Relativistic general-order coupled-cluster method for high-precision calculations: Application to Al+ atomic clock

Abstract

We report the implementation of a general-order relativistic coupled-cluster method for performing high-precision calculations of atomic and molecular properties. As a first application, the static dipole polarizabilities of the ground and first excited states of Al+ have been determined to precisely estimate the uncertainty associated with the BBR shift of its clock frequency measurement. The obtained relative BBR shift is -3.66+-0.44 for the 3s^2 ^1S_0^0 --> 3s3p ^3P_0^0 transition in Al+ in contrast to the value obtained in the latest clock frequency measurement, -9+-3 [Phys. Rev. Lett. 104, 070802 (2010)]. The method developed in the present work can be employed to study a variety of subtle effects such as fundamental symmetry violations in atoms. Comment: 4 pages, 3 tables, submitted

Full text

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arXiv:1010.1231v1 [physics.atom-ph] 6 Oct 2010
Relativistic general-order coupled-cluster method for high-precision calculations:
Application to Al+atomic clock
Mih´ aly K´ allay1, B. K. Sahoo2, H. S. Nataraj1, B. P. Das3, and Lucas Visscher4
1Department of Physical Chemistry and Materials Science,
Budapest University of Technology and Economics, Budapest P.O.Box 91, H-1521 Hungary
2Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India
3Indian Institute of Astrophysics, 560034 Bangalore, India and
4Amsterdam Center for Multiscale Modeling, VU University Amsterdam,
De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
(Dated: October 7, 2010)
We report the implementation of a general-order relativistic coupled-cluster method for performing
high-precision calculations of atomic and molecular properties. As a first application, the static
dipole polarizabilities of the ground and first excited states of Al+have been determined to precisely
estimate the uncertainty associated with the BBR shift of its clock frequency measurement. The
obtained relative BBR shift is −3.66±0.44 for the 3s2 1S0
to the value obtained in the latest clock frequency measurement, −9 ± 3 [Phys. Rev. Lett. 104,
070802 (2010)]. The method developed in the present work can be employed to study a variety of
subtle effects such as fundamental symmetry violations in atoms.
0→ 3s3p3P0
0transition in Al+in contrast
PACS numbers: 31.15.A-,31.15.bw,31.15.V-,32.10.Dk
The role of high precision calculations of various prop-
erties of heavy atoms and molecules which support the
state-of-the-art measurements has gained incredible im-
portance in recent years. This is particularly true in the
context of atomic clocks [1], probes of fundamental sym-
metry violations [2–4], and search for the variation in
the fundamental constants [5]. The relativistic coupled-
cluster (CC) method with single and double excitations
(CCSD) supplemented by the important triple excita-
tions has yielded reasonably accurate results [2–4]. How-
ever, an extension to this method by including higher
order excitations and its application to large systems are
extremely challenging. The general nonrelativistic CC
approach of K´ allay and co-workers provides one of the
most efficient routes to the incorporation of higher order
excitations by exploiting the features of the many-body
diagrammatic techniques and string algebra [6].
In this Letter, we extend the general-order nonrela-
tivistic CC work reported in Ref. [6] to the relativistic
framework aiming to apply it for high-precision studies
in several important areas of fundamental physics; to
mention a few: atomic clocks, parity non-conservation
(PNC), electric dipole moment (EDM) due to parity and
time reversal violations. As a proof-of-principle, we have
employed the method for the calculation of the black-
body radiation (BBR) shift in the 3s2 1S0
clock transition of Al+. This transition provides the ba-
sis for the most accurate atomic clock to date [1, 5, 7],
for which the fractional frequency inaccuracy has recently
been estimated as 8.6×10−18[5]. Although the size of the
BBR shift in the Al+clock is smaller than those in most
of the other ions considered for atomic clocks, the asso-
ciated uncertainty in the estimated BBR shift is about
35% of the total uncertainty. The BBR shift was also ob-
tained using the static polarizabilities calculated from the
0→ 3s3p3P0
0
oscillator strengths taken from different sources [8], and
it was investigated later by Mitroy et al. [9] along simi-
lar lines using the configuration interaction (CI) method
with a semi-empirical core potential. Other nonrelativis-
tic calculations also provide results that agree with each
other [10–12].
With the general-order relativistic CC method, we also
consider the linear response theory which is the first ap-
plication of its kind to atoms for the calculation of static
polarizabilities of the ground and first excited states of
Al+. This method allows for the precise calculations of
the ground-state and one-hole and one-particle excited-
state properties, and it can be suitably modified for ap-
plying it to PNC and EDM studies in the proposed atoms
such as Ra and Yb [13, 14].
The exact wave function in the CC theory involves an
exponential parameterization of the form:
|ΨCC? = eˆ T|0? (1)
where |0? is the Dirac-Fock (DF) reference determinant,
and the cluster operatorˆT can be decomposed as
ˆT =
n
?
k=1
ˆTk
(2)
where
ˆTk=
?
a1<a2...<ak
i1<i2...<ik
ta1a2...ak
i1i2...ik
a+
1i−
1a+
2i−
2...a+
ki−
k
(3)
The convention followed here is that indices i (a) refer
to occupied (virtual) spinors in the reference determi-
nant. Projecting onto the excited determinants defined
by |Ψa1a2...ak
1i−
ear algebraic equations for the correlation energy E and
i1i2...ik? = a+
1a+
2i−
2...a+
ki−
k|0? we get the nonlin-

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unknown cluster amplitudes for the excitation of any or-
der as
?Ψa1a2...ak
i1i2...ik|e−ˆ TˆHNeˆ T|0? = Eδk,0,(k = 1,...n),(4)
whereˆHN is the normal-ordered Dirac-Coulomb (DC)
Hamiltonian and k is the level of excitation. The CC ap-
proaches corresponding to the n = 2,3,4,... values, i.e.,
the CC singles and doubles (CCSD), CC singles, doubles,
and triples (CCSDT), CC singles, doubles, triples, and
quadruples (CCSDTQ), ...methods constitute a hierar-
chy, which converges to the exact solution of the Dirac
equation in the given one-particle basis set.
The excitation energies are obtained invoking the
linear-response CC (LR-CC) theory as given in Refs. [15]
and [16]. In LR-CC theory the excitation energies ω are
calculated by determining the right-hand eigenvalues of
the CC Jacobian as
?Ψa1a2...ak
i1i2...ik|
?
e−ˆ TˆHNeˆ T,ˆR
?
|0? = ω ra1a2...ak
i1i2...ik, (5)
where operatorˆR has the same structure as the cluster
operator with parameters ra1a2...ak
i1i2...ik.
As it is well-known, the energy of the ith state of an
atom placed in an isotropic electric field of strength ε
changes as
Ei(ε) = Ei(0) −αi
2ε2− ...,(6)
where Ei(0) and Ei(ε) are the total energies of the state i
in the absence and the presence of the field, respectively,
and αiis the static dipole polarizability of state i. The
BBR shift for a transition |Ji,Mi? → |Jj,Mj? is the shift
of the corresponding transition energy due to the finite
background thermal radiation. At temperature T, ne-
glecting the dynamic correction factor from the previous
finding [9], in the adiabatic expansion it is given by
∆EBBR
ij
= −1
2(831.9 V/m)2
?T(K)
300
?4
(αi− αj). (7)
Consequently the evaluation of the BBR shift requires
the knowledge of the static polarizabilities for the two
states involved in the clock transition.
It is obvious from Eq. (6) that the static polarizabil-
ity can be evaluated as the second derivative of Ei(ε)
with respect to ε. In our study we followed this ap-
proach and calculated the polarizabilities by numerical
differentiation. The total energies were computed with
and without the perturbation; here the perturbation was
taken to be −D·ε where D is the induced electric dipole
moment, and the values of the electric field ε were fixed
to 1 × 10−3and 2 × 10−3a.u. The polarizabilities were
obtained from the resulting three energy values assuming
that they lie on a quartic polynomial. With the test cal-
culations the numerical error of this procedure was found
to be negligible.
TABLE I: Calculated excitation energies (cm−1) and polariz-
abilities (a.u.)
Excitation
energy
Polarizability
Ground state Excited state Differential
t-aug-cc-pCVDZ (1s, 2s, and virtuals > 5Eh are frozen)
CCSD3722224.215
CCSDT3732424.158
CCSDTQ3732624.156
24.380
24.357
24.358
0.165
0.199
0.202
t-aug-cc-pCVDZ
CCSD
CCSDT
37005
37167
24.203
24.072
24.261
24.208
0.058
0.136
t-aug-cc-pCVTZ
CCSD
CCSDT
37228
37373
24.143
24.017
25.040
24.979
0.897
0.962
t-aug-cc-pCVQZ
CCSD 37160 24.27324.700 0.427
t-aug-cc-pCV5Z
CCSD37186 24.25124.656 0.406
In oder to approach the exact solution of the Dirac-
Coulomb equation for the Al+ion as closely as possible,
the convergent hierarchy of CC methods was combined
with the convergent basis sets in the total energy calcula-
tions. The ground-state energies were obtained using the
CCSD, CCSDT, and CCSDTQ methods, while excited-
state energies were determined by the LR-CC method
in the same excitation manifold. The one electron basis
sets used were Dunning’s triply-augmented correlation
consistent polarized core-valence X-tuple-ζ sets [17, 18]
abbreviated as t-aug-cc-pCVXZ, where X is the so-called
cardinal number of the basis set, X = D, T, Q, and 5 for
double-, triple-, quadruple-, and pentuple-ζ basis sets,
respectively. The basis sets were uncontracted in all the
calculations. The CC calculations were carried out with
our new all-order relativistic CC code implemented in
the Mrcc suite [19]. The transformed molecular orbital
integrals were generated by the Dirac package [20].
To give an accurate estimate of the properties we
studied, we adopted a composite scheme, which is well-
established in quantum chemistry and widely used for
highly-accurate calculations of molecular properties (see,
e.g., Refs.[21–25]), in which the calculations with a
particular method in the CC hierarchy are carried out
with the largest possible basis set and the largest pos-
sible number of correlated electrons. In practice, CCSD
and CCSDT calculations were performed with pentuple-
and triple-ζ basis sets, respectively, correlating all elec-
trons and all orbitals. CCSDTQ calculations were only
feasible with the t-aug-cc-pCVDZ basis set, but further
approximations were necessary even in this basis, and
the 1s and 2s electrons were frozen as well as the virtual
orbitals lying above 5 Ehwere dropped. Our final esti-

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TABLE II: Composite excitation energies (cm−1), polarizabilities (a.u.), and their estimated errors. Note that 1s, 2s, and the
virtual orbitals above 5 Eh were frozen for the CCSDTQ-CCSDT calculations.
Contribution Excitation
energy
37186±25 24.251±0.022 24.656±0.044 0.406±0.021
146±33 −0.126±0.011 −0.061±0.015 0.065±0.026
2±4 −0.002±0.005
−6±6 0.015±0.015
37326±68 24.137±0.053 24.614±0.078 0.477±0.057 Sum of all contributions
Polarizability Source
Ground state Excited state Differential
CCSD
CCSDT-CCSD
CCSDTQ-CCSDT
Breit+QED
Composite
t-aug-cc-pCV5Z
t-aug-cc-pCVTZ
t-aug-cc-pCVDZ
Numerical MCDF
0.001±0.002 0.003±0.007
0.018±0.018 0.003±0.003
mates were obtained by adjusting the pentuple-ζ CCSD
values with the CCSDT-CCSD and CCSDTQ-CCSDT
increments computed with the triple- and double-ζ ba-
sis sets, respectively. The error of our computed values
were estimated on the basis of the convergence pattern
of the results. To improve the results further, we esti-
mated the contributions from Breit interaction and QED
corrections using the numerical multi-configurational DF
(MCDF) method as implemented in the Mcdfgme pro-
gram [26] and the sum-over-states expression for polariz-
abilities [9].
The calculated polarizabilities are compiled in Table I
where we also present the excitation energy of the clock
transition. Since the latter is precisely known from ex-
periments, the performance of our approach can be partly
judged from the agreement of our calculated and the mea-
sured excitation energy.
The convergence of both the polarizabilities and ex-
citation energies with increasing levels of correlation is
rapid. The CCSD values themselves are reliable; further,
the contribution of triple excitations to both properties
is less than 1%. Interestingly the polarizability of the
ground-state is more sensitive to correlation effects than
the excited-state: the effect of triple excitations for the
ground state is twice as large as that for the excited state,
viz. 0.13 a.u. vs. 0.06 a.u. The magnitude of the triples
contribution to the polarizability shift is also moderate,
it only amounts to 0.06 a.u., however, it is more than
10% of the composite value and thus cannot be ignored.
The effect of quadruple excitations is approximately two
orders of magnitude smaller than that of the triples and
can be considered as negligible, which also implies that
higher-order correlation contributions can safely be ig-
nored.
The basis set convergence of the properties we have
studied is in accordance with the usual trend—relatively
slow, but the results are close to the basis set limit when
large basis sets are employed. The polarizabilities and ex-
citation energies are already reliable in the smaller basis
sets, while the polarizability shift, which is a small differ-
ence of two large numbers, requires at least quadruple-
ζ-quality basis set even for a qualitatively correct result.
It is interesting to note that in this case the polarizabil-
ity of the excited-state is more sensitive to the quality of
the basis set than the ground state. From the compari-
son of the quadruple- and pentuple-ζ results we observe
that the CCSD excitation energies and polarizabilities
change on the scales of 10 cm−1and 0.01 a.u., respec-
tively, which means that the relative change is about
0.1% for both properties. Since the basis-set error de-
creases monotonically with the size of the basis set, the
error with respect to the infinite basis set limit is also
expected to be less than 0.1%. Unfortunately the errors
of the ground- and excited-state polarizabilities do not
cancel each other, and consequently the absolute error
of the CCSD polarizability shift is larger. For the afore-
mentioned reason its relative error is also significantly
larger, a couple of percent of the total value. Similar
conclusions can be drawn for the contribution of triple
excitations. The CCSDT-CCSD difference also changes
in the 10 cm−1and 0.01 a.u. range for excitation en-
ergies and polarizabilities, respectively, when going from
the double- to the triple-ζ basis set, and the change in
the polarizability shift is only 0.013 a.u. Thus the error
in our final estimates stemming from the calculations of
the triples contribution is also smaller than 0.1% (3%) for
the excitation energy and polarizabilities (polarizability
shift).
TABLE III: Comparison of theoretical and experimental po-
larizabilities (a.u.), and relative BBR shifts.
Polarizability
3s2 1S0
0
24.19
24.83±5.26 24.63±4.93 −8±3
24.20±0.75
24.12
24.14±0.12 24.62±0.25 −4.18±3.18
24.22±1.21 24.78±1.24 −4.3±2.5
24.14±0.05 24.61±0.08 −3.66±0.44 This work
BBR shift Reference
×1018
3s3p3P0
0
[10]
[8]
[11]
[29]
[9]
[30]
The calculation of the properties that have been inves-
tigated using the composite approach outlined above is
shown in Table II in detail, where we also present our
error estimates based on the convergence of the contri-
butions with the basis set. It has been found in numer-
ous studies that in the higher members of correlation-
consistent basis set family the basis set error for vari-
ous properties is usually reduced by a factor of at least
two when increasing the cardinal number of the basis set

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by one. The reduction of the basis set error would also
be valid for the current properties. In fact, the ratio
of the quadruple-ζ−triple-ζ and pentuple-ζ−quadruple-
ζ differences of CCSD excitation energies, ground- and
excited-state polarizabilities, and polarizability shifts is
2.7, 5.8, 7.8, and 22.1, respectively. Thus we presume
that the entire difference between the pentuple- and
quadruple-ζ results is a conservative estimate for the ba-
sis set error of the CCSD values, and we attach these
numbers as error bars. The estimation of the intrinsic
error of the CCSDT-CCSD contributions is less straight-
forward since the results are not available in the larger
basis sets. Therefore we take twice the difference between
the double- and the triple-ζ triples contributions as a con-
servative choice. The quadruples contribution, i.e., the
CCSDTQ-CCSDT difference is only available in one ba-
sis set, and no conclusion about its basis set dependence
can be drawn. Consequently we take twice the entire
contribution as the error bar. The contribution of Breit
and QED corrections for the excitation energy is ∼ −6
cm−1while for polarizabilities of the ground and excited
states it is 0.015 and 0.018 a.u., respectively. We would
like to remark that these effects are computed using nu-
merical orbitals at the DF level of the theory and hence
they are devoid of any basis set incompleteness errors.
As the missing correlation contribution to these effects is
not expected to exceed its DF value, we have taken the
entire value itself as the upper limit of the error.
For the excitation energy a highly-accurate experimen-
tal value, 37393±0 cm−1is available [1], thus the agree-
ment between the experimental and our best calculated
excitation energy, 37326±68 cm−1is very good and the
deviation is within 0.2% of the experimental energy.
We compare our polarizabilities and the BBR shift to
the previous theoretical and empirical results in Table
III. Our results are in good agreement with the previous
computational results, however, more accurate than the
latter. In contrast, there is a considerable discrepancy be-
tween the present and the experimental BBR shift. There
is a brief discussion on various approaches employed to
calculate the polarizabilities and the BBR shift by Mitroy
et al. [9], hence we do not repeat them here, however we
would like to emphasize that our results are the first ab
initio values based on a relativistic framework.
In conclusion, we have developed a general-order
relativistic coupled-cluster method for high-precision
calculations in atoms and molecules.
method the ground-state, excited-state, and differen-
tial polarizabilities of the Al+ion are obtained to be
24.14±0.05, 24.61±0.08, and 0.48±0.06 a.u., respec-
tively.From the latter value and the measured clock
frequency of 1.121015393207851 × 10−15Hz [5] we ob-
tain −0.0041±0.0005Hz for the absolute and −3.66±0.44
for the relative BBR shift. It is the most accurate esti-
mate of the BBR shift in Al+, using which the system-
atic shift in the above frequency measurement can be
Using this
obtained as (−1112.46± 6.04) × 10−18against the value
of (−1117.8± 8.6) × 10−18considered in Ref. [5].
Financial support to M.K. has been provided by the
European Research Council (ERC) under FP7, ERC
Grant Agreement No. 200639, and by the Hungarian
Scientific Research Fund (OTKA), Grant No. NF72194.
M.K. and B.P.D. acknowledge the Indo-Hungarian (IND
04/2006) project. M.K. acknowledges the Bolyai Re-
search Scholarship of the Hungarian Academy of Sci-
ences. B.K.S. thanks T. Rosenband for useful discus-
sions. L.V. has been supported by NWO through the
VICI programme.
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