Long range interaction coefficients for ytterbium dimers
S.G. Porsev, M.S. Safronova, A. Derevianko, Charles W. Clark
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Long range interaction coefficients for ytterbium dimers
S. G. Porsev , , M. S. Safronova , , A. Derevianko , and Charles W. Clark Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA Petersburg Nuclear Physics Institute, Gatchina, Leningrad District, 188300, Russia Joint Quantum Institute, National Institute of Standards and Technology and theUniversity of Maryland, Gaithersburg, Maryland, 20899,USA Physics Department, University of Nevada, Reno, Nevada 89557, USA (Dated: October 29, 2018)We evaluate the electric-dipole and electric-quadrupole static and dynamic polarizabilities for the6 s S , 6 s p P o , and 6 s p P o states and estimate their uncertainties. A methodology is developedfor an accurate evaluation of the van der Waals coefficients of dimers involving excited state atomswith strong decay channel to the ground state. This method is used for evaluation of the longrange interaction coefficients of particular experimental interest, including the C coefficients forthe Yb–Yb S + P o , and P o + P o dimers and C coefficients for the S + S and S + P o dimers. PACS numbers: 34.20.Cf, 32.10.Dk, 31.15.ac
I. INTRODUCTION
The ytterbium atom has two fermionic and five bosonicisotopes, a S ground state, a long-lived metastable6 s p P o state, and transitions at convenient wavelengthsfor laser cooling and trapping. All this makes Yb a su-perb candidate for a variety of applications such as devel-opment of optical atomic clocks [1], study of degeneratequantum gases [2], quantum information processing [3],and studies of fundamental symmetries [4]. The bestlimit to date on the value of the electron electric-dipolemoment (EDM) which constrains extensions of the stan-dard model of electroweak interactions, was obtained us-ing the YbF molecule [5]. YbRb and YbCs moleculeshave also been proposed for searches for the electronEDM [6] since they can be cooled to very low tempera-tures and trapped in optical dipole traps, leading to verylong coherence times in comparison to molecular beamEDM experiments.Yb is of particular interest for studying quantum gasmixtures [2, 7–15]. Significant progress has been achievedin studying the properties of Yb-Yb photoassociationspectra at ultralow temperatures [7]. Photoassociationspectroscopy has been performed on bosons [2, 8] andfermions [9]. The use of optical Feshbach resonances forcontrol of entangling interactions between nuclear spinsof Yb atoms for quantum information processing ap-plications has been proposed in [16]. A p-wave opti-cal Feshbach resonance using purely long-range molec-ular states of a fermionic isotope of ytterbium
Yb wasdemonstrated in [11]. Recent work [17] theorizes that thecase of
Yb may have sufficiently small direct back-ground interaction between the atoms to support twobound states that represent attractively and repulsivelybound dimers occurring simultaneously.The excited molecular states asymptotically connectedto the S + P o separated Yb atom limit were inves-tigated by Takasu et. al. in [12]. They reported the successful production of a subradiant 1 g state of a two-atom Yb system in a three-dimensional optical lattice.The properties of the long-range potential were studiedand the van der Waals coefficients C , C , and C werepredicted. However, fit of the C and C coefficients forthe 1 g state was rather uncertain, with strong correlationbetween the C and C fit parameters [18].Knowledge of the C and C long-range interactioncoefficients in Yb-Yb dimers is critical to understandingthe physics of dilute gas mixtures. Recently, we evalu-ated the C coefficient for the Yb-Yb S + S dimer andfound it to be C = 1929(39) [19], in excellent agreementwith the experimental result C = 1932(35) [10]. How-ever, the same method cannot be directly applied to thecalculation of the van der Waals coefficients with Yb-Yb S + P o dimer owing to the presence of the P o → S decay channel.In this work, we develop the methodology for an accu-rate evaluation of the van der Waals coefficients of dimersinvolving excited state atoms with a strong decay channelto the ground state and evaluate C and C coefficientsof particular experimental interest. We carefully studythe uncertainties of all quantities calculated in this workso the present values can be reliably used to analyse ex-isting measurements and to facilitate planning of the fu-ture experimental studies. The methodology developedin this work can be used for evaluation of van der Waalscoefficients in a variety of systems. II. GENERAL FORMALISM
We investigate the molecular potentials asymptoticallyconnecting to the | A i + | B i atomic states. The wavefunction of such a system constructed from these statesis | M A , M B ; Ω i = | A i I | B i II , (1)where the index I(II) describes the wave function locatedon the center I(II) and Ω = M A + M B . Here, the M A ( B ) is the projection of the appropriate total atomic angularmomentum J A ( B ) on the internuclear axis. We assumethat Ω is a good quantum number for all calculations inthis work (Hund’s case (c)).The molecular wave functions can be obtained by di-agonalizing the molecular Hamiltonianˆ H = ˆ H A + ˆ H B + ˆ V ( R ) (2)in the model space. Here, ˆ H A and ˆ H B represent theHamiltonians of the two noninteracting atoms and ˆ V ( R )is the residual electrostatic potential defined as the fullCoulomb interaction energy in the dimer excluding inter-actions of the atomic electrons with their parent nuclei.Unless stated otherwise, throughout this paper we useatomic units (a.u.); the numerical values of the elemen-tary charge, | e | , the reduced Planck constant, ¯ h = h/ π ,and the electron mass, m e , are set equal to 1. The atomicunit for polarizability can be converted to SI units via α/h [Hz/(V/m) ]=2.48832 × − α (a.u.), where the con-version coefficient is 4 πǫ a /h , a is the Bohr radius and ǫ is the dielectric constant.The potential V ( R ) may be expressed as an expansionin the multipole interactions: V ( R ) = ∞ X l,L =0 V lL /R l + L +1 , where V lL are given by [20] V lL ( R ) = l s X µ = − l s ( − L ( l + L )! { ( l − µ )! ( l + µ )! ( L − µ )! ( L + µ )! } / × (cid:16) T ( l ) µ (cid:17) I (cid:16) T ( L ) − µ (cid:17) II . (3)Here, l s = min( l, L ) and the multipole spherical tensorsare T ( K ) µ = − X i r Ki C ( K ) µ (ˆ r i ) , (4)where the summation is over atomic electrons, r i is theposition vector of electron i , and C ( L ) µ (ˆ r i ) are the reducedspherical harmonics [21].We now restrict our consideration to the dipole-dipoleand dipole-quadrupole interactions. Introducing desig-nations d µ ≡ T (1) µ , Q µ ≡ T (2) µ , V dd ≡ V /R , and V dq ≡ V /R , we obtain from Eq. (3): V dd ( R ) = − R X µ = − w (1) µ ( d µ ) I ( d − µ ) II , (5) V dq ( R ) = 1 R X µ = − w (2) µ [( d µ ) I ( Q − µ ) II − ( Q µ ) I ( d − µ ) II ] , where the dipole and quadrupole weights are w (1) µ ≡ δ µ ,w (2) µ ≡ p (1 − µ )! (1 + µ )! (2 − µ )! (2 + µ )! . (6)Numerically, w (2) − = w (2)+1 = √ w (2)0 = 3.The energy E ≡ E A + E B , where E A and E B are theatomic energies of the | A i and | B i states, is obtainedfrom (cid:16) ˆ H A + ˆ H B (cid:17) | M A , M B ; Ω i = E | M A , M B ; Ω i . (7)The molecular wave function Ψ g/u Ω can be formed as a lin-ear combination of the wave functions given by Eq. (1).Ψ g/u Ω poses a definite gerade/ungerade symmetry and def-inite quantum number Ω. It can be represented byΨ p Ω = (cid:26) √ ( | A i I | B i II + ( − p | B i I | A i II ) , A = B | A i I | A i II , A = B, (8)where we set p = 0 for ungerade symmetry and p = 1 forgerade symmetry. We have taken into account that thestates A and B that are of interest to the present workare the opposite parity states of Yb atom (when A = B ).Applying the formalism of Rayleigh-Schr¨oedinger per-turbation theory in the second order [22] and keeping theterms up to 1 /R in the expansion of V ( R ) we obtain thedispersion potential in two-atom basis: U ( R ) ≡ h Ψ p Ω | V ( R ) | Ψ p Ω i≈ h Ψ p Ω | ˆ V dd | Ψ p Ω i + X Ψ i =Ψ p Ω " h Ψ p Ω | ˆ V dd | Ψ i ih Ψ i | ˆ V dd | Ψ p Ω iE − E i + h Ψ p Ω | ˆ V dq | Ψ i ih Ψ i | ˆ V dq | Ψ p Ω iE − E i , (9)The intermediate molecular state | Ψ i i with unperturbedenergy E i runs over a complete set of two-atom states,excluding the model-space states, Eq. (1).The dispersion potential can be approximated as U ( R ) ≈ − C R − C R − C R . (10) A. First-order corrections
The first-order correction, which is determined by thefirst term on the right-hand side of Eq. (9), is associ-ated with the C coefficient in Eq. (10). For the statesconsidered in this work, this coefficient is nonzero onlyfor the molecular potential asymptotically connecting tothe S + P o atomic states. It depends entirely on thereduced matrix element (ME) of the electric-dipole oper-ator |h P o || d || S i| and is given by a simple formula C (Ω p ) = ( − p +Ω (1 + δ Ω , ) |h P o || d || S i| . (11)Specifically, C (0 g/u ) = ∓ |h P o || d || S i| ,C (1 g/u ) = ± |h P o || d || S i| , (12)where the upper/lower sign corresponds to ger-ade/ungerade symmetry. B. Second-order corrections
The second-order corrections, associated with the C and C coefficients, are given by the second and thirdterms on the r.h.s. of Eq. (9), − C (Ω p ) R = X Ψ i =Ψ Ω p h Ψ Ω p | ˆ V dd | Ψ i ih Ψ i | ˆ V dd | Ψ Ω p iE − E i − C (Ω p ) R = X Ψ i =Ψ Ω p h Ψ Ω p | ˆ V dq | Ψ i ih Ψ i | ˆ V dq | Ψ Ω p iE − E i , where E = E A + E B and the complete set of doubledatomic states satisfies the condition X Ψ i | Ψ i ih Ψ i | = 1 . After angular reduction, the C coefficient can be ex-pressed as C (Ω) = J A +1 X J α = | J A − | J B +1 X J β = | J B − | A J α J β (Ω) X J α J β , (13)where A J α J β (Ω) = X µM α M β (cid:20) w (1) µ (cid:18) J A J α − M A µ M α (cid:19) (cid:18) J B J β − M B − µ M β (cid:19)(cid:21) ,X J α J β = X α,β = A,B |h A || d || α i| |h B || d || β i| E α − E A + E β − E B (14)with fixed J α and J β .If A and B are the spherically symmetric atomic statesand there are no downward transitions from either ofthem, the C and C coefficients for the A + B dimersare given by well known formulas (see, e.g., [23]) C AB = C AB (1 , ,C AB = C AB (1 ,
2) + C AB (2 , , (15)where the coefficients C AB ( l, L ) ( l, L = 1 ,
2) are quadra-tures of electric-dipole, α ( iω ), and electric-quadrupole, α ( iω ), dynamic polarizabilities at an imaginary fre-quency: C AB (1 ,
1) = 3 π Z ∞ α A ( iω ) α B ( iω ) dω,C AB (1 ,
2) = 152 π Z ∞ α A ( iω ) α B ( iω ) dωC AB (2 ,
1) = 152 π Z ∞ α A ( iω ) α B ( iω ) dω. (16)For the Yb–Yb S + P o dimer considered in this work,the expressions for C and C are more complicated dueto the angular dependence, the P o → S decay channeland non-vanishing quadrupole moment of the P o state.After some transformations, we arrive at the followingexpression for the C coefficient in the S + P o case: C (Ω p ) = X J =0 A J (Ω) X J , (17)where the angular dependence A J (Ω) is represented by A J (Ω) = 13 X µ = − (cid:26) w (1) µ (cid:18) J − Ω − µ Ω + µ (cid:19)(cid:27) (18)with the dipole weights w (1) µ given by Eq. (6) and Ω =0 ,
1. It is worth noting that A J (Ω) (and, consequently,the C coefficients) do not depend on gerade/ungeradesymmetry.The quantities X J for the S + P o dimer are givenby X J = 272 π Z ∞ α A ( iω ) α B J ( iω ) dω + δX δ J, . (19)where A ≡ S and B ≡ P o and δX is defined below.The possible values of the total angular momentum J are 0, 1, and 2; α A ( iω ) is the electric-dipole dynamicpolarizability of the S state at the imaginary argument.The quantity α Φ KJ ( iω ) is a part of the scalar electric-dipole ( K = 1) or electric-quadrupole ( K = 2) dynamicpolarizability of the state Φ, in which the sum over theintermediate states | n i is restricted to the states withfixed total angular momentum J n = J : α Φ KJ ( iω ) ≡ K + 1)(2 J Φ + 1) × X γ n ( E n − E Φ ) |h γ n , J n = J || T ( K ) || γ Φ , J Φ i| ( E n − E Φ ) + ω . (20)Here, γ n stands for all quantum numbers of the interme-diate states except J n .The correction δX to the X term in Eq.(19) is dueto a downward P o → S transition and is given by thefollowing expression: δX = 2 |h P o || d || S i| X n = P o ( E n − E S ) |h n || d || S i| ( E n − E S ) − ω + |h P o || d || S i| ω , (21)where ω ≡ E P o − E S . The expression for the C ( S + P o ) coefficient is sub-stantially more complicated, so it is discussed in the Ap-pendix. III. METHOD OF CALCULATION
All calculations were carried out by two methods whichallows us to estimate the accuracy of the final val-ues. The first method combines configuration interaction(CI) with many-body perturbation theory (MBPT) [24].In the second method, which is more accurate, CI iscombined with the coupled-cluster all-order approach(CI+all-order) that treats both core and valence correla-tion to all orders [25–27].In both cases, we start from a solution of the Dirac-Fock (DF) equations for the appropriate states of theindividual atoms, ˆ H ψ c = ε c ψ c , where H is the relativistic DF Hamiltonian [24, 26] and ψ c and ε c are single-electron wave functions and energies.The calculation was performed in the V N − approxima-tion, i.e, the self-consistent procedure was done for the[1 s , ..., f ] closed core. The B-spline basis set, consist-ing of N = 35 orbitals for each of partial wave with l ≤ − s , 6 − p , 5 − d , 5 − f , and 5 − g .The wave functions and the low-lying energy levels aredetermined by solving the multiparticle relativistic equa-tion for two valence electrons [28], H eff ( E n )Φ n = E n Φ n . (22)The effective Hamiltonian is defined as H eff ( E ) = H FC + Σ( E ) , (23)where H FC is the Hamiltonian in the frozen-core ap-proximation. The energy-dependent operator Σ( E )which takes into account virtual core excitations is con-structed using the second-order perturbation theory inthe CI+MBPT method [24] and using linearized coupled-cluster single-double method in the CI+all-order ap-proach [26]. Σ( E ) = 0 in the pure CI approach. Con-struction of the effective Hamiltonian in the CI+MBPTand CI+all-order approximations is described in detailin Refs. [24, 26]. The contribution of the Breit interac-tion is negligible at the present level of accuracy and wasomitted.The dynamic polarizability of the 2 K -pole operator T ( K ) at imaginary argument is calculated as the sumof three contributions: valence, ionic core, and vc . The vc term subtracts out the ionic core terms which are for-bidden by the Pauli principle. Then α K ( iω ) = α vK ( iω ) + α cK ( iω ) , (24)where both the core and vc parts are included in α cK ( iω ). A. Valence contribution
The valence part of the dynamic polarizability, α vK ( iω ),of an atomic state | Φ i is determined by solving the inho-mogeneous equation in the valence space. If we introducethe wave function of intermediate states | δ Φ i as | δ Φ i ≡ Re ( H eff − E Φ + iω X i | Φ i ih Φ i | T ( K )0 | Φ i ) = Re (cid:26) H eff − E Φ + iω T ( K )0 | Φ i (cid:27) , (25)where “Re” means the real part, then α v ( iω ) is given by α v ( iω ) = 2 h Φ | T ( K )0 | δ Φ i . (26)Here, T ( K )0 is the zeroth component of the T ( K ) tensor.We include random-phase approximation (RPA) correc-tions to the 2 K -pole operator T ( K )0 . The Eqs. (25) and(26) can also be used to find α vKJ , i.e, the part of the va-lence polarizability, where summation goes over only theintermediate states with fixed total angular momentum J . We refer the reader to Ref. [29] for further details ofthis approach. B. Core contribution
The core and vc contributions to multipole polariz-abilities are evaluated in the single-electron relativis-tic RPA approximation. The small α vc term is cal-culated by adding vc contributions from the individualelectrons, i.e., α vc (6 s ) = 2 α vc (6 s ) and α vc (6 s p ) = α vc (6 s ) + α vc (6 p ).A special consideration is required when we need tofind the core contribution to α Φ KJ ( iω ) of a state Φ. Ifwe disregard possible excitations of the core electrons tothe occupied valence shells, the valence and core subsys-tems can be considered as independent. Then, the totalangular momenta J Φ and J n of the states Φ and Φ n , re-spectively, can be represented as the sum of the valenceand core parts J = J v + J c . In our consideration, the coreof the Φ state consists of the closed shells, and J c Φ = 0. Ifwe assume that the electrons are excited from the core,while the valence part of the wave function remains thesame, we can express the reduced matrix element of theoperator T ( K ) as h J Φ || T ( K ) || J n i = h J c Φ = 0 , J v Φ , J Φ || T ( K ) || J cn = K, J v Φ , J n i . (27)If T ( K ) acts only on the core part of the system, we arriveat (see, e.g., [21]) h J c Φ = 0 , J v Φ , J Φ || T ( K ) || J cn = K, J v Φ , J n i = r J n + 12 K + 1 h J c Φ = 0 || T ( K ) || J cn = K i . (28) TABLE I: Theoretical and experimental [30] energy levels (in cm − ). Two-electron binding energies are given in the first row,energies in other rows are counted from the ground state. Results of the CI, CI+MBPT, and CI+all-order calculations aregiven in columns labeled “CI”, “CI+MBPT”, and “CI+All”. Corresponding relative differences of these three calculations withthe experiment are given in cm − and in percentages.State Exper. CI CI+MBPT CI+All Differences (cm − ) Differences (%)CI CI+MBPT CI+all CI CI+MBPT CI+All6 s S − − d s D d s D d s D d s D − − s s S − − s s S − − s p P o − −
19 5.6 2.76 s p P o − −
18 5.3 2.56 s p P o − −
18 5.0 2.76 s p P o − − s p P o − − s p P o − − s p P o − − s p P o − − − − − − Then, using Eq. (20), we can write the core contributionto α KJ ( iω ) of the Φ state as α cKJ ( iω ) = 2 (2 J + 1)(2 K + 1) (2 J Φ + 1) × X γ cn ( E n − E Φ ) |h J c Φ = 0 || T ( K ) || J cn = K i| ( E n − E Φ ) + ω . (29)Taking into account that the core polarizability α cK ( iω )of the operator T ( K ) in a single-electron approximationcan be written as α cK ( iω ) = 22 K + 1 × X a,n ε n − ε a ( ε n − ε a ) + ω |h n || T ( K ) || a i| , (30)where | a i and | n i are the single-electron core and virtualstates, we arrive at α cKJ ( iω ) = 2 J + 1(2 K + 1)(2 J Φ + 1) α cK ( iω ) . (31)Finally, α KJ ( iω ) of the Φ state can be approximated as α KJ ( iω ) = α vKJ ( iω ) + 2 J + 1(2 K + 1)(2 J Φ + 1) α cK ( iω ) , (32)where possible values of J are from min(0 , | J Φ − K | ) to J Φ + K . IV. RESULTS AND DISCUSSIONA. Energy levels
We start from the calculation of the low-lying en-ergy levels of atomic Yb. The calculations were car-ried out using CI, CI+MBPT, and CI+all-order meth-ods. The results are listed in Table I (see also the Sup-plemental Material to Ref. [19]) in columns labeled “CI”,“CI+MBPT”, and “CI+All”. Two-electron binding en-ergies are given in the first row, energies in other rows arecounted from the ground state. Corresponding relativedifferences of these three calculations with experimentare given in cm − and in percentages. The even- andodd-parity levels are schematically presented in Fig. 1.Table I illustrates that the difference between the the-ory and the experiment are as large as 19% for the odd-parity states at the CI stage. When we include the core-core and core-valence correlations in the second order ofthe perturbation theory (CI+MBPT method), the ac-curacy significantly improves. Further improvement isachieved when we use the CI+all-order method includ-ing correlations in all orders of the MBPT. B. Polarizabilities
In Table II we give a breakdown of the main con-tributions from the intermediate states to the staticelectric-dipole and electric-quadrupole polarizabilities ofthe 6 s S , 6 s p P o , and 6 s p P o states in the CI+all-order approximation. For the P o state the contributionsto the scalar parts of the polarizabilities are presented.While we do not explicitly use the sum-over-states to TABLE II: A breakdown of the contributions to the 6 s S ,6 s p P o , and 6 s p P o electric-dipole, α , and electric-quadrupole, α , static polarizabilities in the CI+all-order ap-proximation. For the P o state, the scalar polarizabilities aregiven. The row labeled “Other” gives the contribution of allother valence states not explicitly listed in the table. The rowlabeled “Core+vc” gives the contributions from the core and vc terms. The row labeled “Total” lists the final values ob-tained as the sum of all contributions. |h n || T ( K ) || m i| are thereduced matrix elements; T (1) = d and T (2) = Q stand forthe electric-dipole and electric-quadrupole operators, respec-tively. The theoretical and experimental transition energiesare presented in columns ∆ E th and ∆ E exp (in cm − ). Thecontributions to the polarizabilities are given in the columnlabeled “ α ”.Polarizability Contrib. |h n || T ( K ) || m i| ∆ E th ∆ E exp αα ( P o ) 5 d s D s s S s d D α s ( P o ) 6 s S d s D d s D d s D s s S s s S s d D s d D s d D α ( S ) 5 d s D d s D s d D s d D α ( P o ) 6 s p P o s p P o α s ( P o ) 6 s p P o s p P o s p P o s p P o s p P o (cid:1)(cid:0)(cid:2) (cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17) (cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24) (cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30) (cid:31) ! " FIG. 1: (Color online) Low-lying energy levels of Yb. Otherstates of the 4 f d s configuration are not shown. calculate the polarizabilities, we can separately computecontributions of individual intermediate states. The rowlabeled “Other” lumps contributions of all other valencestates not explicitly listed in the table. The row labeled“Core+vc” gives the contributions from the core and vc terms and the row labeled “Total” is the final value ob-tained as the sum of all contributions. The theoreti-cal and experimental transition energies are presented incolumns ∆ E th and ∆ E exp (in cm − ). We used the theo-retical energies when calculating the contributions of theindividual terms to the polarizabilities. These contribu-tions as well the total values of the polarizabilities aregiven in the column labeled “ α ”.The role of different contributions to the 6 s p P o po-larizability was analyzed in Ref. [19] (see the Supplemen-tal Material). We compare the P o case with the contri-butions to the scalar part of the P o polarizability givenin Table II. We find that the main contributions to the6 s p P o and 6 s p P o polarizabilities are similar in everyrespect. In particular, the 5 d s D J states contribute ∼
57% to both polarizabilities. The contributions of the6 s d D J states are at the level of 7-10%. The higher-excited states not explicitly listed in the table, labeled as“Other”, contribute ∼
21% in both cases.To the best of our knowledge, there are no experimen-tal data for the electric-quadrupole polarizabilities listedin the table or any transitions that give dominant con-tributions to α . For instance, the main contribution(76%) to α ( S ) comes from the 5 d s D state. Anyaccurate experimental data for the 5 d s D state (life-time, oscillator strengths, etc) would provide an impor-tant benchmark relevant to the ground state quadrupolepolarizability.We also give the breakdown of the 6 s p P o and thescalar part of 6 s p P o electric-quadrupole polarizabil-ities. The main contribution (80%) comes from the6 s p P o state in both cases. We note that the remainder TABLE III: The values of the D ≡ |h s p P o || d || s S i| matrix element (in a.u.) and C coefficients in the CI+MBPTand CI+all-order approximations.CI+MBPT CI+all-order Experiment D .
581 0 .
572 0 . a . b C (0 u ) 0 .
225 0 .
218 0 . b C (0 g ) − . − . C (1 u ) − . − . C (1 g ) 0 .
113 0 .
109 0 . ca Reference [31]. The experimental number was obtainedfrom the weighted P o lifetime τ ( P o ) = 845(12) ns; b Reference [7] (this error is pure statistical); c Reference [12]. contribution (listed in rows “Other”) is significant for allpolarizabilities considered here. These contributions areat the level of 15–18%. The uncertainties of the polariz-ability values are discussed later in Section V. C. C coefficients The values of the C coefficients obtained in theCI+MBPT and CI+all-order approximations for the S + P o dimer are given in Table III (also seethe Supplemental Material [32]). We calculated the |h s p P o || d || s S i| matrix element (ME) and thenfound C coefficients using Eq. (12). The C (0 g ) and C (1 u ) have the same numerical values as C (0 u ) and C (1 g ), but the opposite sign. Our CI+all-order value forthis ME differs from the experimental results by 4-5%. Itis not unexpected, because the S − P o transition is anintercombination transition and due to cancelation of dif-ferent contributions its amplitude is relatively small. Itmay be also affected by the mixing with the core-excitedstates that are outside of our CI space as is discussed indetail in [19]. As a result, the accuracy of calculation ofsuch MEs is lower. Using Eq. (12) we can estimate theaccuracy of C coefficients at the level of 8-10%. D. C and C coefficients To find the van der Waals coefficients for the S + P o and P o + P o dimers we computed the dynamicelectric-dipole and electric-quadrupole polarizabilities ofthe S and P o states at imaginary frequency and thenused Eqs. (15) and (16). In practice, we computed the C AB coefficients by approximating the integral (17) byGaussian quadrature of the integrand computed on thefinite grid of discrete imaginary frequencies [33]. The C coefficient for the S + S dimer was obtained inRef. [19].The calculation of the C and C coefficients for the S + P o dimer was carried out according to the expres- TABLE IV: A breakdown of the contributions to the C (Ω)coefficient for Yb-Yb ( S + P o ) dimer. The expressions for X J and A J are given by Eqs. (18,19). The δX term is givenseparately in the second row; it is included in J = 0 contri-bution. The CI+MBPT and CI+all-order values for X J aregiven in columns labeled “MBPT” and “All”. J X J A J C (Ω)MBPT All HO Ω = 0 Ω = 1 Ω = 0 Ω = 10 1107 1135 2.5% 4/9 1/9 504 126 δ
248 253 2.0% 4/9 1/9 112 281 4564 4480 -1.9% 1/9 5/18 498 12442 6752 6702 -0.7% 11/45 19/90 1638 1415Sum 2753 2814TABLE V: A breakdown of the contributions to the C (Ω)coefficient for Yb–Yb ( S + P o ) dimer. The expressions for X J α J β k and A J α J β k are given in the Appendix A. The δX term (designated as δ ) is given separately in the first row;it is included in the X contribution. The δX term (desig-nated as δ ) is given separately in the fifth row; it is includedin the X contribution. The CI+all-order values are givenfor X J α J β k and C ; the relative differences of the CI+all-orderand CI+MBPT values are given in columns labeled “HO” in%. The + / − sign corresponds to the ungerade/gerade sym-metry, respectively. J α J β , k X J α J β k HO A J α J β k C (Ω)Ω = 0 Ω = 1 Ω = 0 Ω = 1 δ δ ± ± ± ± ± ± ± ± C (Ω u ) 320130 411067 C (Ω g ) 318461 410511 sions given by Eqs. (17)-(19) and in the Appendix A.A breakdown of the contributions to the C (Ω) coeffi-cient for Yb–Yb ( S + P o ) dimer is given in Table IV.We list the quantities X J and coefficients A J given byEqs. (18) and (19) for allowed J = 0 , ,
2. The δX termis presented separately in the second row to illustratethe magnitude of this contribution. It is relatively small,4% of the total for Ω = 0 and 1% for Ω = 1. It is in-cluded in the X value given in the table. We note thatthe C ( S + P o ) coefficient do not depend on u/g sym-metry. The CI+MBPT and CI+all-order values for X J are given in columns labeled “MBPT” and “All”. Therelative differences between these values, which give anestimate of the higher-order contributions, are listed inthe column labeled “HO”. We find that the higher orderscontribute with a different sign to J = 0 and J = 1 , TABLE VI: The 6 s S , 6 s p P o , and 6 s p P o electric-dipole, α , and electric-quadrupole, α , static polarizabilities in theCI+MBPT and CI+all-order approximations (in a.u.). For the P o state the scalar parts of the polarizabilities are presented.The values of C (Ω u/g ) and C (Ω u/g ) coefficients for the A + B dimers in the CI+MBPT and CI+all-order approximations arelisted in the second part of the table. The (rounded) CI+all-order values are taken as final.Level Property CI+MBPT CI+all HO Final Other6 s S α a1 b c d s p P o α a1 b s p P o α s s S α s p P o α s p P o α s S + S C a6 e C × × S + P o C b3 P o + P o C b1 S + P o C (0 u/g ) 2649 2640 -0.3% 2640(103) 2410(220) f C (1 u/g ) 2824 2785 -1.4% 2785(109) 2283.6 g C (0 u ) 321097 320130 -0.3% 3.20(14) × C (1 u ) 412779 411067 -0.4% 4.11(18) × C (0 g ) 319300 318461 -0.3% 3.18(14) × C (1 g ) 412180 410511 -0.4% 4.11(18) × Safronova et al. [19], theory. b Dzuba and Derevianko [34], theory. c Zhang and Dalgarno [35], based on experiment. d Sahoo and Das [36], theory. e Kitagawa et al. [10], experiment. f Borkowski et al. [7], experiment; the error includes only uncertainty of the fit. g Takasu et al. [12], experiment.
A breakdown of the contributions to the C (Ω) coeffi-cients for Yb–Yb S + P o dimer is given in Table V. Welist the quantities X J α J β k and coefficients A J α J β k (the ana-lytical expressions for them are given in the Appendix A).The δX and δX terms are given separately in the firstand fifth rows; they are included in the X and X con-tributions, respectively. For calculation of δX we usedthe values |h P o || Q || P o i| = 17 .
75 a.u. and the static S polarizability α A (0) = 140 . A and A con-tain ( − p , therefore their sign is different for gerade andungerade symmetry resulting in slightly different valuesfor C (Ω u ) and C (Ω g ). In Table V, the + / − sign cor-responds to the ungerade/gerade symmetry, respectively.The CI+all-order values are given for X J α J β k and C ; therelative differences of the CI+all-order and CI+MBPTvalues are given in column labeled “HO” in %.Our final results for polarizabilities and the van derWaals C and C coefficients are summarized in Table VI.The 6 s S , 6 s p P o , and 6 s p P o electric-dipole, α , and electric-quadrupole, α , static polarizabilitiesin the CI+MBPT and CI+all-order approximations arelisted in a.u.. For the P o state the scalar parts of the polarizabilities are presented. The values of C (Ω u/g )and C (Ω u/g ) coefficients for the A + B dimers in theCI+MBPT and CI+all-order approximations are listedin the second part of the table. The (rounded) CI+all-order values are taken as final. The relative contributionof the higher-order corrections is estimated as the dif-ference of the CI+all-order and CI+MBPT results, it islisted in column labeled “HO” in percent. V. DETERMINATION OF UNCERTAINTIES
We compare frequency-dependent polarizabilities cal-culated in the CI+MBPT and CI+all-order approxima-tions for all ω used in our finite grid to estimate theuncertainties of the C and C coefficients. We find thatthe difference between the CI+all-order and CI+MBPTfrequency-dependent polarizability values is largest for ω = 0 and decreases significantly with increasing ω . Thisis reasonable because for large ω the main contribution tothe polarizability comes from its core part. But the coreparts are the same for both CI+all-order and CI+MBPTapproaches.Therefore, the fractional uncertainty δC AB ( l, L )( l, L = 1 ,
2) may be expressed via fractional uncertain-ties in the static multipole polarizabilities of the atoms A and B [37], δC AB ( l, L ) = q(cid:0) δα Al (0) (cid:1) + (cid:0) δα BL (0) (cid:1) . (33)The absolute uncertainties induced in C AB and C AB ( A = B ) are given by∆ C AB = ∆ C AB (1 , , ∆ C AB = q (∆ C AB (1 , + (∆ C AB (2 , . (34)The polarizabilities and their absolute uncertaintiesare presented in Table VI. The uncertainties of theelectric-dipole S and P o polarizabilities were discussedin detail in Ref. [19]; the uncertainty of the P o polar-izability was determined to be 3.4%. Table I illustratesthat the accuracy of calculation of the P o and P o energylevels is practically the same ( ∼ .
5% at the CI+all-orderstage). We use the same method of solving the inho-mogeneous equation to determine both the P o and P o polarizabilities. The main contributions to these polariz-abilities are also very similar. Based on these arguments,we assume that the uncertainty of the scalar part of the P o polarizability can be estimated at the level of 3.5%.Our estimates of the uncertainties of the electric-quadrupole polarizabilities are based on the differencesbetween the CI+MBPT and CI+all-order values. Be-sides that we take into account that in all cases the dom-inant contribution comes from the low-lying state whichenergies we reproduce well (see Table I). Based on thesize of the higher-order correction, we assign the uncer-tainties 3-4% to these polarizabilities. These results, aswell as the final (recommended) values of the polarizabil-ities, are presented in Table VI (see also Ref. [32]).Using Eqs. (33) and (34) we estimated the fractionaluncertainties of the C coefficient for the S + P o , dimers at the level of 4–4.5% . The uncertainty of the C ( S + S ) coefficient is 3.2% and the uncertainties ofthe C ( S + P o ) coefficients are ∼ C ( P o + P o ) coefficient. VI. CONCLUSION
To conclude, we evaluated the electric-dipole andelectric-quadrupole static and dynamic polarizabilitiesfor the 6 s S , 6 s p P o , and 6 s p P o states and esti-mated their uncertainties. The C and C coefficientsare evaluated for the Yb-Yb dimers. The uncertaintiesof our calculations of the van der Waals coefficients donot exceed 5%. Our result C = 1 . × for the S + S dimer is in excellent agreement with the ex-perimental value C = 1 . × [10]. The quantitiescalculated in this work allow future benchmark tests ofmolecular theory and experiment. Most of these quanti-ties are determined for the first time. Methodology de-veloped in this work can be used to evaluate propertiesof other dimers with excited atoms that have a strongdecay channel. Acknowledgement
We thank P. Julienne for helpful discussions. This re-search was performed under the sponsorship of the U.S.Department of Commerce, National Institute of Stan-dards and Technology, and was supported by the Na-tional Science Foundation under Physics Frontiers Cen-ter Grant No. PHY-0822671 and by the Office of NavalResearch. The work of S.G.P. was supported in part byUS NSF Grant No. PHY-1212442 and RFBR Grant No.11-02-00943. The work of A.D. was supported in part bythe US NSF Grant No. PHY-1212482.
Appendix A: C coefficients for the S + P o dimer Following formalism of Section II, the C coefficient may be expressed as: C (Ω p ) R = X A,B = α,β h AB | ˆ V dq | αβ ih αβ | ˆ V dq | AB i + ( − p h AB | ˆ V dq | αβ ih αβ | ˆ V dq | BA i E α + E β − E , which can be further reduced to: C (Ω p ) = X k =1 X J α J β A J α J β k (Ω p ) X J α J β k , A J α J β (Ω) = X µM α M β (cid:26) w (2) µ (cid:18) J A J α − M A µ M α (cid:19) (cid:18) J B J β − M B − µ M β (cid:19)(cid:27) ,X J α J β = X αβ |h A || d || α i| |h B || Q || β i| E α − E A + E β − E B ; A J α J β (Ω) = X µM α M β (cid:26) w (2) µ (cid:18) J A J α − M A µ M α (cid:19) (cid:18) J B J β − M B − µ M β (cid:19)(cid:27) ,X J α J β = X αβ |h A || Q || α i| |h B || d || β i| E α − E A + E β − E B ; A J α J β (Ω p ) = ( − p X µλM α M β ( − J A − J α + J B − J β +1 w (2) µ w (2) λ × (cid:18) J A J α − M A µ M α (cid:19) (cid:18) J A J β − M A λ M β (cid:19) (cid:18) J B J β − M B − µ M β (cid:19) (cid:18) J B J α − M B − λ M α (cid:19) ,X J α J β = X αβ h A || d || α ih α || Q || B ih B || Q || β ih β || d || A i E α − E A + E β − E B ; A J α J β (Ω p ) = ( − p X µλM α M β ( − J A − J α + J B − J β +1 w (2) µ w (2) λ × (cid:18) J A J α − M A µ M α (cid:19) (cid:18) J A J β − M A λ M β (cid:19) (cid:18) J B J β − M B − µ M β (cid:19) (cid:18) J B J α − M B − λ M α (cid:19) ,X J α J β = X αβ h A || Q || α ih α || d || B ih B || d || β ih β || Q || A i E α − E A + E β − E B . The total angular momenta J α and J β of the intermediatestates α and β are fixed in all of the equations above.We are interested in the case when A ≡ S and B ≡ P o . Then, J A = 0, J B = 1, and Ω = M B = 0 , k = 1, we have J α = 1 and J β = 1 , ,
3. Thecoefficients A J β (Ω) are listed in Table V. The quantities X J β are given by X J β = 452 π Z ∞ α A ( iω ) α B J β ( iω ) dω + δX δ J β , δX = 32 |h P o || Q || P o i| α A (0) . (1)For k = 2, we have J α = 2 and J β = 0 , ,
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