Electron correlation effects in the dipole polarizabilities of the ground states of Be, Mg, Ca, Sr, Ba and Yb

0.13 MB·English

by  B. K. Sahoo

February 2, 2008
Electron correlation effects in the dipole polarizabilities of the ground states of Be,
Mg, Ca, Sr, Ba and Yb
B. K. Sahoo ∗
Max-Planck Institute for the Physics of Complex Systems
N¨othnitzer Straße 38, D-01187 Dresden, Germany
B. P. Das †
Non-Accelerator Particle Physics Group, Indian Institute of Astrophysics, Bangalore-560034, India
8
0
0
2 (Dated: February 2, 2008)

n We investigate the role of electron correlation effects in the electric dipole polarizabilities of the
a ground states of the alkaline earth and ytterbium atoms by employing the relativistic coupled-
J cluster (RCC) theory. These effects are incorporated via the residual Coulomb interaction to all

1 orders in the RCC singles and doubles approximation. The perturbed wavefunctions used in the
calculations of the polarizabilities are obtained by directly solving the first order perturbed RCC

] equations, thereby avoiding the sum-over-states approach. Our results are compared with other
h
calculations and available experimental data.
p
-
m PACSnumbers: 31.15.Ar,31.15.Dv,31.25.Jf,32.10.Dk
o
t I. INTRODUCTION sev and Derevianko have performed calculations of the
a
. dipole polarizabilities of Mg, Ca, Sr and Yb by a hybrid
s
c A knowledgeof electricdipole polarizabilitiesis neces- approach involving many-body perturbation theory and
si sary in many areas of physics and chemistry. In partic- the configuration interaction method [10]. A calculation
y ular it is requiredin studies of collisionsinvolving atoms of the dipole polarizability of Yb based on the time de-
h and molecules [1]. In recent years, the pre-eminent role pendent density functional theory (TDDFT) has been
p ofpolarizabilitiesinthedeterminationofinteratomicin- reported recently [16].
[
teractionshasassumedspecialsignificanceinthecontext

1 of researchon ultra-cold atoms [2].
v Calculationsofatomicpolarizabilitieshavecomealong II. THEORY AND METHOD OF
5
waysincetheclassicworkofDalgarnoandLewis[3]. Fol- CALCULATIONS
9
lowingaseriesofcalculationsusingthe coupledHartree-
2
0 Fock method (see for example, [4]), a number of state- In a DC electric field E = Eˆz, the energy shift ∆E
1. othf-etohreyahratvembeetehnodusseidnctloudcainlcgultahteeactooumpliecda-ncldusmteorle(cCulCa)r of the ground state |Ψ(0)(γ,J0,M0) > with the parity
0 eigenvalueγandangularmomentumJ anditsazimuthal
polarizabilities [5, 6, 7]. A few calculations of the polar- 0
8 value M is given by
izabilities of heavy atomic systems have been performed 0
0
: in the past few years using the linearized [8] as well as 1
v the non-linearized [9] relativistic CC theory. These are ∆E =− αE2, (2.1)
i 2
X based on approachesthat sum over a set of intermediate
r states. In contrast, we have obtained the first order per- whereα is the static polarizabilityandcanbe defined as
a turbed wavefunctionby solving the firstorderperturbed
CC equation and used it to obtain the dipole polariz-
abilities of the closed-shell alkaline earth atoms and yt- α=−2X|hΨ(0)(γ,J0,M0)|Dz|Ψ(0)(γ′,JI,MI)i|2(,2.2)
terbium (Yb). Information on these quantities is useful I E0−EI
for the frequency standards experiments that have been
proposedforMg[10],Ca[11],Sr [12]andYb [13]aswell where Dz is the zth componentof the electric dipole op-
as the search for the electric dipole moment in Yb [14]. erator depending upon the applied field E, subscript I
There has been considerable interest in accurate cal- denotes the intermediate states and JI, MI are their an-
culations of the dipole polarizabilities of alkaline earth gular momentum quantum numbers, γ and γ′ are parity
atoms and Yb. Sadlej et al have calculated these quan- quantum numbers for states of opposite parity and E0
tities for Ca, Sr and Ba using a quasi relativistic ap- andEI aretheenergyvaluesofthegroundstateandthe
proachbuttheirtreatmentofcorrelationisrigorous[15]. corresponding intermediate states I, respectively.
Their calculations have been carriedout at the finite or- In a more explicit form, the above expression can be
der many-body perturbation theory and CC levels. Por- written as
2
∗E-mail: [email protected]
†E-mail: [email protected]
α = −2XhΨ(0)(γ,J0,M0)|Dz|Ψ(0)(γ′,JI,MI)ihΨ(0)(γ′,JI,MI)|Dz|Ψ(0)(γ,J0,M0)i
I E0−EI
hΨ(0)(γ,J0,M0)|Dz|Ψ(0)(γ′,JI,MI)ihΨ(0)(γ′,JI,MI)|Dz|Ψ(0)(γ,J0,M0)i
= −X
I E0−EI
hΨ(0)(γ′,JI,MI)|Dz|Ψ(0)(γ,J0,M0)ihΨ(γ′,JI,MI)|Dz|Ψ(γ,J0,M0)i
−X
I E0−EI
= −hΨ(0)(γ,J0,M0)|Dz|Ψ(1)(γ′,J0,M0)i+hΨ(1)(γ′,J0,M0)|Dz|Ψ(0)(γ,J0,M0)i
= −2hΨ(0)(γ,J0,M0)|Dz|Ψ(1)(γ′,J0,M0)i
= −hΨ(γ,J0,M0)|Dz|Ψ(γ,J0,M0)i
(2.3)
where we define |Ψ(1)(γ′,J ,M )i as the first order Our aim in this work is to obtain the exact wavefunc-
0 0
correction to the original unperturbed wavefunction tion |Ψ(γ,J ,M )i by calculating both |Ψ(0)(γ,J ,M )i
0 0 0 0
|Ψ(0)(γ,J0,M0)i due to the operator Dz and hence the and |Ψ(1)(γ,J0,M0)i wavefunctions using an approach
totalwavefunctioninthepresenceofanexternalDCelec- whichcanrigorouslyincorporatethe relativisticandcor-
tric field is given by relation effects . In other words, we would like to ob-
tain|Ψ(1)(γ′,J ,M )ifortheDirac-Coulomb(DC)wave-
0 0
|Ψ(γ,J0,M0)i = |Ψ(0)(γ,J0,M0)i+|Ψ(1)(γ′,J0,M0)i. function|Ψ(0)(γ,J0,M0)iasthesolutionofthefollowing
(2.4) equation
(H(DC)−E(0))|Ψ(1)(γ′,J ,M )i=(E −H )|Ψ(0)(γ,J ,M )i, (2.5)
0 n 0 0 0 int 0 0
whereH(DC) andH correspondtotheDCHamiltonian explicitly, we can write the unperturbed and perturbed
0 int
and interaction due to Dz, respectively. wavefunctions in terms of the RCC excitation operators
Using the CC ansatz, we express wavefunctions as
|Ψ(γ,J ,M )i by [17]
0 0 |Ψ(0)(γ,J ,M )i = eT(0)|Φ (γ,J ,M )i, (2.8)
0 0 0 0 0
|Ψ(γ,J0,M0)i = eT|Φ0(γ,J0,M0)i, (2.6) |Ψ(1)(γ,J0,M0)i = eT(0)T(1)|Φ0(γ,J0,M0)i, (2.9)
where|Φ0(γ,J0,M0)iaretheDirac-Fock(DF)wavefunc- where the exponential function of T(1) reduces to the
tionsdeterminedusingthemean-fieldapproximationand linear term since we have considered only one order of
T are the electron excitation operators from the corre- Dz operator in Eq. (2.5).
sponding DF states. In the present work, we have considered all possible
To obtain both the unperturbed and perturbed wave- single and double excitations (CCSD approach) in the
functionsofEq. (2.4)separately,wedividetheexcitation calculations. WeobtainfirsttheunperturbedT(0)ampli-
operators T as tudes by solving the usual (R)CC equations, then these
amplitudes are used to determine the T(1) amplitudes.
T = T(0)+λT(1) (2.7) The corresponding equations to solve these amplitudes
are given by
where T(0) andT(1) arethe allorderexcitationoperator
of the relativistic coupled-cluster (RCC) method and its hΦ∗|H(DC)|Φ i = 0 (2.10)
firstordercorrectionarisinginthepresenceofDz,respec- 0 N 0
tively. λ represents the perturbation parameter . More hΦ∗|H(DC)T(1)|Φ i = −hΦ∗|H |Φ i (2.11)
0 N 0 0 int 0
3
where the subscript N represents normal order form of
TABLEI:Staticdipolepolarizabilitiesindivalentatoms: Be,
the operators and we have defined H = e−T(0)HeT(0) =
Mg, Ca, Sr, Ba and Yb(in au).
(HeT(0))con, with the subscript ’con’ representing con-
nected terms.
Atoms Expts Others This work
The polarizabilities are determined by evaluating the
following expression Be 37.755a, 37.69b, 37.9c 37.80
α = hΦ0|eT†DeT|Φni Mg 71.5(3.1)d 71.35b, 72.0c, 71.3(7)e 73.41
hΦ |eT†eT|Φ i
0 0 Ca 169(17)d 159.4b, 152.7c, 157.1(1.3)e 154.58
= hΦ0|(T(1)†D(0)+D(0)T(1))|Φ0 >, (2.12) 152f, 158.0g,
hΦ |eT(0)†eT(0)|Φ i
0 0
Sr 186(15)d 201.2b, 193.2c, 197.2(2)e 199.71
190f, 198.9g,
where for computational simplicity we define D(0) =
eT(0)†DeT(0). We compute these terms after expressing Ba 268(22)h 264i, 273f, 273.9g 268.19
themaseffectiveone-bodyandtwo-bodytermsusingthe
generalized Wick’s theorem [18]. Yb 142(36)j 111.3(5)e, 141.7k, 157.30l 144.59
aReference: [25]
III. RESULTS AND DISCUSSIONS bReference: [19]
cReference: [26]
dReference: [27]
In Table I, we present our results of electric dipole eReference: [10]
polarizabilities and compare with those available in the fReference: [15]
literature. The errorbarsinthe experimentalresultsare gReference: [22]
large for all the systems and our results lie within them. hReference: [28]
Theresultsofthecalculationsthataregivenintheabove iReference: [21]
tableareobtainedusingavarietyofmany-bodytheories. jReference: [29]
Ourresultsareinreasonableagreementwiththemexcept kReference: [23]
for a few cases where they differ by more than 5%. lReference: [16]
MitroyandBromley[19]haveusedoscillatorstrengths
from a semiempirical approach to obtain these quanti-
ties. Some results based on an ab initio method that TABLE II:Contributions from DF and important perturbed
CC terms (in au) for thedipole polarizabilities.
combines the configuration interaction (CI) method and
many-body perturbation theory (MBPT) are available
[10, 20, 21]. In these calculations, the valence elec- Atoms DF (DT(1)+cc)−DF DT(1)+cc Norm
1 2
trons correlation effects are evaluated by the CI method
whereas the core electrons correlation effects are calcu- Be 45.82 −7.94 −0.09 0.02
Mg 82.44 −8.77 −0.21 0.03
lated using the MBPT method. However, these calcula-
Ca 184.14 −29.23 −0.07 −0.26
tions consider the core-polarization effects in the frame-
Sr 234.41 −34.46 −0.17 −0.08
work of finite order MBPT, while our CC method takes Ba 328.32 −61.18 0.09 0.81
them into account to all orders. Lim and Schwerdt- Yb 183.32 −39.86 0.032 1.10
feger had employed the scalar-relativistic Douglas-Kroll
Hamiltonian to determine some of these quantities [22].
In these calculations, they had shown the importance
of the relativistic effects. There are also a few calcula-
RCC contributions in Table II for the electric dipole po-
tions available for the Yb polarizabilities using the CC
larizabilities. Forallthecasesthathavebeenconsidered,
method [23, 24], where the atomic orbitals are evaluated
the DF results are larger than the total results. From
using the molecular symmetries. Recently TDDFT was
the individual RCC contributions, we find that only the
used for calculating the same quantities [16], but this
method treats exchange and correlation effects via local terms arising from DT(1) and its conjugate (cc) are sig-
1
potentials. The main difference between the methods on nificant. Given that these terms include the DF, leading
whichallthesecalculationsarebasedandoursisthatwe core-polarizationand other important correlationeffects
calculate both the unperturbed and the first order per- to all orders, it is not surprising that they should col-
turbedwavefunctionsusingaRCCapproachthatimplic- lectively make up the largest contribution. In Fig 1, we
itly takes into consideration all the intermediate states. give the breakdown of DT(1) in terms of the DF, ran-
1
To emphasize the importance of correlation effects in dom phase approximation (RPA) and other diagrams.
these calculations, we present the DF and the leading All the above mentioned calculations where these terms
4
in the residual Coulomb interaction. The 3P and 1P
1 1
TABLE III: Contributions from important nsmp 3P10 and configurations are of crucial importance for the leading
nsmp1P10statesintheformofsingleparticleorbitalsthrough termofDT(1) anditsconjugate. InTableIII,wepresent
the(DT(1)+cc) CC terms to thedipole polarizabilities. 1
1 the contributions arising from different combinations of
single particle orbitals.
Atoms ns mp DF DT(1)+cc D
1 D
D
D
V
Be 2s1/2 2p1/2 1.154 0.893 D VN N D
2s1/2 3p1/2 8.261 6.596 D
T(1) (i) (ii) (iii)
2s1/2 4p1/2 5.529 4.583 D D D D D
2s1/2 2p3/2 2.308 1.816
V
2s1/2 3p3/2 16.521 13.484 N
2s1/2 4p3/2 11.059 9.390 VN VN DVN
Mg 3s1/2 3p1/2 3.521 2.292 D (iv) D (v) D (vi)
33ss11//22 45pp11//22 175..778001 150..225749 D VN VN DVN
3s1/2 3p3/2 7.009 6.742 VN VND VN
3s1/2 4p3/2 31.351 31.308 (vii) (viii) (ix)
3s1/2 5p3/2 15.605 17.030
FIG. 1: Breakdown of the DT(1) CC diagram in terms of
Ca 4s1/2 4p1/2 20.550 17.790 1
lower order MBPT and RPA diagrams that contribute sig-
4s1/2 5p1/2 32.290 29.712
nificantly to the polarizability calculations. Here, D and VN
4s1/2 6p1/2 6.928 8.073
represent the dipole and normal order Coulomb interaction
4s1/2 4p3/2 40.358 32.079
operators which areshown assingle dotted anddashed lines,
4s1/2 5p3/2 64.805 53.218
respectively.
4s1/2 6p3/2 14.097 13.437
Sr 5s1/2 5p1/2 30.288 25.795
5s1/2 6p1/2 38.465 35.555 IV. CONCLUSION
5s1/2 7p1/2 6.838 8.402
5s1/2 5p3/2 56.608 45.432
We have carried out calculations of electric dipole po-
5s1/2 6p3/2 78.344 67.307
larizabilities for many alkaline earth atoms and ytter-
5s1/2 7p3/2 14.665 15.247
bium using the RCC method and highlightedthe impor-
Ba 6s1/2 6p1/2 55.019 47.128 tance ofthe correlationeffects. The novelty in these cal-
6s1/2 7p1/2 45.796 45.569 culations is that it avoids the sum-over-states approach
6s1/2 8p1/2 4.631 8.292 indeterminingpolarizabilitiesandimplicitlyconsidersall
6s1/2 6p3/2 95.404 71.484 the intermediate states by solving the perturbed RCC
6s1/2 7p3/2 98.613 81.924 wavefunctionto firstorderinthe dipoleandallordersin
6s1/2 8p3/2 11.420 11.623
the residual Coulomb interaction.
Yb 6s1/2 6p1/2 26.578 19.824
6s1/2 7p1/2 29.112 24.629
6s1/2 8p1/2 3.234 3.925
V. ACKNOWLEDGMENT
6s1/2 6p3/2 43.498 32.953
6s1/2 7p3/2 61.077 52.816
6s1/2 8p3/2 8.518 9.522 We thank ProfessorA.DalgarnoandDr. P.Zhangfor
usefuldiscussions. Thecomputationsreportedinthispa-
perwereperformedusingC-DAC’sTeraFlopSuperCom-
putingfacilityatBangalore. WethankMrN.Mohanram,
have been evaluated are based on finite order MBPT, Dr. Subroto Chattopadhyay and their colleagues for co-
but in our present method, we treat them to all orders operation.
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