Communication: Theoretical study of ThO for the electron electric dipole moment search

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 (eEDM) at the level significantly greater than 10⁻³⁸e cm would indicate the presence of a “new physics” beyond the Standard model. Popular extensions of the Standard model predict the magnitude of the eEDM at the level of 10⁻²⁶ to 10⁻²⁹e cm.¹ The most rigid upper bound on the eEDM is attained in the experiments on a beam of YbF molecular radicals² (1.05 × 10⁻²⁷e cm) and in the measurements with atomic Tl beam³ (1.6 × 10⁻²⁷e cm).

Nowadays a number of other prospective experiments are suggested and in part prepared^4–6 which promise to achieve a sensitivity to eEDM up to 10⁻²⁹ to 10⁻³⁰e cm. One of the most promising experiments towards the measurement of eEDM is proposed and now prepared on the metastable ³Δ₁ state of the thorium monoxide (ThO) molecule by ACME Collaboration (groups of DeMille, Gabrielse, and Doyle).^7,8 A very high sensitivity to eEDM 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 eEDM 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 eEDM 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 eEDM, Ψ is the wave function of the considered state of ThO, and

$Ω=⟨Ψ|J·n|Ψ⟩$⁠, J is the total electronic momentum, n is the unit vector along the molecular axis directed from Th to O (Ω = 1 for the considered ³Δ₁ state of ThO),

E is the inner molecular electric field, and

$σ$ are the Pauli matrices. In these designations 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, γ₀ and γ₅ are the Dirac matrices, and

$n(r)$ is the nuclear density normalized to unity. To extract the fundamental 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 ³Δ₁ state of ²²⁹ThO, we have also performed calculations of the H³Δ₁ → X¹Σ transition energy and the molecule-frame dipole moment for the ³Δ₁ 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 ³Δ₁ state of ThO were performed in Ref. 5. The experimental measurements of molecule-frame dipole moments of the ground X¹Σ⁺ 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.

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.

Recently²⁴ we have developed a code of nonvariational restoration which has been interfaced to the DIRAC12²⁵ and the MRCC²⁶ 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 

The 1s–4f inner-core electrons of Th were excluded from molecular correlation calculations using the valence (semi-local) version of GRECP¹⁷ 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 set²⁹ 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 X¹Σ⁺ and excited H³Δ₁ states as well as the molecule-frame dipole moment, E_eff, W_{T, P}, and A_|| constants for the ³Δ₁ 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 X¹Σ⁺ state and 3.511 a.u. for H³Δ₁ 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 X¹Σ⁺ and 3.515 a.u. for H³Δ₁. The harmonic frequencies are also in a very good agreement with experiment:^31,32 for X¹Σ⁺ the calculated value is 895 cm⁻¹ and experimental is 896 cm⁻¹, while for H³Δ₁ the calculated value is 857 cm⁻¹ and experimental is 864 cm⁻¹.

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 ²²⁹Th nucleus determined earlier³⁴ 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 eEDM search experiment on the ThO molecule, the eEDM 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 H³Δ₁ state has a very small magnetic moment, μ_H[ThO] = 8.5(5) × 10⁻³μ_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, Δg, comes from interactions of H³Δ₁ with 0⁺ and 0⁻ electronic states.²¹ Our calculations show that being presented in the ΛS coupling scheme, the spin-orbit mixed H state of ThO has the main contribution (more than 95%) from the ³Δ₁ configurations (see also Ref. 16). Therefore, due to the identity

$⟨ΨΔ13|S+e|Ψn0±⟩≡0$⁠, where S^e is the electronic spin operator, for pure ΛS state, the inequality

$|\langle \Psi _{H^3\Delta _1}|S^e_+ |\Psi _{n0^\pm }\rangle | \ll 1$

$|⟨ΨH3Δ1|S+e|Ψn0±⟩|≪1$ holds with a good accuracy. This inequality gives sufficient condition for Δg 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) − H³Δ₁ and O(Ω = 0) − H³Δ₁ bands has shown that the Ω-doublet spacing (Δ = |E(J = 1⁻) − E(J = 1⁺)|) in H³Δ₁ 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 H³Δ₁.

The microwave spectroscopy data³⁷ obtained later confirm our conclusion about the levels ordering and finally give

$E(J=1−)−E(J=1+)=362±10$ kHz. In Fig. 1 the calculated g factors for the J = 1 levels of ThO H³Δ₁ state are given as functions of the laboratory electric field. The lowest value, Δg = 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 eEDM 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 eEDM shift added. According to Eqs. (1) and (2), the eEDM shift for a completely polarized molecule is determined by the expression +d_eE_effΩ, 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 eEDM shift, since the E_eff with n directed from O to Th will have the opposite sign (negative).

The parameters E_eff and W_{T, P}, which are required to interpret experimental measurements on the H³Δ₁ 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 eEDM estimate in the ongoing ThO experiment compared to the measured upper bounds on eEDM in Tl³ and YbF² experiments.

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.”