Anatoly V. Titov

1. Heavy-atom molecules as key objects to study nonconservation of time-reversal symmetry (EDM of electron) [qchem.pnpi.spb.ru] PNPI QChem Group: Anatoly V. Titov A.N.Petrov, L.V. Skripnikov and N.S. Mosyagin B.P. Konstantinov PNPI RAS, St.-Petersburg State University, St.-Petersburg, RUSSIA

2. Outline • Why measure electric dipole moment (EDM) of the electron? • Milestones in studying PNC effects • Current status of the electron EDM (eEDM) search • How to measure eEDM • Recent experiments with heavy-atom molecules & solids • Electronic structure modeling for eEDM studies • Why heavy-atom systems and why relativistic effects are important • Semi-empirical; RECP / one-center restoration; four-component • Recent calculations of electronic properties in heavy-atom molecules for the eEDM experiments • …

3. d S Why measure EDMs? • EDM is the electric dipole moment of an elementary particle. • Dipole momentd is a polar vector (P-odd, where P is the space parity); Nonzero EDMs can exist only due to the both space Parity, P, and Time reversal, T, symmetries violated (P,T-odd interactions) [L. Landau, Pis’ma ZhETP 32, 405(1957)]: The only vector for a particle at rest is its spin, therefore, dshould be directed along to S, (moreover, d = deSsimilar to magnetic moment, where de is a fixed real number once the coordinate system is chosen)! • Spin S is a T-oddpseudo-vector (P-even axial vector), where T is the time-reversal symmetry. Therefore, for P-even and / or T-even world d should be zero!

4. Hamiltonian for an EDM in electric field: /2 ~

5. Status Experimental limit on the electron EDM: |de| < 1.610-27ecm [B. Regan, E. Commins, C. Schmidt, D. DeMille,PRL 88, 071805 (2002)] Estimated with the KL0 (T-odd!) decay exptl data

6. Basic detection scheme of an EDM (for a neutral system with S=1/2, magnetic moment  and EDM d) Single system with coherence time : Nuncorrelated systems measured for time The wavepacket e -iE t | + e-iE t | can be prepared using the electronic spin resonance method Statistical sensitivity: T ~ m

7. P,T-odd effects in heavy-atom molecules: • 1965:Sandars suggests to use heavy atoms to search for EDMs(In the nonrelativistic case Eeff is zero in accord to the Schiff theorem;relativistic eEDM enhancementEeff/Eextα2Z3[V.Flambaum, Sov.J.Nucl.Phys. 24 (1976)]) EDMs of charged particles e-, p etc. can be studied ! • 1967:Sandars: in polar heavy-atom molecules Emol/Eext >> 1. He initiated the search for the P,T-odd effects on 205TlF and estimated the these effects semiempirically (Eeff 20 kV/cm on a valence proton). • 1991: The last series of the 205TlF experiments is finished by Hinds group at Yale (USA) and the best limitation on the proton EDM, dp=(-4 ± 6)x10-23 ecm, is obtained. • 2002: Petrov et al. recalculated it with RCC as dp=(-1.7 ± 2.8)x10-23 ecm.

8. P,T-odd effects in heavy-atom molecules (cont.): • 1978:Labzowsky: ideas to use diatomic radicals CuO, CuS, CuSe due to additional enhancement of P-odd, because of the closeness of levels of opposite parity in -doublets having a 2Π1/2 ground state,Emol/Eext105. • 1978:Sushkov & Flambaum, and in 1979Gorshkov, Labzowsky & Moskalev: ideas to use diatomic radicals (-doublets) to search for P,T-odd effects including EDM of electron due to additional enhancement. 1984:Sushkov, Flambaum & Khriplovich; Flambaum & Khriplovich,1985 : Kozlov suggest to use diatomics with a 2Σ1/2 ground state. • Many new molecules, molecular cation and solids are considered up-to-date for the eEDM search, mainly by Novosibirsk & SPb groups. • 2002: The last series of the 205Tl beam experiment is finished at Berkeley (USA) and the best to-date limitation on de, |de| < 1.610-27ecm, is obtained • 2002: The first results are obtained by Hinds group on the 174YbF molecular beam at Sassex (UK) for the electron EDM, de=(-0.2 ± 3.2)x10-26 ecm; • 2010 (???): some new limitation on de is obtained on YbF.

9. Experiments on the electron EDM Search Heavy-atom polar molecules and cations: • YbF-radical beam (E.Hinds: Imperial college, London,UK); • ThO*beam [&PbO* in optic cell ] (ACME collaboration: D.DeMille:Yale Uni.; J.Doyle & G.Gabrielse: Harvard); • PbFradicals in a Stark trap (N.Shafer-Ray:Oklahoma); • HfF+(&ThF+, PtH+ …) trapped cations (E.Cornell: JILA, Boulder); • WC (3Δ1 – ground state)molecular beam (A.E.Leanhard: Michigan U.) Solids: • Gd-Ga Garnet (S. Lamoreaux: LANL ; C.-Y. Liu: Indiana) • Gd-Iron Garnet (L. Hunter: Amherst), • Eu0.5Ba0.5TiO3 (perovskite, ferroelectric structure) (S. Lamoreaux: Yale Uni; J.Haase: Leipzig Uni; O.Sushkov: UNSW). • Nuclear EDM / Schiff moment: • Liquid Xe experiment(M.Romalis: Princeton Uni.) • ??? RaO (Flambaum 2008: 500 times enhancement compared to TlF)

10. What should be calculated ? • HP,T-odd = Wd de (Je n), where de=| de|, (Jen)= is projection of the electron momentum on the molecular axis (n); Wd||/ Elabcharacterizes the eEDM enhancement. • The value of Wd ||can be considered as some effective electric field on the electron,Eeff  Wd ||. It is non-zero only due to the relativistic effects! • This field is strongly localized near the heavy nuclei, so the only one-electron-states with small je contribute to Wd: For point nucleus:

11. Calculations of PNC effectsin heavy-atom molecules: • First ab initio nonrelativistic calculations of P,T-parity nonconservation effects in TlF followed by the relativistic scaling were performed by Hinds & Sandars in 1980 and by Coveney & Sandars in 1983 (Oxford, UK). • A series of semiempirical calculations was performed since 1978 by Kozlov& Labzowskii(St.Petersburg); Sushkov, Flambaum & Khriplovich (Novosibirsk) for many heavy-atom molecules. • Two-step (RECP /one-center-restoration) relativistic calculations at SPbSU, PNPI: RECP = Relativistic Effective Core Potential methodwithout correlations: on PbF & HgF (1985-1991); with correlations: on YbF (1996,1998), BaF (1997), TlF (2002), PbO*(2004), HI+(2005), liquid Xe & HfF+(2006+); PtH+(2009) • First Dirac-Fock calculations on TlF (1997) and YbF (1998) are performed by Parpia (USA) and by Quiney et al. (EU). In 2006, correlation four-component calculation of BaF andYbFareperformed by Indian group (Nayak & Chaudhuri). • … PtH+, ThO & ThF+ (2008) are performed “semi-ab-initio” by Meyer & Bohn (JILA, Boulder, USA).

12. Methods of calculations • Effective Hamiltonian(s): Generalized RECP / NOCR methods (SPbSU-PNPI): A.V. Titov & N.S. Mosyagin, IJQC 71, 359(1999); A.V. Titov et al., PTCP B15, 253 (2006). • Correlation Methods: RCC: U.Kaldor, E.Eliav, A. Landau, Tel-Aviv Uni., Israel; SODCI: R.Buenker et al., Uni. of Wuppertal, Germany); Developments: A.V. Titov et al., IJQC 81, 409(2001); T.A. Isaev et al, JPB 33, 5139 (2000); A.N.Petrov et al., PRA , 72, 022505 (2005). • Basis Sets: GC-basis: N.S.Mosyaginet al., JPB, 33 (2000); T.A. Isaev et al, JPB, 33 (2000); ANObasis sets for light atoms.

13. Что делает псевдопотенциал (ПП) Задачей метода ПП является сведение расчета электронной структуры системы к явному рассмотрению в расчете только валентных электронов, т.е. • исключение химически неактивных (остовных) электронов из расчета при сохранении достаточно точного описания электронной структуры и взаимодействий в валентной области; • обеспечение «ортогональности» (принципа Паули) по отношению к занятым (но явно исключенным) остовным состояниям, т.е. предотвращение «провала» валентных электронов в эти состояния; • эффективный учет релятивистских эффектов (scalar + SO + Breit); • сглаживание псевдоспиноров для минимизации размеров атомных базисов и вычислительных издержекв зависимости от задачи: • «large-core» ПП (наиболее экономичные, плохая точность) • «small-core» ПП (менее экономичные, хорошая точность) • корреляционный псевдопотенциал • возможность восстановления электронной структуры в остовах. При универсальности метода ПП он является наиболее гибким в расчетах электронной структуры.

14. Radial parts of large components of spinors 5s1/2 and 6s1/2 and ofcorresponding pseudospinors for the Thallium atom.

15. «Согласованные-по-форме» ПП • Наиболее важные особенности СФ ПП являются следствием двух естественныхограничений при его построении: • требования «жесткости» ПП в остове (r<Rc) (с учетом свойства жесткости исходного атомного потенциала по сравнению с амплитудами взаимодействий в валентной областии валентными (орбитальными) энергиями); • требования «физичности» ПП в валентной области (r>Rc) (т.е. взаимодействия, смоделированные посредством ПП должны с высокой точностью отслеживать исходные атомные потенциалы в валентной области). • как валентные, так и виртуальные псевдоорбитали будут с высокой точность отслеживать исходные атомные орбитали в валентной области вместе с их орбитальными энергиями даже при введении возмущения в валентной области (хим.связь, внешние поля и т.п.); • точность расчетов с ПП становится прогнозируемой иуправляемой, а процедура восстановления орбиталей в остове – обоснованной.

16. Radial parts of the 7s1/2 spinor (all-electron Dirac-Fock) and pseudospinor32-electron GRECP/SCF) of Uranium for the state averaged over thenonrelativistic 5f26d17s2configuration andtheirdifference multiplied by 1000.

17. Nonvariational One-Center Restoration (NOCR)of electronic structure in cores of heavy-atoms in a molecule: [A.Titov, PhD Thesis (1985)]

18. Advantages & disadvantages of GRECP / NOCR scheme: • Удается естественным образом разделить задачу на две части – атомную (с большим числом электронов и численными функциями) и молекулярную (с минимальным числом электронов и гауссовыми функциями); • «Естественное» представление в расчете остовных функций как спиноров, а валентных – как спин-орбиталей–за счет «приближенного» учета их орто-гональности вПП-расчете, что невозможно в полноэлектронном расчете; • Выполнение молекулярного расчета в спин-орбитальном базисе дает очень большую экономию ресурсов, позволяет существенно повысить точность; • Хотя молекулярные псевдоорбитали не ортогональны (строго!) к остовным, но при их восстановлении также восстанавливается иих точная ортогональность; при этом восстановленные валентные функции уже являются не спин-орбиталями, но спинорами! • Спин-орбитальным взаимодействием в валентном расчете с ПП часто можно пренебречь (или учесть приближенно), и «включить» его только при восстановлении в остовах, что очень важно в расчетах сложных соединений; • Не учитывается поляризация (релаксация) остова, кот. обычно невелика; она может быть учтена в «вариационной» схеме восстановления. [A.Titov, PhD Thesis (1985)]

19. First two-step calculations of 199HgF and 207PbF

20. HI+ model: • Iodine (Z=53): [Kr] 4s24p64d10 5s25p5 + H+: 1s0 [ outer core ] [valence] [valence] • HI+ground state: 23/2; configuration: […] 2 1/22 3/21(derived from 5p5) • Highestdoublyoccupied-orbitalis bondingandmost “mixed”:  5p0(I)+1s(H) • isnothighest-by-energyamong theoccupiedorbitals,butit gives77%tothemolecule-frame dipole moment.

21. HI+: <Basis sets>: I:[5s5p3d2f] + H:[4s3p2d]

22. HI+: <Basis sets>: I:[5s5p3d2f] + H:[4s3p2d]

23. Calculations of PNC effectsin heavy-atom molecules (continued): Old (2006) (1 GV/cm = 0.24210241 Hz/e∙cm) (2008) [См. постер К. Бакланова] [См. постер А. Петрова] ?? 60 (2010) PtH+28 (2009) 73

24. Неэмпирический расчёт Eu++ во внешнем электрическом поле [См. постер Л.Скрипникова] Eu++: 4s24p6 4d10 5s2 5p64f7 = -4.6 Вклады в K от матричных элементов s-p, p-d, d-f :

25. Thanks to: • L.Labzowskii – initiator & supervisor of the PNC study at SPbSU & PNPI • M. Kozlov (PNPI) • Yu.Yu. Dmitriev, A.Mitrushchenkov (SPbSU) • I.Khriplovich, O.Sushkov & V.Flambaum (Novosibirsk & Sydney, Australia) • D.DeMille (Yale, USA) • E.Cornell (Boulder, USA) • E. Eliav & U.Kaldor (Tel Aviv, Israel) • R.Buenker & A.Alekseyev (Wuppertal, Germany) • A.Zaitsevskii (Kurchatovskii institute, Moscow)

26. Concluding remarks: • The eEDM experiments on heavy-atom molecules (and solids ?) are of key importance for modern theory of fundamental interactions and symmetries – windowfor a new physics beyond the Standard model. • High-accuracy calculations of prospective heavy-atom systems are of increasing interest for the eEDM experiment. • The two-step method – RECP / one-center-restoration – has better flexibility than the four-component approaches and good prospects for further improvement of accuracy Accuracy is limited by present possibilities of correlation methods rather than by basis set limitations, RECP and other approximations. Extension of the method to study more complicated systems (solids etc.) is simple (in contrast to four-component ones); applicability to study other physical-chemical properties is straightforward. • Further development of accurate effective Hamiltonians, correlation methods and new basis sets is highly desirable for actinides/lanthanides! [A.V.Titov et al., PTCP B15, 253 (2006)]

27. The end.

28. Gadolinium Gallium Garnet (Gd3Ga5O12) • Gd3+ in GGG - 4f75d06s0 (7 unpaired electrons) • Atomic enhancement factor = -4.91.6 • Langevin paramagnet • Dielectric constant ~ 12 • Low electrical conductivity and high dielectric strength • Volume resistivity = 1016 -cm • Dielectric strength = 10 MV/cm for amorphous sample • Garnet Structure: • {A3}[B2](C3)O12 • A {dodecahedron}: M3 • Ca, Mn, Fe, R (La,..Gd,..Lu) • B [octahedron],C (tetrahedron): • Fe, Ga, …

29. Methods of calculations: GRECP & NOCR

30. Why use molecular ions? [R.Stutz & E.Cornell,Bull.Am.Phys.Soc.49, 46 (2004)] • Ions are easy to trap (in RF quadruple trap); • Potential for long spin coherence times (ion-ion repulsion); • Can get Eeff/Elab= 109(for Ω>1/2 have closely spaced levels of opposite parity fully polarized with E ~ 10 V/cm); • Rotating external electric field can be used for eEDM measurements keeping the cold ions in the trap.

33. Mass-spectrometry:

35. HfF+ model Proposal:HfH+:[L.Sinclair et al.,Bull.Am.Phys.Soc.450, 134 (2005)]; HfF+& ThF+: [E.Cornell & A.Leanhardt, private communication]. Calculation: HfF+: [A.N.Petrov et al.,PRA76, 030501(R) (2007)+ …] • HfF+working state - 31; config.: […] 12 21 1, • Hf2+: […4f14 ]5s25p6 5d16s1 + F– : 1s2 2s2 2p6 • [outer core] [ valence ] [core] [ valence ] • 1st question: which state is the ground one, • 31 or 11 (config.: […] 12 22 )?! • (and if 31 is not the ground one, how to populate it?) • 2nd question: which is effective field on e-, Eeff ? • 3rd question: which transitions to excited states(3, 1) can be used to measure the EDM signals? , 2 1 30-.0+,1,2, 32,3

36. Our SODCI calculations withHfF+

39. HfF+

45. Our SODCI calculations withHfF+ The GRECP (with 60 e- in core) and basis set for Hf (Z=72) is generated and used in 10e- & 20e-SODCI calculations; basis sets: Hf: (12s,16p,16d,10f,10g) / [6s,5p,5d,3f,1g] our GC basis; F: (14s,9p,4d,3f) / [4s,3p,2d,1f] ANO basis set. Up to 12×106selected SAFs are used in SODCI calculations. • Effective electric field one- : Eint = 2.51010 V/cm (5.81024 Hz/e∙cm); • Hyperfine constants: A|| [177Hf] = -1239 MHz ; A|| [19F] = -58 MHz ; • The ground state is 11 and 31 is the long-lived (1/2~0.5 sec) lying only about 2000 cm-1 higher [calc-n by Skripnikov L., Dec.2008]; • Spectroscopic constants and curves, electric dipole moments (molecule-frame and transition), radiative lifetimes are calculated for ten lowest states. Errors for energetic properties are about 500 cm-1; • 4f14relaxation is shown to be not important for these studies.