Dynamic dipole polarizabilities for the ground 41S

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 2313–2323

www.iop.org/Journals/jb

PII: S0953-4075(01)19638-0

Dynamic dipole polarizabilities for the ground 4 1 S and the low-lying 4 1,3 P and 5 1,3 S excited states of Zn. Calculation of long-range coefficients of Zn2 K Ellingsen1 , M Mérawa2 , M Rérat2 , C Pouchan2 and O Gropen1 1 2

Department of Chemistry, Faculty of Science, University of Tromsø, N-9037 Tromsø, Norway Laboratoire de Chimie Structurale, UMR 5624, IFR rue Jules Ferry, F-64000 Pau, France

Received 30 November 2000, in final form 24 April 2001 Abstract Dynamic dipole polarizabilities for the ground 4 1 S and the low-lying 4 1,3 P and 5 1,3 S excited states of Zn are calculated by the time-dependent gaugeinvariant method and compared with other experimental and theoretical results. The wavefunctions are obtained from multi-reference configuration-interaction calculations using a two-electron relativistic pseudopotential. Core–valence polarization is accounted for by the use of a semi-empirical core–valence potential. Core-polarization effects are also considered when calculating the oscillator strengths using a modified dipole transition operator. Long-range coefficients for the molecular states of Zn2 dissociating into: 4 1 S + 4 1 S; 4 1 S + 4 3 P; 4 1 S + 4 1 P; 4 1 S + 5 1 S and 4 1 S + 5 3 S are presented.

1. Introduction There is great interest in the determination of the electrical properties of atoms in their ground and excited states because these properties are involved in many physical and chemical processes [1] and intermolecular interactions [2]. Accurate experimental data are, however, rather scarce, and often theoretical polarizabilities are considered to provide the sole available or the most reliable information about the system, particularly for a description of the excited states [3, 4]. Knowledge of potential energy curves and electronic transition dipole moments of diatomic systems near the dissociation limit plays, in particular, an important role in laser cooling and trapping technology and has greatly renewed the interest in theoretical predictions of long-range interactions between atoms. These may be strongly affected or even dominated by the induction interaction which depends directly on the atomic dynamic polarizabilities at imaginary frequencies [5–9]. There has been renewed experimental and theoretical interest in dipole polarizabilities of the group IIB metals Zn, Cd and Hg [10–21]. Possessing only two valence electrons but a large and polarizable core, these systems cannot be accurately described at the selfconsistent field (SCF) level of approximation. Complete and precise all-electron correlated calculations are difficult owing to the complete set of electrons in the (n − 1)d shell. Reliable and accurate information on the properties of such systems has been obtained using relativistic 0953-4075/01/122313+11$30.00

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K Ellingsen et al

electron pseudopotentials, while core–valence interactions have effective semi-empirical core polarization potentials. Zn2 , Cd2 and Hg2 have also attracted considerable interest as possible laser systems in analogy with lasers based on the rare-gas dimers. The group IIB dimers have repulsive (except for weak van der Waals forces) ground state potentials and a variety of bound excited curves. The possibility of bound–free transitions in these systems is therefore of great interest in gas laser development. The purpose of this paper is to present theoretical calculations of the dynamic (at real and imaginary frequencies) dipole polarizabilities for the 4 1 S ground state and for the four lowlying 4 1,3 P and 5 1,3 S excited states of Zn using our gauge-invariant (TDGI) method [22–24]. With regard to the dissociative channels, we also report some interaction coefficients for all the molecular states of Zn2 dissociating into 4 1 S + 4 1 S, 4 1 S + 4 3 P, 4 1 S + 4 1 P, 4 1 S + 5 1 S and 4 1 S + 5 3 S, respectively. Some methodological and computational details are given in section 2. Results on the atomic Zn spectrum, polarizabilities and long-range coefficients are presented and discussed in section 3. Atomic units are used throughout the paper. 2. Methodological and computational details Static and dynamic polarizabilities of the Zn atom in the 4 1 S (1s2 2s2 2p6 3s2 3p6 3d10 4s2 ) ground state and the low-lying excited states 4 1,3 P (1s2 2s2 2p6 3s2 3p6 3d10 4s1 4p1 ) and 5 1,3 S (1s2 2s2 2p6 3s2 3p6 3d10 4s1 5s1 ) were calculated by the TDGI method, a variation-perturbation approach where the polarizability is defined as the second-order perturbation energy, −α(ω) = 0|(H1 − E1 |1± (1) ±

using the following first-order perturbation function: N M ± |1± = a ± g(r )| 0 + bn± | n + cm |φm . n=0

(2)

m=0

In this expression (2), g(r ) is a first-degree polynomial of the electronic coordinates which ensures gauge invariance. n are multiconfigurational wavefunctions of the first true spectral states and φm a quasi-spectral series obtained from monoexcitations of the determinantal subspace characterizing the ground state. The expansion coefficients, a, b, c, are determined variationally by projecting the frequency-dependent first-order perturbation equation on equation (2). The polarizability results are dependent upon the quality of the wavefunctions of the first true spectral states, e.g. accurate transition energies and oscillator strengths. The zerothorder wavefunctions were obtained from two-electron relativistic pseudopotential (RPP) multireference configuration-interaction (MRCI) calculations, using the CIPSI algorithm [25, 26]. Concerning the first-order function, the gauge factor has not been used since it gives rise to a quadrupole moment of the ground state, which has not been calculated with the core–valence polarization correction. However, we will see later that this latter correction has been taken into account in the determination of oscillator strengths. A description of the extraction of the RPP as well as the parameters are given in [27]. An effective semi-empirical core polarization potential, formulated by Foucrault et al [28], was used to improve the description of the large and polarizable core and to include core–valence correlation: VCPP = − 21 αc fc (i) · fc (i) (3) c

Dynamic dipole polarizabilities for the ground 4 1 S and the low-lying 4 1,3 P and 5 1,3 S excited states of Zn

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where αc is the polarizability of the core and fc (i) is the electric field which acts on the core due to the charges of the valence electrons (i). The experimental core polarizability, α = 0.34 Å3 [29], and cut-off radii ls = 1.5582, lp = 1.7400, ld = 1.4000 and lf = 1.4000 au were used. The calculated atomic spectrum was sensitive to small changes in s and p cutoff radii, so the cut-off radii from Jamorski et al [27] were reoptimized by fitting on the experimental excitation energies from the ground state to 5 1 S and 4 1 P [30], respectively. The form of the field fc (i) is more explicitly discussed in [27]. A (7s7p5d2f) primitive, Gaussian basis set contracted to [5s5p5d2f] was utilized. The basis set was taken from [27] and slightly modified to improve the correspondence with the experimental Zn spectrum. Core-polarization effects were also considered when calculating the oscillator strengths. We used the modified transition operator proposed by Hameed et al [31] and given in [32] µi = −ri + αc fc (i) (4) c

giving an effective valence electron expression for the oscillator strengths: f0k = φ0 |

(val) i

µi |φk × 2E0k

(5)

where φ0 and φk are the valence electron wavefunctions of the considered states. The experimental core polarizability, αc , was added to the calculated static polarizabilities. At frequencies ω 0 αc is considered to depend on the transition from the highest occupied core orbital (3d) with energy c to a virtual zero-energy state and can be calculated using the following formula: αc (ω) = αc /(1 − ω2 /c2 ). The oscillator strengths and corresponding transition energies were also involved in the calculation of the long-range coefficients C6 of Zn2 through the quantities R, which describe the interaction between two atoms at long distances. Between two atoms a and b, RL1111 [33] a Lb is given as

RL1111 = −(−1)La +La +Lb +Lb a Lb

(3gLa fγ0 La γr L a /2)(3gLb fγ0 Lb γs L b /2) a b a b r,s (Eγ0 La γr L a + Eγ0 Lb γs L b )Eγ0 La γr L a Eγ0 Lb γs L b

.

(6)

fγ0 Lγr L is the oscillator strength corresponding to the |γ0 L → |γr L transition with the transition energy Eγ0 Lγr L and a degeneracy gL of the initial |γ0 L state. The sum excludes the term where E a + E b equals zero. Dynamic polarizabilities at imaginary frequencies are related to oscillator strengths and transition energies in a sum over states formula: 3gL fγ0 Lγr L αLL (iω) = (−1)L+L . (7) 2 2 r (Eγ0 Lγr L + ω ) By fitting our calculated dynamic polarizabilities at imaginary frequencies between 0 and 2 au we deduced new values for oscillator strengths and transition energies (f, E). The quality of the fit was controlled by verifying that the sum of the oscillator strengths is close to the total number of electrons (Thomas–Reiche–Kuhn rule) and that the least-squares fit value as well as the slope of the gradients are small (less than 1×10−8 ). The R-coefficients were then evaluated by using these new values of (f, E) taking the initial calculated ones as a guess. The C6 were easily obtained from the R coefficients by the set of formulae given in [34]. To compare and to validate the values obtained at the TDGI level, the static polarizability of the ground state was also calculated using the finite field method at several levels of

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K Ellingsen et al Table 1. Transition energies relative to the 4 1 S ground state (cm−1 ). Config.

State

Calculated

Experimental [30]a

Jamorski et al [27]

4s2

1S

4s1 4p1 4s1 4p1 4s1 5s1 4s1 5s1 4s1 5p1 4s1 4d1 4s1 4d1 4s1 5p1 4s1 6s1 4s1 6s1 4s1 6p1 4s1 5d1 4s1 5d1 4s1 6p1

3P

0 32 708.32 46 801.60 54 575.47 55 853.66 60 958.10 62 408.14 62 684.67 62 648.90 65 474.14 65 801.15 67 798.55 68 114.15 68 356.23 68 611.25

0 32 696.34 46 745.37 53 672.24 55 789.22 61 302.16 62 548.51 62 774.39 62 910.00 65 432.32 66 037.60 68 091.31 68 338.48 68 581.52 68 607.26

0 32 649.61 46 665.22 54 676.24 55 980.14 61 040.33 63 075.29 62 717.55 62 646.66 65 652.14 65 974.32 67 795.08

1P 3S 1S 3P 1D 3D 1P 3S 1S 3P 1D 3D 1P

a Statistical averaging over the experimental j –j states was performed in order to obtain values for LS states.

theory using the GAUSSIAN 94 set of programs [35]. A 9s7p3d2f basis optimized for polarizability calculations by Kellö and Sadlej [13] was used in all-electron calculations at the self-consistent field and single reference single and double excited configurationinteraction (CISD) levels, as well as in single and double excited coupled-cluster (CCSD) and its perturbative triple correction extension (CCSD(T)) calculations. Calculations were equally performed with a 20-electron non-relativistic pseudopotential (PP) [36], a 20-electron relativistic pseudopotential [36] and the two-electron RPP also used in the TDGI calculations. In order to obtain a basis set as diffuse as that used in the two-electron RPP calculations in the 20-electron pseudopotential calculations, a 6s5p3d basis [36] was augmented with 2s, 3p and 4d even tempered functions in addition to 2f functions from the basis of Kellö and Sadlej [13]. 20 electrons were correlated in the all-electron and the 20-electron RPP calculations. As in the TDGI calculations the experimental core polarizability was added to the two-electron RPP results, while the polarizability of the core electrons was considered negligible when using the 20-electron pseudopotentials. 3. Results and discussion 3.1. The atomic Zn spectrum As the quality of the wavefunction is crucial for the results of the TDGI method, we compared our calculated atomic spectrum with the experimental spectrum [30] presented in table 1. The agreement is satisfactory, the discrepancy being between 4 and 344 cm−1 for all transitions except for the transition to 5 3 S (4s1 5s1 ) which is off by 903 cm−1 compared with experiment. The experiment predicts the 4 1 P (4s1 5p1 ) state to be 136 cm−1 below the 4 3 D (4s1 4d1 ) state. Our calculations, however, predict 1 P to be 36 cm−1 above 4 3 D. The discrepancies are probably due to limitations in the basis sets as well as an incomplete description of core–valence correlation due to the use of the effective semi-empirical core polarization potential. Our results are in line with the values from [27] which are given in table 1 and calculated at the same level of theory as in this work.


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Table 2. Average static polarizabilities for the ground state. α0 (au)

Finite field, all electron Finite field, 20-el. PP Finite field, 20-el. RPP Finite field, two-el. RPP TDGI, two-el. RPP TDGI, two-el. RPP + VCPP TDGI, two-el. RPP + VCPP + eff. transition operator Experimental

SCF

CISD

CCSD

CCSD(T)

53.39 52.83 49.66 52.54

42.30 42.31 39.57 47.96 47.83 45.19 39.12

41.31 41.40 38.74 47.94

40.11 40.33 37.69

38.80 ± 0.80 [12]

3.2. Polarizability of the 4 1 S ground state The polarizability of the Zn ground state is reported by several authors at different levels of theory [10–21], and was also recently measured experimentally to be 38.8 ± 0.8 au [12]. The calculated average static polarizabilities are given in table 2. The finite field polarizability calculations show the importance of electron correlation with SCF values 5–13 au higher than the corresponding correlated ones. Our all-electron SCF value of 53.39 au is in line with the numerical RHF value of Stiehler and Hinze [14] of 54.10 au. There are only minor changes of the polarizabilities going from all-electron to 20-electron PP calculations. The 20-electron relativistic pseudopotential (RPP) SCF polarizability, being 3.17 au lower than the corresponding non-relativistic pseudopotential result, demonstrates the importance of including relativistic effects. The all-electron CCSD result is 1 au lower than the CISD result and the triple excitations improve the result by 1.2 au giving a polarizability of 40.11 au. The basis set limit was reached adding one diffuse f-function to the basis set. The CCSD(T) polarizability augmented by 0.4% in the basis set limit. The coupled cluster results are in agreement with previously presented works [12, 13]. The 20-electron pseudopotential calculations show the same tendencies with a CCSD(T) value of 37.69 au. Yu and Dolg obtained 38.3 au from corresponding calculations [11]. The two-electron RPP calculations give a polarizability of about 47.96 au at all levels of correlation indicating a significant loss of correlation energy due to the exclusion of the 3d-orbitals. Using the two-electron RPP, the TDGI method gives a polarizability of 47.83 au in excellent agreement with the finite-field results. The inclusion of core–valence effects are of great importance. The polarizability is 2.6 au lower when taking into account core–valence correlation in the calculation of the wavefunctions (table 2). The result is further improved by 6.07 au using the effective transition operator (equation (4)) giving a final TDGI value of 39.12 au, which is within the range of the experimental value. Calculated transition energies from the ground state to the low-lying excited states and corresponding oscillator strengths involved in the two-electron RPP-TDGI calculation are given in table 3. The agreement with experiment is excellent, being off by only 0– 0.0012 au. The oscillator strengths for the 4 1 S ← 4 1 P and 4 1 S ← 5 1 P transitions are 1.570 and 0.119 compared with the experimental values of 1.55 ± 0.08 [37] and 0.122 [38], respectively. The effective transition operator reduces the first oscillator strength by 0.23 and gives the largest correction to the polarizability. The oscillator strengths for the other two transitions are only slightly reduced when the effective transition operator is taken into account.

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Figure 1. (a) Dynamic average polarizabilities α0 (au) as a function of frequency (au) for the 1 S (4s2 ) ground state up to the first resonance. Experimental values from Goebel et al [12] are shown with open diamonds. (b) Dynamic average polarizabilities α0 (au) as a function of frequency (au) for the 1 S (4s2 ) ground state up to the third resonance. The vertical lines show calculated transition energies.


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Table 3. Singlet → singlet transition energies and oscillator strengths (au). E Transitions

Calculated

Experimental [30]a

Oscillator strengths

4s2 (1 S) → 4s1 4p1 (1 P) 4s2 (1 S) → 4s1 5p1 (1 P) 4s2 (1 S) → 4s1 6p1 (1 P)

0.213 248 0.285 458 0.312 622

0.213 000 0.286 655 0.312 622

1.570 (1.55 ± 0.08 [37]) 0.119 (0.122 [38])a 0.049

−0.213 248 4s1 4p1 (1 P) → 4s2 (1 S) 0.041 245 4s1 4p1 (1 P) → 4s1 5s1 (1 S) 0.086 570 4s1 4p1 (1 P) → 4s1 6s1 (1 S) 4s1 4p1 (1 P) → 4s1 4d1 (1 D) 0.071 153 4s1 4p1 (1 P) → 4s1 5d1 (1 D) 0.099 007

−0.213 000 0.041 909 0.087 906 0.072 008 0.098 391

−0.523 0.152 0.007 0.451 (0.49 ± 0.04 [37]) 0.026

4s1 5s1 (1 S) → 4s1 4p1 (1 P) −0.041 245 4s1 5s1 (1 S) → 4s1 5p1 (1 P) 0.030 965 0.058 129 4s1 5s1 (1 S) → 4s1 6p1 (1 P)

−0.041 909 0.032 446 0.058 406

−0.456 1.331 0.121

a Statistical averaging over the experimental j –j states was performed in order to obtain values for LS states.

Table 4. Dynamic average TDGI-polarizabilities of the 4 1 S (4s2 ) ground state, the 4 3 P (4s1 4p1 ) and the 4 1 P (4s1 4p1 ) excited states for frequencies up to the first resonance. 1 S (4s2 ) Frequency (au) α0 (au)

3 P (4s1 4p1 ) Frequency (au) α0 (au)

1 P (4s1 4p1 ) Frequency (au) α0 (au)

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750 0.0875 0.1000 0.1125 0.1250 0.1375 0.1500 0.1625 0.1750 0.1875 0.2000 0.2125

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.038

39.13 39.25 39.63 40.27 41.21 42.48 44.16 46.34 49.15 52.82 57.69 64.32 73.72 87.91 111.54 158.26 293.26 4936.98

66.50 66.58 66.82 67.22 67.80 68.56 69.53 70.72 72.18 73.93 76.04 78.60 81.72 85.58 90.47 96.90 105.84 119.53 144.62 215.98

209.92 210.20 211.06 212.52 214.62 217.40 220.94 225.35 230.77 237.38 245.47 255.40 267.74 283.33 303.52 330.61 368.85 427.17 525.76 746.89

Dynamic polarizabilities calculated up to the first resonance are given in table 4, and plotted in figure 1(a) together with recent experimental results [12]. Our calculated polarizabilities agree well with experiment, deviating by only 0.70, 0.89 and 2.83 au from the experimental values for the three measured frequencies (0.071 981 15, 0.083 830 75 and 0.140 138 90 au). The difference between experiment and calculations becomes more pronounced (as expected) for the highest frequencies, changing from 1.6% to 4% over the described range of frequencies. The frequency dependence of the calculated polarizabilities up to the third resonance is displayed in figure 1(b).

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K Ellingsen et al Table 5. Static TDGI polarizabilities in au for the 4 1,3 P and 5 1,3 S excited states.

State

Config.

4s1 4p1

3P

4s1 4p1

1P

4s1 5s1 4s1 5s1

3S

a b

1S

α01

α10

α11

α12

α0

−147.48

151.56

−299.40

−738.64

20.74

−1128.95

66.50 67.59a , 74.04b 209.92 373.43b 1626.8 1170.0

−4880.5 −3510.1

α2 −11.29 −13.27a , −12.93b −93.41 −166.01b

Large-core relativistic ECP SCF [19]. Large-core relativistic ECP CI [19]. Table 6. Triplet → triplet transition energies and oscillator strengths (au). E Calculated

Experimental [30]a

Oscillator strengths

4s1 5s1 (3 S)

→ 4s1 4p1 (3 P) → 4s1 6s1 (3 S) 4s1 4p1 (3 P) → 4s1 4d1 (3 D) 4s1 4p1 (3 P) → 4s1 5d1 (3 D) 4s1 4p1 (3 P) → 4p2 (3 P)

0.099 636 0.149 295 0.136 585 0.162 427 0.219 915

0.095 579 0.149 165 0.137 053 0.163 514 0.218 334

0.118 0.014 0.421 (0.52 ± 0.06 [37], 0.42 ± 0.03 [39]) 0.128 0.583

4s1 5s1 (3 S) → 4s1 4p1 (3 P) 4s1 5s1 (3 S) → 4s1 5p1 (3 P) 4s1 5s1 (3 S) → 4s1 6p1 (3 P)

−0.099 636 0.029 082 0.060 250

−0.095 579 0.034 766 0.065 702

Transitions 4s1 4p1 (3 P)

−0.354 1.331 0.121

a Statistical averaging over the experimental j –j states was performed in order to obtain values for LS states. Experimental energies for the 4p2 (3 P) state were only given for J = 0, 1. The J = 2 level was deduced using the Landé interval rule, and averaging over the three J states was performed.

3.3. Polarizability of the 4 1,3 P and 5 1,3 S excited states Due to the success in describing the ground state, the polarizabilities of the excited states were all calculated using the TDGI method with the semi-empirical core polarization potential and the modified transition operator. To the best of our knowledge there are very few works on the polarizability of the excited states of Zn. Rozenkrantz et al [19] report a two-electron RPP polarizability of 4 3 P (4s1 4p1 ) and 4 1 P (4s1 4p1 ), but they have used a limited basis set and the core–valence correlation was not included in their approach. To check the credibility of our results we compare the transition energies and oscillator strengths with experimental values [30]. Calculated average (α0 ) and tensor (α2 ) polarizabilities are displayed together with the reduced spherical components αLL in table 5. αLL shows the contribution to the polarizability of a state with angular quantum number L from states with angular quantum number L (L = L ± 1). For an atom in an S (L = 0) state there is only one component of the polarizability, α01 , representing contributions from P (L = 1) states. In a Cartesian frame αxx = αyy = αzz and α01 = −3αxx . For an atom in a P (L = 1) state there are three different spherical components corresponding to S ← P, P ← P and D ← P transitions. In the Cartesian representation the three components are no longer degenerate and have to be calculated independently to determine the α1L values. The average and tensor polarizabilities are defined as α0 = −(α10 − α11 + α12 )/9 = (αxx + 2αzz )/3 and α2 = (α10 + α211 + α1012 )/9 = (αxx − αyy )/3.


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Table 7. C3 and C6 coefficients for Zn2 in au. Interaction

Molecular states −C3

1 S(4s2 )

1(+ g 3) g,u 3(+ g,u 1) g,u 1(+ g,u 3(+ g,u 1(+ g 1(+ u

1 S(4s2 )

1 S(4s2 )

+ + 3 P(4s1 4p1 )

1 S(4s2 )

+ 1 P(4s1 4p1 )

1 S(4s2 )

+ 3 S(4s1 5s1 ) + 1 S(4s1 5s1 )

1 S(4s2 )

±3.68 ±7.36

−C6 282 (270–296 [12]) 370 435 674 1139 3091 3790 3108

Energies for transitions from 4 3 P(4s1 4p1 ) deviate from experimental energies by 0.0001– 0.0094 au (0.1–4.2%). The oscillator strength for the 4 3 P ← 4 3 D transition is 0.421 compared with the experimental values of 0.52±0.06 [37] and 0.42±0.03 [39]. The average polarizability of the 4 3 P state is found to be 66.50 au and the tensor polarizability was determined to be −11.29 au. Taking into account their poor basis, we obtain a reasonable agreement with [19] which reports an average polarizability of 74.04 au and a tensor polarizability of −12.93 au at the CI level. The differences between calculated and experimental transition energies from 4 1 P are 0.0002–0.0013 au (0.1–1.6%). The oscillator strength for the 4 1 P ← 4 1 D transition of 0.451 is in good agreement with the experimental value of 0.49 ± 0.04 [37]. We obtain an average polarizability of 209.92 au and a tensor polarizability of −93.41 au in contradiction with the CI results of Rozenkrantz et al of 373.43 au (average polarizability) and −166.01 au (tensor polarizability) [19]. The difference between the two approaches is considerable, but with a successful description of the ground state polarizability and well described transitions we expect our results to be of the right order of magnitude. Finally, we calculated the 5 3 S(4s1 5s1 ) polarizability to be 1626.8 au and that of 1 5 S(4s1 5s1 ) to be 1170.0 au. Energies for transitions from 5 3 S(4s1 5s1 ) are off by 0.0040– 0.0057 au (4.2–16.4%) and for transitions from 5 1 S(4s1 5s1 ) by 0.0003–0.0015 au (0.5–4.6%) from experimental energies. Dynamic average polarizabilities up to 0.095 au and 0.38 au for 4 3 P and 4 1 P, respectively, are shown in table 4. Dynamic polarizabilities up to the fifth and fourth resonances for these two states and up to the second and third resonances for 5 3 S and 5 1 S can be obtained on request from the authors. 3.4. van der Waals coefficients The van der Waals coefficients involved in the interaction energy expanded in powers of R −n are valid only if the interaction distances R are larger than the Le Roy radius [40] defined as 1/2 RLR = 2 r 2 a1/2 + r 2 b . (8) We report C3 coefficients for interactions between an atom in the ground state and an atom in the 4 1 P state in table 7 along with the dispersion coefficients C6 for interactions between an atom in the ground 4 1 S state (r 2 1/2 = 4.20 au) and the lowest excited 4 3 P (r 2 1/2 = 4.96 au), 4 1 P (r 2 1/2 = 6.02 au), 5 3 S (r 2 1/2 = 9.59 au) and 5 1 S (r 2 1/2 = 9.91 au) states. The Le Roy radii are 16.8, 18.3, 20.4, 27.6 and 28.2, respectively. The C6 value for the ground state of Zn2 , which was calculated to be 282 au, is in excellent agreement with a recent experimentally derived value [12]. The experimental estimate of

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257 au is expected to be 5–15% too small [12] giving a C6 value in the range of 270–296 au. The C6 coefficient can also be calculated directly from dynamic polarizabilities at imaginary frequencies using the Casimir–Polder formula [8]: 3 ∞ C6 = α(iω)2 dω. (9) ) 0 The value obtained using our fitting procedure was checked against the value from the more direct Casimir–Polder formula. This was done by roughly estimating the above integral employing trapezoidal numerical integration from 0 to 2 au with step lengths from 0.01 to 1.0 au. The result (283 au) is within 1% of the value obtained using the fitting procedure. No reference C6 or C3 values for the excited states of Zn2 are available in the literature as far as we know. The first interaction coefficient C3 calculated for the 4 1 S + 4 1 P interaction is due to a resonant dipole–dipole interaction. This term is not related to the polarizabilities but depends only upon the accuracy of the atomic functions of the S and P states. According to the ungerade or gerade symmetry the sign of the C3 term is equal or opposite with respect to C6 . C3 /R 3 is clearly dominant for the interaction and the C6 /R 6 term gives only small contributions. For the ungerade states we find an energy maximum of 2609 cm−1 at 6.76 au for the ( state and 1102 cm−1 at 7.15 au for the ) state. These maxima are, however, found at distances which are significantly shorter than the Le Roy radii, where the influence of the exchange-interaction should be taken into account. The C6 coefficients between atoms in their ground 4 1 S and the excited 4 1 P and 4 3 P states are of the same order of magnitude due to the values of α0 . C6 interactions 4 1 S + 5 3 S and 4 1 S + 5 1 S are much larger. This is expected since the open-shell character of the 5 1,3 S states gives a very large value for the static polarizabilities. Note that for the 4 1 S + 5 1 S interaction the R 11(∗) term (given by equation (18) in [41]) corresponding to n 1 S ← 4 1 S transitions via a p-state contributes in the calculation of C6 , while this term is zero for the 4 1 S + 5 3 S interaction. 4. Conclusions Static and dynamic dipole polarizabilities are calculated using the time-dependent gaugeinvariant method. Inclusion of core-polarization effects is shown to be crucial to describe the ground-state polarizability. Using a semi-empirical core polarization potential and the modified dipole transition operator brings the calculated ground-state polarizability within the range of the experimental value. The effective transition operator reduces the first oscillator strength and gives the largest correction to the polarizability. Static and dynamic polarizabilities for the first 3 P, 1 P, 3 S and 1 S excited states are reported and discussed as well as long-range coefficients for all the molecular states of Zn2 dissociating into 4 1 S + 4 1 S, 4 1 S + 4 3 P, 4 1 S + 4 1 P, 4 1 S + 5 1 S and 4 1 S + 5 3 S, respectively. We obtained a C6 -coefficient for the ground state dissociation of Zn2 in very good agreement with a recent experimental result. Acknowledgments KE acknowledges the award of a grant from Elf Aquitaine and is grateful to have participated in the programme of ‘co-tutelle de Thèse’ financed by the French government. This work has received support from The Research Council of Norway (Programme for Supercomputing) through a grant of computing time. Support from the ‘Ministère de l’Education Nationale de l’Enseignement Supèrieur de la Recherche et de la Technologie’ (MENRT) and the ‘Centre


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National de la Recherche Scientifique’ (CNRS) as well as financial support from the Centre Informatique National de l’Enseignement Supérieur (CINES) is acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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