P , T -Violating and Magnetic Hyperfine Interactions in Atomic Thallium

Abstract

We present state-of-the-art string-based relativistic general-excitation-rank configuration interaction and coupled cluster calculations of the electron electric dipole moment, the nucleon–electron scalar-pseudoscalar, and the magnetic hyperfine interaction constants ( α d e , α C S , A | | , respectively) for the thallium atomic ground state 2 P 1 / 2 . Our present best values are α d e = − 558 ± 28 , α C S = 6.77 ± 0.34 [ 10 − 18 e cm], and A | | = 21172 ± 1059 [MHz]. The central value of the latter constant agrees with the experimental result to within 0.7% and serves as a measurable probe of the P , T -violating interaction constants. Our findings lead to a significant reduction of the theoretical uncertainties for P , T -odd interaction constants for atomic thallium but not to stronger constraints on the electron electric dipole moment, d e , or the nucleon–electron scalar-pseudoscalar coupling constant, C S .

Full text

symmetry
S
S
Article
P,T-Violating and Magnetic Hyperfine Interactions
in Atomic Thallium
Timo Fleig 1,* and Leonid V. Skripnikov 2
1Laboratoire de Chimie et Physique Quantiques, IRSAMC, Université Paul Sabatier Toulouse III, 118 Route
de Narbonne, F-31062 Toulouse, France
2B.P. Konstantinov Petersburg Nuclear Physics Institute of National Research Centre “Kurchatov Institute”,
Gatchina, 188300 Leningrad District, Russia; leonidos239@gmail.com
*Correspondence: timo.fleig@irsamc.ups-tlse.fr
Received: 5 March 2020; Accepted: 17 March 2020; Published: 30 March 2020


Abstract:
We present state-of-the-art string-based relativistic general-excitation-rank configuration
interaction and coupled cluster calculations of the electron electric dipole moment,
the nucleon–electron scalar-pseudoscalar, and the magnetic hyperfine interaction constants
(
αde
,
αCS
,
A||
, respectively) for the thallium atomic ground state
2P1/2
. Our present best values are
αde=−
558
±
28,
αCS=
6.77
±
0.34
[
10
−18e
cm
]
, and
A|| =
21172
±
1059 [MHz]. The central value of
the latter constant agrees with the experimental result to within 0.7% and serves as a measurable
probe of the
P,T
-violating interaction constants. Our findings lead to a significant reduction of the
theoretical uncertainties for
P,T
-odd interaction constants for atomic thallium but not to stronger
constraints on the electron electric dipole moment,
de
, or the nucleon–electron scalar-pseudoscalar
coupling constant, CS.
Keywords:
electron electric dipole moment; scalar-pseudoscalar interaction; magnetic hyperfine
interaction; relativistic many-body theory
1. Introduction
Electric dipole moments (EDM) of elementary particles, atoms and molecules give rise to spatial
parity (
P
) and time-reversal (
T
) violating interactions [
1
] and are a powerful probe for physics beyond
the standard model (BSM) [
2
]. Current single-source limits [
3
–
5
] on the electron EDM, for instance,
can probe New Physics (NP) up to an energy scale of 1000 TeV [
6
] (radiative stability approach) or
even greater [
7
], surpassing the current sensitivity of the Large Hadron Collider for corresponding
sources of NP.
Until today no low-energy EDM experiment has delivered a positive result. However, the obtained
EDM upper bounds are useful for constraining
CP
-violating parameters [
8
] of BSM models, cast as
effective field theories [6,9] at different energy scales.
Open-shell atomic and molecular systems are particularly sensitive probes of leptonic and
semi-leptonic
CP
-violation [
10
]. In most BSM models [
11
] the dominant
CP
-odd sources are the
electron EDM,
de
, and the nucleon–electron scalar-pseudoscalar (Ne-SPS) coupling,
CS
. Figure 1
(Courtesy: Martin Jung, Torino, Italy (2019)) shows the constraints (yellow surface) on
de
and
CS
using
the combined information from measurements [
3
,
12
–
14
] and calculations [
4
,
5
,
15
–
23
], including the
associated experimental and theoretical uncertainties, on the open-shell systems ThO (green), YbF (red;
the red surface underlies the others and its extent is indicated by the thin red line), HfF
+
(orange)
and Tl (blue) through a global fit in the
de
/
CS
plane. Results from a single system do, therefore,
not constrain
de
or
CS
individually at all in this multiple-source interpretation [
24
], but lead to a
Symmetry 2020,12, 498; doi:10.3390/sym12040498 www.mdpi.com/journal/symmetry

Symmetry 2020,12, 498 2 of 16
fan-shaped surface of allowed combinations. The width of this surface is a function of the experimental
and theoretical uncertainties.
Figure 1. Constraints on deand CSwithout the results of this work. See text for details.
This means that a substantial reduction of an uncertainty for an individual system could lead to
more stringent constraints on the unknown
CP
-violating parameters. The main reason for this is that
the surfaces for different systems are not fully aligned, which is due to the different dependency of
electron EDM and Ne-SPS atomic interactions on the electric charge of the respective heavy nuclei
[25]
.
A substantial part of the width of the surface for the Tl atom is due to the great spread of theoretical
values for the electron EDM atomic enhancement,
R
, calculated in the past by various groups using
different electronic-structure approaches [
1
,
19
–
23
,
26
]. Strikingly, Nataraj et al. [
22
] used a high-level
many-body approach, the Coupled Cluster (CC) method, and produced a value for
R
that strongly
disagrees with the results from all other groups that have included electron correlation effects, on the
order of 20%.
The purpose of this paper is two-fold:
1.
We use state-of-the-art relativistic Configuration Interaction (CI) and Coupled Cluster
approaches for large-scale applications to determine the mentioned atomic interaction constants.
Our calculations represent the most elaborate treatment of electron correlation effects to date
on the discussed properties of the thallium atom ground state. We put particular emphasis on
the electron EDM enhancement
R
and a conclusive resolution of the major discrepancy between
literature values. Claims about physical effects that purportedly underlie these discrepancies
are scrutinized.
2.
We investigate whether a reduced uncertainty for
R
(Tl) impacts the above-described constraints
on deand CS.
The paper is structured as follows. In Section 2we lay out the theory underlying the atomic
electron EDM, Ne-SPS, and magnetic hyperfine interaction constants. Electron EDM and Ne-SPS
interactions are both sensitive to electron spin density in the vicinity of the atomic nucleus. The same is
true for the magnetic hyperfine interaction. For this reason, the latter is an experimentally measurable
probe for the New Physics atomic interaction constants (that cannot be measured by experiment),
and we thus include it in our study as a validating property. The following Section 3contains technical
details about our calculations, results, and a discussion of these results in comparison with literature
values. The final Section 4concludes on our findings.

Symmetry 2020,12, 498 3 of 16
2. Theory
An atomic EDM is defined [27] (p. 16) as
da=−lim
Eext→0∂(∆ε)
∂Eext (1)
where
∆ε
is a
P,T
-odd energy shift and
Eext
is an external electric field. In atoms with nuclear spin
I≤1
2
[
28
] and in an electronic state with unpaired electrons, this energy shift is dominated by and
originates from either the electron EDM,
de
, or a
P,T
-odd nucleon–electron (Ne) interaction, or a
combination of the two [10,11]. The two cases are presented separately.
2.1. Atomic Edm Due to Electron Edm
The Hamiltonian for the interaction of the electron electric dipole moment,
de
, is for an
atomic system
HEDM =−∑
j
dj·E(rj) = −de∑
j
γ0
jΣj·E(rj)(2)
where
γ0
is a Dirac matrix,
Σ= σ0
0σ!
is a vector of spin matrices in Dirac representation,
j
is an
electron index,
E(rj)
the electric field at position
rj
and the bare fermion’s electric dipole moment is
expressed as d=deγ0Σ, necessarily linearly dependent on the particle’s spin vector Σ[1,29].
Supposing a non-zero electron EDM de, the resulting energy shift can be evaluated as
∆εEDM =de*−∑
j
γ0
jΣj·E(rj)+ψ(Eext)
(3)
where
ψ(Eext)
is the field-dependent atomic wavefunction of the state in question. The expectation
value in Equation (3) has the physical dimension of electric field and can be regarded as the mean
interaction of each electron EDM with this field in the respective state. Following stratagem II of
Lindroth et al. [
30
] the expectation value is recast in electronic momentum form as an effective
one-body operator
*−∑
j
γ0
jΣj·E(rj)+ψ(Eext)≈2ıc
e¯h*∑
j
γ0
jγ5
jp2
j+ψ(Eext)
(4)
where the approximation lies in assuming that
ψ
is an exact eigenfunction of the field-dependent
Hamiltonian of the system. This momentum-form EDM operator has already been used as early as in
1986, by Johnson et al. [
26
]. In the present work the field-dependent Hamiltonian is the Dirac-Coulomb
Hamiltonian (in a.u.,e=me=¯h=1)
ˆ
H:=ˆ
HDirac-Coulomb +ˆ
HInt-Dipole
=
n
∑
j"cαj·pj+βjc2−Z
rjK
114#+
n
∑
k>j
1
rjk
114+∑
j
rj·Eext 114(5)
with
Eext
weak and homogeneous, the indices
j
,
k
run over
n
electrons,
Z
the proton number with the
nucleus
K
placed at the origin, and
α
are standard Dirac matrices.
Eext
is not treated as a perturbation
but included a priori in the variational optimization of the atomic wavefunction. Furthermore, the final
results reported in this work include high excitation ranks in the correlation expansion of
ψ
. For these
reasons, the approximation in Equation (4) is considered very good in the present case.

Symmetry 2020,12, 498 4 of 16
Within the so-defined picture and using Equations (1), (3), and (4) the atomic EDM becomes
da=−lim
Eext→0
∂
∂Eext
2ıc de
e¯h*∑
j
γ0
jγ5
jp2
j+ψ(Eext)
. (6)
The (dimensionless) atomic EDM enhancement factor is defined as
R:=da
de
. Denoting
Eeff =
2ıc
e¯h*∑
j
γ0
jγ5
jp2
j+ψ(Eext)
for the sake of simplicity, the enhancement factor is
R=−lim
Eext→0∂Eeff
∂Eext . (7)
The external field used in the experiment on Tl [
14
] was
Eext =
1.23
×
10
7hV
mi≈
0.2392
×
10
−4
a.u. In the present work
Eext =
0.24
×
10
−4
a.u. is used. This is a very small field which is well within
the linear regime considering the derivative in Equation (7). The enhancement factor may under these
circumstances be written as a function of two field points
Rlin =−∆Eeff
∆Eext
=−Eeff(2)−Eeff(1)
Eext(2)−Eext (1). (8)
We set
Eext(
1
):=
0, in which case atomic states are parity eigenstates. Since the EDM operator is
parity odd, it follows that Eeff(1) = 0, and so
R≈Rlin =−Eeff
Eext . (9)
Eeff
is calculated as described in reference [
31
].
ψ
is an approximate configuration interaction (CI)
eigenfunction of the Dirac-Coulomb Hamiltonian including
Eext
. Alternatively,
Eeff
can be calculated
within the finite-field approach [32,33]. The latter has been used in coupled cluster calculations.
The electron EDM enhancement factor Ris in the particle physics literature often denoted as
αde:=R, (10)
the atomic-scale interaction constant of the electron EDM.
2.2. Nucleon–Electron Scalar-Pseudoscalar Interaction
The effective Hamiltonian for a
P,T
-odd nucleon–electron scalar-pseudoscalar interaction is
written as [34]
HNe-SPS =ıGF
√2ACS∑
j
γ0
jγ5
jρN(rj)(11)
and the resulting atomic energy shift is accordingly
∆εNe-SPS =GF
√2ACS*ı∑
j
γ0
jγ5
jρN(rj)+Ψ(Eext)
, (12)
where
A
is the nucleon number,
CS
is the S-PS nucleon–electron coupling constant,
GF
is the Fermi
constant (A comment on units: Its value is
GF
(¯hc)3=
1.166364
×
10
−5[GeV]−2=
0.86366
×
10
−20E−2
H
.
With
¯h=
1 a.u. and
c=
137.036 a.u., the Fermi constant is also expressed as
GF=
2.2225
×
10
−14
a.u.)
and
ρN(rj)
is the nucleon density at the position of electron
j
. Please note that in the present work we
define
γ5:=ıγ0γ1γ2γ3
, whereas Flambaum and co-workers [
21
,
25
] define
γ5:=−ıγ0γ1γ2γ3
which

Symmetry 2020,12, 498 5 of 16
explains the sign difference between the present Ne-SPS atomic interaction constants and those of
Flambaum and co-workers.
Next, we define (see also reference [
35
]) in analogy with Equation (7) an Ne-SPS ratio (The physical
dimension of the
S
ratio is dim
(S) =
dim
ρN
E=hQ T2
M L4i
. This is consistent with the dimension of
S
in
the definition, Equation (13), where dim(S) = dimda
CSGF=hQ T2
M L4i.)
S:=da
ACSGF
√2
(13)
and so one can write, using Equation (1),
S=−lim
Eext→0
∂
∂Eext *ı∑
j
γ0
jγ5
jρN(rj)+Ψ(Eext)
(14)
and in the linear regime
S=−*ı∑
j
γ0
jγ5
jρN(rj)+Ψ(Eext)
Eext . (15)
The initial implementation of this expectation value in the latter expression has been described in
reference [
36
]. The independent implementation of the matrix elements of the Hamiltonian (11) has
been developed in ref. [4].
For comparison with literature results we also define the S-PS nucleon–electron
interaction constant
αCS:=da
CS
=S A GF
√2. (16)
2.3. Magnetic Hyperfine Interaction
Minimal substitution according to
p−→ p−q
cA
in the Dirac equation and the representation of
the vector potential in magnetic dipole approximation as
AD(r) = m×r
r3
with
m
the nuclear magnetic
dipole moment leads to the magnetic hyperfine Hamiltonian
ˆ
HHF =cα·−q
c
m×r
r3=qm·α×r
r3(17)
for a single point charge
q
at position
r
outside the finite nucleus. Given the nuclear magnetic dipole
moment vector as
m=µ
IµNI=gIµNI
where
µ
is the magnetic moment in nuclear magnetons (
µN
),
gIis the nuclear g-factor, and Iis the nuclear spin, Equation (17) for a single electron is written as
ˆ
HHF =−eµ
IµNI·α×r
r3(18)
Based on Equation (18) we now define the magnetic hyperfine interaction constant for
n
electrons
in the field of nucleus K(in a.u.)
A||(K) = −µK[µN]
2cI mpMJDΨJ,MJ
n
∑
i=1 αi×riK
r3
iK !zΨJ,MJE(19)
where
1
2cmp
is the nuclear magneton in a.u., and
mp
is the proton rest mass. The term
1
MJ
in the prefactor
of Equation (19) is explained as follows.

Symmetry 2020,12, 498 6 of 16
The vector operator
αi×riK
r3
iK z
can be regarded as the
q=
0 component of a rank
k=
1 irreducible
tensor operator
ˆ
T(k)
q
. Application of the Wigner-Eckart Theorem to the diagonal matrix element in
Equation (19) yields
Dα,J,MJ|ˆ
T(1)
0|α,J,MJE=J,MJ; 1, 0|J, 1; J,MJDα,J|| ˆ
T(1)||α,JE
√2J+1
where the Clebsch–Gordan coefficient is—using the general definition in Ref. [
37
], p. 27—evaluated as
J,MJ; 1, 0|J, 1; J,MJ=MJ
1
pJ(J+1), (20)
which depends linearly on the total electronic angular momentum projection quantum number
MJ
.
However, the magnetic hyperfine energy must be independent of
MJ
which is assured by the above
prefactor
1
MJ
. Magnetic hyperfine interaction matrix elements have been calculated based on the
implementations in references [
4
,
38
] which do not make direct use of the Wigner-Eckart theorem and
reduced matrix elements.
3. Results and Discussion
3.1. Technical Details
Gaussian atomic basis sets of double-, triple-, and quadruple-
ζ
quality [
39
–
41
] (including
correlating functions for 4
f
and 5
d
shells in the case of CI and cvDZ/CC) [
42
] have been used
in the present work.
The atomic spinor basis is obtained in Dirac-Coulomb Hartree–Fock (DCHF) approximation where
the Fock operator is defined by averaging over 6
p1
j=1/2
and 6
p1
j=3/2
open-shell electronic configurations.
A locally modified version of the
DIRAC
program package [
43
] has been used for all
electronic-structure calculations. Interelectron correlation effects are taken into account through
relativistic Configuration Interaction (CI) theory as implemented in the
KRCI
module [
44
] of
DIRAC
. Kramer’s unrestricted CC calculations have been carried out within the MRCC code [
45
–
47
].
Both implementations are based on creator-string driven algorithms and can treat expansions of
general excitation rank.
The nomenclature for both CI and CC models is defined as: S, D, T, etc. denotes Singles,
Doubles, Triples etc. replacements with respect to the reference determinant. The following number
is the number of correlated electrons and encodes which occupied shells are included in the CI
or CC expansion. In detail we have 3
b= (
6
s
, 6
p)
, 13
b= (
5
d
, 6
s
, 6
p)
, 21
b= (
5
s
, 5
p
, 5
d
, 6
s
, 6
p)
,
29
b= (
4
s
, 4
p
, 5
s
, 5
p
, 5
d
, 6
s
, 6
p)
, 31
b= (
4
d
, 5
s
, 5
p
, 5
d
, 6
s
, 6
p)
, 35
b= (
4
f
, 5
s
, 5
p
, 5
d
, 6
s
, 6
p)
. 81
b=
(
1
s
, 2
s
, 2
p
, 3
s
, 3
p
, 3
d
, 4
f
, 5
s
, 5
p
, 5
d
, 6
s
, 6
p)
. The notation type S10_SD13, as an example, means that
the model SD13 has been approximated by omitting Double excitations from the
(
5
d)
shells. CAS3in4
means that an active space is used with all possible determinant occupations distributing the 3 valence
electrons over the 4 valence Kramers pairs. Further details about active-space-based correlation
expansions are given in Ref. [36].
The use of a Kramer’s unrestricted formalism allows performance of coupled cluster calculations
of the hyperfine structure constant as well as other properties considered in this paper within the
finite-field approach. This method is equivalent to the analytical evaluation of energy derivatives
within the Lambda-equation technique [
48
]. However, some additional uncertainty of the finite-field
approach can be expected due to numerical differentiation. To estimate this uncertainty we have
compared the values of the effective electric field acting on the electron EDM in Tl placed in the
(experimental) external electric field using these two approaches at the CCSD level and cvTZ basis set.
The finite-field value differs from the analytical value only by 0.02%. The advantage of the finite-field

Symmetry 2020,12, 498 7 of 16
approach is that one can use CC models for which the analytical evaluation of energy derivatives is
not implemented, in particular in the four-component relativistic domain.
We use the experimental value [
49
] for the nuclear magnetic moment of
205
Tl with nuclear
spin
I=1
2
,
µ=
1.63821[
µN
], in calculations of the magnetic hyperfine interaction constant. In all
calculations the Tl nucleus is described by a Gaussian distribution for the nuclear density with exponent
taken from Ref. [50].
3.2. Results for Atomic Interaction Constants
The results from the systematic study of many-body effects on atomic EDM enhancement (
R
),
Ne-SPS interaction ratio (
S
) and magnetic hyperfine interaction constant (
A
) are compiled in Table 1.
The general strategy is to first qualitatively investigate the relative importance of various many-body
effects on the properties using a rather small atomic basis set. Then, in a second step, accurate models
are developed that include all important many-body effects using the insight from the first step
and larger atomic basis sets. Since EDM enhancement and Ne-SPS interaction ratio are analytically
related [25] it is sufficient to discuss the trends for Ronly.
3.3. Step 1: Many-Body Effects in cvDZ Basis
3.3.1. Valence Electron Correlation
The result of
R=−
388 for CAS1in3 which is a singles CI expansion for the electronic ground
state can be regarded as close to a DCHF result. The Full CI (FCI) result including only the three
valence electrons (CAS3in4_SDT3/60au) of
R=−
487 shows that valence correlation effects lead to a
considerable change by more than 25% (in the large cvQZ basis by more than 35%). The valence FCI
enhancement in cvQZ basis of
R=−
587 is, therefore, a benchmark. This value is closely reproduced
using the universal basis set of reference [
22
]. Further effects can be considered to be modifications of
this benchmark result and will be studied one by one.
3.3.2. Subvalence Electron Correlation
Subvalence electrons of the Tl atoms are those occupying the 5
s
, 5
p
, and 5
d
shells. All other
electrons will be considered core electrons. Correlations among the 5
d
electrons and in particular of
the 5
d
and the valence electrons lead to a strong decrease of
R
, on the absolute, on the order of 10%
(for instance, compare models SD10_CAS3in4_SD13 and CAS3in4_SD3). Corresponding contributions
from the 5
s
and 5
p
electrons are significantly smaller (compare SD18_CAS3in4_SD21 with
SD10_CAS3in4_SD13).
3.3.3. Outer-Core Electron Correlation
Outer-core-valence correlations have been evaluated by allowing for one hole in the respective
outer core spinors along with excitations from the subvalence and valence electrons (compare,
for instance, S8_SD18_CAS3in4_SDT29 with SD18_CAS3in4_SDT21). In sum for the shells with
effective principal quantum number n=4 these effects amount to about 1.5%.

Symmetry 2020,12, 498 8 of 16
Table 1.
R, S, and A for Tl atom. By default, calculations were performed using DCHF spinors for
the neutral Tl atom (
VN
potential) and, for comparison in selected cases, with the Tl
+
cation (
VN−1
potential) and Tl3+cation (VN−3potential) spinors.
Model/Virtual Cutoff R S [a.u.] A||(205 Tl) [MHz]
Dyall cvDZ
CAS1in3 −388 269 18,800
CAS3in4 −415 288 18,800
CAS3in4_SD3/60au −487 339 19,092
CAS3in4_SDT3/60au −487 339 19,103
S10_CAS3in4_SD13/10au −458 321 20,003
SD10_CAS3in4_SD13/10au −442 309 19,502
SD10_CAS3in4_SD13/30au −441 309 19,575
SD10_CAS3in4_SDT13/10au −465 326 19,357
SD10_CAS3in4_SDTQ13/10au −464 326 19,345
SDT10_CAS3in4_SDT13/10au −460 323 19,254
SDT10_CAS3in4_SDTQ13/10au −460 323 19,341
SD18_CAS3in4_SD21/10au −437 307 19,445
SD18_CAS3in4_SD21/10au(Tl+)−428 300 18,934
S8_SD18_CAS3in4_SD29/10au −438 308 19,536
SD18_CAS3in4_SD21/30au −443 311 19,758
SD18_CAS3in4_SD21/60au −443 311 19,759
SD8_SD18_CAS3in4_SD29/30au −449 315 19,980
SD18_CAS3in4_SDT21/10au −473 331 19,439
SD18_CAS3in4_SDT21/10au(Tl+)−467 328 19,228
SDT18_CAS3in4_SDT21/10au −461 325 19,274
SD18_CAS3in4_SDT21/30au −483 338 19,761
SD18_CAS3in4_SDT21/60au −483 338 19,763
S10_SD18_CAS3in4_SDT31/10au −469 329 19,423
S14_SD18_CAS3in4_SDT35/10au −469 330 19,448
S8_SD18_CAS3in4_SDT29/30au −484 340 19,999
SD8_SDT10_CAS3in4_SDT21/10au −471 331
SD18_CAS3in4_SDTQ21/10au −469 329 19,395
Dyall cvTZ
CAS3in4 −460 323
CAS3in4_SD3/10au −565 397 19,027
CAS3in4_SD3/50au −565 397 19,041
CAS3in4_SDT3/50au −566 398 19,050
SD18_CAS3in4_SD21/10au −481 340 19,619
SD18_CAS3in4_SD21/30au −484 342 19,751
SD18_CAS3in4_SDT21/10au −542 383 19,995
SD18_CAS3in4_SDT21/10au(Tl3+)−524 371
SD18_CAS3in4_SDT21/20au −541 383
Dyall cvQZ
CAS1in3 −429 301 18,806
CAS3in4 −476 334 18,806
CAS3in4_SD3/10au −587 412 19,023
CAS3in4_SD3/35au −587 412 19,050
CAS3in4_SDT3/35au −587 413 19,060
SD18_CAS3in4_SD21/35au −459 322 17,442
SD18_CAS3in4_SDT21/10au −555 391 20,432
SD18_CAS3in4_SDT21/35au −562 397 20,592
Nataraj universal
CAS3in4 −483 339 18,800
CAS3in4_SD3/Nat100 −595 418 19,060
CAS3in4_SD3/200au −595 418 19,060
SD18_CAS3in4_SD21/45au −510 361 19,864
cvQZ/SD18_CAS3in4_SDT21/35au + ∆corr −539 388 20,614

Symmetry 2020,12, 498 9 of 16
3.3.4. Effect of Higher Excitation Ranks
Allowing for three holes in the shells with effective principal quantum number
n=
5 and up to
four particles in the virtual spinors (i.e., adding combined quadruple excitations) leads to a total change
of around 3.5%. Of particular importance are triple excitations into the virtual space, compare models
SD18_CAS3in4_SDT21 and SD18_CAS3in4_SD21.
3.4. Step 2: Accurate CI Results
Subsets of important CI models based on the findings of the previous subsection have been
repeated using the larger atomic basis sets, cvTZ and cvQZ. The single best values from these
calculations are given by the model SD18_CAS3in4_SDT21/35au. These latter values
V
are then
corrected by a “correction shift”, calculated as follows (all corrections using cvDZ basis):
∆corr :=V(S10_SD18_CAS3in4_SDT31/10au)−V(SD18_CAS3in4_SDT21/10au)
+V(S14_SD18_CAS3in4_SDT35/10au)−V(SD18_CAS3in4_SDT21/10au)
+V(S8_SD18_CAS3in4_SDT29/30au)−V(SD18_CAS3in4_SDT21/30au)
+V(SDT18_CAS3in4_SDT21/10au)−V(SD18_CAS3in4_SDT21/10au)
+V(SD18_CAS3in4_SDTQ21/10au)−V(SD18_CAS3in4_SDT21/10au)
The final best CI values are obtained by adding the above sum of individual corrections to the
value from the model SD18_CAS3in4_SDT21/35au.
3.5. Accurate CC Results
Table 2gives values of R, S and
A||
(
205
Tl) constants obtained within the all-electron coupled
cluster with single, double and non-iterative triple cluster amplitudes, CCSD(T), method employing
several basis sets. One can see a good convergence of the results in the series of the Dyall’s DZ, TZ and
QZ basis sets: values of R obtained within the QZ and TZ basis sets differ by about 2%. Table 2also
gives values of the constants obtained within the Nataraj’s universal basis set [
22
]. Please note that the
latter basis set is the even-tempered basis set (geometry progression). One can see a good agreement
of the results obtained within the QZ basis set and Nataraj’s universal basis set.
Table 2.
R, S, and A for Tl atom calculated within the 81e-CCSD(T) method in different basis sets.
In the case denoted “
VN
” the atomic spinors are obtained for the neutral Tl atom and the external field
perturbs both the spinor coefficients and the CC amplitudes. In the case denoted “
VN−1
” the atomic
spinors are obtained for the Tl
+
cation and the external electric field only perturbs the CC amplitudes
but not the atomic spinors.
Basis Set/Virtual Cutoff R S [a.u.] A||(205 Tl) [MHz]
Nataraj universal/103au (VN)−559 397 21,087
Nataraj universal/103au (VN−1)−550 390 21,071
Dyall cvDZ/104au (VN)−493 347 20,626
Dyall cvTZ/104au (VN)−545 387 20,760
Dyall cvQZ/104au (VN)−558 397 21,172
Table 3gives values of
R
calculated with different number of correlated electrons. As can be
seen contributions from subvalence and outer-core electrons are close to those obtained within the CI
approach above.

Symmetry 2020,12, 498 10 of 16
Table 3. R for Tl atom calculated within the CCSD(T) method in Dyall’s cvQZ basis set.
Method/Virtual Cutoff R
3e-CCSD(T)/10au −589
21e-CCSD(T)/150au −527
53e-CCSD(T)/150au −542
81e-CCSD(T)/104au −558
To check the convergence with respect to electron correlation effects we performed a series of
successive 21-electron coupled cluster calculations within the TZ basis set (see Table 4). In these
calculations two sets of atomic bispinors were used. The first one was obtained within the DCHF
approximation where the Fock operator is defined by averaging over 6
p1
j=1/2
and 6
p1
j=3/2
open-shell
electronic configurations as in the CI case above. The second one was obtained within the closed-shell
DCHF method for the Tl
+
cation. One can see that CC values gives almost identical result for each set
at any level. Moreover, the contribution of correlation effects beyond the CCSD(T) model is almost
negligible in the considered case. We considered models up to coupled cluster with Single, Double,
Triple and perturbative Quadruple cluster amplitudes, CCSDT(Q).
Contribution of the effect of the Breit interaction on R has been estimated in reference [
23
] as
0.36%. Based on the uncertainties discussed above we conservatively estimate the uncertainty of
our final CC value for
R
to be less than 5%. For CI the expected residual uncertainties for basis set,
inner-core correlations, and inclusion of higher excitation ranks have been added to obtain a final total
uncertainty of 6%.
Table 4.
Values of
R
calculated at different level of theory with correlation of 21 electrons of Tl,
cvTZ basis set. Calculations were performed using DCHF spinors for the neutral Tl atom (
VN
potential)
and for the Tl
+
cation (
VN−1
potential) cases. In both cases the external field perturbs both the spinor
coefficients and the CC amplitudes.
VNVN−1
DCHF −418 −402
CCSD −531 −530
CCSD(T) −521 −522
CCSDT −523 −523
CCSDT(Q) −522 −522
3.6. Discussion in Comparison with Literature Results
Our present best results are shown in Table 5in comparison with previous work. The earlier
controversy between different groups over results for
R
(Tl) can be condensed into three main points
which we address one by one.
3.6.1. Basis Sets
From the results in Tables 1and 2it is evident that a large atomic basis set, at least of
quadruple-zeta quality, must be used for obtaining very accurate interaction constants. The results
in Tables 1and 2obtained with our correlation methods demonstrate that the basis set used by
Nataraj et al., in ref. [
22
] fulfills this requirement, yielding interaction constants that are very close to
those obtained with Dyall’s cvQZ basis set and the same correlation expansion. The earlier suggestion
of Porsev et al., about an inadequate basis set used in ref. [
22
] can, therefore, be excluded as a possible
reason for the outlier result in ref. [22].

Symmetry 2020,12, 498 11 of 16
Table 5. Comparison with Literature Values.
Work αdeαCS[10−18ecm]A||(205Tl) [MHz]
Literature values
Khriplovich et al. [1] 5.1
Flambaum [51] (semi-empirical) −500
Kraftmakher [52] (Hartree–Fock) −300
Johnson et al. [26] (Norcross potential) −562 −18764
Mårtensson–Pendrill et al. [19,53] (estimate) −600 ±200 7 ±2
Liu et al. [20]−585 ±(30 −60)
Dzuba et al. [21]−582 ±20 7.0 ±0.2 21067
Nataraj et al., (CCSD(T)) [22]−466 ±10 21053
Sahoo et al., (CCSD(T)) [54] 4.06 ±0.14 21026
Kozlov et al. [55] 21663
Porsev et al. [23]−573 ±20 22041
This work CI −539 ±33 6.61 ±0.4 20614
This work CC −558 ±28 6.77 ±0.34 21172
Experiment [56,57] 21310.835 ±0.005
3.6.2. Treatment of Correlation Effects by the Many-Body Method
It is claimed in reference [
22
] that the treatment of electron correlation effects was more
complete than in references [
20
,
21
]. We have therefore first attempted to reproduce the electron
EDM enhancement calculated by Nataraj et al., by using the same many-body Hamiltonian and EDM
operator, the same atomic basis set (“Nataraj universal”) and the same method, CCSD(T). A persisting
difference with the approach of Nataraj et al., is the use of CC amplitudes for the closed shells of
neutral Tl (our case) or the closed shells of the singly ionized Tl
+
(Nataraj case). These results are
shown in Table 2under the label “
VN−1
”. Our calculation of the hyperfine constant
A|| =
21071 [MHz]
reproduces the value of Nataraj et al., which is
A|| =
21053 [MHz] almost precisely (residual difference
of less than 0.1%). However, using the same wavefunction we obtain
R=−
550 which differs from
the value of Nataraj et al., by 17%. Our CC result for
R
is in accord with similar calculations using the
large cvQZ basis set, in accord with the present best CI result (
R=−
539 which after correction for core
correlations from the innermost 28 electrons, according to the results in Table 3, becomes
R=−
555)
and in good agreement with the best results of Liu et al., [
20
], Dzuba et al. [
21
], and Porsev et al. [
23
],
see Table 5. The correct evaluation of the electron EDM enhancement in our codes has been assured by
comparative tests of the independent implementations of present CI and CC, as well as with the
DIRRCI
module [
31
,
58
] in the DIRAC program package. All three independent implementations produce
the same values of
R
for small test cases using Full CI/Full CC expansions. These findings strongly
suggest that the CC wavefunctions used by us and by Nataraj et al., are almost identical, but that there
is a problem in the evaluation of Rin reference [22].
Since correlation effects have been treated at a very similar (but physically more accurate) level in
the present work as in ref. [
22
] and the result is very different, the claim of correlation effects being
responsible for the large difference between previous results is untenable.
3.6.3. Use of VN,VN−1, and VN−3Potentials
First, given a fixed atomic basis set and a fixed many-body Hamiltonian (In the present and
previous works the Dirac-Coulomb picture is employed where negative-energy states are implicitly
or explicitly excluded from the orbital/spinor space which is used as a basis for the many-body
expansion.), the Full CI expansion delivers the exact solution in the
N
-particle sector of Fock space [
59
],
independent of the orbital/spinor basis used for this Full CI expansion. This implies that a many-body
expansion that closely approximates the Full CI expansion, such as CCSDT or CCSDT(Q), must also be
nearly independent of the employed Dirac-Fock potential.

Symmetry 2020,12, 498 12 of 16
Our results in Table 4clearly confirm this conjecture and demonstrate that even in the more
approximate CCSD expansion the electron EDM enhancement factor
R
is almost independent (0.2%
difference) of the underlying spinor set. As the many-body expansion becomes more approximate,
such as in the CI model SD18_CAS3in4_SD21 (see Table 1) basic theory leads us to expect that the
difference in
R
should increase which is indeed the case (roughly 2% difference). Adding external Triple
excitations to the CI expansion, model SD18_CAS3in4_SDT21, quenches the difference to a mere 1.2%,
again in accord with expectation. Even the use of a
VN−3
potential (i.e., spinors optimized for the Tl
3+
system) changes
R
by less than 3% relative to spinors for the neutral atom in the SD18_CAS3in4_SDT21
model. This difference is expected to be even smaller in CC models.
Despite the unimportance of the employed spinor set in highly correlated calculations, we have
used the physically most accurate spinors for the neutral Tl atom in obtaining our best final
results. The ratios of our calculated
P,T
-odd interaction constants are
αde
αCS
(CI)
=
81.5
1
10−18[ecm]
and
αde
αCS
(CC)
=
82.6
1
10−18[ecm]
which agree well with the analytical value of Dzuba et al. [
25
] of
αde
αCS
(an.)
=
89
1
10−18[ecm]
. The validity of this analytical relationship has been confirmed numerically
in numerous electronic-structure studies of EDM enhancements and Ne-SPS interactions on other
systems, for instance in Refs. [17,18,60–65].
4. Conclusions
In the present study we have carried out a systematic and elaborate treatment of electron
correlation effects for the
P,T
-violating and magnetic hyperfine interaction constants of atomic
Tl. This present treatment of electron correlation effects surpasses the one by Nataraj et al., in ref. [
22
]
in that we include higher CC excitation ranks in the wavefunction expansion. Our findings recommend
excluding the result of Nataraj et al., from the dataset used to constrain the
CP
-odd parameters
de
and
CS
. Likewise, the result by Sahoo et al. [
54
] (see Table 5) – presumably obtained with a similar
code as R(Tl) by Nataraj et al., – is also to a great degree too small. Our CC ratio for the eEDM and
Ne-SPS interaction constants differs from the analytical ratio developed by Dzuba et al. in Refs. [
25
] by
7.1%. This is within the combined uncertainties of the analytical/numerical approaches used for this
comparison. Both the results of Nataraj et al., and Sahoo et al., being too small, the last test is a check
for internal consistency of the ratio for those two interaction constants. That ratio amounts to
αde
αCS
(CC
Nataraj/Sahoo)
=
115.8
1
10−18[ecm]
which deviates from the analytical ratio by 30%, so those two results
are even inconsistent with each other.
Figure 2displays the updated version of the global fit shown in the introduction, using the dataset
of reliable calculations of
αde
and
αCS
for the Tl atom. The strongly reduced uncertainty of atomic
interaction constants for Tl leads to a discernable shrinking of the associated parameter surface (blue),
but does not lead to modified constraints. The essential reason for this is the extremely high sensitivity
of the experiments on ThO (green) and HfF
+
(orange) and the fact that the surface for Tl is too well
aligned with the surfaces for these latter two molecules. However, tighter constraints on
de
and
CS
can be obtained by including experimental and theoretical results for closed-shell atomic systems as
discussed in ref. [66].

Symmetry 2020,12, 498 13 of 16
Figure 2. Constraints on deand CSincluding the results of this work. See text for details.
Author Contributions:
Conceptualization, T.F.; Investigation, T.F. and L.V.S.; Methodology, T.F. and L.V.S.;
Resources, T.F. and L.V.S.; Software, T.F. and L.V.S.; Writing— original draft, T.F. and L.V.S. All authors have read
and agreed to the published version of the manuscript.
Funding: The CC research was funded by Russian Science Foundation Grant No. 19-72-10019.
Acknowledgments:
We thank Martin Jung (Torino) for providing updated plots and for helpful discussions.
Huliyar Nataraj is thanked for sharing many technical details of his calculations with us. Electronic structure
calculations were partially carried out using resources of the collective usage center Modeling and predicting
properties of materials at NRC “Kurchatov Institute” – PNPI.
Conflicts of Interest: The authors declare no conflict of interest.
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