PHYSICAL REVIEW A 95, 042513 (2017)
Theoretical analysis of effective electric fields in mercury monohalides
V. S. Prasannaa,1,2 M. Abe,3,4 V. M. Bannur,2 and B. P. Das5
1Indian Institute of Astrophysics, Koramangala II Block, Bangalore 560034, India
2Department of Physics, Calicut University, Malappuram, Kerala 673 635, India
3Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji City, Tokyo 192-0397, Japan
4JST, CREST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
5International Education and Research Center of Science and Department of Physics, Tokyo Institute of Technology,
2-12-1-H86 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
(Received 11 August 2016; published 21 April 2017)
Mercury monohalides are promising candidates for electron electric dipole moment searches. This is due
to their extremely large values of effective electric fields, besides other attractive experimental features. We
have elucidated the theoretical reasons of our previous work. We have also presented a detailed analysis of our
calculations, by including the most important of the correlation effects’ contributions. We have also analyzed the
major contributions to the effective electric field, at the Dirac-Fock level, and identified those atomic orbitals’
mixings that contribute significantly to it.
DOI: 10.1103/PhysRevA.95.042513
I. INTRODUCTION II. THEORY
The electron electric dipole moment (eEDM) is a conse- The eEDM Hamiltonian, HeEDM, is given by
quence of parity and time-reversal violations [1–4]. It is an
Ne
important nonaccelerator probe of physics beyond the standard ∑
H = −d βσ .E (1)
model [5,6]. Ibrahim et al. make a case that the eEDM can be eEDM e j intl,j
j=1
a sensitive probe of PeV physics [7]. There is a large body of
work on eEDMs and CP violation in supersymmetric models where de is the eEDM. The summation is over the number of
(for example, see Ref. [8]). A knowledge of eEDMs also electrons in the molecule, Ne. β is one of the Dirac matrices,
provides insights into the baryon asymmetry in the universe σ refers to the Pauli matrices, and Eintl is the internal electric
(BAU) [9,10]. One of the Sakharov conditions [11], which field.
gives the necessary prerequisites for BAU, is CP violation. If The shift in energy due to the eEDM is given by
the CPT theorem [12] is true, then T violation must correspond
to CP violation, to preserve CPT symmetry. This correspon- E = 〈ψ |HeEDM|ψ〉 (2)
dence is what connects the two seemingly disparate phenom- = −deEeff. (3)
ena, eEDMs and BAU. The importance of this connection is
demonstrated in the work by Fuyuto et al. [10], who argue Here, |ψ〉 is the ground-state wave function of mercury
that the relationship between the BAU-related CP violations monohalides. Comparing Eqs. (1) and (3), we obtain the
and eEDMs is important for the test of the electroweak following expression for Eeff:
baryogenesis (EWBG) scenario. They proceed to show that ∑Ne
if BAU-related CP violation does exist then the EWBG region Eeff = 〈ψ | βσj .Eintl,j |ψ〉. (4)
might be entirely verified by the future eEDM experiments. j=1
Heavy polar diatomic molecules are currently the preferred
To obtain the wave function, we employ a fully relativistic
candidates to look for a shift in the energy of a molecule in a
coupled cluster method. The wave function is given by
particular state, due to the presence of the eEDM (for example,
see Ref. [13]). The electric field corresponding to that shift in |ψ〉 = eT |0〉. (5)
energy, with the proportionality constant being the eEDM, is
called the effective electric field, . It is the electric field that T is called the cluster operator. |0〉 is the Dirac-Fock waveEeff
an electron experiences, due to all other electrons and nuclei function. We use the relativistic coupled cluster singles and
in the molecule [14]. These calculations warrant a relativistic doubles (CCSD) approximation in our paper. More details
treatment to compute this quantity, asEeff completely vanishes
about the relativistic CCSD method and its salient features
in the nonrelativistic limit [15]. can be found in Refs. [14,17].
The expectation value of any operator, O, in an RCCM, can
We had calculated the effective electric fields of mercury
be expressed as [18,19]
monohalides and identified them as promising candidates for
eEDM searches [16]. The main thrust of this paper is to 〈 〉 = 〈ψ |O|ψ〉O
elaborate on the theoretical aspects of our previous one [16]. 〈ψ |ψ〉
In particular, we analyze and elucidate the contributions T † T
to the effective electric fields, at the Dirac-Fock (DF) and = 〈0|e ONe |0〉C + 〈0|O|0〉. (6)
correlation levels. We employ a relativistic coupled cluster The subscript, N, means that the operator is normal or-
method (RCCM), for our computations. dered [20], and C means that each of the terms are connected
2469-9926/2017/95(4)/042513(6) 042513-1 ©2017 American Physical Society
V. S. PRASANNAA, M. ABE, V. M. BANNUR, AND B. P. DAS PHYSICAL REVIEW A 95, 042513 (2017)
[21]. Therefore, TABLE I. Summary of the basis sets employed in our calculations.
E T † eff T effeff = 〈0|e HeEDM,Ne |0〉C + 〈0|HeEDM|0〉 (7) Atom Basis (DZ) Basis (TZ)
≈ 〈 † eff0|(1 + T1 + T2) HeEDM,N (1 + T1 + T2)|0〉C Hg 22s,19p,12d,9f,1g 29s,24p,15d,11f,2g
+〈 | | F 9s,4p,1d 10s,5p,2d,1f H eff0 eEDM 0〉. (8) Cl 12s,8p,1d 15s,9p,2d,1f
Br 14s,11p,6d 20s,13p,9d,1f
We replace the usual eEDM operator by an effective
eff I 21s,15p,11d 28s,21p,15done [14], HeEDM, given by
N
2 eic ∑
βγ5p
2
j (9)
way:
e = ⎡j 1 ∑MO (MO∑−1)/2
eff ⎣ eff
where c is the speed of light, e is the charge of the electron, 〈ϕi |heEDM|ϕi〉 = 〈ϕi ′ |heEDM|ϕi ′ 〉
′
Ne refers to the number of electrons in the molecule, β is one i i
of the Dirac matrices, γ5 is the product of the Dirac matrices,
⎤
(MO∑−1)/2
and pj is the momentum of the jth electron. This is done, + 〈ϕ ′ |heff |ϕ ′ 〉⎦
so that the Hamiltonian is rewritten in terms of only one- i eEDM i′
i
body operators. The term Eintl [from Eq. (4)] has a two-body
eff
operator in it. Although it can be calculated, in principle, it +〈ϕv|heEDM|ϕv〉. (11)
is very time demanding and complicated. Using an effective In the above expression, we have decomposed the left-hand
one-body operator simplifies the computations by a significant side into three terms. The first and the second summation
amount. Further details can be found in Ref. [14] and the terms on the right-hand side denote the contributions from the
references therein. We consider only the linear terms in the
T doubly occupied orbitals in the Kramers pairs, ϕ ′ and ϕ ′ . Theexpansion of e , both on the bra and the ket sides, in the first i ithird term is the contribution from SOMO. The Kramers pair
term of Eq. (6), as shown in Eq. (8). This is a reasonable orbitals are related by the time-reversal operator (τ ) [24]:
approximation, and we can see this from the accuracy of our
results from our previous works, where we compare them |ϕ ′ 〉 = τ |ϕ ′ 〉 (12)
with experimental values [14,17,22,23]. This approximation, i i
hence, not only saves computational cost by only taking into − |ϕ ′ 〉 = τ |ϕ ′ 〉. (13)
account only the linear terms but also provides very accurate i i
results. Therefore,
Since the dominant contribution to Eeff is at the DF 〈ϕ ′ |heff |ϕ ′ 〉 = 〈ϕ ′ |τ † efflevel [16], we analyze the terms that constitute it. The i eEDM i heEDMτ |ϕi i ′ 〉
contribution, EDFeff , can be rewritten as = −〈ϕi ′ |heffeEDM|ϕi ′ 〉. (14)
EDF = 〈 |H effeff 0 eEDM|0〉 Hence, the first two terms in Eq. (11) cancel out pairwise, and
∑MO only the SOMO remains.
= 〈ϕ |heffi eEDM|ϕi〉
i III. RESULTS AND DISCUSSIONS
= 〈ϕ |heffv eEDM|ϕv〉 In this section, we present the method of calculations used
∑NB 2∑NB 〈 ∣ ∣ 〉 in this paper, followed by a detailed discussion of the results.= 4ic C∗SCL χS ∣p2∣χL . (10) We used and modified the UTCHEM code [25], for the DF
e k l v,k v,l= = + and atomic orbital to MO integral transformations [26]. Wek 1 l NB 1
performed the CCSD calculations in the DIRAC08 program [27].
Here, ϕv refers to the singly occupied molecular orbital The details of the basis sets are given in Table I (uncon-
(SOMO).heffeEDM is the single-particle effective eEDM operator. tracted [14]; kinetically balanced [24] Gaussian-type orbitals;
Summation over the number of molecular orbitals (MO) is cc-pV DZ (correlation consistent polarization valence double
indicated by i, while summations over the number of large zeta) and TZ (triple zeta) basis for F, Cl, and Br [28]; Dyall’s
and small components of the basis sets are given by k and basis for I [29]; and Dyall’s c2v and c3v basis sets for Hg [29]).
l, respectively. NB refers to the number of large component We did not freeze any of our occupied orbitals. We chose the
basis functions. Ck and Cl refer to the coefficients, obtained by following bond lengths (in angstroms): HgF (2.00686) [30],
solving the DF equations, and their superscripts L and S stand HgCl (2.42), HgBr (2.62), and HgI (2.81) [31].
for large and small components, respectively. The χs refer Table II shows the terms from Eq. (8), and also the total
to the atomic orbitals (basis sets) of the constituent atoms. Eeff, for HgX. In our previous work, we had performed this
The mixing between large and small components is due to the analysis for only HgF. Extending it to all HgX enables us
fact that the eEDM operator is off-diagonal. Only the SOMO to study the trends in Eeff, for these molecules. Also, in
survives in the expression for EDFeff , because the remaining our previous work, we had identified HgBr as a promising
terms cancel out. This can be understood in the following candidate, among the HgX molecules. Hence, theoretical
042513-2
THEORETICAL ANALYSIS OF EFFECTIVE ELECTRIC . . . PHYSICAL REVIEW A 95, 042513 (2017)
TABLE II. Contributions, from the individual terms, to the
effective electric field of HgX; cc refers to complex conjugate, of
the term that it accompanies.
Term HgF HgCl HgBr HgI
DF 104.25 103.57 97.89 96.85
H effEDMT1+ cc 20.16 19.34 22.18 24.78
†
T H eff1 EDMT1 −3.91 −3.58 −
(a) (b) (c)
4.07 −4.77
†
T1 H
eff
EDMT2+ cc 0.44 0.194 −0.2 −0.30
†
T2 H
eff
EDMT2 −5.52 −5.96 −6.5 −7.26
Total 115.42 113.56 109.29 109.30
details about HgX other than HgF become important. In the
notation used in the table, H effeEDMT1, for example, actually (d) (e)
refers to 〈0|{H effeEDM}T1|0〉C , where the curly brackets refer
to a normal-ordered operator. This is done for the purpose FIG. 1. Goldstone diagrams for E eff
〈 |{ }| 〉 eff
: (a) DF term, (b) H
eff eEDM
T1,
of brevity. Note that the 0 H 0 is zero, since (c) † †eEDM C T H eff1 eEDMT1 term, and (d) Direct diagrams of T H
eff
1 eEDMT2 term and
the effective eEDM operator is normal ordered [22]. The †T H eff2 eEDMT2 term, respectively.
H effeEDMT2 term, and its complex conjugate, are zero, due to
the Slater-Condon rules [20]. Hence, we are left with seven
nonzero terms. YbF, the correlations account for about 20% [14], while in
The DF term is the largest, and it decreases from HgF RaF [33] it is close to 30%. Again, all of these systems have
to HgI. Correlation effects account for about 9% of the total one unpaired electron, and their heavier atoms have atomic
effective field. This indicates that for these molecules both Eeff numbers fairly close to one another, but their effective fields
and the amount of correlation do not significantly vary with Z and their correlation effects are very different. In HgI, for
of the lighter halide atom. Among the correlation terms, the example, the correlation effects are 10%, owing to the fact
eff effHeEDMT1 term is the largest. The second and the third largest that nearly half of the HeEDMT1 term is canceled out by the
correlation contributions come from the † eff other correlation terms. In RaF, this is not so. In fact, theT2 HeEDMT2 term and eff
† eff HeEDMT1 term adds to about 20 GV/cm, and the rest close tothe T1 HeEDMT1 term. Their effect is to reduce Eeff. The overall −0.5 GV/cm. The values that we finally obtain are a conse-
values of Eeff decrease from HgF to HgBr. HgBr and HgI have quence of several cancellations at work, among the various
almost the same values of Eeff, although the DF value of HgI DF and the correlation terms. We shall attempt to understand
is smaller in comparison with HgBr. This can be understood further these cancellations for HgX in the rest of this paper.
from the fact that the difference between the H effeEDMT1 term
† † This brief discussion illustrates that further detailed theoreticaland the T H eff T + T H eff1 eEDM 1 2 eEDMT2 term is larger for HgI. studies are required to understand better the correlation effects
We shall remark briefly about how the correlation trends and trends, of these class of polar molecules.
vary in the Eeffs of HgX, as compared to those in our previous Figure 1 shows some of the dominant Goldstone diagrams
and ongoing works. In our previous work on the permanent involved in the expectation value expression, given by Eq. (8),
dipole moment (PDM, also known as the molecular electric specifically the DF, the H eff T1, the
†
T H eff T1 term, the
dipole moment) of SrF [17], and subsequently on the PDMs eEDM 1 eEDM
direct counterparts of the †T H eff † effof the other alkaline-earth monofluorides (AEMs) [22], we 1 EDMT2, and the T2 HeEDMT2
had performed the same analysis. Although and PDM are terms. The conjugate diagrams are not given, since they giveEeff
different properties, they do share similarities; for example, the same result.
both the properties depend on the mixing of orbitals of opposite The physical interpretation of these diagrams is discussed
parity. Hence, it is worthwhile to check if there are any in detail in another work on PDMs [22]. For the sake ofeff
similarities in their correlation trends. We first compare the completeness, we choose a representative diagram, HeEDMT1,
correlation trends between the of HgX and the PDMs of the to explain its physical significance. We choose this diagram,Eeff
AEMs. Both of them are systems with one unpaired electron, since it contributes the most to the effective electric field. This
but we see that in AEMs, while correlation can decrease (for term can be expanded as
example, BeF, by around 20%) or increase (for example, BaF, ∑[〈ϕ |{ }heff |ϕ 〉〈ϕ | ]t |ϕ 〉 (15)
by around 20%) the PDM, the effect of correlation on the i eEDM a a i C
i,a
Eeffs of HgX is almost the same throughout, from HgF to HgI.
The PDMs of HgX follow an entirely different trend, where where the summation is over i and a, where i refers to the oc-
the correlations decrease the PDM drastically [32]. We now cupied orbitals (holes) and a corresponds to the virtual orbitals
compare the correlations in theEeffs of HgX with those in YbF, (particles). We obtain this expression, if we apply the Slater-
the candidate that currently sets the second best limit on eEDM, Condon rules to the original expression, 〈0|{H effeEDM}T1|0〉C .
and RaF, a promising candidate for eEDM experiments. In all Mathematically, the H effeEDMT1 term represents an all-order
the HgX molecules, correlations account for about 10%. In residual Coulomb interaction, resulting in an electron from
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V. S. PRASANNAA, M. ABE, V. M. BANNUR, AND B. P. DAS PHYSICAL REVIEW A 95, 042513 (2017)
TABLE III. Summary of the DF results, of the contributions from can expect the matrix elements of the eEDM operator between
various orbitals’ mixings, at the TZ level. these opposite parity orbitals to be large.
The importance of s − p1/2 and p1/2 − s mixing of the
Atom Mixing HgF HgCl HgBr HgI heavier atom in the Eeff of HgF has been understood in the
Hg − −266.29 −262.07 −249.39 −242.34 past, for example, Ref. [34]. We shall compare our resultss p1/2
Hg − 373.37 367.74 349.42 339.56 with the previous ones later in this paper. In our paper, we havep1/2 s
Hg p − d 31.22 25.22 21.84 20.99 taken into account not only the s − p1/2 and p1/2 − s mixing3/2 3/2
Hg d but also that of the other orbitals of both the atoms, and then3/2 − p3/2 −32.26 −26.35 −22.48 −21.84
Hg d − f −0.91 −0.51 −0.39 −0.33 demonstrate that it is the s − p1/2 mixing of the heavier atom5/2 5/2
Hg f − d 0.92 0.52 0.4 0.33 that dominates. In the table, we have only shown the s − p5 2 5 2 1/2/ /
X s − p −2.78 −4.85 −10.58 −17.19 and p1/2 − s mixing of the lighter atom, but that is because we1/2
X p − s 2.79 4.92 11.17 19.87 found the other mixings to be negligibly small. Also, note that1/2
Total: 106.06 104.62 99.99 99.05 our analysis is not only for HgF but for the heavier mercury
DF 105.47 104.03 99.55 98.99 monohalides too. For example, in HgI, the “lighter” atom,
s − p1 2 and p1 2 − s 107.08 105.67 100.03 97.22 iodine, is sufficiently heavy. In spite of that, we see that the/ /
s − p1/2 and p1/2 − s mixing from I is surprisingly small.
Finally, we observe that not only the magnitude of the s −
p1/2 and the p1/2 − s mixings (of the heavier atom) are large
an occupied orbital, i, being excited to a virtual orbital, a, and but so is the remainder when these terms cancel each other’s
then falling back into the same state, i, due to the interaction of contributions. This illustrates the importance of cancellations
the particle with the eEDM. This diagram represents several that occur in ab initio calculations. In the case of iodine (in
correlation effects, like the Brueckner pair correlation [20], HgI), the two terms themselves are non-negligible, but they
among others, but is not obvious from the coupled cluster cancel each other out, leaving behind a very small contribution
diagram, since the T1 part embodies in it the residual Coulomb to the DF Eeff from the lighter atom.
interaction, to all orders of perturbation. We shall now attempt to explain why the H
eff
eEDMT1 term
Table III presents the various contributions to the DF value is large, among the correlation terms. The DF contribution
of E , due to the mixing between various orbitals (or basis dominates among the others, due to the significantly higheff
sets) [Eq. (10)], at the TZ level. We have not presented the anal- difference between the large values of s − p1/2 and p1/2 − s
ysis for the DZ basis sets, since TZ is closer to the actual wave of the Hg atom (the notation is the same as that in Table III).
function, and the results from both the basis sets show the same We wish to reiterate that s is an occupied orbital, while
trend. In the seco∑nd c∑olumn, s − p , for example, is actually p1/2 is a virtual one. Now, let us focus only on the matrix1/2
a shorthand for C∗SCL 〈χS |p2|χL 〉. The first elements. Their values are several orders larger than theirs p1/2 s p1/2 v,s v,p1/2
summation is over all the small component basis sets of the s corresponding coefficients. These large matrix elements, ofeff
angular momentum function, and the second is over the large the form 〈o|heEDM|v〉 (where we abbreviate occupied orbitals
component basis sets of the p1/2 angular momentum function.
as o and virtual orbitals as v), also occur in the expression for
eff
The mixing between the same parity orbitals is zero, and hence HeEDMT1 (and hence it also contains matrix elements between
eff
those terms that contain matrix elements between s and d, for s and p1/2), except that HeEDMT1 has accompanying it a t1
example, are ruled out. Only s, p amplitude, which indicates the “weight” associated with a1/2, p3/2, d3/2, d5/2, and f5/2
orbitals for Hg, and s and p one-hole one-particle excitation. The t amplitudes are like1/2 for X, have been considered in
Eq. (10), since the terms involving the mixing between other probability amplitudes, and thereby lesser than 1 always.
orbitals contribute negligibly to EDF. This can also be recog- Hence, we can view the amplitude as having a “reducing”eff
nized from the difference between the rows labeled “Total,” effect on the H
eff
eEDMT1 term, for each i and a. This is probably
which gives the sum of the mixings associated with the orbitals why the H
eff
eEDMT1 term is not as large as the DF one; the large
considered, and “DF,” which gives the total DF contribution. matrix elements are accompanied by the smaller values of
The difference between the two decreases from F to Br. the t1 amplitudes. Obviously, this is not the only reason why
The combined s − p1 2 and p1 2 − s contribution is clearly the DF term is substantially larger than the other terms. The/ /
the highest among all others, contributing to over 100% of the term also contains matrix elements of the type 〈o|heffeEDM|o〉
total DF value of Eeff in all cases, except HgI. The “anomaly” and 〈v|heffeEDM|v〉, which cancel each other in a way that results
in HgI is due to the halide atom’s contribution becoming in the final value of the DF term. There are also cancellations
important. between various terms that constitute H
eff
eEDMT1, since not all
We observe that the absolute magnitude of all the terms for matrix elements or the t1 amplitudes are positive.
eff
Hg decreases from F to I. However, in X, we see the opposite The matrix elements, of the form 〈o|heEDM|v〉, occur only in
trend. In fact, for HgI, the contribution from X increases the †H effeEDMT1 and T
eff
2 HeEDMT1 (and its complex conjugate term).
effective field by over 2.5 GV/cm. The latter, however, is not as large as the former and is, in
The angular momentum functions are strictly not atomic fact, very small, probably due to two reasons. The first is the
orbitals, but the terms from the basis sets that we employ. cancellations that arise among the four terms that constitute
Hence, we cannot, from our results alone, deduce those prin- †T H eff2 eEDMT1:
cipal quantum numbers that contribute significantly. However, ∑ [
ab∗ a eff ab∗ b eff
we can intuitively guess that the major contribution is from the tij ti 〈b|heEDM|j 〉 + tij tj 〈a|heEDM|i〉
6s and the virtual 6p1/2 orbitals of the Hg atom, since they lie i>j,a>b ]
close in energy, and their radial overlap is large. Moreover, we −tab∗ta〈b|heff |i〉 − tab∗tbij j eEDM ij i 〈a|heffeEDM|j 〉 . (16)
042513-4
THEORETICAL ANALYSIS OF EFFECTIVE ELECTRIC . . . PHYSICAL REVIEW A 95, 042513 (2017)
TABLE IV. The terms that contribute to †T H eff T1. briefly discuss them. The first work on the Eeff of HgF was by1 eEDM
Kozlov [35]. It was a relativistic, semiempirical calculation.
Term Contribution (GV/cm) The focus of the paper was the nuclear anapole moment, and
− electron-nucleus P and T violating interactions. The table ofI 10 3 results gives the final result of Eeff. Note that since it is aII 2.94
− semiempirical calculation it cannot break the final value ofIII 4.44
Eeff into its constituent DF and correlation parts.IV 2.4
Dmitriev et al. [36] computed the Eeff of HgF, using their
calculated bond length of 2.11 Å. They chose the minimal
We have not explicitly mentioned that the operator is normal atomic basis set for F, while for Hg they used five relativistic
ordered, or that each term is connected. The four terms are 0.72, valence orbitals, 5d3/2, 5d5/2, 6s1/2, 6p1/2, and 6p3/2. They
0.24, −0.03, and 0.77, respectively, for HgF, which we choose obtained a value of about 100 GV/cm. Their calculation can
as a representative case (we expect the trends to be similar be described as quasirelativistic, since it requires the addition
for the other monohalides). All the four terms have matrix of the spin-orbit interaction to a nonrelativistic Hamiltonian.
elements of the form 〈o|heffeEDM|v〉, of which two are dominant, Our work is fully relativistic (we do not resort to an effective
and they almost cancel each other out. The second reason is Hamiltonian, but the Dirac-Coulomb one) and has the spin-
that in this term there is another t amplitude, t2, which is also orbit interaction and other effects built into it. They did not
less than 1, and hence “reduces” the net contribution further. account for mixing between orbitals beyond d5/2, the effect of
The †T1 H
eff
eEDMT1 consists of terms of the type 〈o|heff |o〉 F was ignored, and also only the principal quantum numbers 5〈 | | 〉 eEDMand v heff v . To discern how these terms contribute, we and 6 were chosen. We have made no such restrictions. Finally,eEDM
expand the †T H eff T term: they had adopted CI, exciting only three outer electrons. In our1 eEDM 1∑ ∑ CCSD calculation, all the 89 electrons were excited. Hence,
− ta∗ta〈j |heff |i〉 + ta∗tb〈a|heff |b〉 the Hilbert space that we considered to capture the correlationi j eEDM i i eEDM
i,j,a i,a,b effects is larger than that in their work. Our error estimate of 5%
∑ ∗ ∑ ∗ is better than their estimate of 20% . Their estimate ofE does− tai tai 〈 |
eff
i heff a a effeEDM|i〉 + ti ti 〈a|heEDM|a〉 (17) not contain in it information on the DF or correlation contri-
i,a i,a butions. We have provided a detailed breakdown of Eeff in our
= I + II + III + IV. (18) paper. Also, their close agreement with our results forEeff may
be a result of fortuitous cancellations. For example, their work
The first term corresponds to Fig. 1(c). The second term is computed the PDM of HgF to be 4.15 D, which is close to our
similar to the figure, except that the eEDM vertex is between DF value of 3.9 D. But, our relativistic CCSD result is 2.61 D.
two particles, instead of two holes. The third term is the same Meyer et al. [34] calculated Eeff for HgF, among sev-
as the first, except that in its Goldstone diagram both the holes eral other molecules, using their nonrelativistic software,
are the same orbitals, that is, the interaction of the hole with the to compare the accuracy of their method. Later, in 2008,
eEDM leaves it unchanged. The fourth is again the same as the they improved upon their approach further, to obtain a more
second, but with the two particles being the same orbital. We
† accurate value, of 95 GV/cm [37].have expanded T H eff1 eEDMT1 this way, so that we can understand In our previous work on HgX [16], we had taken recourse
which types of matrix elements contribute to it dominantly. to the relativistic coupled cluster method. We had shown that
Table IV summarizes the contributions to this term. Eeff, for all the HgX molecules, is substantially larger than that
We observe from the table that, magnitude-wise, the terms for all the current eEDM molecular candidates. However, we
that contain the eEDM operator between the same orbitals had not performed a detailed analysis of the physical effects
(terms II and III) contribute significantly. Note that these at the DF and correlation levels, which is what we have done
matrix elements are nonzero, although heffeEDM is P-odd. This in the present paper. We wish to emphasize that besides the
is because the orbitals are MOs, and each MO is expanded as fact that we use a fully relativistic coupled cluster approach
a linear combination of basis functions, of different angular and extend the results to all HgX we also break the final value
momenta. The major contributions to the †T H eff2 eEDMT2 term of Eeff down into its constituent terms, both at the DF and
are also from matrix elements between the same orbitals correlation levels.
(−4.7 GV/cm). Since we are elaborating on the theoretical aspects of
Table V summarizes the results obtained from previous our previous work, the error estimates are the same (see
works. Only results for HgF are available, and we proceed to Ref. [16]). We recently improved upon our earlier results
on HgX Eeffs, where we take into account the effect of the
TABLE V. Effective electric field, Eeff , in the HgF molecule, neglected nonlinear coupled cluster terms in the expectation
calculated in earlier literature. value [38]. Therefore, the nonlinear terms, in fact, contribute
far less than our earlier estimate of 3.5% .
Work Eeff (GV/cm)
Dmitriev et al. [36] 99.26
Meyer et al. [34] 68 IV. CONCLUSION
Meyer and Bohn [37] 95
This paper 115.42 We have calculated the effective electric fields of mercury
monohalides. We have not frozen any of the core orbitals. We
042513-5
V. S. PRASANNAA, M. ABE, V. M. BANNUR, AND B. P. DAS PHYSICAL REVIEW A 95, 042513 (2017)
employed Dyall’s basis sets for Hg and I, and cc-pV basis ACKNOWLEDGMENTS
sets for the other halides. The DF term contributes the most
Computations were performed on the GPC supercomputer
to Eeff (about 90%). We have reported the trends in some at the SciNet HPC Consortium. SciNet is funded by the Canada
of the correlation terms for these molecules, at the DZ level. Foundation for Innovation under the auspices of Compute
We observe that the dominant contribution to the correlation Canada; the Government of Ontario; Ontario Research Fund—
effects is from a one hole-one particle excitation coupled Research Excellence; and the University of Toronto [39]. We
cluster diagram. We present one example of a physical effect also used the Hydra cluster, in Indian Institute of Astrophysics
that is included in this diagram. We have also reported on (IIA). This research was supported by Japan Science and
those mixings of atomic orbitals that significantly contribute Technology Agency, CREST. M.A. thanks The Ministry
to the DF value of Eeff and observed their trends, at the TZ
− of Education, Culture, Sports, Science and Technology forlevel. We recognize that the s p1/2 mixing in Hg contributes financial support. The DiRef database was extremely useful in
dominantly to EDFeff . searching for literature [40].
[1] L. Landau, Nucl. Phys. 3, 127131 (1957). [23] A. Sunaga, M. Abe, M. Hada, and B. P. Das, Phys. Rev. A 93,
[2] L. E. Ballentine, in Quantum Mechanics: A Modern Develop- 042507 (2016).
ment (World Scientific, Singapore, 1998), Chap. 13, pp. 372– [24] K. G. Dyall and K. Faegri Jr., Introduction to Relativistic
373, 384–386. Quantum Chemistry (Oxford University Press, New York,
[3] P. G. H. Sandars, J. Phys. B 1, 499 (1968). 2006).
[4] I. B. Kriplovich and S. K. Lamoureaux, CP Violation Without [25] T. Yanai, M. Kamiya, Y. Kawashima, T. Nakajima, H. Nakano,
Strangeness (Springer, London, 2011), Chap. 2. Y. Nakao, H. Sekino, J. Paulovic, T. Tsuneda, S. Yanagisawa,
[5] T. Fukuyama, Int. J. Mod. Phys. A 27, 1230015 (2012). and K. Hirao, in International Conference on Computational
[6] M. Pospelov and A. Ritz, Ann. Phys. 318, 119 (2005). Science ICCS 2003, Melbourne, Australia, edited by P. M.
[7] T. Ibrahim, A. Itani, and P. Nath, Phys. Rev. D 90, 055006 A. Sloot, D. Abramson, A. V. Bogdanov, Y. E. Gorbachev, J.
(2014). J. Dongarra, and A. Y. Zomaya, Lecture Notes in Computer
[8] A. Arbey, J. Ellis, R. M. Godbole, and F. Mahmoudi, Eur. Phys. Science Vol. 2660 (Springer, Berlin, 2003), pp. 84–95; M. Abe,
J. C 75, 1 (2015). H. Iikura, M. Kamiya, T. Nakajima, J. Paulovic, S. Yanagisawa,
[9] A. M. Kazarian, S. V. Kuzmin, and M. E. Shaposhnikov, and T. Yanai, Comput. Sci. ICCS 2003, 6526 (2001); T. Yanai, T.
Phys. Lett. B 276, 131 (1992). Nakajima, Y. Ishikawa, and K. Hirao, J. Chem. Phys. 116, 10122
[10] K. Fuyuto, J. Hisano, and E. Senaha, Phys. Lett. B 755, 491 (2002).
(2016). [26] M. Abe, T. Yanai, T. Nakajima, and K. Hirao, Chem. Phys. Lett.
[11] A. D. Sakharov, Sov. Phys. Usp. 34, 392 (1991) 388, 68 (2004).
[JETP 5, 24 (1967)]. [27] L. Visscher, T. J. Lee, and K. G. Dyall, J. Chem. Phys. 105, 8769
[12] G. Luders, Ann. Phys. 281, 1004 (2000). (1996).
[13] The ACME Collaboration: J. Baron, W. C. Campbell, D. [28] K. L. Schuchardt, B. T. Didier, T. Elsetha-gen, L. Sun, V.
DeMille, J. M. Doyle, G. Gabrielse, Y. V. Gurevich, P. W. Hess, Gurumoorthi, J. Chase, J. Li, and T. L. Windus, J. Chem. Inf.
N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Model. 47, 1045 (2007).
Panda, M. F. Parsons, E. S. Petrik, B. Spaun, A. C. Vutha, and [29] K. G. Dyall, Theor. Chem. Acc. 99, 366 (1998); 108, 365 (2002);
A. D. West, Science 343, 269 (2014). Addendum: 115, 441 (2006).
[14] M. Abe, G. Gopakumar, M. Hada, B. P. Das, H. Tatewaki, and [30] S. Knecht, S. Fux, R. van Meer, L. Visscher, M. Reiher, and T.
D. Mukherjee, Phys. Rev. A 90, 022501 (2014). Saue, Theor. Chem. Acc. 129, 631 (2011).
[15] P. G. H. Sandars, J. Phys. B 1, 511 (1968). [31] N.-H. Cheung and T. A. Cool, J. Quantum Spectrosc. Radiat.
[16] V. S. Prasannaa, A. C. Vutha, M. Abe, and B. P. Das, Phys. Rev. Transfer 21, 397 (1979).
Lett. 114, 183001 (2015). [32] V. S. Prasannaa, M. Abe, and B. P. Das (unpublished).
[17] V. S. Prasannaa, M. Abe, and B. P. Das, Phys. Rev. A 90, 052507 [33] V. S. Prasannaa et al. (unpublished).
(2014). [34] E. R. Meyer, J. L. Bohn, and M. P. Deskevich, Phys. Rev. A 73,
[18] J. Cizek, in Advances in Chemical Physics, Volume XIV: 062108 (2006).
Correlation Effects in Atoms and Molecules, edited by W. C. [35] M. G. Kozlov, Zh. Eksp. Teor. Fiz. 89, 1933 (1985).
Lefebvre and C. Moser (Interscience Publishers, New York, [36] Y. Y. Dmitriev, Y. G. Khait, M. G. Kozlov, L. N. Labzovsky,
1969). A. O. Mitrushenkov, A. V. Shtoff, and A. V. Titov, Phys. Lett.
[19] I. Shavitt and R. J. Bartlett, in Many Body Methods in Chemistry A 167, 280 (1992).
and Physics (Cambridge University Press, Cambridge, UK, [37] E. R. Meyer and J. L. Bohn, Phys. Rev. A 78, 010502(R) (2008).
2009), Chap. 11, p. 347. [38] V. S. Prasannaa, M. Abe, V. M. Bannur, and B. P. Das
[20] I. Lindgren and J. Morrison, in Atomic Many-Body Theory, (unpublished).
2nd ed. (Springer-Verlag, Berlin, 1986), p. 232. [39] C. Loken, D. Gruner, L. Groer, R. Peltier, N. Bunn, M. Craig, T.
[21] V. Kvasnicka, V. Laurinc, and S. Biskupic, Phys. Rep. 90, 159 Henriques, J. Dempsey, C.-H. Yu, J. Chen, L. J. Dursi, J. Chong,
(1982). S. Northrup, J. Pinto, N. Knecht, and R. Van Zon, J. Phys.: Conf.
[22] V. S. Prasannaa, S. Sreerekha, M. Abe, V. M. Bannur, and B. P. Ser. 256, 012026 (2010).
Das, Phys. Rev. A 93, 042504 (2016). [40] P. F. Bernath and S. McLeod, J. Mol. Spectrosc. 1, 287 (2001).
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