Towards the electron EDM search. Theoretical study of PbF $^{2}$Petersburg Nuclear Physics Institute, Gatchina, Leningrad district 188300, Russia A.V. Titov and M.G. Kozlov We report ab initio relativistic correlation calculations of potential curves and spectroscopic con- stants for four lowest-lying electronic states of the lead monofluoride. We also calculated parameters of the spin-rotational Hamiltonian for the ground $^{2}$Π$_{1}$$_{/}$$_{2}$ and the first excited A $^{2}$Σ + 1 / 2 states including P,T -odd and P -odd terms. In particular, we have obtained hyperfine constants of the $^{207}$Pb nucleus. For the $^{2}$Π$_{1}$$_{/}$$_{2}$ state A$_{⊥}$ = − 6859 . 6 MHz, A$_{‖}$ = 9726 . 9 MHz and for the A $^{2}$Σ + 1 / 2 A$_{⊥}$ = 1720 . 8 MHz, A$_{‖}$ = 3073 . 3 MHz. Our values of the ground state hyperfine constants are in good agreement with the previous theoretical studies. We discuss and explain seeming disagreement in the sign of the con- stant A$_{⊥}$ with the recent experimental data. The effective electric field on the electron E eff , which is important for the planned experiment to search for the electric dipole moment of the electron, is found to be 3 . 3 × 10 $^{10}$V / cm. The Hamiltonian H sr can be written as [2] A$_{⊥}$ . The interest in theoretical study of $^{207}$PbF is caused by the planned experiment to search for the simultaneous violation of the time-reversal invariance (T) and space parity (P) [1–4]. The latter may be caused either by the P,T -odd electron-nucleus interaction or by the electric dipole moment of the electron (eEDM, or d e ), which can exist also only due to fundamental P,T -odd interactions [5]. The possible experiments to search for P,T -odd ef- fects in atoms and molecules were discussed in several books and reviews [6–12]. where the first line corresponds to the rotational struc- ture with Ω-doubling and the hyperfine interaction of the effective electron spin S $^{′}$, S ′ = 1 $_{2}$, with the spin of the nucleus I ; B ′ is the rotational constant, J is the electron- rotational angular momentum, Δ is the Ω-doubling con- stant, A is the hyperfine structure tensor. The second line describes interaction of the molecule with the exter- nal fields B and E , n is the unit vector along the molecu- lar axis, ζ , directed from Pb to F; G is the G-factor ten- sor. The last line corresponds to the interaction with the P -forbidden anapole moment of the nucleus, the P,T -odd weak interactions of the electrons with the heavy nucleus and the interaction of eEDM with the internal electric field of the molecule; κ$_{P}$ and κ$_{P,T}$ are the anapole con- stant and the P,T -odd neutral current constant for Pb nucleus. Recently McRaven et al. [4] have measured hyperfine structure parameters for the ground and the first excited states of $^{207}$PbF and found them to be in disagreement with previous predictions [1, 2]. Taking into account that theoretical studies of the lead monofluoride took place decades ago within rather simple theoretical models we recalculate spectroscopic constants and parameters of the spin-rotational Hamiltonian, H sr for PbF molecule. In particular, we focus on the calculations of the effective electric field on the electron E eff and the hyperfine struc- ture constants for the lead nucleus. Our new results are in reasonable agreement with previous calculations [1, 2]. We discuss the discrepancy between the theory and ex- periment [4] and argue that it is caused by the incorrect phase factor in the spin-rotational wave function, rather than in the difference in the sign of the hyperfine constant Parameters W$_{i}$ in (3) can not be measured experimen- tally and have to be found from the molecular electronic structure calculations. Their accurate knowledge is cru- cial for extracting constants κ$_{P}$ , κ$_{P,T}$ , and d$_{e}$ from the experiment. Expressions for calculating parameters W$_{i}$ can be found in Refs.[1, 6, 11]. In the literature the ef- fective electric field seen by the unpaired electron is usu- ally used instead of the parameter W$_{d}$ . It is defined as: E eff = W$_{d}$ | Ω | , where Ω = 〈 Ψ e Ω | S ′ · n | Ψ e Ω 〉 = ± 1 / 2 and Ψ e Ω is the electronic wavefunction for the $^{2}$Π$_{1}$$_{/}$$_{2}$ or $^{2}$Σ + 1 / 2 states. We used the generalized relativistic effective core po- tential (GRECP) [14] to simulate the interaction of va- lence electrons (4 electrons of Pb) with the explicitly ex- arXiv:1001.2260v1 [physics.atom-ph] 13 Jan 2010 H$_{sr}$ = B ′ J 2 + Δ S ′ · J + S ′ AI + (1) + µ $_{0}$S ′ GB − D n · E + (2) + W$_{P}$κ$_{P}$ n × S ′ · I + ( W$_{P,T}$ κ$_{P,T}$ + W$_{d}$d$_{e}$ ) S ′ · n , (3) K.I. Baklanov and A.N. Petrov 2 Institute of Physics, Saint Petersburg State University, Saint Petersburg, Petrodvoretz 198904, Russia Experiment to search for eEDM with PbF molecule is currently under preparation by the group of Shafer- Ray [3]. PbF molecule has one of the strongest effective electric field E eff on the valence electron, which deter- mines experimentally measured frequency shifts [1, 2]. This experiment can test “new physics” beyond the Stan- dard Model including different supersymmetric models. Many such models predict eEDM in the range between 10 − $^{28}$− 10 − 29 e · cm ( e is the electric charge of the elec- tron), within the reach of the new generation of experi- ments [12, 13].