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

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 –…

Nonzero EDMs can exist only due to the both space Parity, P, and Time reversal, T, symmetries violated (P,T-odd interactions) [L. Landau, Pisma ZhETP 32, 405 (1957)]: Why measure EDMs? EDM is the electric dipole moment of an elementary particle. Dipole moment d is a polar vector (P-odd, where P is the space parity); d S The only vector for a particle at rest is its spin, therefore, d should be directed along to S, (moreover, d = d e S similar to magnetic moment, where d e is a fixed real number once the coordinate system is chosen)! Therefore, for P-even and / or T-even world d should be zero! Spin S is a T-odd pseudo-vector (P-even axial vector), where T is the time-reversal symmetry.

Hamiltonian for an EDM in electric field: ~ /2

Milestones in studying PNC effects: 1949: Analyzing Einsteins relativity theory (physical dynamic theories or Poincaré group), Dirac states that P- and T-invariance is not necessary attribute of the nature laws. 1950: Purcell & Ramsey state that the validity of P- or/and T-odd theories must be confirmed experimentally. They initiated search for the neutron EDM. 1956: Puzzle of τ–θ mesons: θ2π(L=0); τ3π(L=0), but m τ m θ ; T τ 1/2 T θ 1/2. Analyzing the decays, Lee and Yang have suggested nonconservation of space parity, P, in decay of K + (J P =0 ) on π-mesons (J P =0 ) / K π3 τ ;K π2 θ /. They suggest to study spirality (p·S) in decay experiments. 1957: Wu et al. discover the P-violation in β-decay of 60 Co ( n p + e + ν e ). To save the world from the left-handed asymmetry, Landau, Lee & Yang suggest invariance of the nature laws with respect to the combined CP-parity. Landau: d S, i.e. the only P,T-odd forces can give nonzero d, otherwise d=0.

Milestones in studying PNC effects (cont.): 1964: Christenson, Cronin, Fitch & Turlay discover nonconservation of the combined inversion, CP, in decay K 0 L (CP=-1) 2π (CP=+1 !). 1978: Barkov A & Zolotarev (Novosibirsk, Russia, idea of Khriplovich) have found that the atomic Bi vapor rotate the polarization plane of the laser ray (in search for the neutral weak interaction of electrons with nuclei). Similar experiments were performed by Sandars et al. (Oxford. UK), Fortson et al. (Seattle, USA), and by Sobelmann (FIAN, Russia). A few months later it was confirmed by Atwood et al. (Stanford, USA) in deeply-inelastic scattering of electrons on deuterium & hydrogen. 1983: Discovery of W ± and Z 0 bozons by Rubbia et al. in CERN confirmed the Standard electroweak model.

Standard model (SM): electro-weak interactions s p i n o r s v e c t o r s scalar leptons quarks (q) photon bosons boson Standard models fundamental particles: Charged weak current of quarks: h e W, Z, W + … u c t d s b PNPN V ud V us V ub V cd V cs V cb V td V ts V tb U [ Sh.Glashow (1961), S.Weinberg (1967, 1972), A.Salam (1968) ] J = 2 P L UN L (L left chirality, N L 1/2(1 5 )N ) Only one parameter in U can be interpreted as CP (or T)-violating phase! [ M.Kobayashi & T.Maskawa (1973) ] (Noble Prize 2008) Search for the Higgs boson is a key experiment at LHC in CERN Dirac matrices: 5 = - 5 ; N L 1/2(1 5 )N; P L 1/2 P (1 + 5 ) Source for W ± fields; H = J A[ W ± ] Z: +2/3 -1/3

Нобелевскую премию по физике 2008 г. получили японские ученые: Лауреатом Нобелевской премии в области физики за 2008 год стал гражданин США 87- летний Йисиро Намбу, который живет в Америке с 1971 года и сейчас работает в Институте Энрико Ферми при университете Чикаго, а также представители Японии - 64-летний Макото Кобаяси (университет Цукуба) и 68-летний Тосихидэ Маскава (университет Киото). Японцы удостоились награды за открытие "причин симметричного расщепления, согласно которому природа должна обладать как минимум тремя семействами кварков". Йисиро Намбу еще в 1960 году сформулировал математическое описание спонтанного нарушения симметрии в области физики элементарных частиц. Его идеи пронизывают наиболее развитую теорию физики элементарных частиц - Стандартную Модель (она описывает электромагнитное, слабое и сильное взаимодействие всех элементарных частиц и не включает в себя гравитацию). Здесь соединяются самые малые частицы материи и три из четырех сил природы, говорится в решении Нобелевского комитета. Явление нарушения Т-симметрии, стало полной неожиданностью, когда было открыто в ходе экспериментов с элементарными частицами в 1964 году. Японским ученым Макото Кобаяси и Тосихидэ Маскава удалось ввести нарушение временной симметрии в рамки Стандартной Модели, однако они обнаружили, что саму модель необходимо расширить до трех семейств кварков. Лишь в последние годы их открытие смогло получить полное подтверждение благодаря мощнейшим ускорителям элементарных частиц.ускорителям элементарных частиц Нарушения симметрии, подобные открытым нобелевскими лауреатами, лежат в основе происхождения Вселенной в результате "Большого взрыва" 14 млрд лет назад. Детали этого процесса по-прежнему остаются загадкой. Возможно, новые пути ее решения ученые узнают с помощью ускорителя LHC в швейцарском Церне.

P,T-violation = window to physics beyond the SM: e w q -loop w e = 0; = 0 ww ee W q q × [M. Pospelov & I. Khriplovich, J.Nucl.Phys. 53(4) (1991)] Standard model: T-violation Matrix U can be rewritten as: s ud s us r ub e -i s cd +r cd e i s cs +r cs e i s cb s td +r td e i s ts +r ts e i s tb K 0 L (CP=1) 2π (CP=+1) Phase can be phenomenologically fixed within the Standard Model using exptl T-violating decay data: [Christenson, Cronin, Fitch & Turlay (1964)]

Experimental limit on the electron EDM: |d e | < e cm [B. Regan, E. Commins, C. Schmidt, D. DeMille, PRL 88, (2002)] Physics model|d e | (e·cm) Standard Model

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

P,T-odd effects in heavy-atom molecules: 1965: Sandars suggests to use heavy atoms to search for EDMs ( In the nonrelativistic case E eff is zero in accord to the Schiff theorem; relativistic eEDM enhancement E eff /E ext α 2 Z 3 [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 E mol /E ext >> 1. He initiated the search for the P,T-odd effects on 205 TlF and estimated the these effects semiempirically (E eff 20 kV/cm on a valence proton). 1978: Labzowsky and then Sushkov & Flambaum: ideas to use diatomic radicals (PbF, BiS) due to additional enhancement of P-odd and P,T-odd effects, respectively, because of the closeness of levels of opposite parity in -doublets having a 2 Π 1/2 ground state, E mol /E ext Then Sushkov, Flambaum & Khriplovich (1984); Flambaum & Khriplovich (1985); Kozlov (1985) suggest to use diatomics with a 2 Σ 1/2 ground state. Many new molecules, molecular cation and solids are considered up-to-date for the eEDM search.

P,T-odd effects in heavy-atom molecules: 1965: Sandars suggests to use heavy atoms to search for EDMs ( In the nonrelativistic case E eff is zero in accord to the Schiff theorem; relativistic eEDM enhancement E eff /E ext α 2 Z 3 [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 E mol /E ext >> 1. He initiated the search for the P,T-odd effects on 205 TlF and estimated the these effects semiempirically (E eff 20 kV/cm on a valence proton). 1991: The last series of the 205 TlF experiments is finished by Hinds group at Yale (USA) and the best limitation on the proton EDM, d p =(-4 ± 6)x e cm, is obtained. 2002: Petrov et al. recalculated it with RCC as d p =(-1.7 ± 2.8)x e cm.

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 2 Π 1/2 ground state, E mol /E ext : 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 2 Σ 1/2 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 205 Tl beam experiment is finished at Berkeley (USA) and the best to-date limitation on d e, |d e | < e cm, is obtained 2002: The first results are obtained by Hinds group on the 174 YbF molecular beam at Sassex (UK) for the electron EDM, d e =(-0.2 ± 3.2)x e cm; 2010 (???): some new limitation on d e is obtained on YbF.

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); PbF radicals in a Stark trap (N.Shafer-Ray: Oklahoma); HfF + (& ThF +, PtH + …) trapped cations (E.Cornell: JILA, Boulder); WC ( 3 Δ 1 – 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), Eu 0.5 Ba 0.5 TiO 3 (perovskite, ferroelectric structure) (S. Lamoreaux: Yale Uni; J.Haase: Leipzig Uni; O.Sushkov: UNSW). Experiments on the electron EDM Search Nuclear EDM / Schiff moment: Liquid Xe experiment (M.Romalis: Princeton Uni.) ??? RaO (Flambaum 2008: 500 times enhancement compared to TlF)

H P,T-odd = W d d e (J e n), where d e =| d e |, (J e n)= is projection of the electron momentum on the molecular axis (n); W d | | / E lab characterizes the eEDM enhancement. The value of W d | | can be considered as some effective electric field on the electron, E eff W d | |. 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 j e contribute to W d : What should be calculated ? For point nucleus:

Spin-rotational Hamiltonian for YbF, HgF etc.

P- and P,T-odd terms in H sr for YbF, HgF etc. For point nucleus:

Calculations of PNC effects in 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 method without correlations: on PbF & HgF ( ); 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 and YbF are performed by Indian group (Nayak & Chaudhuri). … PtH +, ThO & ThF + (2008) are performed semi-ab-initio by Meyer & Bohn (JILA, Boulder, USA).

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, (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. Methods of calculations

The inner core (IC), outer core (OC) and valence (V) electrons are first treated employing different approximations for each (including relaxation of IC shells). GRECP involves both radially-local, separable and Huzinaga- type potentials (shifting the core energies) as its components. The GRECP operator includes terms of other types (self-consistent and term-splitting) for economical treatment of transition metals, lanthanides and actinides. The outermost core pseudoorbitals (nodeless) together with valence pseudoorbitals (nodal) are used for constructing the GRECP components. Quantum electrodynamics effects (Breit etc.) and correlations with the IC shells can be efficiently treated within GRECPs. [Titov & Mosyagin, IJQC 71, 359 (1999)] Generalized relativistic ECP [Petrov et al. JPB (2004); Mosyagin et al., PTCP (2006), JCP (2006)]

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 издержек в 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.

Inner-core / outer-core / valence regions Valence Outer Core IC IC is Inner Core

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

Преимущества метода ПП Точность приближения ПП может быть очень высокой – в пределах химической точности (1 kcal/mol 350 cm eV) – и даже выше, чем в приближении замороженного остова точность расчетов, как правило, лимитируется возможностями корреляционных методов. Осцилляции валентных орбиталей (спиноров) обычно сглаживаются в остовной области тяжелого атома (с одновременным исключением малых компонент четырех компонентных дираковских спиноров из расчетов) значительное сокращение числа примитивных базисных функций. Метод ПП позволяет использовать стандартные программные пакеты для нерелятивистских квантовохимических расчетов и учитывать релятивистские эффекты с помощью оператора ПП. Возможность использования спин-орбитального базиса в расчетах, тогда как взаимодействия с исключенными из расчетов спинорами описываются с помощью спин-зависящих компонент ПП значительное сокращение вычислительных затрат (или улучшение базиса) по сравнению со спинорным базисом в корреляционных расчетах.

«Согласованные-по-форме» ПП Наиболее важные особенности СФ ПП являются следствием двух естественных ограничений при его построении: требования «жесткости» ПП в остове (rR c ) (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 и управляемой, а процедура восстановления орбиталей в остове – обоснованной.

Пропорциональность исходных атомных спиноров

Пропорциональность (псевдо)спиноров (continued)

Generalized RECP operator with separable correction: conventional radially-local RECP operator The separable terms (the second and third lines in Eq.(1)) The separable terms take into account the difference between the potentials acting on the outercore and valence electrons with the same l & j.

The GRECPs provides the level of chemical accuracy (1 kcal/mol or 350 cm -1 ) for valence energies. The GRECP accuracy can be even higher than the accuracy of the frozen core approximation when accounting for the inner core relaxation terms. The cumulative computational precision is limited by current possibilities of correlation methods and codes. The expenses of correlation treatment can be seriously reduced as compared to Dirac-Coulomb-Breit methods when using basis of spin-orbitals instead of spinors. GRECP accuracy

Radial parts of the 7s 1/2 spinor (all-electron Dirac-Fock) and pseudospinor 32-electron GRECP/SCF) of Uranium for the state averaged over the nonrelativistic 5f 2 6d 1 7s 2 configuration and their difference multiplied by 1000.

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

Advantages & disadvantages of GRECP / NOCR scheme : [A.Titov, PhD Thesis (1985)] 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 в ПП-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 и их 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.

First two-step calculations of 199 HgF and 207 PbF

Parameters of the spin-rotational Hamiltonian for 171 YbF.

Why is accurate accounting for correlation important?

Iodine (Z=53): [Kr] 4s 2 4p 6 4d 10 5s 2 5p 5 + H + : 1s 0 [ outer core ] [valence] [valence] HI + ground state: 2 3/2 ; configuration : […] 2 1/2 2 3/2 1 (derived from 5p 5 ) Highest doubly occupied -orbital is bonding and most mixed: 5p 0 (I) +1s(H) is not highest-by-energy among the occupied orbitals, but it gives 77% to the molecule-frame dipole moment. HI + model:

HI + model

HI + : : I:[5s5p3d2f] + H:[4s3p2d]

HI + : : I:[5s5p3d2f] + H:[4s3p2d]

Calculations of PNC effects in heavy-atom molecules (continued): Old (2006)(2008) (1 GV/cm = Hz/ecm) PtH + 28 (2009) (2010) ?? [См. постер А. Петрова] [См. постер К. Бакланова]

Eu ++ : 4s 2 4p 6 4d 10 5s 2 5p 6 4f 7 Вклады в K от матричных элементов s-p, p-d, d-f : spdf s p d f - = -4.6 Неэмпирический расчёт Eu ++ во внешнем электрическом поле [См. постер Л.Скрипникова]

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)

The eEDM 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 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! Concluding remarks: [A.V.Titov et al., PTCP B15, 253 (2006 )]

Thank you!

The end.

(1 V/cm = Hz/ecm)

53 Shapiros proposal -- using a solid state system to measure eEDM Usp. Fiz. Nauk., (1968) B.V. Vasilev and E.V. Kolycheva, Sov. Phys. JETP, 47 [2] 243 (1978) /Ni-Zn ferrite/ d e =( ) e-cm ||

Gadolinium Gallium Garnet (Gd 3 Ga 5 O 12 ) Gd 3+ in GGG - 4f 7 5d 0 6s 0 (7 unpaired electrons) Atomic enhancement factor = Langevin paramagnet Dielectric constant ~ 12 Low electrical conductivity and high dielectric strength Volume resistivity = 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

P- and P,T-odd terms in H sr for YbF, HgF etc.

First two-step calculations of 199 HgF and 207 PbF

Why is accurate accounting for correlation important?

Parameters of the spin-rotational Hamiltonian for 171 YbF.

Contributions from triple and quadruple CC amplitudes to total energies of the Pb terms and errors for the VCIC-corrected transition energies.

PbO* is a Novel System for Measuring Electron EDM |d e | a(1) has very small -doublet splitting complete polarization with small fields (>15 V/cm) –equivalent to E ~ >10 10 V/cm on an atom! PbO is thermodynamically stable –a(1) populated via laser excitation can work in vapor cell. MUCH larger density than beam: PbO Cell (Yale): Tl Beam (Berkeley): N = nV ~ N = nV ~ 10 8

Calculation of PbO*: Starting point PbO* crude model of a(1): two valence electrons, one is excited both nominally in -orbitals: no s-wave component Pb: [Xe] 5s 2 5p 6 5d 10 6s 2 6p 2 ; O: 1s 2 2s 2 2p 4. [ outer core ] [ valence ] [OC] [ V ] any admixture of s-wave due to relativistic & correlation effects can dramatically influence on the calculated HFS and PNC values. Accurate calculation of electronic structure both in the valence and core regions of PbO is required: Combination of SO-CI (nondynamic or V correlations) & RCC-SD (dynamic or OC-V correlations) is applied: W = W [30-el. RCC-SD] - W [10-el. RCC-SD] + W [10-el. SO-CI] Pb:[4s7p5d3f] + O:[4s3p2d1f] : : Pb:[5s7p4d2f] + O:[4s3p2d1f]

Calculated parameters A ( in MHz ) & W d (in Hz / ecm ) Experiment: A [a(1)]= -4113; A [B(1)]= 5017 (MHz) State Parameters Internuclear distance R = 3.8 a.u. 10e-RCC-SD e-RCC-SD Outecore e-CI FINAL Internuclear distance R = 4.0 a.u. 10e-CI FINAL (1 V/cm = Hz/ecm)

A ( in MHz ) & W d (in Hz / ecm ) calculated by SODCI with different thresholds (T, in mHartree): State Parameters T (number of SAFs)Internuclear distance R=4.0 a.u. Reference ( 2 500) T=0.1 ( ) T=0.01 ( ) T= ( ) T=0.001 ( ) T=0 ( ) T=0 + FCI

Semi-empirical model of wave functions using experimental data to constrain partial waves near Pb [M.Kozlov and D.DeMille, PRL 89, (2002)] E int > V/cm ( Hz / ecm ) ; Ab initio RCCSD + SODCI calculations of QChem PNPI RCCSD: [T.Isaev et al., PRA 69, (R) (2004) ] SODCI: [A.Petrov et al., PRA 72, (2005)] E int ~ V/cm ( Hz / ecm ) ! Calculation of E int in PbO*: Results (1 V/cm = Hz/ecm)

Why use molecular ions? Ions are easy to trap (in RF quadruple trap); Potential for long spin coherence times (ion-ion repulsion); Can get E eff /E lab = 10 9 (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. [R.Stutz & E.Cornell, Bull.Am.Phys.Soc. 49, 46 (2004)]

Mass-spectrometry:

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., PRA 76, (R) (2007)+ …] HfF + working state ; config. : […] , Hf 2+ : […4f 14 ]5s 2 5p 6 5d 1 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] [core] [ valence ] 1 st question: which state is the ground one, 3 1 or 1 1 (config. : […] )?! (and if 3 1 is not the ground one, how to populate it?) 2 nd question: which is effective field on e -, E eff ? 3 rd question: which transitions to excited states ( 3, 1 ) can be used to measure the EDM signals? ,1,2, 3 2,3,2 1

Our SODCI calculations with HfF +

HfF +

Our SODCI calculations with HfF + Effective electric field on e - : E int = V/cm ( Hz / ecm ); Hyperfine constants: A || [ 177 Hf] = 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 (molecule- frame and transition), radiative lifetimes are calculated for ten lowest states. Errors for energetic properties are about 500 cm -1 ; 4f 14 relaxation is shown to be not important for these studies. 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 12×10 6 selected SAFs are used in SODCI calculations.

Calculated spectroscopic parameters for HfF +

Liquid Xenon: spherical cell model [B.Ravaine & A.Derevianko PRA 69, (R) (2004)]

Lattice Model of Liquid Xenon [T.Isaev et al., PRA 75, (2007)]

Lattice Model of Liquid Xenon (2)

Lattice Model of Liquid Xenon (3)

Lattice Model of Liquid Xenon (4) Series a and b : AREP/SCF X value against cell size.

Concluding remarks (valence properties):

HI + :

Milestones in studying PNC effects: 1949: Analyzing Einsteins relativity theory, Dirac states that P- and T-invariance is not necessary attribute of the nature laws (i.e. of physical dynamic theories). 1950: Purcell & Ramsey state that the validity of P- or/and T-odd theories must be confirmed experimentally. The search for the neutron EDM is initiated. 1956: Puzzle of τ–θ mesons: θ2π(L=0); τ3π(L=0), but m τ m θ ; T τ 1/2 T θ 1/2. Analyzing the decays, Lee and Yang have suggested nonconservation of space parity, P, in decay of K(J P =0 ) on π-mesons (J P =0 ) / K π3 τ ;K π2 θ /. They suggest to study spirality (p·S) in decay experiments. 1957: Wu et al. discover the P-violation in β decay of 60 Co ( n p + e + ν e ). To save the world from the left-handed asymmetry, Landau, Lee & Yang suggest invariance of the nature laws with respect to the combined CP-parity. Landau: d S, i.e. the proportionality coefficient is P,T-odd, otherwise d= : Christenson, Cronin, Fitch & Turlay discover nonconservation of the combined inversion, CP, in decay K 0 L (CP=-1) 2π (CP=+1 !).

P,T-odd interactions in TlF:

Volume effect:

Magnetic effect:

The parameters X and M (in a.u.) for the ground state of 205 TlF in Dirac-Fock (DF) 1,2 and GRECP/RCC 3 calculations.

Effective relativistic Hamiltonians

The most popular Hamiltonians used in calculations of heavy-atom molecules: Dirac-Coulomb(-Breit) Hamiltonian is the most accurate relativistic approximation that is used in practice when calculating many-electron systems. Two-component all-electron approaches: – Douglas-Kroll transformation (of 2nd & 3rd orders); – zero/first-order relativistic approximations (ZORA/FORA). Relativistic effective core potentials (RECPs) employing operators of types: – radially-local (semi-local) pseudopotentials; – Huzinaga-type (ab initio) model potentials (using level-shift terms for freezing core shells); – separable pseudopotentials (applied to many-atomic systems); – core polarization potentials (containing one- & two-electron terms).

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

Variational one-center restoration Valence Outer Core IC IC is Inner Core [A.Titov, IJQC (1996)]

The Coupled-Cluster Approaches

Solution of Coupled-Cluster equations: Three basic Coupled Cluster categories:

The Coupled-Cluster Approaches (cont.)

The PT2/CI method

The PT2/CI method (cont.)

Spin-Orbit Configuration Interaction

The SO-CI computational scheme:

Perturbative corrections to CI:

Spin-Orbit Conguration Interaction (cont.)

Errors in all-electron transition energies of the Pb atom obtained by the RCC-SD and PT2/CI methods for states with the 6s 2 6p 2 configuration.

Contributions from triple and quadruple CC amplitudes to total energies of the Pb terms and errors for the VCIC-corrected transition energies.