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arXiv:2204.13015v2 [physics.atom-ph] 2 Nov 2022
Refined nuclear magnetic dipole moment of rhenium: 185Re and 187 Re
L.V. Skripnikov1,2, S.D. Prosnyak1,2
1Petersburg Nuclear Physics Institute named by B.P. Konstantinov of National
Research Center “Kurchatov Institute” (NRC “Kurchatov Institute” - PNPI),
1 Orlova roscha, Gatchina, 188300 Leningrad region, Russia and
2Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia∗
The refined values of the magnetic dipole moments of 185 Re and 187Re nuclei are obtained. For
this, we perform a combined relativistic coupled cluster and density functional theory calculation
of the shielding constant for the ReO−
4anion. In this calculation, we explicitly include the effect of
the finite nuclear magnetization distribution in the single-particle nuclear model using the Woods-
Saxon potential for the valence nucleon. By combining the obtained value of the shielding constant
σ= 4069(389) ppm with the available experimental nuclear magnetic resonance data we obtain the
values: µ(185Re) = 3.1567(3)(12)µN, µ(187Re) = 3.1891(3)(12)µN, where the first uncertainty is the
experimental one and the second is due to theory. The refined values of magnetic moments are in
disagreement with the tabulated values, µ(185Re) = 3.1871(3)µN, µ(187Re) = 3.2197(3)µN, which
were obtained using the shielding constant value calculated for the atomic cation Re7+ rather than
the molecular anion. The updated values of the nuclear magnetic moments resolve the disagreement
between theoretical predictions of the hyperfine structure of H-like rhenium ions which were based
on the tabulated magnetic moment values and available experimental measurements. Using these
experimental data we also extract the value of the parameter of nuclear magnetization distribution
introduced in [J. Chem. Phys. 153, 114114 (2020)], which is required to predict hyperfine structure
constants for rhenium compounds.
I. INTRODUCTION
Nuclear magnetic dipole moments are of wide inter-
est for many physical problems. They can be used to
test predictions of the nuclear theory. They are required
as external parameters to predict the hyperfine structure
(HFS) of neutral atoms, and molecules. Such data are re-
quired to probe the accuracy of calculated electronic wave
functions, which are used for calculation of characteris-
tics of symmetry-violation interactions in atoms [1–6] and
molecules [7–16]. Such characteristics cannot be directly
measured, but they are required to extract the value of
the T,P-violating nuclear Schiff and magnetic quadrupole
moments, the electron electric dipole moment and other
similar effects from the experimental data [2, 6, 17–20].
Magnetic dipole moments of stable isotopes can be com-
bined with the experimental and theoretical data on hy-
perfine structure for stable and short-lived isotopes to
obtain magnetic moments of short-lived isotopes [21–28].
Magnetic moments are used to predict hyperfine split-
tings in highly-charged ions, which can be used to test
predictions of the bound state quantum electrodynam-
ics [29].
Magnetic dipole moments of stable nuclei can be ob-
tained from nuclear magnetic resonance (NMR) experi-
ments on molecules, though there are suggestions to ex-
tract them from precise g-factor experiments on highly
charged ions [30–32]. In molecular NMR experiments,
one usually obtains so-called uncorrected values of the
nuclear magnetic moment. To obtain the intrinsic value
∗skripnikov lv@pnpi.nrcki.ru,leonidos239@gmail.com;
http://www.qchem.pnpi.spb.ru;
of the nuclear magnetic moment, one has to apply a cor-
rection on the shielding effect. It is induced by electrons
surrounding the nuclei of interest in a given atom or a
molecule. Accurate calculation of the shielding constant
in molecules containing heavy atoms is rather compli-
cated. Therefore, one often uses shielding corrections
calculated for the corresponding atomic ions. Such ap-
proach can lead to serious errors [33–35].
In the present paper we study nuclear magnetic mo-
ments for two stable isotopes of rhenium, 185Re and
187Re, both having nuclear spin I= 2.5. Nuclear mag-
netic resonance experiments with the aqueous solution
of the NaReO4molecule were carried out in 1951 [36].
The tabulated values of the nuclear magnetic moments
of 185Re and 187Re [37] are based on those experimental
data combined with the shielding constant calculated for
the Re7+ atomic ion [38–40]. In Ref. [41] it has been
noted that such interpretation is not free from possi-
ble errors due to neglect of the chemical shift effect, i.e.
contribution of molecular environment. The authors of
Ref. [34] have used a combination of the nonrelativis-
tic coupled cluster theory and the relativistic density
function theory to calculate the molecular shielding con-
stant. Here we perform a precise study of the shielding
effect within the relativistic coupled cluster and relativis-
tic density functional theories and show that the com-
pletely relativistic treatment allows one to significantly
reduce the uncertainty of the shielding constant. In the
present paper, we also explore the influence of the finite
nuclear magnetization distribution effect in the single-
particle approximation on the shielding constant value.
Using the refined value of the shielding constant we ob-
tain the updated values of the nuclear magnetic moments
of 185Re and 187Re.
2
The structure of the article is as follows. In Section II,
we give a brief overview of the theory for calculating the
shielding constant and the nuclear magnetization distri-
bution effect. In Section III A, we discuss the shield-
ing constant calculation scheme. Section III B provides
an analysis of possible uncertainties. These two sections
contain some technical details. In Section II I C, we derive
the nuclear magnetic moments from experimental NMR
data and the theoretical value of the shielding constant
obtained in sections III A and III B. Section IV analyzes
the available experimental data on hyperfine splitting in
H-like rhenium ions in various aspects.
The relativistic units (m=~=c= 1) and the charge
units α=e2/(4π) are used in this paper.
II. THEORY
One can use the following definition of the shielding
tensor corresponding to the nucleus jin a given molecule:
σj
a,b =∂2E
∂µj,a ∂Bbµj=0,B=0
.(1)
Here Eis the energy of the system, µj,a is the a’th com-
ponent of the nuclear magnetic moment vector µjof
j’th nucleus, Bbis b’th component of the uniform ex-
ternal magnetic field vector B. For the interpretation of
the molecular NMR experiments, performed in a solu-
tion, we need the isotropic part σof the shielding tensor,
σ= 1/3Paσa,a. From the nuclear magnetic resonance
experiment, it is possible to obtain the uncorrected value
µuncorr.of the nuclear magnetic dipole moment value, i.e.
the value, which is not corrected for the magnetic shield-
ing. The intrinsic value of the magnetic moment µcan
be obtained as:
µ=µuncorr./(1 −σ).(2)
The interaction of electrons in a molecule with an ex-
ternal uniform magnetic field Bcan be described by the
following term included in the Dirac-Coulomb Hamilto-
nian:
HB=B·|e|
2(rG×α),(3)
where αare the Dirac matrices and rG=r−RG,RG
is the gauge origin [42], i.e. the origin for the coordinate
system that describes the electron radius-vector in this
equation. In principle, the choice of RGcan influence the
results obtained in the modest basis sets (see below). In
the point magnetic dipole approximation, the hyperfine
interaction of an electron with the magnetic moment µj
of the jth nucleus can be written in the following way:
Hhyp =|e|
4πµj·(rj×α)
r3
j
,(4)
where rj=r−Rj,Rjis the position of the nucleus
j. Note, that the interaction (4) does not take into ac-
count the finite nuclear magnetization distribution effect.
In the theory of atomic hyperfine structure this effect is
called the Bohr-Weisskopf (BW) effect [43–45]. One can
use the following substitution to consider this effect [46–
48]:
µ→µ(r) = µF(r).(5)
Function F(r) takes into account the nuclear magnetiza-
tion distribution inside the finite nucleus. In the point
magnetic dipole moment approximation F(r) = 1. In the
finite distribution case F(r) can significantly differ from
1 inside the nucleus. Expressions for different models
can be found in Refs. [46, 48–51]. In the simplest uni-
formly magnetized ball model function F(r) = (r/rn)3
for rinside the sphere of radius rn=p5/3rc(rcis the
root-mean-square charge radius) and is equal to 1 out-
side [49]. In studies of neutral atoms, this model is most
widely used to calculate the BW correction [25, 52–55].
In the present paper, we mainly use the model which
implies that magnetization can be ascribed to the single-
particle structure of the nucleus. In this model function
F(r) is given by [46]:
F(r′) = µN
µ(Zr′
0
r2|u(r)|2dr 1
2gS+
I−1
2+2I+ 1
4(I+ 1) mpφSO (r)r2gL+
Z∞
r′
r2r′
r3
|u(r)|2dr −2I−1
8(I+ 1) gS+
I−1
2+2I+ 1
4(I+ 1) mpφSO (r)r2gL
(6)
for I=L+ 1/2, and
F(r′) = µN
µ(Zr′
0
drr2|u(r)|2−I
2(I+ 1) gS+
I(2I+ 3)
2(I+ 1) −2I+ 1
4(I+ 1)mpφSO (r)r2gL+
Z∞
r′
r2r′
r3
|u(r)|2dr 2I+ 3
8(I+ 1) gS+
I(2I+ 3)
2(I+ 1) −2I+ 1
4(I+ 1)mpφSO (r)r2gL
(7)
for I=L−1/2. Here µNis the nuclear magneton, mp
is the proton mass, Iis the nuclear spin, |u(r)|2is the
density of the valence nucleon, φSO is the radial part of
the spin–orbit interaction VSO =φS O σ
σ
σ·l
l
l,l
l
lis the angu-
lar moment operator and σ
σ
σis the vector of Pauli matri-
ces. In the Woods-Saxon (WS) model of the nucleus, the
wave function of the valence nucleon is determined as a
solution of the Schr¨odinger equation with the WS poten-
tial. A detailed description of the implementation and
3
parameters of the potential can be found in Ref. [28] and
references therein. For the valence proton we set gL= 1,
for the valence neutron gL= 0. Parameter gSis obtained
from the following equations:
µ
µN
=1
2gS+I−1
2+2I+ 1
4(I+ 1) mphφSO r2igL(8)
for I=L+ 1/2, and
µ
µN
=−I
2(I+ 1) gS+I(2I+ 3)
2(I+ 1) −2I+ 1
4(I+ 1) mphφSO r2igL
(9)
for I=L−1/2. In the simple single-particle model
with the uniform distribution of the valence nucleon, the
density of the valence nucleon |u(r)|2is a constant inside
the nucleus volume and there is no spin-orbit term in this
model [56].
In the one-electron case, the tensor (1) can be calcu-
lated using the sum-over-states method corresponding to
the second-order perturbation theory with perturbations
(3) and (4):
σa,b =(10)
X
n6=0
h0||e|
4π((rj×α)
r3
j
)a|nihn|(|e|
2(rG×α))b|0i
E0−En
+h.c.,
where |0iis the unperturbed one-particle state of inter-
est, |niis the unoccupied n’th unperturbed state (orbital)
and h.c. is the Hermitian conjugate. For the case of
the four-component Dirac theory, the summation should
include both positive energy and negative energy states
|ni[57]. The part of the sum associated with the positive
energy states is called “paramagnetic” term. The part
associated with the negative energy states is called “dia-
magnetic term” [57]. For the cases of the Dirac-Hartree-
Fock (DHF) and density functional theory (DFT) many-
electron methods, one can use the response technique
to calculate both terms [57–60]. The result of the ap-
plication of this technique is equivalent to (and derived
from) the analytical calculation of the DHF/DFT energy
derivative (1). In the present paper we have used the
implementation of the method within the dirac [60, 61]
code.
In calculations of the shielding constant, we have used
the following Gaussian-type basis sets to describe elec-
tronic wave functions. The first one corresponds to the
uncontracted Dyall’s AE4Z [62, 63] basis set for all atoms
and will be called QZQZ below. This basis set con-
tains [34s30p19d14f10g5h1i] primitive Gaussian func-
tions for Re and [18s10p5d3f1g] functions for each
oxygen. The second one, TZTZ, corresponds to the
uncontracted AE3Z [62, 63] basis set on rhenium and
contracted aug-cc-pVTZ [64, 65] on oxygen. This ba-
sis set contains [30s24p15d11f5g1h] functions for Re
and [5s4p3d2f]/([11s6p3d2f) for each oxygen, where
in the [...]-brackets the numbers of contracted functions
are given and in the (...)-brackets the corresponding
numbers of primitive functions are given (e.g. each
of five contracted s−type functions of oxygen is a lin-
ear combination of 11 primitive functions). We have
also used the DZDZ basis set which corresponds to the
uncontracted Dyall’s AE2Z [62, 63] basis set on rhe-
nium and aug-cc-pVDZ [64, 65] on oxygen. This ba-
sis set contains [24s19p12d9f1g] functions for Re and
[4s3p2d]/([10s5p2d) for oxygen. The quality of basis
sets increases in the series: DZDZ, TZTZ, QZQZ.
Formally, the interaction of a molecule with an external
uniform magnetic field should not depend on the choice
of the origin RGin Eq. (3). But for finite-size basis
sets there may be some dependence [42, 58, 59] which
can affect the shielding constant value. To minimize
such a dependence one can use the London atomic or-
bitals (LAOs) method, developed at the four-component
DFT level in Refs. [58, 59]. In this approach, basis func-
tions are replaced by the so-called London atomic orbitals
which are obtained from the original basis functions by
applying a magnetic field-dependent factor [42, 58, 59].
This corresponds to the transformation of the wave func-
tion due to the gauge transformation of the vector po-
tential in Eq. (3). The use of London orbitals guarantees
the gauge-origin invariance of results in a finite basis ap-
proximation [42, 58, 59]. Even for usual basis set, the
gauge-origin problem should decrease with the basis set
size increase. In the present case, we are interested in the
shielding constant for the rhenium nucleus. Therefore, it
is natural to place the origin at this nucleus. Accord-
ing to our DFT estimates, the values of the shielding
constant calculated for the QZQZ basis set (i) with such
choice of the origin and employing usual basis functions
or (ii) within the LAOs technique coincide within 7 ppm.
This value is negligible in comparison with the total un-
certainty of the present calculation (see below).
Geometry structure parameters of the ReO−
4anion
have been optimized using the four-component den-
sity functional theory with the Perdew-Burke-Ernzerhof,
PBE0, functional [66] and using the TZTZ basis set. No
solvent effects were considered at this stage. The opti-
mized value of the Re–O bond length in the ReO−
4anion
with the regular tetrahedral symmetry was found to be
1.723˚
A. This value is in good agreement (within 0.003˚
A)
with the study [34].
Relativistic four-component calculations were per-
formed within the locally modified dirac15 [60, 61] code.
High-order correlation effects have been calculated using
the mrcc code [67]. The code for calculating the BW
matrix elements in the WS model has been developed in
Ref. [28] for atoms and generalized to the molecular case
in the present paper. Taking into account that the ac-
tion of the corresponding operator is localized inside the
nucleus we have neglected the contribution of basis func-
tions centered on oxygen atoms in the present implemen-
tation. In the molecular electronic structure calculations,
the Gaussian nuclear charge distribution model [68] has
been used.
4
TABLE I. Calculated values of rhenium shielding constant σ
contributions for ReO−
4in ppm.
Contribution Value
Diamagnetic:
QZQZ-LAO/PBE0 7633
Paramagnetic:
TZTZ/108e-CCSD −3741
TZTZ/108e-CCSD(T) - 108e-CCSD 350
DZDZ/24e-CCSDT - 24e-CCSD(T) -81
Basis set correction −10
Gaunt 15
Solvent effect, from Ref. [34] −25
Finite magn. distribution (WS) −73
Total 4069
III. RESULTS AND DISCUSSION
A. Shielding constant calculation
We have used the following scheme to calculate the
shielding constant for ReO−
4and its contributions (see
Table I). The diamagnetic part has been calculated at
the four-component PBE0 method [66]. As in previ-
ous studies [33, 69] we have found that this contribution
is almost independent of the choice of the functional or
method used. For example, the values calculated within
the Dirac-Hartree-Fock (7630.8 ppm) and PBE0 (7633.3
ppm) response theories coincide within a few ppm. More-
over, the same value within a few ppm can be obtained
even using the uncoupled Dirac-Hartree-Fock approach
(7633.2 ppm), i.e. simple orbital perturbation theory.
The latter corresponds to calculation using Eq. (10) with
an additional summation over all occupied molecular or-
bitals, |0i, which are included in the Slater determinant,
and the sum over nin Eq. (10) is limited to the negative
energy orbitals.
The most challenging part of the problem is the cal-
culation of the paramagnetic contribution to the shield-
ing constant which is strongly affected by both corre-
lation and relativistic effects. The authors of Ref. [34]
have used the nonrelativistic coupled cluster theory com-
bined with the relativistic correction calculated within
the DFT approach to calculate the shielding constant for
ReO−
4. As it has been analysed in Ref. [34] the dom-
inant source of the uncertainty of such an approach is
“the systematic error of correlation and relativistic ef-
fects nonadditivity”. To avoid such an error, we avoided
the use of non-relativistic theory at all and used the
four-component relativistic coupled clusters theory as the
main approach for calculating the paramagnetic contri-
bution to the shielding constant. We have directly calcu-
lated the mixed derivative (1) within the standard numer-
ical finite-difference technique. For this we needed the
(numerical) dependence of the energy Eon the magnetic
moment and external magnetic field values. To calculate
this dependence we have added perturbations (3) and (4)
with required (small) values of the nuclear magnetic mo-
ment and the external uniform magnetic field amplitude
to the molecular relativistic Hamiltonian and solved cou-
pled cluster equations with these perturbed Hamiltonian
to obtain perturbed values of the energy E. This elec-
tronic correlation calculation has been performed within
the TZTZ basis set using the relativistic coupled clus-
ter with single, double and perturbative triple cluster
amplitudes method, CCSD(T)[70]. All 108 electrons of
ReO−
4were included in the correlation calculation and
no virtual energy cutoff has been applied. In Table Iwe
separate the CCSD(T) value into the CCSD value (line
“TZTZ/108e-CCSD”) and a contribution of perturbative
triple cluster amplitudes (line “TZTZ/108e-CCSD(T) -
108e-CCSD”). Thus the “TZTZ/108e-CCSD(T) - 108e-
CCSD” line gives the difference of the shielding constants
calculated within the CCSD(T) and CCSD methods (in
both cases all electrons were correlated and the TZTZ
basis set has been used).
To explore even higher-order correlation effects, we
have performed correlation calculations within the rel-
ativistic coupled cluster with single, double and itera-
tive triple cluster amplitudes method, CCSDT, and com-
pared it with the CCSD(T) one. Due to extremely
high complexity of the CCSDT approach (e.g. in the
present calculation our cluster operator included 2.3 bil-
lions cluster amplitudes), we have included 24 valence
electrons of ReO−
4in these two calculations and employed
the DZDZ basis set. One should note that the contri-
bution of perturbative triple cluster amplitudes in the
DZDZ/24e-CCSD(T) calculation reproduces such contri-
bution obtained in the main calculation (350 ppm, see
line “TZTZ/108e-CCSD(T) - 108e-CCSD” in Table I)
within 80%. As one can see, the difference between
CCSDT and CCSD(T) results (-81 ppm) is rather small
(see the “DZDZ/24e-CCSDT - 24e-CCSD(T)” line in Ta-
ble I).
To take into account the effect of the extended basis
set with respect to the main TZTZ one, we have cal-
culated basis set correction within the relativistic PBE0
approach. In this calculation, we have increased the ba-
sis set up to the QZQZ one, i.e. we have calculated the
difference of the paramagnetic contributions to shielding
constants calculated within the QZQZ and TZTZ basis
set using the PBE0 approach. One can see from Table I
that this correction is small (-10 ppm). Note, that there
is no guarantee, that DFT can reasonably take into ac-
count basis set correction accurately. Therefore, we have
estimated the influence of the basis set size increase from
the DZDZ basis set to the TZTZ one on the paramagnetic
part of the shielding constant. Here we studied how DFT
(PBE0) can reproduce this effect, calculated within the
wave function-based relativistic CCSD(T) method. Ob-
tained corrections are: -30 ppm within DFT vs. -71 ppm
within CCSD(T). As expected, DFT underestimated the
5
effect of the basis set size increase (by a factor of 2.4).
Thus, the mentioned correction (-10 ppm) in Table Ican
be underestimated. We take this fact into account in the
uncertainty estimation below.
The effect of solvent has been extensively analyzed in
Ref. [34]. It seems that the “δ4” scheme used in Ref. [34]
is the most elaborate study of this effect at present. It ex-
plicitly takes into account the effect of the first solvation
shell and approximately takes into account the influence
of the solution on the shielding constant under consid-
eration (within the polarizable continuum model) at the
DFT level [34]. Therefore, we include this contribution
in our final value.
The Gaunt interaction contribution has been calcu-
lated at the relativistic DFT (PBE0) level using the
TZTZ basis set, i.e. it has been calculated as a dif-
ference between the shielding constant values obtained
within the relativistic DFT method with inclusion and
without inclusion of the Gaunt interaction into the elec-
tronic Hamiltonian.
In the present paper, we have studied the influence
of the finite nuclear magnetization distribution on the
shielding constant. For this, we have used the substi-
tution given by Eq. (5) in the hyperfine interaction op-
erator (4) and have used the single-particle WS model,
described above. Calculation of the paramagnetic part
has been performed at the relativistic CCSD(T) level
using the DZDZ basis set, while the diamagnetic part
has been calculated at the uncoupled Dirac-Hartree-Fock
level (this method is described above). The latter contri-
bution, 8 ppm, to the considered correction was found to
be much smaller than the paramagnetic part, −81 ppm,
of this correction. As one can see from Table Ithe fi-
nite nuclear magnetization distribution effect (-73 ppm
or 1.8% of the total σ) is more important than the sol-
vent effect for the system under consideration. For both
isotopes considered, nuclear magnetization distribution
effects were found to be almost identical and they are
not distinguished in Table I. We did not find previous
attempts to take into account the influence of the finite
nuclear magnetization distribution effect on the shield-
ing constants in many-electron molecules, although such
studies have been performed for H-like ions [71, 72].
B. Shielding constant uncertainty estimation
The total uncertainty (δ) of the present calculation
of shielding constant can be estimated as a square
root of the sum of squares of electronic correlation
calculation uncertainty (δel.corr.), basis set uncertainty
(δBS), Gaunt interaction effects inclusion uncertainty
(δG), finite nuclear magnetization distribution uncer-
tainty (δFMD), quantum electrodynamics (δQED ), uncer-
tainty due to geometry structure uncertainty (δgeom) and
solvent effects uncertainty (δsol):
δ=qδ2
el.corr.+δ2
FMD +δ2
G+δ2
BS +δ2
sol +δ2
QED +δ2
geom.
(11)
Uncertainty of the calculated electronic correlation ef-
fect δel.corr.can be estimated as the contribution of the
perturbative triple cluster amplitudes given in Table I
in the line “TZTZ/108e-CCSD(T) - 108e-CCSD”, i.e.
δel.corr.=350 ppm. For a more detailed analysis of this
contribution, we also calculated it within the smaller
DZDZ basis set, i.e. calculated the difference between
shielding constants calculated using the DZDZ/108e-
CCSD(T) and DZDZ/108e-CCSD approaches. The cal-
culated value, 355 ppm, reproduces with high accuracy
the contribution of perturbative triple clusters ampli-
tudes within the TZTZ basis set given above, 350 ppm.
This suggests a small contribution from the interference
effect between the high-order correlation effects defined
by perturbative triple amplitudes and the size of the basis
set (the uncertainty associated with the size of the ba-
sis set itself is discussed below). Note that the indicated
uncertainty δel.corr.seems to be quite conservative, given
that the estimate of the contribution of higher-order cor-
relation effects (outside the considered CCSD(T) approx-
imation) calculated within the framework of the CCSDT
approach (see “DZDZ/24e-CCSDT - 24e-CCSD(T)” line
in Table Iand description in section I) is several times
smaller than δel.corr..
The uncertainty δFMD of the calculated value of the fi-
nite nuclear magnetization distribution contribution (-73
ppm) can be estimated as about 30% of the value of this
contribution (see also the analysis of the HFS data for H-
like Re below): δFMD = 23 ppm. This value is obtained
by comparing contributions of the finite nuclear magneti-
zation distribution to the shielding constant calculated in
the WS model, -73 ppm, and in the uniformly magnetized
ball model, -96 ppm. The latter value has been calculated
using the same approach as that employed for the first
one (described in the previous section) with only replace-
ment of the nuclear magnetization distribution function
F(r).
As a measure of the uncertainties of the Gaunt inter-
action and solvent effects calculated within one method
(DFT), we have used corresponding values of these ef-
fects given in Table I, i.e. δG= 15 ppm, δsol = 25 ppm.
This means that the (conservative) uncertainties of these
corrections are suggested to be 100%. A more accurate
calculation of these effects and their uncertainties will
be required when the remaining uncertainties are made
smaller (e.g. the uncertainty δel.corr.considered above is
more than an order of magnitude bigger than the effects
under consideration).
Let us estimate the uncertainty due to the basis set in-
completeness δBS. For this we can compare shielding con-
stant calculated within the best employed QZQZ basis
set and a smaller one, TZTZ basis set. These differences
for the diamagnetic contributions, paramagnetic contri-
bution and the total shielding constant values calculated
6
within the QZQZ and TZTZ basis sets using the DFT
(PBE0) method are +38 ppm, −10 ppm and +28 ppm,
respectively. Thus, we can suggest that δBS is about
28 ppm. However, as we have mentioned above, the DFT
approach can underestimate the effect of the basis set size
increase for the paramagnetic contribution by a factor of
2.4. This will lead to the estimation for the uncertainty
of the basis set incompleteness as 38 −10 ·2.4 =14 ppm.
For the conservative estimate we choose the largest of
these two possible values, i.e. we set δBS = 28 ppm.
The geometry parameters, i.e. the Re–O distances in
the ReO−
4anion have been optimized at the relativis-
tic PBE0 level (see above). To check the uncertainty
of the optimized geometry we have also performed ge-
ometry optimization within another popular functional –
B3LYP [73]. Within this approach the optimized Re–O
distance was found to be 1.736 ˚
A, i.e. the estimation for
the geometry parameters uncertainty of ReO−
4is about
0.013 ˚
A. According to our DFT-based estimation, this
uncertainty in the geometry structure parameters leads
to contribution to the uncertainty of the shielding con-
stant of about δgeom = 158 ppm. This is about six times
bigger than the solvent effect contribution (and is about
an order of magnitude bigger than the Gaunt interaction
effect).
In a recent paper [74] it was estimated that contribu-
tion of quantum electrodynamics effects to the shielding
constant is about 0.5% for such many-electon atoms as
astatine (atomic number 85). For H-like ions, ab-initio
calculations are available [71, 72]. According to these
Refs., the QED contribution to the shielding constant of
H-like Bi (atomic number 83) is 0.7% and generally in-
creases with atomic number. In the present work, we
do not take into account QED effects but according to
the notes above suggest that their contribution can be
about 1% of the total value of the shielding constant and
include this value in the uncertainty, i.e δQED=41 ppm.
The final value of the uncertainty of the shielding con-
stant is dominated by the correlation contribution uncer-
tainty δel.corr.. Substituting all estimated values of the
uncertainties in Eq. (11) we obtain the total uncertainty
value: δ=389 ppm, i.e. the final value of the shielding
constant is σ=4069(389) ppm. It is in reasonable agree-
ment with the previous calculation 3698(927) ppm [34]
but has reduced uncertainty due to the use of the rela-
tivistic coupled cluster approach for the most challenging
part of the calculation. Note also, that in the present pa-
per we consider more sources of the uncertainties.
C. New values of magnetic dipole moments of
185Re and 187 Re
To obtain the nuclear magnetic dipole moments of
185Re and 187Re we need corresponding uncorrected val-
ues of these moments, i.e. magnetic dipole moments
which are not corrected for the magnetic shielding, see
Eq. (2). The nuclear magnetic resonance experiment on
the ReO−
4anion has been performed in the aqueous so-
lution of NaReO4with the magnetic dipole moment of
23Na as the reference [36]. In the experiment the two res-
onances of 185Re and 187Re were located near a frequency
of 6.4 Mc in an external field of 6700 gauss. The follow-
ing results for the resonance frequencies νwere obtained
in the experiment [36]:
ν(185Re)/ν(23Na) = 0.85114(9),(12)
ν(187Re)/ν (23Na) = 0.85987(9).(13)
As in Ref. [75] we use the uncorrected NMR value
µuncorr.(23Na) from Ref. [76] to obtain the uncorrected
values of the nuclear magnetic moments of 185Re and
187Re [75] according to the relation [75] µuncorr.(Re) =
µuncorr.(Na)(ν(Re)/ν(Na))(IRe/INa) (where the 23Na nu-
clear spin I23 Na=1.5):
µuncorr.(185Re) = 3.1439(3)µN,(14)
µuncorr.(187Re) = 3.1761(3)µN.(15)
Using these uncorrected values and our theoretical shield-
ing constant value σwe obtain the final values of the
nuclear magnetic moments according to Eq. (2):
µ(185Re) = 3.1567(3)(12) µN,(16)
µ(187Re) = 3.1891(3)(12) µN.(17)
Here the first uncertainty is due to the experiment and
the second is due to the present theory.
The obtained value of the shielding constant for ReO−
4
molecular anion is about three times smaller than the
shielding constant for the Re7+ atomic cation [38–40]
which was used in some of the previous interpretations
of the molecular NMR data [37, 75, 77]. It means that
in previous studies the uncertainty of the shielding cor-
rection used for interpretation of the molecular NMR ex-
periment has been substantially underestimated [75].
IV. HYDROGEN-LIKE RHENIUM IONS
A. Nuclear magnetic moments from HFS data for
H-like Re
Rhenium is one of several elements for which measure-
ments of hyperfine splitting for H-like (Re74+) ion was
carried out [41]. In principle, it is possible to extract the
magnetic moment value from these HFS data if the val-
ues of BW and QED contributions are known [78]. For
this one can use the following expression for the hyperfine
structure constant A[79]:
A=A(0) −ABW +AQED =A(0)(1 −ε) + AQED ,(18)
where A(0) is the HFS constant calculated in the
point magnetic dipole approximation, ABW is the Bohr-
Weisskopf contribution to the HFS constant, εis the rel-
ative Bohr-Weisskopf correction, and AQED is the QED
7
TABLE II. Calculated values of the relative BW correction ǫ
to hyperfine structure constants of H-like rhenium in different
nuclear models in %. Both rhenium isotopes 185 Re and 187Re
have nuclear spin I= 2.5.
185Re 187 Re
Ball 1.69 1.69
UD 1.35 1.36
WS without SO 1.30 1.30
WS with SO 1.32 1.32
contribution. Constants A(0), ABW and AQED are pro-
portional to the nuclear magnetic moment µ=gII µN,
where gIis the g-factor of nucleus with spin I.A(0) and
AQED can be accurately calculated [47, 80] and ABW can
be estimated within some nuclear magnetization distri-
bution model. In Ref. [28] we have calculated the rel-
ative BW correction εfor H-like 185Re. Here we have
also estimated BW effect for the 187 Re isotope using four
different nuclear magnetization distribution models (see
Table II): uniformly magnetized ball model (Ball), single-
particle model with the uniform distribution of the va-
lence nucleon (UD), WS model with and without spin-
orbit interaction in Schr¨odinger equation for valence nu-
cleon (see the Theory section). The obtained values for
both considered isotopes are almost identical, since these
nuclei have similar single-particle structures and charge
radii. Using the experimental values of A[41], the values
of BW corrections εcalculated within different nuclear
magnetization distribution models and given in Table II,
the values of the ratio A(0)/gII= 0.2926(3) eV calculated
in [47, 56] [81] and QED effect calculated in Refs. [47, 80]
AQED/gII=−0.00158(3) eV we have determined the
corresponding values of magnetic moments according to
the equation µ=A/[A(0)(1 −ε)/gII+AQED/gII]. The
values of µdeduced in such a way using different models
of nuclear magnetization distribution are given in Ta-
ble III. In such approach the main uncertainty is due to
the BW effect, as it is hard to reliably treat many-body
nuclear structure effects. We suppose that this uncer-
tainty can be estimated by comparing different nuclear
magnetization distribution models given in Table II. Us-
ing this approach, the relative Bohr-Weisskopf correction
can be estimated as ε=1.32(37)% for both isotopes [82].
It corresponds to the following value of the magnetic mo-
ments derived from the experimental HFS data [41] for
H-like ions:
µ(HFS)(185 Re) = 3.156(2)(3)(12) µN,(19)
µ(HFS)(187 Re) = 3.187(2)(3)(12) µN.(20)
Here the first uncertainty is due to the experimental de-
termination of A[41], the second one is due to uncertain-
ties of A(0) (the uncertainty of AQED is negligible) and
the third one corresponds to the uncertainty of the cal-
culated BW effect. These values are in agreement with
Ref. [78] and in agreement with the values (16) and (17)
derived from the NMR data above but have an order
TABLE III. Values of the nuclear magnetic moments (in units
of µN) extracted from the experimental data on hyperfine
structure constants of H-like rhenium [41] using QED correc-
tions from Refs. [47, 80] and calculated BW corrections within
different nuclear magnetization distribution models. For an
uncertainty estimation, see the main text.
185Re 187 Re
Ball 3.168 3.199
UD 3.157 3.188
WS without SO 3.155 3.186
WS with SO 3.156 3.187
of magnitude larger uncertainty. It is mainly due to the
BW effect uncertainty. The influence of the BW effect on
the shielding constant (1.8%) in the considered molecular
anion and on the hyperfine structure of H-like rhenium
ion (1.3%) are similar (see also Tables Iand II). How-
ever, the influence of the BW effect on the final value of
the magnetic moment, extracted from the NMR data is
much smaller than in the case of H-like HFS data. In the
first case, the finite nuclear magnetization distribution
effect is a correction to the shielding effect, which is 4069
ppm, i.e. only 0.4% itself (see above). The uncertainty
of the molecular electronic structure shielding constant
calculation can be controlled, see section II I B. In the
second case, the BW effect directly contributes to the
magnetic moment, and an estimation of its uncertainty
is complicated due to the absence of direct many-body
calculations of this effect for the rhenium nucleus.
It is possible to employ the obtained results also as
follows. If one uses the tabulated values [37] of the nu-
clear magnetic moments of 185Re and 187 Re, the fol-
lowing theoretical values of the HFS constant Ath for
H-like rhenium can be obtained according to the equa-
tion A=µ[A(0)(1 −ε)/gII+AQED /gII]: Ath(185 Re) =
0.9152(34) eV and Ath(187 Re) = 0.9245(34) eV. For the
new values of magnetic moments (16), (17), extracted
from the NMR data above one obtains: Ath(185 Re) =
0.9065(34) eV and Ath(187 Re) = 0.9157(34) eV. The ex-
perimental values Aexp are [41]: Aexp (185Re) = 0.9063(6)
eV and Aexp(187 Re) = 0.9150(6) eV. Thus, the updated
values of the nuclear magnetic moments resolve the dis-
agreement between theoretical predictions of HFS con-
stants (or HFS splittings) of H-like rhenium ions (based
on the old values of magnetic moments) [47, 83, 84] and
experimental values [41].
B. BW effect from HFS data for H-like ions and
magnetic moments from molecular NMR data
As mentioned in the Introduction, accurate theoret-
ical prediction of the hyperfine structure of atoms and
molecules can be used to probe the accuracy of the elec-
tronic wave function. However, such predictions depend
on the nuclear magnetic dipole moment value and the
8
function of the nuclear magnetization distribution F(r)
in Eq. (5). If both of these components are accurately
known, then one can predict the hyperfine structure of
the compound or ion under consideration. It was shown
in Ref. [15] that for many-electron heavy-atom molecules
and heavy atoms to a good approximation it is possible
to factorize the Bohr-Weisskopf contribution to the hy-
perfine structure constant into a pure electronic part and
just one universal numerical parameter, which depends
on the nuclear magnetization distribution, see Eq. (29) in
Ref. [15] (recently a related approach has been considered
for atoms in sand p1/2states in Ref. [85]). The latter pa-
rameter Bs[15] is proportional to the BW contribution
ABW to the hyperfine structure constant in Eq. (18) for
the H-like ion in the ground electronic state and can be
calculated as Bs=ABW/2gIin this case; actually, the
constant of interest is the product BSgI=ABW /2 [15].
Using the experimental values of the H-like rhenium HFS
constants [41], QED corrections from Refs. [47, 80] and
the values of the nuclear magnetic moments (16) and (17)
refined in the present paper above, we obtain the follow-
ing values of the BW contribution to the HFS constants,
ABW(“exp′′), according to Eq. (18):
ABW(“exp′′)(185 Re) = 0.0124(6)(8)(4) eV,(21)
ABW(“exp′′)(187 Re) = 0.0130(6)(9)(4) eV.(22)
Here the first uncertainty is due to the experimental
HFS data for H-like ions [41], the second one is due
to uncertainty of A(0) and the third one corresponds
to nuclear magnetic moment uncertainties in (16) and
(17). From these values one can obtain for the prod-
uct BSgI=ABW/2: BSgI(185 Re) = 0.0062(3)(4)(2) eV,
BSgI(187Re) = 0.0065(3)(5)(2) eV. Here the uncertain-
ties correspond to the uncertainties in Eqs. (21) and (22)
above.
V. CONCLUSION
In the present paper, we have obtained refined values
of the magnetic moments of 185 Re and 187Re nuclei. For
this, we have calculated the shielding constant for the
ReO−
4anion using a combination of the relativistic cou-
pled cluster and relativistic density functional theories.
We have studied the influence of the finite nuclear mag-
netization distribution effect on the shielding constant
value. Such effect is usually omitted in molecular cal-
culations. However, according to our study, this effect
can be more important than the solvent effect which is
often estimated. Updated values of the nuclear magnetic
moments resolve the disagreement between theoretical
predictions [47, 83, 84] and experimental values [41] for
the hyperfine splittings of H-like rhenium ions. In addi-
tion to the nuclear magnetic moment values, we have also
used H-like data for rhenium HFS constants to extract
the universal parameter [15] of the nuclear magnetization
distribution. The values of the nuclear magnetic moment
and this parameter are necessary ingredients for the the-
oretical prediction of HFS constants in different rhenium
ions and compounds.
ACKNOWLEDGMENTS
We are grateful to V.M. Shabaev for useful discussions.
Electronic structure calculations have been carried out
using computing resources of the federal collective us-
age center Complex for Simulation and Data Processing
for Mega-science Facilities at National Research Centre
“Kurchatov Institute”, http://ckp.nrcki.ru/, and partly
using the computing resources of the quantum chemistry
laboratory.
Molecular electronic structure calculations performed
at NRC “Kurchatov Institute” – PNPI have been sup-
ported by the Russian Science Foundation Grant No.
19-72-10019. Calculations of nucleon wave function per-
formed at SPbSU were supported by the foundation for
the advancement of theoretical physics and mathematics
“BASIS” grant according to Project No. 21-1-2-47-1.
[1] R. Alarcon, J. Alexander, V. Anastassopoulos, T. Aoki,
R. Baartman, S. Baeßler, L. Bartoszek, D. H. Beck,
F. Bedeschi, R. Berger, et al. (2022), arXiv:2203.08103
[hep-ph](2022).
[2] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,
A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90,
025008 (2018).
[3] S. G. Porsev, K. Beloy, and A. Derevianko, Phys. Rev.
Lett. 102, 181601 (2009).
[4] J. S. M. Ginges, A. V. Volotka, and S. Fritzsche, Phys.
Rev. A 96, 062502 (2017).
[5] T. Fleig and L. V. Skripnikov, Symmetry 12, 498 (2020).
[6] J. S. M. Ginges and V. V. Flambaum, Phys. Rep. 397,
63 (2004).
[7] M. Kozlov and L. Labzowsky, J. Phys. B 28, 1933 (1995).
[8] H. M. Quiney, H. Skaane, and I. P. Grant, J. Phys. B 31,
L85 (1998).
[9] A. V. Titov, N. S. Mosyagin, A. N. Petrov, T. A. Isaev,
and D. P. DeMille, Progr. Theor. Chem. Phys. 15, 253
(2006).
[10] L. V. Skripnikov and A. V. Titov, Phys. Rev. A 91,
042504 (2015).
[11] L. V. Skripnikov and A. V. Titov, J. Chem. Phys. 142,
024301 (2015).
9
[12] A. Sunaga, M. Abe, M. Hada, and B. P. Das, Phys. Rev.
A93, 042507 (2016).
[13] T. Fleig, Phys. Rev. A 96, 040502(R) (2017).
[14] P. A. B. Haase, E. Eliav, M. Iliaˇs, and A. Borschevsky,
J. Phys. Chem. A 124, 3157 (2020).
[15] L. V. Skripnikov, J. Chem. Phys. 153, 114114 (2020).
[16] L. V. Skripnikov, A. V. Titov, and V. V. Flambaum,
Phys. Rev. A 95, 022512 (2017).
[17] L. V. Skripnikov, A. D. Kudashov, A. N. Petrov, and
A. V. Titov, Phys. Rev. A 90, 064501 (2014).
[18] L. V. Skripnikov, J. Chem. Phys. 147, 021101 (2017).
[19] L. V. Skripnikov, A. N. Petrov, N. S. Mosyagin, A. V.
Titov, and V. V. Flambaum, Phys. Rev. A 92, 012521
(2015).
[20] L. V. Skripnikov, D. E. Maison, and N. S. Mosyagin,
Phys. Rev. A 95, 022507 (2017).
[21] J. Persson, Eur. Phys. J. A 2, 3 (1998).
[22] B. Cheal and K. T. Flanagan, Journal of Physics G: Nu-
clear and Particle Physics 37, 113101 (2010).
[23] S. Schmidt, J. Billowes, M. L. Bissell, K. Blaum, R. F. G.
Ruiz, H. Heylen, S. Malbrunot-Ettenauer, G. Neyens,
W. N¨ortersh¨auser, G. Plunien, et al., Phys. Lett. B 779,
324 (2018).
[24] A. E. Barzakh, L. K. Batist, D. V. Fedorov, V. S. Ivanov,
K. A. Mezilev, P. L. Molkanov, F. V. Moroz, S. Y. Orlov,
V. N. Panteleev, and Y. M. Volkov, Phys. Rev. C 86,
014311 (2012).
[25] S. D. Prosnyak, D. E. Maison, and L. V. Skripnikov, J.
Chem. Phys. 152, 044301 (2020).
[26] B. M. Roberts and J. S. M. Ginges, Phys. Rev. Lett. 125,
063002 (2020).
[27] A. E. Barzakh, D. Atanasov, A. N. Andreyev, M. Al Mon-
thery, N. A. Althubiti, B. Andel, S. Antalic, K. Blaum,
T. E. Cocolios, J. G. Cubiss, et al., Phys. Rev. C 101,
034308 (2020).
[28] S. D. Prosnyak and L. V. Skripnikov, Phys. Rev. C 103,
034314 (2021).
[29] V. M. Shabaev, A. N. Artemyev, V. A. Yerokhin, O. M.
Zherebtsov, and G. Soff, Phys. Rev. Lett. 86, 3959
(2001).
[30] G. Werth, H. H¨affner, N. Hermanspahn, H.-J. Kluge,
W. Quint, and J. Verd´u, in The Hydrogen Atom
(Springer, 2001), pp. 204–220.
[31] W. Quint, D. L. Moskovkhin, V. M. Shabaev, and M. Vo-
gel, Phys. Rev. A 78, 032517 (2008).
[32] A. Volchkova, A. Varentsova, N. Zubova, V. Agababaev,
D. Glazov, A. Volotka, V. Shabaev, and G. Plunien,
Nuclear Instruments and Methods in Physics Research
Section B: Beam Interactions with Materials and Atoms
408, 89 (2017), ISSN 0168-583X, proceedings of the
18th International Conference on the Physics of Highly
Charged Ions (HCI-2016), Kielce, Poland, 11-16 Septem-
ber 2016.
[33] L. V. Skripnikov, S. Schmidt, J. Ullmann, C. Geppert,
F. Kraus, B. Kresse, W. N¨ortersh¨auser, A. F. Privalov,
B. Scheibe, V. M. Shabaev, et al., Phys. Rev. Lett. 120,
093001 (2018).
[34] A. Antuˇsek and M. Repisky, Phys. Chem. Chem. Phys.
22, 7065 (2020).
[35] A. Antuˇsek, M. Repisky, M. Jaszu´nski, K. Jackowski,
W. Makulski, and M. Misiak, Phys. Rev. A 98, 052509
(2018).
[36] F. Alder and F. C. Yu, Phys. Rev. 82, 105 (1951).
[37] N. Stone, Table of nuclear magnetic dipole and electric
quadrupole moments, INDC(NDS)–0658, International
Atomic Energy Agency (IAEA) (2014).
[38] F. D. Feiock and W. R. Johnson, Phys. Rev. Lett. 21,
785 (1968).
[39] F. D. Feiock and W. R. Johnson, Phys. Rev. 187, 39
(1969).
[40] D. Kolb, W. R. Johnson, and P. Shorer, Phys. Rev. A
26, 19 (1982).
[41] J. R. Crespo L´opez-Urrutia, P. Beiersdorfer, K. Wid-
mann, B. B. Birkett, A.-M. M˚artensson-Pendrill, and
M. G. H. Gustavsson, Phys. Rev. A 57, 879 (1998).
[42] K. G. Dyall and K. Fægri Jr, Introduction to relativistic
quantum chemistry (Oxford University Press, 2007).
[43] A. Bohr and V. F. Weisskopf, Phys. Rev. 77, 94 (1950).
[44] A. Bohr, Phys. Rev. 81, 134 (1951).
[45] L. Sliv, Zh. Eksp. Teor. Fiz. 21, 770 (1951).
[46] O. M. Zherebtsov and V. M. Shabaev, Can. J. Phys. 78,
701 (2000).
[47] V. M. Shabaev, M. Tomaselli, T. K¨uhl, A. N. Artemyev,
and V. A. Yerokhin, Phys. Rev. A 56, 252 (1997).
[48] I. I. Tupitsyn, A. V. Loginov, and V. M. Shabaev, Optics
and Spectroscopy 93, 357 (2002).
[49] A. V. Volotka, D. A. Glazov, I. I. Tupitsyn, N. S. Ore-
shkina, G. Plunien, and V. M. Shabaev, Phys. Rev. A
78, 062507 (2008).
[50] E. Malkin, M. Repisk´y, S. Komorovsk´y, P. Mach, O. L.
Malkina, and V. G. Malkin, J. Chem. Phys. 134, 044111
(2011).
[51] B. M. Roberts and J. S. M. Ginges, Phys. Rev. A 104,
022823 (2021).
[52] J. Sapirstein and K. T. Cheng, Phys. Rev. A 67, 022512
(2003).
[53] E. A. Konovalova, M. G. Kozlov, Y. A. Demidov,
and A. E. Barzakh, Rad. Appl. 2, 181 (2017), URL
arXiv:1703.10048.
[54] J. Ginges and A. Volotka, Phys. Rev. A 98, 032504
(2018).
[55] M. G. Kozlov, S. G. Porsev, and W. R. Johnson, Phys.
Rev. A 64, 052107 (2001).
[56] V. M. Shabaev, J. Phys. B 27, 5825 (1994).
[57] G. A. Aucar, T. Saue, L. Visscher, and H. J. A. Jensen,
J. Chem. Phys. 110, 6208 (1999).
[58] M. Olejniczak, R. Bast, T. Saue, and M. Pecul, J. Chem.
Phys. 136, 014108 (2012).
[59] M. Ilias, H. J. A. Jensen, R. Bast, and T. Saue, Mol.
Phys. 111, 1373 (2013).
[60] DIRAC, a relativistic ab initio electronic structure pro-
gram, Release DIRAC15 (2015), written by R. Bast, T.
Saue, L. Visscher, and H. J. Aa. Jensen, with contribu-
tions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ek-
stroem, E. Eliav, T. Enevoldsen, E. Fasshauer, T. Fleig,
O. Fossgaard, A. S. P. Gomes, T. Helgaker, J. Henriks-
son, M. Ilias, Ch. R. Jacob, S. Knecht, S. Komorovsky,
O. Kullie, J. K. Laerdahl, C. V. Larsen, Y. S. Lee, H.
S. Nataraj, M. K. Nayak, P. Norman, G. Olejniczak, J.
Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, R.
Di Remigio, K. Ruud, P. Salek, B. Schimmelpfennig, J.
Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen,
S. Villaume, O. Visser, T. Winther, and S. Yamamoto
(see http://www.diracprogram.org).
[61] R. Saue, T. Bast, A. S. P. Gomes, H. J. A. Jensen,
L. Visscher, I. A. Aucar, R. Di Remigio, K. G. Dyall,
E. Eliav, E. Fasshauer, T. Fleig, et al., J. Chem. Phys.
10
152, 204104 (2020).
[62] K. G. Dyall, Theor. Chem. Acc. 117, 491 (2007).
[63] K. G. Dyall, Theor. Chem. Acc. 131, 1217 (2012).
[64] T. H. Dunning, Jr, J. Chem. Phys. 90, 1007 (1989).
[65] R. A. Kendall, T. H. Dunning, Jr, and R. J. Harrison, J.
Chem. Phys. 96, 6796 (1992).
[66] C. Adamo and V. Barone, J. Chem. Phys. 110, 6158
(1999).
[67] M. K´allay, P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J.
Csontos, J. Cs´oka, P. B. Szab´o, L. Gyevi-Nagy, B. H´egely,
I. Ladj´anszki, L. Szegedy, B. Lad´oczki, K. Petrov, M.
Farkas, P. D. Mezei, and ´a. Ganyecz: The mrcc pro-
gram system: Accurate quantum chemistry from water
to proteins, J. Chem. Phys. 152, 074107 (2020).” “mrcc,
a quantum chemical program suite written by M. K´allay,
P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J. Cson-
tos, J. Cs´oka, P. B. Szab´o, L. Gyevi-Nagy, B. H´egely,
I. Ladj´anszki, L. Szegedy, B. Lad´oczki, K. Petrov, M.
Farkas, P. D. Mezei, and ´a. Ganyecz. See www.mrcc.hu.
[68] L. Visscher and K. G. Dyall, Atomic Data and Nuclear
Data Tables 67, 207 (1997).
[69] V. Fella, L. V. Skripnikov, W. N¨ortersh¨auser, M. R.
Buchner, H. L. Deubner, F. Kraus, A. F. Privalov, V. M.
Shabaev, and M. Vogel, Phys. Rev. Research 2, 013368
(2020).
[70] L. Visscher, T. J. Lee, and K. G. Dyall, J. Chem. Phys.
105, 8769 (1996).
[71] V. A. Yerokhin, K. Pachucki, Z. Harman, and C. H. Kei-
tel, Phys. Rev. Lett. 107, 043004 (2011).
[72] V. A. Yerokhin, K. Pachucki, Z. Harman, and C. H. Kei-
tel, Phys. Rev. A 85, 022512 (2012).
[73] A. Becke, J. Chem. Phys. 98, 5648 (1993).
[74] K. Kozio l, I. A. Aucar, and G. A. Aucar, J. Chem. Phys.
150, 184301 (2019).
[75] M. G. H. Gustavsson and A.-M. M˚artensson-Pendrill,
Phys. Rev. A 58, 3611 (1998).
[76] O. Lutz, Phys. Lett. A 25, 440 (1967), ISSN 0375-9601.
[77] P. Raghavan, Atomic Data and Nuclear Data Tables 42,
189 (1989).
[78] J. R. Crespo L´opez-Urrutia, P. Beiersdorfer, D. W. Savin,
and K. Widmann, AIP Conference Proceedings 392, 87
(1997).
[79] Hyperfine splittings given in Ref. [41] are equal to 3Afor
the case of considered H-like 185 Re and 187Re ions in the
ground electronic states.
[80] A. Artemyev, V. Shabaev, G. Plunien, G. Soff, and
V. Yerokhin, Phys. Rev. A 63, 062504 (2001).
[81] The uncertainty of A(0)/gIIis estimated as the differ-
ence between the values of A(0)/gIIcalculated within the
Fermi [47] and uniform charge distribution models [56].
A(0)/gII= 0.2926(3) eV was calculated in [47] for 185Re.
Constant for 187Re was obtained here using the constant
for 185Re and the known dependence [47] of the charge
distribution correction on the charge radius. Difference
of the A(0)/gIIvalues for 185Re and 187 Re was found to
be negligible.
[82] So, the estimated uncertainty of the used single-particle
model of the nuclear magnetization distribution is about
30%. Similar uncertainty δFMD (in %) has been found for
the contribution of the nuclear magnetization distribu-
tion to the shielding constant in section I II B.
[83] T. Beier, Physics Reports 339, 79 (2000).
[84] S. Boucard and P. Indelicato, The European Physical
Journal D 8, 59 (2000).
[85] B. M. Roberts, P. G. Ranclaud, and J. S. M. Ginges,
Phys. Rev. A 105, 052802
