An experiment to search for the electron electric dipole moment (*e*EDM) on the metastable *H*^{3}Δ_{1} state of ThO molecule was proposed and now prepared by the ACME Collaboration [http://www.electronedm.org]. To interpret the experiment in terms of *e*EDM and dimensionless constant *k*_{T, P} characterizing the strength of the T,P-odd pseudoscalar–scalar electron–nucleus neutral current interaction, an accurate theoretical study of an effective electric field on electron, *E*_{eff}, and a parameter of the T,P-odd pseudoscalar–scalar interaction, *W*_{T, P}, in ThO is required. We report our results for *E*_{eff} (84 GV/cm) and *W*_{T, P} (116 kHz) together with the hyperfine structure constant, molecule frame dipole moment, and *H*^{3}Δ_{1} → *X*^{1}Σ^{+} transition energy, which can serve as a measure of reliability of the obtained *E*_{eff} and *W*_{T, P} values. Besides, our results include a parity assignment and evaluation of the electric-field dependence for the magnetic *g* factors in the Ω-doublets of *H*^{3}Δ_{1}.

## INTRODUCTION

One of the most intriguing fundamental problems of modern physics is the search for a permanent electric dipole moment (EDM) of elementary particles. A nonzero value of EDMs implies manifestation of interactions which are not symmetric with respect to both time (T) and spatial (P) inversions (T,P-odd interactions). Particularly, the observation of electron EDM (*e*EDM) at the level significantly greater than 10^{−38}*e* cm would indicate the presence of a “new physics” beyond the Standard model. Popular extensions of the Standard model predict the magnitude of the *e*EDM at the level of 10^{−26} to 10^{−29}*e* cm.^{1} The most rigid upper bound on the *e*EDM is attained in the experiments on a beam of YbF molecular radicals^{2} (1.05 × 10^{−27}*e* cm) and in the measurements with atomic Tl beam^{3} (1.6 × 10^{−27}*e* cm).

Nowadays a number of other prospective experiments are suggested and in part prepared^{4–6} which promise to achieve a sensitivity to *e*EDM up to 10^{−29} to 10^{−30}*e* cm. One of the most promising experiments towards the measurement of *e*EDM is proposed and now prepared on the metastable ^{3}Δ_{1} state of the thorium monoxide (ThO) molecule by ACME Collaboration (groups of DeMille, Gabrielse, and Doyle).^{7,8} A very high sensitivity to *e*EDM is expected in the nearest future, up to an order of magnitude and more than that attained in the YbF and Tl experiments, due to some unique combination of experimental advantages of the molecule. Even the value for *e*EDM compatible with zero will lead to serious consequences for the modern theory of fundamental symmetries.

To interpret the results of the corresponding molecular ThO (or other diatomics) experiment in terms of the *e*EDM one should know a parameter usually called “the effective electric field on electron,” *E*_{eff}, which cannot be measured. To obtain *E*_{eff} one can evaluate an expectation value of some T,P-odd operator (discussed in Refs. 9–11):

where *d*_{e} is the value of *e*EDM, Ψ is the wave function of the considered state of ThO, and

**is the total electronic momentum,**

*J***is the unit vector along the molecular axis directed from Th to O (Ω = 1 for the considered**

*n*^{3}Δ

_{1}state of ThO),

** E** is the inner molecular electric field, and

*E*

_{eff}=

*W*

_{d}|Ω|.

In addition to the interaction given by operator (2) there is a T,P-odd pseudoscalar–scalar electron–nucleus neutral currents interaction with the dimensionless constant *k*_{T, P}. The interaction is given by the following operator (see Ref. 12):

where G is the Fermi constant, γ_{0} and γ_{5} are the Dirac matrices, and

*k*

_{T, P}constant from an experiment, one needs to know the factor

*W*

_{T, P}that is determined by the electronic structure of a studied molecule on a given nucleus:

Similarly to *E*_{eff}, the *W*_{T, P} parameter cannot be measured and have to be obtained from a molecular electronic structure calculation.

A commonly used way of verification of the theoretical *E*_{eff} and *W*_{T, P} values is to calculate “on equal footing” (the same approximation for the wave function) those molecular characteristics (properties or effective Hamiltonian parameters) which have comparable to *E*_{eff} and *W*_{T, P} sensitivity to variations of wave function in core region but, in contrast, can be measured. Similar to *E*_{eff} and *W*_{T, P} these parameters should be sensitive to a change of densities of the *valence* electrons in atomic cores. The hyperfine structure constant, *A*_{||}, is traditionally used as such a parameter (e.g., see Ref. 13) and this is a valid touchstone for the ThO case as well. To obtain *A*_{||} on Th in the ThO molecule theoretically, one can evaluate the following matrix element:

where μ_{Th} is the magnetic moment of an isotope of Th nucleus having spin *I*.

To validate our present study of *E*_{eff}, *W*_{T, P}, and the hyperfine structure constant *A*_{||} for the ^{3}Δ_{1} state of ^{229}ThO, we have also performed calculations of the *H*^{3}Δ_{1} → *X*^{1}Σ transition energy and the molecule-frame dipole moment for the ^{3}Δ_{1} state.

Recently, spectroscopy of ThO was investigated both theoretically and experimentally in a series of papers.^{5,14–16} First estimates of *E*_{eff} in the ^{3}Δ_{1} state of ThO were performed in Ref. 5. The experimental measurements of molecule-frame dipole moments of the ground *X*^{1}Σ^{+} and excited *E* states were performed in Ref. 14. In Ref. 15 a theoretical study of the potential energy curve and electric properties of the ground ThO state were performed. A series of relativistic calculations of the spectroscopic parameters of the ground and excited states were performed in Ref. 16. In the present paper we concentrate on the precise two-component relativistic coupled clusters calculations of the complementary data (*E*_{eff}, *W*_{T, P}) which are most relevant for the direct interpretation of ThO experiment in terms of the electron EDM and constant of T,P-odd pseudoscalar–scalar electron–nucleus interaction.

## THEORETICAL DETAILS

The evaluation of *E*_{eff}, *W*_{T, P}, and *A*_{||} is usually a challenging problem for modern *ab initio* methods when studying systems containing heavy transition metals, lanthanides and, particularly, actinides (such as Th in the present consideration). An accurate theoretical investigation of such systems should take into account both the relativistic and correlation effects with the best to-date accuracy. It follows from Eqs. (2)–(5) that the operators related to *E*_{eff}, *W*_{T, P}, and *A*_{||} are essentially localized in the atomic core region. On the other hand, the main contribution to the corresponding matrix elements is due to the valence electrons since contributions from the closed inner-core shells compensate each other in most cases of practical interest for the operators dependent on the total angular momentum and spin. It was shown by our group (see Ref. 11 and references therein) that the problem of computation of such characteristics can be significantly simplified by splitting the calculation into two steps. In the first step the electron correlation for valence electrons (and outer-core electrons for better accuracy) is taken into account in a molecular calculation using some method of electron correlation treatment (such as coupled clusters, configuration interaction, etc.), whereas the core (inner-core) electrons are excluded from this calculation using the generalized relativistic effective core potential (GRECP),^{17,18} which yields an accurate valence region wave function in the most economical way. Second, since the inner-core parts of the valence one-electron “pseudo-wavefunctions” are smoothed within the GRECP method, they have to be recovered using some core-restoration method.^{11} The non-variational restoration is based on a proportionality (scaling) of valence and virtual spinors in the inner-core region of heavy atoms (e.g., see Ref. 18 for details). The two-step approach has been recently used in Refs. 19–23 for calculation of a number of characteristics, such as hyperfine structure constants, electron electric dipole moment enhancement factor, etc., in molecules and atoms. Besides it has been extended to the case of crystals in Ref. 24.

Recently^{24} we have developed a code of nonvariational restoration which has been interfaced to the DIRAC12^{25} and the MRCC^{26} codes. These codes are used in the present paper. Scalar-relativistic calculations (i.e., without spin-orbit terms in the GRECP operator) were performed using CFOUR code.^{27}

## RESULTS AND DISCUSSIONS

The 1*s*–4*f* inner-core electrons of Th were excluded from molecular correlation calculations using the valence (semi-local) version of GRECP^{17} operator. Thus, the outermost 38 electrons were treated explicitly. The basis set for Th was constructed using the generalized correlated scheme.^{28} It can be written as (30,20,17,11,4,1)/[30,8,6,4,4,1], so the only *p*, *d*, and *f* Gaussians are contracted. For oxygen the aug-ccpVQZ basis set^{29} with removed two g-type basis functions was employed, i.e., we used the (13,7,4,3)/[6,5,4,3] basis set.

Calculations of the transition energy between the ground *X*^{1}Σ^{+} and excited *H*^{3}Δ_{1} states as well as the molecule-frame dipole moment, *E*_{eff}, *W*_{T, P}, and *A*_{||} constants for the ^{3}Δ_{1} state of ThO were performed using the single reference two-component relativistic coupled clusters with single, double, and perturbative treatment of triple cluster amplitudes (CCSD(T)).^{30,39} In addition, the basis set enlargement corrections to the considered parameters were also calculated. For this we have performed (i) scalar-relativistic CCSD(T) calculation using the same basis set as used for the two-component calculation; (ii) scalar-relativistic CCSD(T) calculation using the extended basis set on Th (with added *f*, *g*, *h*, and *i* Gaussians). Corrections were estimated as a difference between the values of the corresponding parameters. The experimental equilibrium internuclear distances^{31,32} (3.478 a.u. for *X*^{1}Σ^{+} state and 3.511 a.u. for *H*^{3}Δ_{1} state) were used in these calculations. The results are given in Table I. Optimized (at the scalar-relativistic CCSD(T) level and the enlarged basis set on Th) internuclear distances are close to the experimental ones: 3.482 a.u. for *X*^{1}Σ^{+} and 3.515 a.u. for *H*^{3}Δ_{1}. The harmonic frequencies are also in a very good agreement with experiment:^{31,32} for *X*^{1}Σ^{+} the calculated value is 895 cm^{−1} and experimental is 896 cm^{−1}, while for *H*^{3}Δ_{1} the calculated value is 857 cm^{−1} and experimental is 864 cm^{−1}.

. | T_{e}
. | d
. | E_{eff}
. | W_{T, P}
. | A_{||}
. |
---|---|---|---|---|---|

Method . | (cm^{−1})
. | (debye) . | (GV/cm) . | (kHz) . | ( $\frac{\mu _{\rm Th}}{\mu _{\rm N}}\,$ $\mu Th \mu N$MHz)
. |

2c-CCSD | 5510 | 4.27 | 87 | 118 | −2958 |

2c-CCSD(T) | 6120 | 4.22 | 84 | 116 | −2885 |

2c-CCSD(T) | 5808 | 4.32 | 84 | 116 | … |

+ basis corr. | |||||

Experiment | 5321 | 4.24 ± 0.1 | … | … | … |

31 and 32 | 8 |

. | T_{e}
. | d
. | E_{eff}
. | W_{T, P}
. | A_{||}
. |
---|---|---|---|---|---|

Method . | (cm^{−1})
. | (debye) . | (GV/cm) . | (kHz) . | ( $\frac{\mu _{\rm Th}}{\mu _{\rm N}}\,$ $\mu Th \mu N$MHz)
. |

2c-CCSD | 5510 | 4.27 | 87 | 118 | −2958 |

2c-CCSD(T) | 6120 | 4.22 | 84 | 116 | −2885 |

2c-CCSD(T) | 5808 | 4.32 | 84 | 116 | … |

+ basis corr. | |||||

Experiment | 5321 | 4.24 ± 0.1 | … | … | … |

31 and 32 | 8 |

In the scalar-relativistic CCSD(T) calculations we have found that *E*_{eff} only very slightly depends on the internuclear distance: it has a maximum at 3.48 a.u., while at the interatomic distances of 3.4 a.u. and 3.56 a.u. it decreases by 0.1 GV/cm with respect to the maximal value.

The calculated value of transition energy is in a very good agreement with experimental datum, the deviation is on the level of accuracy attained earlier by our group for compounds of transition metals and lanthanides. It is also in a good agreement with previous calculation performed in Ref. 16.

It was recently shown in Ref. 33 that the magnetic moment of ^{229}Th nucleus determined earlier^{34} is inaccurate. Therefore, *A*_{||} is given in Table I in the units of (μ_{Th}/μ_{N}) MHz (where μ_{N} is the nuclear magneton) in Table I to exclude the uncertainty of μ_{Th} from our result. One can see from Table I that a good convergence of *E*_{eff} with respect to both the basis set enlargement and correlation level is achieved. Taking into account the results from Table I as well as our earlier studies within the two-step procedure (e.g., see Ref. 22) with calculating the *E*_{eff}, *W*_{T, P}, and *A*_{||} we expect that the theoretical uncertainty for our final values of the constants is smaller than 15%. Unfortunately, there are no experimental data on *A*_{||} up to now. Therefore, the corresponding indirect experimental verification of accuracy of *E*_{eff} (see above) cannot be performed to-date and further experimental measurements of *A*_{||} are required.

In the *e*EDM search experiment on the ThO molecule, the *e*EDM induced Stark splitting between the *J* = 1, *M* = ± 1 states of *e* (parity is ( − 1)^{J}) or *f* (parity is −( − 1)^{J}) levels of the Ω-doublet is measured. The *H*^{3}Δ_{1} state has a very small magnetic moment, μ_{H[ThO]} = 8.5(5) × 10^{−3}μ_{B},^{8} where μ_{B} is the Bohr magneton. The latter is a benefit for suppressing systematic effects due to spurious magnetic field. In a polarized molecule the *e* and *f* levels have opposite signs of *E*_{eff} and almost identical *g* factors. Therefore, when taking the difference between the splitting for *e* and *f* levels further suppression of the systematics is possible.^{35} The small difference between *g* factors, Δ*g*, comes from interactions of *H*^{3}Δ_{1} with 0^{+} and 0^{−} electronic states.^{21} Our calculations show that being presented in the Λ*S* coupling scheme, the spin-orbit mixed *H* state of ThO has the main contribution (more than 95%) from the ^{3}Δ_{1} configurations (see also Ref. 16). Therefore, due to the identity

**S**

^{e}is the electronic spin operator, for pure Λ

*S*state, the inequality

*g*to be determined by the energy splitting between the top and bottom levels of the Ω-doublet.

^{21,22}The rotational analysis given in Ref. 32 for

*P*(Ω = 0) −

*H*

^{3}Δ

_{1}and

*O*(Ω = 0) −

*H*

^{3}Δ

_{1}bands has shown that the Ω-doublet spacing (Δ = |

*E*(

*J*= 1

^{−}) −

*E*(

*J*= 1

^{+})|) in

*H*

^{3}Δ

_{1}is 350–470 kHz. At the moment, the parity assignment for electronic states

*P*and

*O*is yet unclear. According to Ref. 36,

*P*is 0

^{−}and

*O*is 0

^{+}. The latter, as can be shown, indicates that

*e*states are the top levels whereas the

*f*states are the bottom levels of the Ω-doublets for

*H*

^{3}Δ

_{1}.

The microwave spectroscopy data^{37} obtained later confirm our conclusion about the levels ordering and finally give

*g*factors for the

*J*= 1 levels of ThO

*H*

^{3}Δ

_{1}state are given as functions of the laboratory electric field. The lowest value, Δ

*g*= 2.7 × 10

^{−6}, is attained at the electric field 4.4 V/cm. Note that the molecule is almost completely polarized at the electric field larger than 3 V/cm.

It should be noted that there is a discrepancy in sign conventions of *E*_{eff} in different publications. Therefore, to avoid such an ambiguity, the energy levels of ThO in the *e*EDM experiment with *J* = 1 are shown in Fig. 2 for the case of a positive *d*_{e} value. Solid lines are the levels in the external electric field, dotted lines are the levels with *e*EDM shift added. According to Eqs. (1) and (2), the *e*EDM shift for a completely polarized molecule is determined by the expression +*d*_{e}*E*_{eff}Ω, where Ω = *M*⟨**n**⟩_{z}, *M* is the electron-rotational angular momentum projection on the laboratory axis *z*. Note that for the opposite direction of ** n** (from O to Th), the signs of all the Ω in Fig. 2 will be opposite. This does not affect the directions of

*e*EDM shift, since the

*E*

_{eff}with

**directed from O to Th will have the opposite sign (negative).**

*n*## CONCLUSION

The parameters *E*_{eff} and *W*_{T, P}, which are required to interpret experimental measurements on the *H*^{3}Δ_{1} state of ThO molecule in terms of fundamental quantities are calculated. Though the previous estimation of *E*_{eff} made in Ref. 5 is only 25% more than our final value, the good agreement can be rather considered as “fortunate” since the accuracy of semiempirical estimates for the systems like ThO having a very complicated electronic structure is severely limited, see Ref. 19. In turn, a reliable *ab initio* calculation of ThO is on the threshold of current possibilities of computational methods and we estimate the accuracy of our calculation of *E*_{eff} by 15% only. Nevertheless, even such accuracy is important to establish a reliable *e*EDM estimate in the ongoing ThO experiment compared to the measured upper bounds on *e*EDM in Tl^{3} and YbF^{2} experiments.

## ACKNOWLEDGMENTS

We are grateful to N. S. Mosyagin for providing us with the GRECP for Th.^{38} This work is supported by the SPbU Fundamental Science Research grant from Federal Budget No. 0.38.652.2013, RFBR Grant No. 13-02-01406, and partially by the RFBR Grant No. 13-03-01307a. L.S. is also grateful to the Dmitry Zimin “Dynasty” Foundation. The molecular calculations were performed on the Supercomputer “Lomonosov.”

## REFERENCES

*ab initio*electronic structure program, Release DIRAC12 (2012), written by

^{1}Σ

^{+}and

^{3}Δ

_{1}states the same set of one-electron spinors obtained in the two-component Hartree-Fock calculation of the

^{1}Σ

^{+}state were employed. Particularly, the reference determinant for the single-reference coupled clusters calculation of

^{3}Δ

_{1}state was constructed from this set of spinors. Such choice of one-electron functions is justified for the case of treatment by the coupled clusters approaches which include single clusters amplitudes. The latter are required to account for relaxation effects (see, e.g., Ref. 39).