Calculation of $P_ and $T_ odd effects in $"" sup 205_TIF including electron correlation

4 Pages·0.15 MB·English

by  A. N. Petrov

Calculation of P,T-odd effects in 205TlF including electron correlation
A. N. Petrov,∗ N. S. Mosyagin, T. A. Isaev, A. V. Titov, and V. F. Ezhov
Petersburg Nuclear Physics Institute, Gatchina, Petersburg district 188300, Russia
E. Eliav and U. Kaldor
School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
2 (Dated: February 2, 2008)
0
A method and codes for two-step correlation calculation of heavy-atom molecules have been
0
developed, employing the generalized relativistic effective core potential (GRECP) and relativistic
2
coupled cluster (RCC) methods at the first step, followed by nonvariational one-center restoration

n of proper four-component spinors in the heavy cores. Electron correlation is included for the first
a time in an ab initio calculation of the interaction of the permanent P,T-odd proton electric dipole
J
moment with the internal electromagnetic field in a molecule. Inclusion of electron correlation by

2 GRECP/RCC has a major effect on the P,T-odd parameters of 205TlF, decreasing M by17% and
2 X by22%.

] PACSnumbers: 31.90.+s,35.10.Wb,31.20.-d
h
p
-
m Introduction. Themeasurementofpermanentelectric ρψ(~r) is the electronic density calculated from the wave
dipole moments (EDM) of elementary particles is highly function ψ. Keeping only the dominant diagonal terms
e
importantforthetheoryofP,T-oddinteractions. Exper- of the two-electron operator for the magnetic effect (see
h
c imentsperformedsofarhavegivenonlyupperboundsfor [2]) they have
. the EDMs. The extractionof EDMs from measurements
s
c onmoleculescontainingheavyatomsrequiresknowledge
si ofnuclearandelectronicpropertiesofthemolecule. High dM =2√2d µ + 1 M , (4)
p
y quality calculations of the relevant electronic properties Z 2mc
(cid:18) (cid:19)
h are therefore essential for accurate determination of the
p EDMs [1, 2]. where µ, m and Z are the magnetic moment, mass and
[ Here we consider the interaction of the proton EDM charge of the Tl nucleus, c is the velocity of light,

6 with the internal electromagnetic field of the 205TlF
v molecule. Thismolecule isone ofthe bestcandidatesfor
4 proton EDM measurements. Following Hinds and San- 1 α~i ~li
M = ψ × ψ , (5)
02 dinarTslF[3]i,stwhreitetffenecitnivtehientfoerramction with the proton EDM √2h |Xi ri3 !z| i
8
0 ~li isthe orbitalmomentumoperatorofelectroni,andα~i
1 H =(dV +dM)~σ ~λ , (1) are its Dirac matrices. Heff leads to different hyperfine
0 eff N · splitting of TlF in parallel and antiparallel electric and
/
cs where ~σN is the Tl nuclear spin operator, ~λ is the unit magneticfields. Thelevelshifthν =4(dV+dM)h~σN·~λiis
vectoralongtheinternuclearaxisz fromTltoF,dV and measured experimentally (for the latest data see [5]; an-
i
s dM are constants correspondingto the volume andmag- otherexperimentisnowinpreparationatthePetersburg
y
netic effects according to Schiff’s theory [4]. Hinds and Nuclear Physics Institute).
h
p Sandarsshowed[3]thatthevolumeeffectinacoordinate The parameters X of Eq. (3) and M of Eq. (5) are
: system centered on the Tl nucleus is given by determined by the electronic structure of the molecule.
v They were calculated recently for the X0+ ground state
i
X of TlF by Parpia [6] and by Quiney et al. [7] using the
dV = d XR , (2)
r − p Dirac-Hartree-Fock (DHF) method with large Gaussian
a basis sets (see Table I). No calculation which includes
where d is the proton EDM, R is a factor determined
p
by the nuclear structure of 205Tl, correlation effects is available. The main goal of the
present work is to calculate the X and M parameters
for the molecule with correlationincluded to high order.
2π ∂ Methods. The generalized relativistic effective core
X = ρ (~r) , (3)
ψ
3 ∂z potential (GRECP) method [8] is applied to the TlF
(cid:20) (cid:21)x,y,z=0
molecule. A two-component electronic (pseudo)wave
function is first obtained with the 21-electron GRECP
[9, 10] for Tl, providing proper electronic density in the
∗Electronic address: [email protected]; URL: http://www. valence and outer core regions, followed by restoration
qchem.pnpi.spb.ru of the proper shape of the four-component molecular
2
spinors in the inner core region of Tl. Details of the a.u., whereas previous calculations employed a spherical
method may be found elsewhere [8, 11, 12, 13]. Gaussiannuclearchargedistribution[6,7](therootmean
The correlationspin-orbitalbasisset usedconsistedof squareradius in all calculations is 5.5 fm, in accordwith
26s,25p,18d,12f,and10gGaussian-typeorbitalsonTl, the parametrizationof Johnsonand Soff [18], and agrees
contracted to 6s6p4d2f1g. The basis was optimized in with the experimental value 5.483 fm for the 205Tl nu-
a series of atomic two-component GRECP calculations, cleus[19]). Theall-electronfour-componenthfd[20]and
withcorrelationincludedbytheall-orderrelativisticcou- two-component grecp/hfj [9, 10] codes were employed
pled cluster (RCC) method [14] with single and double to generate the two equivalent [15s12p12d8f] numerical
excitations;theaverageenergyofthetwoloweststatesof basis sets for restoration. These sets, describing mainly
the atom was minimized. The basis set generation pro- the core region, are generated independently of the ba-
cedure is described in Refs. [15, 16]. The basis set was sis set for the molecular GRECP calculations discussed
designed to describe correlationin the outer core 5s and earlier. The molecular pseudospinorbitals are then ex-
5p shells of Tl, in addition to the 5d and valence shells. panded in the basis set of one-center two-component
While 5s and 5p correlation may not be important for atomic pseudospinors,
manyofthechemicalandphysicalpropertiesoftheatom,
itisessentialfordescribingpropertiescomingfrominner
regions, including P,T-odd effects. The (14s9p4d3f)/- Lmaxj=|l+1/2|
φ˜(r) ci f˜ (r)χ . (6)
[4s3p2d1f]basis set fromthe ANO-L library [17] is used i ≈ nljm nlj ljm
for fluorine. Xl=0 j=|Xl−1/2|Xn,m
A one-component self-consistent-field (SCF) calcula- Note that for linear molecules only one value of m sur-
tion of the (1σ...7σ)14(1π2π3π)12(1δ)4 ground state vivesinthesumforeveryφ . Finally,thetwo-component
i
of TlF is performed first, using the GRECP for Tl pseudospinorsinthebasisarereplacedby theequivalent
which simulates the interactions of the valence and four-component spinors and the expansion coefficients
outer core (5s5p5d) electrons with the inner core from Eq. (6) are preserved [11, 12, 13]:
[Kr]4d4 4d6 4f6 4f8 . This is followed by two-
3/2 5/2 5/2 7/2
component RCC calculations, with only single (RCC-
S) or with single and double (RCC-SD) cluster ampli- Lmaxj=|l+1/2| f (r)χ
φ (r) ci nlj ljm . (7)
tudes. The RCC-S calculations with the spin-dependent i ≈ nljm gnlj(r)χl′jm
GRECP operator take into account effects of spin-orbit Xl=0 j=|Xl−1/2|Xn,m (cid:18) (cid:19)
interaction at the level of the one-configurational SCF-
The molecular four-component spinors constructed this
type method. The RCC-SD calculations include, in ad-
wayareorthogonaltotheinnercorespinorsofTl,asthe
dition, the most important electron correlation effects.
atomicbasisfunctionsusedinEq.(7)aregeneratedwith
Theelectrondensityobtainedfromthetwo-component
the inner core electrons treated as frozen.
GRECP/RCC (pseudo)wavefunction in the valence and
The quality of the approximation for the two-center
outercoreregionsisveryclosetothatofthecorrespond-
molecular spinors and, consequently, of the calculated
ing all-electron four-component function. The pseu-
properties increases with the value of L . A series of
max
dospinors are smoothed in the inner core region [8], so
calculationsoftheM parameterwasperformedusingEq.
that the electronic density in this region is not correct.
(7)withbasisfunctionsgoinguptop,dandf harmonics.
The operators in equations (3) and (5) are heavily con-
We found (see Table I) that including only s andp func-
centrated near the nucleus, and are therefore strongly
tionsintheexpansiondeterminesM with90%accuracy.
affected by the wave function in the inner region. The
Because the contribution of f is only about 0.3% and
four-component molecular spinors must therefore be re-
amplitudes of higher harmonics on the nucleus are sup-
stored in the inner region of Tl. All molecular spinors pressed by the leading term r(j−1/2), the error due to
φ are restored as one-center expansions on the Tl nu- ∼
i the neglectofsphericalharmonicsbeyondf is estimated
cleus, using the nonvariational restoration scheme (see
to be below 0.1%. Calculation of the X parameter re-
[8, 11, 12, 13] and references therein).
quiressandpharmonics(seeRef.[7]),although,strictly
Therestorationisstartedbygeneratingequivalentba-
speaking, d harmonics also give nonzero contributions.
sis sets of atomic (one-center) four-component spinors
The restoration procedure implemented here gives a
fnlj(r)χljm and two-component pseudospinors very good description of the wave function in the core
gnlj(r)χl′jm region,which is important for accurate evaluation of the
(cid:26)(cid:18) (cid:19)(cid:27)
f˜ (r)χ by atomic finite-difference all-electron X and M parameters. This is done at a fraction of the
nlj ljm
{ }
DHF and two-component GRECP/SCF calculations of cost necessary for all-electron four-component molecu-
the same valence configurations of Tl and its ions. Here lar calculations with Gaussian basis sets, where a large
nistheprincipalquantumnumber,j andmarethetotal number of additional basis functions must be included
electronic momentum and its projection on the internu- for proper description of the inner core region and small
clearaxis,landl′ aretheorbitalmomenta,andl′=2j l. components of spinors [7]. Here we calculate (restore)
−
The nucleus is modeled as a uniform charge distribution the four-component electronic wave function in the core
within a sphere with radius r =7.1fm 1.34 10−4 region from the (pseudo)wave function obtained in the
nucl
≡ ×
3
molecularGRECPcalculation,whichmaybe considered X,whichisdeterminedbythederivativeoftheelectronic
“frozen” in the valence region at the restoration stage. charge density at a single point, the Tl origin. M is af-
Basis functions describing a large number of chemically fected by ψ in a broader region, and is therefore far less
inertcoreelectronsmaythusbeexcludedfromthemolec- sensitive to the nuclear model.
ular GRECP calculation. Themaingoalofthisworkistheevaluationofelectron
The X and M parameters were calculated by the fi- correlationeffects onthe P,T-oddparameters. Theseef-
nite field method (see, e.g., Refs. [21, 22]). The operator fectsarecalculatedbytheRCC-SDmethodatthemolec-
correspondingtoadesiredproperty[seeEqs.(3)and(5)] ularequilibriumseparationR . Amajorcorrelationcon-
e
is multiplied by a small parameter λ and added to the tribution is observed, decreasing M by 17% and X by
Hamiltonian. The derivative of the energy with respect 22%.
toλgivesthecomputedproperty. Thisisstrictlycorrect Using the correlated values for X and M calculated
onlyatthe limit ofvanishingλ, but itis usuallypossible here and R = 1.036 10−9 a.u. from [23], one obtains
×
tofindarangeofλvalueswheretheenergyislinearwith from Eqs. (2) and (4)
λandenergychangesarelargeenoughtoallowsufficient
precision. The quadratic dependence on λ is eliminated
in the present calculations by averaging absolute energy dV = 7.909 10−6dp a.u. (8)
− ×
changes obtained with λ of opposite signs.
Resultsanddiscussion. Calculationswerecarriedout
at two internuclear separations, the equilibrium Re = dM =1.622×10−6dp a.u. (9)
2.0844 ˚A as in Ref. [6], and 2.1 ˚A, for comparison with
Ref. [7]. The results are collected in Table I. The first The effective electric field interacting with the EDM of
point to notice is the difference between spin-averaged the valence proton of 205Tl in the fully polarized TlF
SCFandRCC-Svalues,whichincludespin-orbitinterac- moleculeisE = dV+dM /dp =6.287 10−6a.u.=32.33
| | ×
tioneffects. TheseeffectsincreaseX by9%anddecrease kV/cm;therevisedprotonEDMlimitfortheexperiment
M by21%. TheRCC-Sfunctionmaybewrittenasasin- of Ref. [5] is dp =( 1.7 2.8) 10−23 ecm.
− ± × ·
gle determinant, and results may therefore be compared ThehyperfinestructureconstantsofTl6p11/2andTl2+
withDHFvalues,eventhoughtheRCC-Sfunctionisnot 6s1, which (like X and M) depend on operators concen-
variational. GRECP/RCC-S values of the M parameter trated near the Tl nucleus, were also calculated. The
areindeedwithin 3%and1%ofcorrespondingDHF val- errorsin the DF values are 10–15%;RCC-SD results are
ues[6,7](Table I). Thisagreementconfirmsthe validity within 1–4% of experiment. The improvement in X and
oftheapproximationsmadebyus. Inparticular,freezing M uponinclusionofcorrelationisexpectedtobesimilar.
the inner core shells is justified, as inner core relaxation Concluding remarks. Note that the codes developed
effects have little influence on the properties calculated for GRECP/RCC calculation followed by nonvariational
here, a conclusion already drawn by Quiney et al. [7]. one-center restoration in heavy cores are equally appli-
Much larger differences occur for the X parameter. cable to calculation of other properties described by op-
HeretherearealsolargedifferencesbetweenthetwoDHF erators singular near nuclei (hyperfine structure, quan-
calculations, which cannot be explained by the small tumelectodynamiceffects,etc.). BecausetheFock-space
change in internuclear separation. The value of X may RCC-SD approach [14] is used, the two-step method is
beexpectedtobelessstablethanM,becauseitisdeter- applicable to both closed-shell and open-shell systems,
minedbythederivativeoftheelectronicdensityattheTl including excited states. In particular, calculations for
nucleus and involves large cancellations [7] between con- the ground state of YbF and for excited states of PbO
tributions of large and small components, each of which are in progress now. Triple and higher cluster ampli-
is about 20 times larger than their sum. Thus, a strong tudes in the valence region are important for chemical
dependence of X on the basis used may be expected. and spectroscopic properties, but not for the effects dis-
The DHF values collected in Table I indeed show such cussed here, as concluded from previous calculations for
dependence. Results obtained in Refs. [6] and [7] with YbF [12]. These excitations are believed to be unim-
comparableeven-temperedbasissets,(28s28p12d8f)and portantinthecoreregiontoo. Wethereforesuggestthat
(28s28p14d8f),areratherclose,differingby340a.u. Im- furtherimprovementinthecorrelationtreatmentwillnot
provingthe Tlbasisto (34s34p16d9f)[7]increasesX by seriously affect our M and X values.
650 a.u. or 8%. Further improvement of the basis may Acknowledgments. We are grateful to M. G. Kozlov,
be expected to yield even higher X values. The numeri- I. A. Mitropolsky, and V. M. Shabaev for valuable dis-
cal basis functions obtained in atomic DHF calculations cussions. This work was supported by INTAS grant No.
andusedfor the restorationarehighly accuratenearthe 96–1266. AP, NM, TI and AT are grateful to RFBR for
nucleus, so that our RCC-S value for X, which is higher grants No. 99–03–33249, No. 01–03–06334, No. 01–03–
thanthatofQuineyetal.[7],seemsreasonable. Thedif- 06335andto CRDFforGrantNo. 10469. WorkatTAU
ferentnuclearmodels usedinthe presentandDHF [6, 7] was supported by the Israel Science Foundation and the
calculations may also contribute to the disagreement in U.S.-Israel Binational Science Foundation.
4
TABLE I: Calculated X and M parameters [Eqs. (3) and (5)] for the 205TlF ground state, compared with DHF values with
differentbasissets[6,7]. Individualshellcontributionsarecalculatedfromspin-averagedGRECP/SCForbitals. GRECP/RCC-
S results includespin-orbit interaction, and GRECP/RCC-SD values also account for electron correlation. All valuesin a.u.
Re=2.0844˚A R=2.1˚A
Expansion s,p s,p,d s,p,d,f s,p s,p s,p,d,f s,p
Shell: maincontribution M X M X
1σ2 : 1s2(F) 0.01 0.02 0.02 3 0.00 0.02 1
2σ2 : 5s2(Tl) −2.49 −2.49 −2.49 -1114 −2.44 −2.44 -1089
3σ2 : 5p2(Tl) 4.21 3.91 3.91 1897 4.10 3.82 1851
4σ2 : 2s2z(F) −0.79 −0.64 −0.64 -358 −0.74 −0.60 -335
56σσ22 :: 5(6ds2z(2T(Tl)l+) 2pz(F))2 −−09..0318 −−100..0045 −−100..0056 -441-42 −−09..0318 −−100..0052 -442-22
7σ2 : (6s(Tl)−2pz(F))2 28.13 27.19 27.19 12954 27.98 27.07 12893
1π4 : 5p25p2(Tl) 0.00 −0.26 −0.26 0 0.00 −0.25 0
x y
2π4 : 5d2 5d2 (Tl) 0.00 0.31 0.30 0 0.00 0.27 0
xz yz
3π4 : 2p22p2(F) 0.00 −0.39 −0.40 0 0.00 −0.38 0
x y
1δ4 : 5d2x2−y25d2xy(Tl) 0.00 0.00 −0.02 0 0.00 −0.02 0
TotalSCF(spin-averaged) 19.67 17.56 17.51 8967 19.52 17.43 8897
GRECP/RCC-S 16.12 13.84 9813 16.02 13.82 9726
DHF[6] Tl:(28s28p12d8f) 15.61 7743
DHF[7] Tl:(25s25p12d8f) 13.64a 8098
Tl:(28s28p14d8f) 13.62a 8089
Tl:(31s31p15d8f) 13.66a 8492
Tl:(34s34p16d9f) 13.63a 8747
GRECP/RCC-SD 11.50 7635
aM iscalculatedinRef.[7]usingtwo-centermolecularspinors,correspondingtoinfiniteLmax inEq.(7).
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