD(T), theory; (II) the inner-core correlation the convergence with respect to correlation effects is
4
achieved. Basis set correction on high-order harmonics † http://www.qchem.pnpi.spb.ru
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M
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