1. HYPERFINE INTERACTION IN DIATOMICS AS A TOOL FOR VERIFICATION OF THEORETICAL VALUES FOR THE EFFECTIVE ELECTRIC FIELD ON ELECTRON A.N.Petrov PNPI QChem Group: B.P. Konstantinov PNPI, St.-Petersburg State University, St.-Petersburg, RUSSIA L.V. Skripnikov, N.S. Mosyagin, and A.V.Titov http://qchem.pnpi.spb.ru

2. Effective electric field (E eff ) on the electron is one of the most important parameters for EDM search E eff can not be obtained in an experiment  Challenging molecular calculation is required What should be calculated ?

3. Hyperfine structure constant(s), A ‖ (A ┴ ) can be obtained both in an experiment and in calculation Similarly to E eff hyperfine structure constant(s) is(are) determined by wave function near heavy nuclei the Ω=1/2 (YbF, PbF …)molecules have an advantage here since their hyperfine structure is determined by two constants, A ‖ and A ┴, whereas hyperfine structure for Ω=1 (HfF +,WC,…) molecules is mainly determined by only one constant, A ‖ How to check accuracy of the E eff ?

4. YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5

5. YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5 Yb 1+ : […4f 14 ]5s 2 5p 6 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ]

6. YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5 Yb 1+ : […4f 14 ]5s 2 5p 6 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] YbF: 2 ∑ 1/2 ; config.: […]  

7. YbF electronic structure Yb : […4f 14 ]5s 2 5p 6 6s 2 + F : 1s 2 2s 2 2p 5 Yb 1+ : […4f 14 ]5s 2 5p 6 6s 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] YbF: 2 ∑ 1/2 ; config.: […]   = A6s + B6p z 

8. YbF molecule studies for eEDM experiment

9. M.G.Kozlov, JPB 30, L607 (1997):

10. YbF molecule studies for eEDM experiment M.G.Kozlov, JPB 30, L607 (1997): Timothy C. Steimle, Tongmei Ma and Colan Linton, JCP 127, 234316 (2007):

11. YbF molecule studies for eEDM experiment M.G.Kozlov, JPB 30, L607 (1997): Timothy C. Steimle, Tongmei Ma and Colan Linton, JCP 127, 234316 (2007):

12. YbF molecule studies for eEDM experiment M.G.Kozlov, JPB 30, L607 (1997): Timothy C. Steimle, Tongmei Ma and Colan Linton, JCP 127, 234316 (2007):

13. HfF + studies for eEDM experiment

14. 3 Δ 1 state, Ω=1

15. HfF + studies for eEDM experiment

16. For Ω=1 molecules A ┴ ≡ 0

17. HfF + studies for eEDM experiment

18. 3 Δ 2 state of HfF +, WC … is close to 3 Δ 1 In this work we take into account interaction between 3 Δ 1 and 3 Δ 2

19. HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 [outer core] [ valence ]

20. HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 [outer core] [ valence ] Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ]

21. HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2 

22. HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2  = A6s + B6p z 

23. HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2  = A6s + B6p z  ↓ = C5d +2 ↓ + D6p +1 ↑ 

24. HfF + electronic structure Hf : […4f 14 ]5s 2 5p 6 6s 2 5d 2 + F : 1s 2 2s 2 2p 5 Hf 2+ : […4f 14 ]5s 2 5p 6 6s 1 5d 1 + F – : 1s 2 2s 2 2p 6 [outer core] [ valence ] HfF+ 3  1 ; config.: […]  ↓  ↓ 3  2 ; config.: […] (  ↑  ↓ +  ↓  ↑) /√ 2  = A6s + B6p z  ↓ = C5d +2 ↓ + D6p +1 ↑  ↑ = E5d +2 ↑ 

25. HFS (MHz) HF Atomic matrix elements

26. HFS (MHz) HF EDM(GV/cm) Atomic matrix elements

27. HF Molecular matrix elements A ‖ = (-2298A 2 + -235B 2 + 308C 2 + 158D 2 )MHz A ┴ /√2 = (2298A 2 + -66B 2 + -87EC)MHz E eff = ( -125AB + 12CD)GV/cm

28. HF Molecular matrix elements A ‖ = (-2298A 2 + -235B 2 + 308C 2 + 158D 2 )MHz A ┴ /√2 = (2298A 2 + -66B 2 + -87EC)MHz Using of two constants A ‖ and A ┴ is important for the accuracy check of E eff

29. Off-diagonal matrix elements

33. HFS splitting (A ┴ =0)

34. Hyperfine energy splitting between F=J+1/2 and F=J-1/2 levels of 3 Δ 1 WC (MHz), theoretical data [J. LEE et al. PRA 87, 022516 (2013)] JA ┴ =0A ┴ ≠0Δ 1882.5 0 2495.5492.43.1 3 347.5342.25.3 5218.7209.29.5 10113.994.919.0 1577.148.728.4 2058.220.637.6 3039.1−17.056.1

35. Observed rotational transition frequencies γ obs (MHz) [R. J. Mawhorter et al. PRA 84, 022508 (2011) ] of 208 Pb 19 F. The values in parentheses give the 1σ experimental error of the last digit of precision(100 Hz) levelsγ obs 1-33922.5065(20) 10-1112277.6822(7) 9-1212540.8465(8) 14-1516428.5160(10) 13-1616688.4929(20) 1-518414.5880(5) 1-618497.1352(5) 3-722691.9306(5)

36. Hamiltonian of PbF( 2 Π) molecule First term describes the rotational motion Second term describes the hyperne structure Therd term gives small corrections to the hyperne structure

37. The subscripts 1 and 2 refer to nuclear spin of the fluorine and lead respectively

38. First line – nuclear spins – rotational interaction Second line - nuclear magnetic dipole-dipole interaction Third line - takes into account interactions with other electronic states

39. Observed rotational transition frequencies γ obs (MHz) [R. J. Mawhorter et al. PRA 84, 022508 (2011) ] of 208 Pb 19 F. The deviation of fit is given by Δ = γ fit - γ obs in units of the last digit of precision (100 Hz) levelsγ obs Δ1Δ1 1-33922.5065(20)-10 10-1112277.6822(7)12 9-1212540.8465(8)-17 14-1516428.5160(10)11 13-1616688.4929(20)-35 1-518414.5880(5)5 1-618497.1352(5)6 3-722691.9306(5)8

41. Observed rotational transition frequencies γ obs (MHz) [ PRA 84, 022508 (2011) ] of 208 Pb 19 F. The deviation of fit is given by Δ = γ fit - γ obs in units of the last digit of precision (100 Hz) levelsγ obs Δ1Δ1 Δ2Δ2 1-33922.5065(20)-10-5 10-1112277.6822(7)123 9-1212540.8465(8)-17-3 14-1516428.5160(10)11-2 13-1616688.4929(20)-3510 1-518414.5880(5)52 1-618497.1352(5)61 3-722691.9306(5)8

42. From experimental data  A D ┴ (F) = 0.53 kHz A D ┴ (Pb) = -6.4 kHz Ab initio calculation  A D ┴ (F) = 0.71 kHz A D ┴ (Pb) = -5.9 kHz Centrifugal correction to hyperfine constant in 208,207 PbF [ 2 Π 1/2 ] [A.N. Petrov, L.V. Skripnikov, A.V. Titov, and R. J. Mawhorter, to be published]

43. Thank you!