1. THEORETICAL STUDY OF THE PbF AND PbO MOLECULES Alexander N. Petrov PNPI QChem Group: B.P. Konstantinov PNPI RAS, St.-Petersburg State University, St.-Petersburg, RUSSIA A.V. Titov, N.S. Mosyagin,K.I.Baklanov L.V. Skripnikov, and M.G. Kozlov

2.  YbF-radical beam (E.Hinds: Imperial college, London,UK);  ThO* beam [& PbO* in optic cell ] (ACME collaboration: D.DeMille:Yale University; 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.) Experiments on the electron EDM Search On heavy-atom polar molecules and cations:

3. Effective electric field on the electron, E eff, is one of the most important parameters. E eff is non-zero only due to the relativistic effects Electron correlation is also important E eff can not be obtained in an experiment  Challenging molecular calculation is required What should be calculated ?

4. Calculated |E eff |, Gv/cm MoleculeOur groupMeyer et. al., PRA 78, 010502(R) 2008 PbO 2623 YbF 2232 PbF 3331 HfF + 2430 PtH + 2873 WC 3354 ThO 60104

5. Calculation E eff must be checked against experimental data on closely related hyperfine constants. Ref.A per (MHz) A par (MHz) Calc. [Yu. Yu. Dmitriev, Yu. G. Khait, M. G. Kozlov, L. N. Labzovsky, A.O. Mitrushenkov, A.V. Shtoff, and A. V. Titov, Phys. Lett. A 167, 280 (1992)] -899010990 Exper.[ C. P. McRaven, P. Sivakumar, and N. E. Shafer-Ray, Phys.Rev. A 78, 054502(R) (2008)] 7200 ± 150 10300 ± 800 Exper. [R. Mawhorter, T. J. Sears, T. Z. Yang, P. M. Rupasinghe, C. P. McRaven, N. Shafer-Ray, and J.-U. Grabow (to be published).] -726410147 This work 10e Calc. [K.I. Baklanov, A.N. Petrov, A.V. Titov, and M.G. Kozlov Phys.Rev. A 82, 060501(R) (2010) ] -68609727 This work 20e Calc [K.I. Baklanov et. el. (to be published)] -754010073 Results for: PbF XП1/2Results for: PbF XП1/2

6. Experiment [ L.R. Hunter et al., PRA 65, 030501(R) (2002)] ]  A || = -4110 MHz ; Ab initio RCCSD + SODCI calculation RCCSD: [ T.Isaev et al., PRA 69, 030501(R) (2004)] SODCI: [A.Petrov et al., PRA 72, 022505 (2005)]  A|| = -3826 MHz Hyperfine constant in PbO* a(1) [ 3 Σ + 1 ]

7. Describing hyperfine structure of the  =1 electronic levels by only one Constant A || we neglect interaction with other electronic states The latter is small effect. It is much more difficult for experimental determination but can be obtained from ab initio calculation with comparable to A || accuracy  Challenging molecular calculation is required Experimental determination of A|| is more precise compared to ab initio calculation. However

8. Example: a(1) [ 3 Σ + 1 ] state of PbO* is close to 3 Σ + 0- In this work we take into account interaction between 3 Σ + 1 and 3 Σ + 0-

9. Parity of the  -doublet levels 3 Σ + 1 “f” state: p = (-1) J+1 3 Σ + 1 “e” state: p = (-1) J 3 Σ + 0- “f” state: p = (-1) J+1 3 Σ + 1 “e” state is not mixed with 3 Σ + 0-

10. Hyperfine energy splitting between F=J+1/2 and F=J-1/2 for “e” and “f” levels of a(1) 207 PbO (MHz) Jef 13187.3188. 21903.1905. 5 858.863. 10449.458. 15304.317. 20230.248. 25185.207. 35133.163.

11. The g factors for “f ” states of 206,208 PbO as a function of J. For “e” states, g e = 1.857 00 and is independent of J. Jf 11.860 74 21.868 22 51.913 07 102.062 55 152.305 37 202.641 42 253.070 55 303.592 56

12. Difference between g factors for “e” and “f” levels of J=1 a(1) PbO Experiment: g f - g e =0.0030(8) [D. Kawall, F. Bay, S. Bickman, Y. Jiang, and D. DeMille PRL 92, 133007 (2004)] This work (Calculation) : g f - g e =0.0037 [A.N. Petrov, PRA 83, 024502 (2011)]

13. D. Kawall, F. Bay, S. Bickman, Y. Jiang, and D. DeMille, PRL 92, 133007 (2004) S. Bickman, P. Hamilton, Y. Jiang, and D. DeMille PRA 80, 023418 (2009)

15. 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.Mosyagin et al., JPB, 33,2000 ; T.A. Isaev et al, JPB, 33, 2000); ANO basis sets for light atoms. Methods of calculations

16.  The eEDM experiments on heavy-atom molecules are of key importance for modern theory of fundamental interactions and symmetries – “window beyond the Standard model”.  High-accuracy calculations of prospective heavy-atom systems are of primary importance for the eEDM experiment.  It is shown that accounting for interaction with the nearest electronic state 3 Σ + 0- is required for accurate description of the hyperfine structure and g factors of the a(1) [ 3 Σ + 1 ] state of PbO*. One can suppose that a similar situation takes place also for other diatomics in the  = 1 states.  It is found that the difference between g f and g e for 207 PbO is converged to zero at E ≈ 11 V/cm. The latter is important for the suppression of systematic effects in the EDM search experiment. Concluding remarks:

17. Thank you!