Heavy-atom molecules as key objects to study nonconservation

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

Outline • Why measure electric dipole moment (EDM) of the electron? • Milestones in studying PNC effects • Current status of the electron EDM (e. EDM) search • How to measure e. EDM – Recent experiments with heavy-atom molecules & solids • Electronic structure modeling for e. EDM 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 e. EDM experiments – …

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

Hamiltonian for an EDM in electric field: /2 ~

Status Experimental limit on the electron EDM: |de| < 1. 6 10 -27 e cm [B. Regan, E. Commins, C. Schmidt, D. De. Mille, PRL 88, 071805 (2002)] Physics model |de| (e·cm) Standard Model <10 -38 Left-right symmetry 10 -26 -10 -28 Multi-Higgs 10 -27 -10 -28 Technicolor ~10 -29 Supersymmetry ~10 -25, 10 -27 … Estimated with the KL 0 (T-odd!) decay exptl data

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

P, T-odd effects in heavy-atom molecules: Ш 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 e. EDM enhancement Eeff /Eext α 2 Z 3 EDMs of charged particles e-, [V. Flambaum, Sov. J. Nucl. Phys. 24 (1976)] p etc. can be studied ! ) Ш 1967: Sandars: in polar heavy-atom molecules Emol /Eext >> 1. He initiated the search for the P, T-odd effects on 205 Tl. F and estimated these effects semiempirically (Eeff 20 k. V/cm on a valence proton). Ш 1991: The last series of the 205 Tl. F experiments is finished by Hinds group at Yale (USA) and the best limitation on the proton EDM, dp=(-4 ± 6)x 10 -23 e cm, is obtained. Ш 2002: Petrov et al. recalculated it with RCC as dp=(-1. 7 ± 2. 8)x 10 -23 e cm.

P, T-odd effects in heavy-atom molecules (cont. ): Ш 1978: Labzowsky: ideas to use diatomic radicals Cu. O, Cu. Se due to additional enhancement of P-odd, because of the closeness of levels of opposite parity in -doublets having a 2Π 1/2 ground state, Emol /Eext 105. Ш 1978: Sushkov & Flambaum, and in 1979 Gorshkov, 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 2Σ 1/2 ground state. Ø Many new molecules, molecular cation and solids are considered up-todate for the e. EDM search, mainly by Novosibirsk & SPb groups. Ø 2002: The last series of the 205 Tl beam experiment is finished at Berkeley (USA) and the best to-date limitation on de, |de| < 1. 6 10 -27 e cm, is obtained Ø 2002: The first results are obtained by Hinds group on the 174 Yb. F molecular beam at Sassex (UK) for the electron EDM, de=(-0. 2 ± 3. 2)x 10 -26 e cm; Ø 2010 (? ? ? ): some new limitation on de is obtained on Yb. F.

Experiments on the electron EDM Search Heavy-atom polar molecules and cations: ь Yb. F-radical beam (E. Hinds: Imperial college, London, UK); ь Th. O* beam [& Pb. O* in optic cell ] (ACME collaboration: D. De. Mille: Yale Uni. ; J. Doyle & G. Gabrielse: Harvard); ь Pb. F radicals in a Stark trap (N. Shafer-Ray: Oklahoma); ь Hf. F+ (& Th. F+, Pt. H+ …) trapped cations (E. Cornell: JILA, Boulder); ь WC (3Δ 1 – ground state) molecular beam (A. E. Leanhard: Michigan U. ) Solids: ь Gd-Ga Garnet (S. Lamoreaux: LANL ; C. -Y. Liu: Indiana) ь Gd-Iron Garnet (L. Hunter: Amherst), ь Eu 0. 5 Ba 0. 5 Ti. O 3 (perovskite, ferroelectric structure) (S. Lamoreaux: Yale Uni; J. Haase: Leipzig Uni; O. Sushkov: UNSW).

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 | | / Elab characterizes the e. EDM 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:

Calculations of PNC effects in heavy-atom molecules: Ш First ab initio nonrelativistic calculations of P, T-parity nonconservation effects in Tl. F followed by the relativistic scaling were performed by Hinds & Sandars in 1980 and by Coveney & Sandars in 1983 (Oxford, UK). Ш A series of semiempirical calculations was performed since 1978 by Kozlov & Labzowskii (St. Petersburg); Sushkov, Flambaum & Khriplovich (Novosibirsk) for many heavy-atom molecules. Ш Two-step (RECP / one-center-restoration) relativistic calculations at SPb. SU, PNPI: RECP = Relativistic Effective Core Potential method without correlations: on Pb. F & Hg. F (1985 -1991); with correlations: on Yb. F (1996, 1998), Ba. F (1997), Tl. F (2002), Pb. O* (2004), HI+ (2005), liquid Xe & Hf. F+ (2006+); Pt. H+(2009) Ш First Dirac-Fock calculations on Tl. F (1997) and Yb. F (1998) are performed by Parpia (USA) and by Quiney et al. (EU). In 2006, correlation four-component calculation of Ba. F and Yb. F are performed by Indian group (Nayak & Chaudhuri). Ш … Pt. H+, Th. O & Th. F+ (2008) are performed “semi-ab-initio” by Meyer & Bohn (JILA, Boulder, USA).

Methods of calculations • Effective Hamiltonian(s): Generalized RECP / NOCR methods (SPb. SU-PNPI): A. V. Titov & N. S. Mosyagin, IJQC 71, 359 (1999); A. V. Titov et al. , PTCP B 15, 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. Mosyagin et al. , JPB, 33 (2000); T. A. Isaev et al, JPB, 33 (2000); ANO basis sets for light atoms.

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 достаточного 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 издержек в зависимости от задачи: Ø «large-core» ПП (наиболее экономичные, плохая точность) Ø «small-core» ПП (менее экономичные, хорошая точность) Ø корреляционный псевдопотенциал Ø возможность восстановления электронной структуры в остовах. При универсальности метода ПП он является наиболее гибким в расчетах электронной структуры.

Radial parts of large components of spinors 5 s 1/2 and 6 s 1/2 and of corresponding pseudospinors for the Thallium atom.

«Согласованные-по-форме» ПП Наиболее важные особенности СФ ПП являются следствием двух естественных ограничений при его построении: Ø требования «жесткости» ПП в остове (r<Rc) (с учетом свойства жесткости исходного атомного потенциала по сравнению с амплитудами взаимодействий в валентной области и валентными (орбитальными) энергиями); Ø 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). ü 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. ); ü точность расчетов с ПП становится прогнозируемой и управляемой, а процедура восстановления орбиталей в остове – обоснованной.

Radial parts of the 7 s 1/2 spinor (all-electron Dirac-Fock) and pseudospinor 32 -electron GRECP/SCF) of Uranium for the state averaged over the nonrelativistic 5 f 26 d 17 s 2 configuration and their difference multiplied by 1000.

Nonvariational One-Center Restoration (NOCR) of electronic structure in cores of heavy-atoms in a molecule:

Advantages & disadvantages of GRECP / NOCR scheme: Ø 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); Ø «Естественное» представление в расчете остовных функций как спиноров, а валентных – как спин-орбиталей – за счет «приближенного» учета их ортогональности в ПП-расчете, что невозможно в полноэлектронном расчете; Ø Выполнение молекулярного расчета в спин-орбитальном базисе дает очень большую экономию ресурсов, позволяет существенно повысить точность; Ø Хотя молекулярные псевдоорбитали не ортогональны (строго!) к остовным, но при их восстановлении также восстанавливается и их точная ортогональность; при этом восстановленные валентные функции уже являются не спин-орбиталями, но спинорами! Ø 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; Ø Не учитывается поляризация (релаксация) остова, кот. обычно невелика; она может быть учтена в «вариационной» схеме восстановления.

First two-step calculations of 199 Hg. F and 207 Pb. F

HI+ model: Iodine (Z=53): [Kr] 4 s 24 p 64 d 10 5 s 25 p 5 + H+: 1 s 0 [ outer core ] [valence] HI+ ground state: 2 3/2; configuration: […] 2 1/22 3/21 (derived from 5 p 5) Highest doubly occupied -orbital is bonding and most “mixed”: 5 p 0(I) +1 s(H) is not highest-by-energy among the occupied orbitals, but it gives 77% to the molecule-frame dipole moment.

HI+: <Basis sets>: I: [5 s 5 p 3 d 2 f] + H: [4 s 3 p 2 d]

Calculations of PNC effects in heavy-atom molecules (continued): (1 GV/cm = 0. 242 1024 Hz/e∙cm) 1 Old (2006) (2008) [См. постер К. Бакланова] [См. постер А. Петрова] ? ? 60 (2010) Pt. H+ 28 (2009) 73

Неэмпирический расчёт Eu++ во внешнем электрическом поле [См. постер Л. Скрипникова] Eu++: 4 s 24 p 6 4 d 10 5 s 2 5 p 64 f 7 = -4. 6 Вклады в K от матричных элементов s-p, p-d, d-f : s p d f s - -3. 3 0 0 p - 0. 3 0 d - -1. 6 f -

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

Concluding remarks: The e. EDM experiments on heavy-atom molecules (and solids ? ) are of key importance for modern theory of fundamental interactions and symmetries – window for a new physics beyond the Standard model. High-accuracy calculations of prospective heavy-atom systems are of increasing interest for the e. EDM 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 [A. V. Titov et al. , PTCP B 15, 253 (2006)] 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!

The end.

Gadolinium Gallium Garnet (Gd 3 Ga 5 O 12) • Gd 3+ in GGG - 4 f 75 d 06 s 0 (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: {A 3}[B 2](C 3)O 12 –A {dodecahedron}: M 3 • Ca, Mn, Fe, R (La, . . Gd, . . Lu) –B [octahedron], C (tetrahedron): • Fe, Ga, …

Methods of calculations: GRECP & NOCR

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 e. EDM measurements keeping the cold ions in the trap.

Mass-spectrometry:

Hf. F+ model Proposal: Hf. H+: [L. Sinclair et al. , Bull. Am. Phys. Soc. 450, 134 (2005)]; Hf. F+ & Th. F+: [E. Cornell & A. Leanhardt, private communication]. Calculation: Hf. F+: [A. N. Petrov et al. , PRA 76, 030501(R) (2007)+ …] Hf. F+ working state - 3 1; config. : […] 12 21 1, , 2 1 Hf 2+: [… 4 f 14 ]5 s 25 p 6 5 d 1 6 s 1 + F– : 1 s 2 2 p 6 [outer core] [ valence ] [core] [ valence ] 1 st question: which state is the ground one, 3 or 1 1 (config. : […] 12 22 )? ! 1 (and if 3 1 is not the ground one, how to populate it? ) 2 nd question: which is effective field on e-, Eeff ? 3 rd question: which transitions to excited states (3 , 1 ) can be used to measure the EDM signals?

Our SODCI calculations with Hf. F+

Hf. F+

Our SODCI calculations with Hf. F+ The GRECP (with 60 e- in core) and basis set for Hf (Z=72) is generated and used in 10 e- & 20 e-SODCI calculations; basis sets: Hf: (12 s, 16 p, 16 d, 10 f, 10 g) / [6 s, 5 p, 5 d, 3 f, 1 g] our GC basis; F: (14 s, 9 p, 4 d, 3 f) / [4 s, 3 p, 2 d, 1 f] ANO basis set. Up to 12× 106 selected SAFs are used in SODCI calculations. • Effective electric field on e- : Eint = 2. 5 1010 V/cm (5. 8 1024 Hz/e∙cm); • Hyperfine constants: A|| [177 Hf] = -1239 MHz ; A|| [19 F] = -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 (moleculeframe and transition), radiative lifetimes are calculated for ten lowest states. Errors for energetic properties are about 500 cm-1; • 4 f 14 relaxation is shown to be not important for these studies.