Mercury Monohalides: Suitability for Electron Electric Dipole Moment Searches
MMercury Monohalides: Suitability for Electron Electric Dipole Moment Searches
V.S. Prasannaa , A.C. Vutha , M. Abe , , and B.P. Das Indian Institute of Astrophysics, Koramangala II block, Bangalore-560034, India Dept. of Physics and Astronomy, York University, Toronto ON M3J 1P3, Canada Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji-city, Tokyo 192-0397, Japan and JST, CREST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
Heavy polar diatomic molecules are the primary tools for searching for the T -violating permanentelectric dipole moment of the electron (eEDM). Valence electrons in some molecules experienceextremely large effective electric fields due to relativistic interactions. These large effective electricfields are crucial to the success of polar-molecule-based eEDM search experiments. Here we report onthe results of relativistic ab initio calculations of the effective electric fields in a series of moleculesthat are highly sensitive to an eEDM, the mercury monohalides (HgF, HgCl, HgBr,and HgI). Westudy the influence of the halide anions on E eff , and identify HgBr and HgI as interesting candidatesfor future electric dipole moment search experiments. Violation of time-reversal ( T ) symmetry is an essentialingredient to explain the matter-antimatter asymmetryof the universe [1, 2]. As Standard Model sources of T -violation are inadequate to explain the observed asym-metry, it is imperative to look beyond it. The strongestlimits on T -violation arising from new particles and inter-actions outside the Standard Model are set by searchesfor the permanent electric dipole moments of fundamentalparticles [3, 4], like that of the electron ( d e ). A strong con-straint on the electron’s electric dipole moment (eEDM), d e < − e cm, has been set by the experiment with ThOmolecules [5], and improvements of a few orders of mag-nitude are forecast in the near future [5, 6]. The eEDMexperiments take advantage of the large effective electricfield (often > ∼ V/cm) experienced by an electron in apolarized heavy polar molecule, which leads to a measur-able energy shift, ∆ E ∝ d e E eff . The effective electric field, E eff , arises from the relativistic interactions of the eEDMwith the electric fields of all the other charged particlesin the molecule. This effect, whereby molecules polar-ized by ∼ kV/cm laboratory fields cause >
10 GV/cmto be applied to a valence electron, is the reason forthe high precision achievable in molecule-based eEDMexperiments.The value of E eff for a molecule has to be obtained fromrelativistic many-body calculations in order to convertexperimentally measured frequency shifts into eEDM val-ues. A common heuristic that is used to estimate E eff inmolecules, motivated from eEDM enhancement scaling inatoms, is that E eff ∝ Z , where Z + is the charge of the(usually heavier) cationic atom’s nucleus. But moleculesare not atoms. This heuristic ignores the anions which canplay an important role. An improved understanding of themechanisms leading to E eff in relativistic polar moleculeswill lead to better choices of candidate molecules for futureeEDM experiments.In this work, we focus on the E eff for a class of heavypolar molecules, the mercury monohalides, in order totest their suitability for eEDM searches. The propertiesof these systems can be evaluated fairly accurately, as they have a single valence electron. The fact that theyare sensitive to eEDMs in their ground electronic states(unlike molecules with metastable eEDM-sensitive stateswhich require more complicated descriptions [7]), makesthem suitable test cases for high-precision calculations.HgF has one of the largest reported E eff [8], making thisseries of HgX molecules particularly interesting as poten-tial candidates for future eEDM experiments. The heavierHg monohalides (HgCl, HgBr, HgI) are more electricallypolarizable than HgF, which translates to a more effectiveuse of E eff and better control over systematic effects. This,in addition to better prospects for their production andefficient detection, makes the investigation of their E eff values very promising for future eEDM experiments.The expression for E eff in terms of an effective eEDMoperator, H effEDM is given by[9] E eff = 2 ice N e (cid:88) j =1 (cid:104) ψ | βγ p j | ψ (cid:105) = 1 d e (cid:104) ψ | H effEDM | ψ (cid:105) (1)Here, c is the speed of light, e is the charge of theelectron, N e refers to the number of electrons in themolecule, β is one of the Dirac matrices, γ is the productof the Dirac matrices, and p j is the momentum of the j th electron. ψ is the wavefunction of a molecular state. Theabove expression casts the eEDM Hamiltonian in termsof one-electron operators, which makes it convenient forcomputations. Further details of the derivation of thisform can be found in [9].To obtain the molecular wavefunction | ψ (cid:105) , we use arelativistic coupled cluster (RCC) method [10, 11]. Thecoupled cluster wavefunction can be written as | ψ (cid:105) = e T | Φ (cid:105) (2)Here, | Φ (cid:105) refers to the Dirac-Fock (DF) wavefunction,which is built from single particle four-component spinors.This is the model wavefunction for the coupled clustercalculations, and is taken to be a single determinant a r X i v : . [ phy s i c s . a t o m - ph ] O c t corresponding to an open shell doublet. T is the clusteroperator. In this work, we use the CCSD (Coupled ClusterSingles and Doubles) approximation, where T = T + T , where T and T are the single (S) and double (D)excitation operators respectively. They are given by T = (cid:88) i,a t ai a † i (3) T = (cid:88) a>b,i>j t abij a † b † ji. (4)Here, t ai and t abij are called the cluster amplitudes. Inour notation, i, j, k, . . . refer to holes and a, b, c, . . . refer toparticles. When a † i acts on a state, a hole i is destroyedfrom that state, and a particle a is created. The action of a † i on a model state, | Φ (cid:105) results in a state denoted by | Φ ai (cid:105) .The CCSD amplitude equations are (cid:104) Φ ai | e − T H N e T | Φ (cid:105) = 0 (5) (cid:10) Φ abij (cid:12)(cid:12) e − T H N e T | Φ (cid:105) = 0 (6)The term e − T H N e T can be written as { H N e T } c , dueto the linked cluster theorem [12, 13]. H N is the normal-ordered Hamiltonian [14]. The subscript c means thateach term in the expression is connected. The effectivefields are calculated by using only the linear terms inthe coupled cluster wavefunction, since the dominantcontributions come from them. Hence we evaluate E eff = (cid:104) Φ | (1 + T + T ) † ( H effEDM d e ) N (1 + T + T ) | Φ (cid:105) c + (cid:104) Φ | H effEDM d e | Φ (cid:105) (7)We note that although the expectation value uses thelinearized expansion of the coupled cluster wavefunction,the amplitudes are evaluated at the CCSD level.We performed our calculations by combining and modi-fying the UTCHEM [15–17] and the DIRAC08 [18] codes.We used the C point group, which reduces the compu-tational time for the atomic-to-molecular orbital integraltransformations. A summary of our calculations, both atthe DF and the CCSD level, are given in Table I, and theresults plotted in Figure 1. We find that the values for E eff are very large for all of the chosen mercury halides,and are typically about one and a half times that ofThO[7] and about five times that of YbF[9]. This can beattributed to the fact that there is strong mixing betweenthe valence 6s and the virtual 6p orbital.We chose uncontracted correlation-consistent, polarizedvalence double zeta (ccpvdz) basis sets for F, Cl and Br[19], and Dyall’s c2v basis sets for Hg [20]. We use Dyall’sbasis for I [20]. We use Gaussian Type Orbitals (GTO),which are kinetically balanced [21]. Our calculations wereperformed without freezing any of the core orbitals. We Molecule Method Basis T , dia E eff (GV/cm)HgF DF Hg:22s,19p,12d,9f,1g - 104.25F:9s,4p,1dHgCl DF Hg:22s,19p,12d,9f,1g - 103.57Cl:12s,8p,1dHgBr DF Hg:22s,19p,12d,9f,1g - 97.89Br:14s,11p,6dHgI DF Hg:22s,19p,12d,9f,1g - 96.85I:8s,6p,6dHgF CCSD Hg:22s,19p,12d,9f,1g 0.0268 115.42F:9s,4p,1dHgCl CCSD Hg:22s,19p,12d,9f,1g 0.0239 113.56Cl:12s,8p,1dHgBr CCSD Hg:22s,19p,12d,9f,1g 0.0255 109.29Br:14s,11p,6dHgI CCSD Hg:22s,19p,12d,9f,1g 0.0206 109.30I:8s,6p,6dTABLE I. Summary of the calculated results ( E eff ) of thepresent work.FIG. 1. Effective electric fields, E eff , for HgX molcules cal-culated using Dirac-Fock wavefunctions (DF, green squares),and using coupled cluster with singles and doubles (CCSD,red triangles). Solid lines are guides to the eye. The differencebetween DF and CCSD values indicates the contribution ofelectron correlation to E eff . used the following bond lengths (in nm) for our calcula-tions: HgF (0.200686) [22], HgCl (0.242), HgBr (0.262),HgI (0.281) [23].We have also reported the T diagnostics (denoted as T , dia ), a small value of which indicates the stability ofsingle reference CCSD calculations. In the table below,we compare our result for E eff in HgF with previous cal-culations. Dmitriev et al. computed the value of E eff inHgF using relativistic effective core potential calculations.They used the minimal atomic basis set for F, and fiverelativistic valence orbitals 5 d / , 5 d / , 6 s / , 6 p / , and6 p / for Hg. Meyer et al. computed E eff for HgF using Work E eff (GV/cm)Y Y Dmitriev et al. [8] 99.26Meyer et al. [24] 68This work 115.42TABLE II. Effective electric field, E eff , in the HgF molecule. non-relativistic software to compare their method withresults obtained by other methods.We observe from the DF and CCSD values of E eff thatcorrelation effects contribute ∼ Term Contribution (GV/cm)DF 104.25 H effEDM T H effEDM T T † H effEDM T † H effEDM T -3.91 T † H effEDM T T † H effEDM T † H effEDM T T † H effEDM T -5.52TABLE III. Contributions from the individual terms to theeffective electric field of HgF. We see that among the correlation terms, the H effEDM T and the T † H effEDM terms together contribute 20.16 GV/cm.But the T † H effEDM T and the T † H effEDM T terms togethercontribute -9.43 GV/cm. The 9 correlation terms henceadd up to 11.17 GV/cm. Note that the H effEDM T term andthe T † H effEDM are zero. This follows fom the application ofslater-Condon rules[14] to an one-body operator, H EDM .The same reasoning applies, for example, also for the H effEDM T term. The H effEDM T term is the off-diagonalmatrix element between the DF reference state, and astate with one electron excited by the electron-electronCoulomb repulsion.The possible sources of error in our calculations stemfrom our choice of basis sets and not taking into accountcertain higher order correlation effects. From the differ-ence in the effective electric field of HgF between the TZand the DZ basis sets, we can estimate the error due tochoice of basis to be around 1.5 percentage for all themercury halides. A conservative estimate of the totalerror due to basis sets and omitting certain higher ordercorrelation effects would be around 5 percentage.The eEDM sensitivity of experiments is ∝ E eff √ N ,where N is the number of molecules whose spin precessionis detected. In addition to their large effective electricfields, there is the particularly interesting possibility thatHgX molecules can be produced in large quantities at ultracold temperatures, e.g. by photo-association of laser-cooled Hg with magnetically trapped halogen atoms [25].An intense and slow beam or fountain of HgX moleculescould result in upto ∼ X Σ state, theyhave a repulsive A Π state which dissociates into Hg ( S )and X ( P ) atoms. State-selective photo-dissociation ofHgX, coupled with laser-induced cycling fluorescence onthe product Hg atom, can be used to detect spin precessionin these molecules with unit efficiency. Molecules used ineEDM experiments must be fully polarized by lab electricfields in order to take full advantage of their effectiveelectric fields. The quantity that sets the scale for therequired lab electric field is E pol = 2 B e /D , where D isthe molecular dipole moment and B e is the rotationalconstant of the molecule. Figure 2 shows the trend forHgX molecules, and picks out HgBr and HgI as attractiveeEDM search candidates due to their combination of large E eff and low E pol . FIG. 2. Polarizing electric field, E pol = 2 B e /D , for HgXmolecules. In summary, we have performed fully relativistic cou-pled cluster calculations of the effective electric fields ina family of molecules, the mercury monohalides. Wefind that these molecules have some of the largest effec-tive electric fields known for polar diatomics, in additionto features that are favourable for experiments. Thiscombination makes the mercury monohalides attractivecandidates for the next generation of eEDM experiments.
ACKNOWLEDGMENTS
The computational results reported in this work wereperformed on the high performance computing facilitiesof IIA, Bangalore, on the Hydra and Kaspar clusters. Weacknowledge Anish Parwage for his help with installingcodes on the clusters. This research was supported byJST, CREST. M.A. thanks MEXT for financial support.The DiRef database was extremely useful in searching forliterature [26]. [1] Michael Dine and Alexander Kusenko. Origin of thematter-antimatter asymmetry.
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