Table II. The CI+MBPT calculation [4] is a fully relativistic calculation with the post-HF ... a CI framework while using a model potential to represe...

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arXiv:physics/0408120v1 [physics.atom-ph] 27 Aug 2004

atoms. J.Mitroy and M.W.J.Bromley∗ Faculty of TIE, Charles Darwin University, Darwin NT 0909, Australia (Dated: August 29, 2018)

Abstract The static and dynamic properties of the alkaline-earth atoms in their metastable state are computed in a configuration interaction approach with a semi-empirical model potential for the core. Among the properties determined are the scalar and tensor polarizabilities, the quadrupole moment, some of the oscillator strengths and the dispersion coefficients of the van der Waals interaction. A simple method for including the effect of the core on the dispersion parameters is described. PACS numbers: 34.20.Cf, 31.25.Jf, 31.15.Pf, 32.10.Dk

∗

now at Department of Physics, Kansas State University, Manhattan, KS 66506 USA; Electronic address:

[email protected]

1

I.

INTRODUCTION

The low-lying triplet metastable states of alkaline-earth atoms have been generating increasing interest in the area of cold-atom physics for a number of reasons. One application is to use the 1 S0e → 3 P1o transition in calcium as a new optical frequency standard [1]. The use of the 1 S0e → 3 P0o transitions for fermionic

87

Sr stored in an optical lattice is expected

to further result in an improved standard [2]. Another possible application is in the formation of Bose-Einstein condensates (BEC) consisting of alkaline-earth atoms [3, 4] in their metastable triplet states. The stability, size and excitation modes of BECs depends on the sign (and magnitude) of the scattering length, and the scattering length depends sensitively on the precise values of the dispersion constants [5, 6]. Taken in conjunction, the desirability of obtaining precise values of the static and dynamic properties of the low-lying nsnp 3 P o metastable state has greatly increased in importance. In this article, properties of these states are computed from valence electron configuration interaction calculations that use a semi-empirical model potential to describe the core-valence interaction [7, 8, 9, 10, 11]. Among the data computed are the oscillator strengths for some of the low-lying transitions, the scalar and tensor polarizabilities, the quadrupole moments, and the dispersion coefficients for the van der Waals interaction between two atoms.

II.

RESULTS OF THE CALCULATIONS A.

Methodology

The properties of these states are computed using configuration interaction (CI) calculations that treat the correlations between the valence particles in an ab-initio manner while using a semi-empirical model potential to describe the core-valence interaction [7, 8, 9, 10, 11]. The details of this calculation are very similar to those reported in [9, 11, 12] apart from some minor changes in the cutoff parameters and the use of an orbital basis of larger dimension. The polarization potentials were initially defined by tuning the potential to reproduce the ns, np, nd and nf binding energies of the respective singly ionized atom. The Hamiltonian was then diagonalized in a basis consisting of all the two-electron basis states that could be formed from a set of about 140-150 single particle orbitals. The basis set contained orbitals up to and including ℓ ≤ 8 and the two-electron basis dimensions ranged from 1000 2

to 4000. For all practical purposes the basis for the two-valence electrons can be regarded as saturated. The initial binding energies obtained by this procedure were not in perfect agreement with experiment with discrepancies for the ground and excited state energies of the order of 0.1-2.0% (refer to [9, 12] to get an indication of the accuracy). Some further tuning of the cutoff parameters was done to improve the accuracy of the energy differences which directly impact on the accuracy of expectation values. Expectation values for multipole operators were computed with a modified operator that allowed for polarization corrections [11, 13, 14]. The model potential is quite realistic since the direct and exchange interactions with the core were computed without approximation from a Hartree-Fock (HF) wave function, only the core polarization potential was described with a model potential. The resulting polarizabilities, and dispersion parameters for homo-nuclear pairs of atoms were generally within 0.1% of the best variational calculations for Li or Be, and for heavier atoms they were generally within 1-2% of results coming from large fully relativistic calculations combining configuration interaction and many-body perturbation theory techniques [11]. The most likely source of error in the present calculations for the heavier species, Ca and Sr, is the neglect of relativistic effects. However, the use of a polarization potential tuned to the experimental binding energy will implicitly take into account the influence of relativistic effects upon the core electron distribution. Further, Greene and Aymar have shown that the spin-orbit interaction does not have major effect on the structure of the alkaline-earth wave functions [15].

B.

Energy levels

The energy levels of the present calculations are given in Table I and compared with experiment. The polarization cut-off parameters were fine-tuned to reproduce the experimental binding energy of the lowest states of each symmetry. In the case for states with L > 0 the parameters were tuned to reproduce the center-of-gravity of the spin-orbit triplets. The spin-orbit splitting of the triplet states is largest for strontium and its magnitude is about 0.001 Hartree. The agreement between the theoretical and experimental energy levels is sufficiently close to discount the possibility that energy level considerations might make a significant contri3

bution to the uncertainty in the oscillator strengths and polarizabilities. TABLE I:

Theoretical and experimental energy levels (in Hartree) of some of the low-lying

metastable states of the alkaline-earth atoms. The energies are given relative to the energy of the doubly ionized core. The experimental energies for the triplet states are averages with the usual (2J + 1) weighting factors. The experimental data are taken from [16, 17]. The md level is the 3d level for Be, Mg and Ca while for Sr it is the 4d level. Systems

Be

Level

Mg

Ca

Sr

Theory

Exp.

Theory

Exp.

Theory

Exp.

Theory

Exp.

ns2 1 S e

-1.011842

-1.011850

-0.833533

-0.833530

-0.660944

-0.660932

-0.614598

-0.614602

nsnp 3 P o

-0.911710

-0.911701

-0.733378

-0.733788

-0.591387

-0.591388

-0.547611

-0.547612

ns(n + 1)s

3Se

-0.774561

-0.774552

-0.645827

-0.645821

-0.517230

-0.517228

-0.482289

-0.482292

np2 3 P e

-0.739862

-0.739855

-0.569906

-0.569929

-0.485477

-0.485478

-0.452720

-0.452717

nsmd 3 D e

-0.729118

-0.729113

-0.615041

-0.615022

-0.568193

-0.568180

-0.531359

-0.531367

C.

Quadrupole moments

The quadrupole moment of the 3 P2o state is a static property of the state. An exact knowledge of its value is important since the quadrupole-quadrupole interaction has a big impact of the cold collision physics in metastable alkaline-earth metal atoms [4, 18]. Defining the LS coupled reduced matrix element as Q(L) = hΨ(3 P o ) k

X

ri2 C2 (ri ) k Ψ(3 P o )i ,

(1)

i

the quadrupole moment for a triplet state is usually defined as the moment of 3 PJ0 state with MJ = J; In the expression above C2 (ˆ r) is the spherical tensor of rank 2. Q = hΨ(3 PJo); MJ = J|

X

ri2 C02 (ri )|Ψ(3 PJo); MJ = Ji ,

(2)

i

This can be written [19] s Q(LJ) =

S L J 4J(2J − 1) 2+S+L+J (2J + 1)(−1) Q(L) , 2 J L (J + 1)(2J + 1)(2J + 3)

(3)

where the Wigner-Eckart theorem has been used twice to collapse the angular factors. The quadrupole moment for a 3 P level is often given for the J = 2 state. The quadrupole moment Q(LJ) for the J = 2 state is equal to 2hQzz i. 4

Some older calculations of hQzz i exist [20, 21]. The finite element Multi Configuration Hartree-Fock (MCHF) calculation of Sundholm and Olsen for Be gave 4.53 au which is in excellent agreement with the present value of 4.54 au. The CI calculations of Ceraulo and Berry [20] consistently underestimated the present quadrupole moments (e.g. 7.944 au for Mg) and are not listed in Table II. The quadrupole moments are compared with the recent calculations by other groups in Table II. The CI+MBPT calculation [4] is a fully relativistic calculation with the post-HF interactions between the valence electrons and the core treated with perturbation theory while the interaction between the two valence electrons are teated with the CI ansatz. The calculation of Santra and Greene [22] (SG-CI) treated the two active electrons within a CI framework while using a model potential to represent the core-valence interaction. The model potential did include a spin-orbit interaction. One limitation with the SG-CI calculation is that it does not include the di-electronic part of the polarization potential. The noticeable feature of Table II is that all three calculations agree with other with a total variation of less than 2%. The present results generally lie closer to the CI+MBPT calculation than the SG-CI calculation. The high level of agreement between three completely independent calculations suggests that the uncertainty ascribed by Derevianko et al to their quadrupole moment was too big by a factor of 2.

D.

Oscillator strengths of low-lying transitions

The oscillator strengths for the transitions to the lowest lying 3 S e , 3 P e and 3 D e states are given in Table II. The absorption oscillator strength from state ψ0 is calculated according to the identity f0n =

2|hψ0 ; L0 S k

ri C1 (ˆ ri ) k ψn ; Ln Si|2 ǫ0n . 3(2L0 + 1)

P

i

(4)

The oscillator strengths for the Be triplet transitions are probably as accurate as any that have previously been published. The basis for the valence electrons is effectively saturated and the semi-empirical approach to core polarization is capable of high accuracy [11]. For example, the present methodology reproduces the dipole and quadrupole polarizability of Be given by a close to exact calculation [23] to an accuracy of 0.2%. The present oscillator strengths agree very well with the experimental values given in Table II. Not shown in the 5

Table are the 3 P o → 3 P e oscillator strengths of CI calculation of Weiss [24] and the MCHF calculation of Jonsson et al [25]. Both of these calculations were very large and incorporated both core and valence excitations. The Weiss f -value was 0.447, while the Jonsson et al result was 0.4463. These could hardly be any closer to the present value of 0.4467. The present oscillator strength for the transition to the 3 S e state in Mg, namely 0.138 is in excellent agreement with that obtained from the low uncertainty experiment of Andra et al [26], 0.139±0.003. Agreement with the large basis CI calculation of Moccia and Spizzo (MS-CI) is also good [27]. The MS-CI calculation is similar to the present calculation in that excitations are only permitted for the valence electrons. It does not allow for core-valence correlations so the present approach, which does, should be regarded as being more reliable. In the case of Ca, good agreement is achieved with the model potential calculations of Hansen et al [28] for the transitions to the 3 S e and 3 P e states. A 7% discrepancy occurs for the transition of the 4s3d 3 D e state. The larger difference here is expected since the 3d orbital does have a tendency to penetrate into the core and therefore degrade the accuracy associated with model potential methods. The best ab-initio calculation is the MCHF calculation by Froese-Fischer and Tachiev [29]. The MCHF calculation allows for core-valence correlations and also includes relativistic effects using the Briet-Pauli Hamiltonian. The MCHF oscillator strengths listed in Table II are a weighted average of the individual lines in the multiplet. The largest difference between the present and MCHF oscillator strengths is less than 4%. The multi-channel quantum defect theory (MQDT) calculations of Werji et al [30] which use an R-matrix calculation to determine the short-range parameters. Their transition rate data was converted to oscillator strengths using experimental energy differences and lie within 2-3 % of the present oscillator strengths. The most precise experiment for Sr is that of Andra et al [31] which gave a lifetime of 7.89 ± 0.05 ns for the 5p2 3 P2e state. This state can decay to the both the 5s5p and 5s6p levels and the lifetime was converted to an oscillator strength by neglecting the transition to the 5s6p state. This assumption is justified since the dipole matrix element will be small due to the h5p|6pi overlap, and the 5s6p2 3 P o → 5p2 3 P e energy difference of 0.0073 Hartree is also small. The comparison with the time-dependent gauge independent (TDGI) calculations of Merewa et al [32] is mainly of interest because these authors also give estimates of the scalar and tensor polarizabilities. A quick comparison of TDGI f -values with other results 6

in Table II reveals that their oscillator strengths do not have the same level of accuracy as the other calculations. The underlying atomic structure information entering the TDGI formalism comes from CI calculations.

E. 1.

The polarizabilities Theoretical treatment of polarizabilities

This analysis is done under the premise that spin-orbit effects are small and the radial parts of the wave functions are the same for the states with different J. The Stark energy shifts for the different L0 levels in an electric field F are written as [49] 1 ∆E = − αL0 M0 F 2 . (5) 2 The Stark shifts for the different M0 states of the 3 P o level are different and the polarizability is written as

3M02 − L0 (L0 + 1) α2 . (6) αL 0 M 0 = α0 + L0 (2L0 − 1) where α2 is taken from the state with M0 = L0 . The total polarizability is written in terms of both a scalar and tensor polarizability. The scalar polarizability represents the average shift of the different M levels while the tensor polarizability gives the differential shift. In terms of second order perturbation theory, the energy shift from an electric field, F

pointing in the z-direction is P P 1 X 2hψ0 ; L0 M0 | i ri C01 (ˆ ri )|ψn ; Ln Mn ihψn ; Ln Mn | i ri C01 (ˆ ri )|ψ0 ; L0 M0 iF 2 ∆E = . 2 n (E0 − En ) (7) The polarizability can therefore be written 2 P 1 X L0 1 Ln 2|hψ ; L k ri ) k ψn ; Ln i|2 0 0 i ri C (ˆ αL 0 M 0 = (E0 − En ) −M0 0 Mn n

(8)

where the Wigner-Eckart theorem has been used to isolate the M-dependent terms. Using the definition of the oscillator strength, eq. (4) and taking the average of the energy shifts leads to the usual definition as a sum rule over the oscillator strengths. It is

α0 =

L0 X

αL0 M0 /(2L0 + 1) =

M0 =−L0

X f0n n

7

ǫ20n

,

(9)

where the sum includes both valence and core excitations and ǫ0n = (E0 − En ). The f -value distribution for the core was estimated using a semi-empirical method [11]. In this approach one writes αcore =

Ni , (ǫi + ∆)2 i∈core X

(10)

where Ni is the number of electrons in a core orbital, ǫi is the Koopman energy, and ∆ is an energy shift parameter chosen so that eq. (10) reproduces an accurate estimate of the core polarizability determined my other, independent means. Since the M-dependent part of the polarizability is a tensor of rank 2 and it is easiest to define it in terms of α2,L0 L0 .

α2,L0 M0 = α2,L0 L0 × (−1)L0 −M0 = α2,L0 L0 ×

L0

2 L0

−M0 0 M0

3M02 − L0 (L0 + 1) , L0 (2L0 − 1)

,

L0 2 L0

−L0 0 L0

(11) (12)

where α2,L0 L0 is

α2,L0 L0

2 # " P 1 X L0 1 Ln 2|hψ ; L k ri ) k ψn ; Ln i|2 1 0 0 i ri C (ˆ = . − 3(2L0 + 1) (E0 − En ) −L0 0 L0 n

In terms of an f -value sum, this reduces to X f0n 1 X f0n 1 X f0n α2,L0 L0 = − − + . ǫ2 2 n,L =1 ǫ20n 10 n,L =2 ǫ20n n,L =0 0n n

n

(13)

(14)

n

The core does not make a contribution to the tensor polarizability since it has an equal impact on all the different M-levels. The development above is for LS coupled states, but it is common to give the tensor polarizability for LSJ states. These can be related to the LS states by geometric factors arising from the application of Racah algebra. The polarizability can be expanded αJ 0 M 0 = α0 +

3M02 − J0 (J0 + 1) α2,J0 J0 . J0 (J0 − 1)

(15)

where α2,J0 J0 is the tensor polarizability of the state with M0 = J0 . The scalar polarizability for the different J levels are the same and equal to the scalar polarizability in the L representation. The tensor polarizability between the L and J representations can be related by

8

α2,J0 J0

J0 S L J −J0 0 0 = α2,L0 L0 (2J0 + 1)(−1)S+L0 +J0 +2 2 J0 L0 L 0 −L0

2 J0 0 J0

2 L0 0 L0

.

(16)

When J = 1 this reduces to α2,J0 J0 = −α2,L0 L0 /2. (This result has been checked by converting our LS coupled f -values into LSJ coupled values and then using the standard expression in terms of the |hJ0 k r k Jn i|2 matrix elements [49].)

2.

Results of calculations

The program logic and associated numerics were initially tested by estimating the polarizabilities of the 1s2p 3 P o level of He. The present α0 of 46.6 a30 is within 0.2% of the close to exact calculation of Yan et al [35]. Agreement with the TDGI α2,L0 L0 of Rerat and Pouchan [36] is not as good, but it should be noted that the TDGI calculation calculation obtains an α0 of 49.5 a30 , indicating that the Rerat-Pouchan calculation is not quite converged. The present estimates of the Be polarizabilities are the most accurate that have been published. The agreement with the Themelis and Nicolaides MCHF calculation [37] for α0 is reasonable, but they give an α2,L0 L0 that is about 20% smaller. This level of agreement is acceptable given that the MCHF calculation was much smaller, the 2s2p 3 P o state was represented by a 3 configuration MCHF wave function while 14 configurations were used to represent the excited states. Only a moderate level of agreement is achieved with the TDGI polarizabilities for Be, Mg and Ca [38, 42, 45]. The static polarizabilities agree at the 10% level while the TDGI estimates of the tensor polarizability are up to 50% different. The lower level of accuracy achieved by the TDGI calculations is consistent with the earlier discussion concerning the accuracy of the oscillator strengths. A recent measurement of the tensor polarizability for the 3 P1o state of Ca using an atomic polarization interferometer gave 2.623 ± 0.015 kHz/(kV/cm)2 or 10.54 ± 0.06 a30 [50]. The tensor polarizability of the J = 1 state is determined from the Ca entry in Table II by multiplying by - 12 according to eq. (16). The present calculation gives α2,J0 J0 = 14.2 a30 for the 3 P1o state. A very early estimate of the tensor polarizability for this state was 12.9 ± 3.2 9

a30 [51] and another independent experiment gave α2,J0 J0 = 12.1 ± 0.8 a30 [52]. The scaler polarizability of the 3 P o state has not been measured directly, but there have been measurements of the difference between the polarizabilities of the 4s2 1 S0e ground state and the 3 P1o (m = 0) ground state. Morinaga et al [53] obtained 90.4 ± 13.5 a30 for the difference in the polarizabilities. Using the polarizability of 159.4 a30 for the Ca ground state [11], and the present 3 P1o (m = 0) polarizability of 295.3 − 2 × 14.2 = 266.9 a30 gives 107.5 a30 for the difference in the polarizability. The Stark frequency shift of Li and van Wijngaarden of 12.314 ± 0.041 kHz/(kV/cm)2 . for the 4s2 1 S0e → 3 P1o (m = 0) transition [54] converts to a polarizability difference of 98.98 ± 0.33 a30 . Taken together, present estimates of α2,J0 J0 are larger than experiment by about 20% while estimates of the α0 (4s2 1 S e ) − α0 (4s4p 3 P1o ) polarizability difference are about 10% too large. Rectifying the situation in a non-relativistic calculation could be problematic since an improvement in α2,J0 ,J0 will result in the theoretical polarizability difference drifting further away from the experimental polarizability difference. The obvious improvement that could eliminate this problem would be the inclusion of the spin-orbit interaction. The largest contribution to the polarizability comes from the transitions to the 3 D e levels. The spin-orbit splitting leads to the excitation energies for 4s3d states with differing J fluctuating by about ±2%. Given the cancellations that occur in the evaluation of eq. (13) it is possible that introduction of spin-orbit splitting could lead to a Ca tensor polarizability in better agreement with experiment. There have been no measurements of the tensor polarizability for the other alkaline-earth atoms. This should be rectified since it would be a very useful diagnostic with which to assess the accuracy of the structure models of the metastable states.

3.

Alternate treatment of core

It is desirable to partition the core f -value some into contributions that arise from excitations to final states with different core+valence angular momentum, LT . Therefore, it is possible to write symbolically αcore =

X LT

10

αcore,LT ,

(17)

where αcore,LT will include all the contributions from the different magnetic sub-levels, i.e. αcore,LT =

X

αcore,LT MT .

(18)

MT

For any of the core dipole excited magnetic sub-levels one can write αcore,LT MT =

X f (00 : LM → n; LT MT ) . ǫ20n n

(19)

The final states, |LT MT i can be expanded in terms of uncoupled states, e.g. |LT MT i =

X

h1mLM|LT MT i|1mLMi ,

(20)

mM

where 1m refers to the angular momentum of the excited core and LM refers to the angular momentum of the 3 P o metastable state which is acting as a spectator. Therefore, it is possible to decompose the oscillator strength as f (0; 00LM → n; LT MT ) =

X

|h1mLM|LT MT i|LT MT i|2f (0; 00LM → n; 1mLM) . (21)

mM

The polarizability can also be expanded in terms of uncoupled states αL T M T =

X

|h1mLM|LT MT i|LT MT i|2 αcore,mM /(2L + 1) .

(22)

mM

The factor of (2L+1) in the denominator arises due to the sum over spectator states. We now assume that the excitations for the core occur independently of the state of valence electrons which act as spectators. Therefore, the contribution to the polarizability is independent of M. Further, the core initially has a net angular momentum of zero and therefore the different magnetic sub-levels of the core excitations should give equal contributions to the polarizability, hence αcore,mM =

αcore . 3

(23)

The final result is αL T M T =

X

|h1mLM|LT MT i|LT MT i|2

mM

αcore , 3(2L + 1)

(24)

which can be simplified by summing the Clebsch-Gordan coefficients to give αcore,LT =

(2LT + 1)αcore . 3(2L + 1) 11

(25)

When particular values are substituted into eq. (25) the distribution of the core f -value sum into the 3 S o , 3 P o and 3 D o manifolds is given in the proportion

1 9

:

3 9

: 59 . This is of

course just the statistical weighting associated with the (2LT + 1) degeneracy factor. It is simple to verify that such a proportion means the net contribution of the core to the tensor polarizability as defined by eq. (14) is zero.

F.

The van der Waals coefficients

The van der Waals coefficients given in Table II are those for a pair of 3 P0o states. The dispersion parameter, C6 is simple to compute since both of the atoms have a net angular momentum of zero. The expression is C6 =

3 X f0,n1 f0,n2 . 2 n ,n ǫ0n1 ǫ0n2 (ǫ0n1 + ǫ0n2 ) 1

(26)

2

The present dispersion parameters are slightly larger than those of the SG-CI calculation of Santra and Greene [22]. Taking the case of Sr, the difference here is about 15%. About half of this difference can be attributed to the core since C6 = 5668 au when core excitations are omitted from eq.(26). So part of the discrepancy arises from the neglect of the core in the SG-CI calculation. One cautionary note should be made. Santra and Greene reported C6 for the 3 P0o state. Since the 3 P0o state is the most tightly bound state of the 5s5p multiplet one expects the present LS coupled calculation to have a slightly larger C6 . The quantitative impact of spin-orbit splitting can best be determined by separate evaluations of C6 for the J = 0, 1 and 2 states. The van der Waals coefficients that are relevant to BEC studies are those between two 3

P2o states. The algebra related to this is somewhat messy and the coefficients are presented

in the formalism of Santra and Greene [19, 22]. The intermediate dispersion coefficient between two 3 PJo states is defined as BJ1 ,J2 = (−1)J1 −J2 +1

X

n1 ,n2

f0,n1 f0,n2 , ǫ0n1 ǫ0n2 (ǫ0n1 + ǫ0n2 )

(27)

where n1 has angular momentum J1 and n2 has angular momentum J2 . This LS coupled oscillator strengths were converted into the LSJ coupling scheme using

12

the identity 2 S L J 0 0 f (J0 → Jn ) = f (L0 → Ln )(2L0 + 1)(2Jn + 1) . 1 Jn Ln

(28)

When the sum, eq. (27), was evaluated, the core f -value distribution was included using eq. (25) to partition it into 3 S e , 3 P e and 3 D e excitations. The results of our calculations are presented in Table III and compared with earlier CI+MBPT calculations of Derevianko et al [4] and the SG-CI calculations [22]. There is no apparent experimental activity on the metastable states of Be and the present data in the Table were only included for reasons of completeness. The present calculation and the CI+MBPT calculation could hardly be in any better agreement for magnesium. The largest disagreement for any of the BJ1 ,J2 coefficients was 1.2% for the B2,2 coefficient. Agreement with the SG-CI calculation is not as good with the occasional discrepancy of 5% and it is noticeable that the present and CI+MBPT results do tend to be larger in magnitude. For calcium there is a tendency for the present results to be from 1% to 5% larger in magnitude with the differences being smaller for the larger values of J1 and J2 . The present dispersion coefficients all lie within the 10% uncertainty that Derevianko et al associate with their results. Agreement with the SG-CI calculations is not so good with discrepancies exceeding 10% being common. The pattern for strontium is similar to that seen for calcium. The present BJ1 ,J2 coefficients are larger than the CI+MBPT data for B1,1 and smaller for B3,3 . The differences with the SG-CI calculation are generally larger than those with the CI+MBPT calculation. Some general trends are noticeable. The SG-CI calculation always gave the smallest P result for B1,1 , B2,1 , B3,1 , B3,2 and B3,3 . Furthermore, the sum Ji ,Jj BJi ,Jj for the present

calculations and CI+MBPT calculations are consistently bigger than the SG-CI calculations, with the difference becoming larger as the atom gets heavier. This could be a manifestation of the increasing importance of the core contribution to the BJi ,Jj coefficients as the atom gets heavier. It is also evident that some of the uncertainty estimates of the SG-CI calculation were somewhat optimistic. For example, they give B1,1 = 139 ± 7 au for strontium. The contribution of the core f -value sum to this dispersion parameter is 12.0 au. So the core contribution, 13

which is not incorporated in the SG-CI calculation, is larger than their estimated uncertainty.

III.

CONCLUSIONS

A systematic study of the properties of the alkaline-earth atoms reveals that the present non-relativistic approach reproduces the results of the CI+MBPT ansatz of Derevianko et al [4] to better than 5%. Agreement with the model potential CI calculation of Santra and Greene [22] is not so good with discrepancies of 10-15% occuring for the spherical part of the C6 dispersion coefficient. Due to the unknown impact of the spin-orbit energy, splitting upon the polarizabilities and dispersion coefficients, it is not possible to make a definitive statement about any reasons for the differing levels of agreement with these two other calculations. However, we do suspect that the omissions of the SG-CI model, i.e. the di-electronic two body polarization potential, the non-usage of a dressed dipole transition operator, and the lack of core excitation terms in the dispersion sum rules all contribute in part to the differences with the SG-CI model. It should be noted that previous studies with the present model for the alkali atoms and singlet states of the alkaline atoms demonstrated that the method could predict a number of expectation values with an overall accuracy of 1-2% or better [11]. The presence of spin-orbit energy splitting, and the existence of a 3 D e state very close in energy to the 3 P o metastable level leads to a decrease in accuracy for atomic properties such as the tensor polarizability that are sensitive to these energy differences. Additional high precision measurements of the tensor polarizabilities for the Mg, Ca and Sr would be certainly be worthwhile since the sensitivity of this parameter to the fine details of the wave function should help in the refinement of the two-body potentials used to characterize ultra-cold collisions. It is interesting to speculate whether the better agreement with the CI+MBPT calculations could be achieved by incorporating a spin-orbit potential into the Hamiltonian and using jj coupling. Alternatively, a fully relativistic treatment, using a relativistic HF wave function might be necessary. Resolution of these questions requires that explicit calculations be made to determine the additional physics needed to eliminate the anomalies between the present calculations and experiment and between the present calculations and the CI+MBPT calculations.

14

IV.

ACKNOWLEDGMENTS

The authors would like to thank Mr J C Nou and Mr C Hoffman of CDU for workstation support.

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17

TABLE II: Properties of the metastable 3 P o levels of the alkaline-earth atoms and He (note, the lowest 3 P o level is not metastable in He). The oscillator strengths to the lowest 3 S e , 3 P e and 3 De

states are given as f (3 Le ). The scalar and tensor dipole polarizabilities are α0 and α2,L0 L0

respectively. The quadrupole moment Q is given for the 3 P2o state while the dispersion parameter C6 is that for two 3 P0o states. The He ”Other Theory” row reports the results of close to exact calculations with the exception of α2,L0 L0 . The present oscillator strength to the He 3 D e state is not to a physical state, rather it is to the lowest energy pseudo-state. All quantities are in atomic units and the numbers in brackets are the uncertainties in the last digits. Method

f (3 S e )

f (3 P e )

f (3 D e )

Q

α0

α2,L0 L0

C6

5102

He Present Other Theory

-0.1797

0.6251

10.264

46.66

69.62

-0.1797 [33]

0.6102 [33]

10.265 [34]

46.71 [35]

67.09 [36]

4.54

39.02

0.558

39.33

0.47

36.08

1.04

101.9

-14.24

Be Present

0.08187

0.4467

0.2948

MCHF [37] TDGI [38]

0.026

B-spline CI [39]

0.0823

0.453

0.295

0.089(3) [40]

0.44(2) [41]

0.29(1)[40]

0.1383

0.6167

0.6287

Experiment

0.154

220.3

Mg Present CI+MBPT [4]

1004

8.46(8)

SG-CI [22]

8.38

TDGI [42]

0.136

MS-CI [27]

0.1354

0.6383

0.6336

0.139(3) [26]

0.55(4) [43]

0.62(4) [44]

Experimental

8.44

0.625

980(30) 90.7

-19.64

295.3

-28.36

Ca Present

0.1582

0.5071

0.08136

CI+MBPT [4]

12.7

TDGI [45]

0.163

CI+model [28]

0.1526

Experimental

3363

12.9(4)

SG-CI [22]

MCHF [29]

12.96

0.051 0.5030

0.0873

0.161

0.525

0.0806

0.12(2) [46]

0.522(13) [47]

3020(200) 276

-50.0

494.8

-53.84

Sr Present

0.1788

0.4727

0.08254

CI+MBPT [4] SG-CI [22] MQDT [30] Experimental

15.51

6074

15.6(5) 15.4 0.173 0.188(10) [48]

0.0849 0.438(4) [31]

18

5260(500)

TABLE III: The intermediate dispersion coefficients, BJ1 ,J2 for two alkaline-earth-metal atoms P |BJi ,Jj | column sums the absolute value of all the entries in in the metastable 3 P2o state. The

each row (with off-diagonal elements added twice). The numbers in brackets after the data are the uncertainties ascribed to the CI+MBPT and SG-CI calculations. Method

B1,1

B2,1

B2,2

-6.901

11.29

-27.77

|BJi ,Jj |

B3,2

B3,3

-13.52

22.31

-27.21

220.3

Be Present

P

B3,1

Mg Present

-37.46

43.90

-52.73

-76.49

90.76

-157.2

1004

CI+MBPT [4]

-37.8(38)

43.9(44)

-52.1(52)

-76.7(77)

90.1(90)

-156.4(156)

1002

SG-CI [22]

-35.6(2)

42.5(2)

-51.9(2)

-73.4(7)

88.6(7)

-152(2)

976

-96.95

130.7

-176.8

-233.5

317.5

-604.8

3363

Ca Present CI+MBPT [4]

-91.7(92)

123(12)

-167(17)

-225(23)

306(31)

-600(60)

3250

SG-CI [22]

-81(3)

119(5)

-176(8)

-203(10)

302(20)

-553(70)

3087

Present

-165.9

213.9

-278.9

-416.9

556.0

-1231

6074

CI+MBPT [4]

-158(16)

203(20)

-264(26)

-415(42)

555(56)

-1290(130)

6090

SG-CI [22]

-139(7)

196(9)

-280(10)

-370(30)

546(50)

-1210(200)

5780

Sr

19