## Abstract

A systematic *ab initio* study is performed on the ground state static dipole polarizabilities of the ^{2S+1}S, *S*>1/2, atoms from N to Bi, Cr, Mo, Mn, Tc and Re. The benchmark scalar-relativistic values of the scalar polarizability components are obtained using the coupled cluster method. The spin–orbit configuration interaction calculations are carried out for the anisotropic (tensor) polarizability components of these atoms (except Mn and Tc) that arise from the second-order spin–orbit interaction. The tensor polarizabilities are calculated for the first time and found to increase from 10^{−5} (N) to 3.8 atomic units (Bi) approximately as the fourth power of the nuclear charge. The simple correlations and implication to magnetic trapping of cold atoms are discussed.

## 1. Introduction

Anisotropy of weak interatomic interactions manifests itself in the integral transport and collision properties and determines their spatial aspects, as well as the details of the collision-induced light absorption. The notion of anisotropy implies the dependence of the potential energy on the orientation of electronic cloud with respect to the collision (body-fixed) axis. For atoms, the main source of anisotropy is the orbital electronic angular momentum **L**, whose orientation is determined by the body-fixed projection *Λ*. Anisotropy is, therefore, pertinent to atoms in the non-S states (*L*≠0), which, in the simplest case of interaction with a structureless particle, give distinct molecular states specified by the absolute values of the projections |*Λ*| (Aquilanti & Grossi 1980; Krems *et al.* 2004). Spin–orbit (SO) interaction complicates the picture: the spin electronic angular momentum **S** (if any) couples with **L** to form the total electronic angular momentum **J**. The anisotropy is defined by *Ω*, the body-fixed projection of **J**, and therefore contains the spin contribution (as is well known in electron spin resonance spectroscopy and molecular magnets; e.g. Gatteschi *et al.* 2006).

Recent studies of atomic collisions in cold and ultracold regimes gave rise to another quite practical interest in the interaction anisotropy. Cooling and trapping techniques use external fields that break the spatial isotropy. Interplay between the internal anisotropy of atomic collision in the body-fixed frame and the external anisotropy in the space-fixed frame related to the field leads to new important aspects of the collision event, especially prominent at low temperatures. For instance, in a general method of buffer-gas loading into magnetic traps (Doyle *et al.* 1995; Ketterle & Van Druten 1996; Weinstein *et al.* 1998*a*), the atoms bearing non-zero magnetic moment (spin **S**) are loaded into the magnetic trap with the cold He atoms. Elastic (momentum transfer) collisions effectively quench their kinetic energy. Entering the trap, the atoms start to experience magnetic field discrimination of their Zeeman levels into the low-field-seeking (LFS), pulled towards the trap, and the high-field-seeking (HFS), pushed out and condemned to escape. Inelastic collisions with He can convert higher lying LFS levels to HFS ones, thus enhancing the trap loss. As follows from the analysis by Krems *et al.* (2004), the main source of this Zeeman relaxation is the anisotropy of interatomic interaction. This would suggest that the method is applicable only to the ^{2S+1}S atoms with *S*>0.

Fortunately, many exceptions have been found from this oversimplified rule (Buchachenko *et al.* 2009). Even if *L*≠0, interaction anisotropy can be too weak to make the Zeeman relaxation competitive with the momentum transfer. This is the case of the open-shell lanthanides, the majority of which has been trapped with high efficiency (Hancox *et al.* 2004*a*). Their outer spherical 6s^{2} electronic shell screens the *L*-bearing 4*f* shell and suppresses the resulting interaction anisotropy (Hancox *et al.* 2004*a*; Buchachenko *et al.* 2006*a*,*b*). Similar suppression was found for the 3d transition metal atoms Ti and Sc (Hancox *et al.* 2005; Krems *et al.* 2005). It has been recently demonstrated that the first-order (intra-multiplet) SO interaction can also reduce the interaction anisotropy. Cancellation of the spin and orbital electronic angular momenta in their vectorial sum **J** can create the fine-structure levels that exhibit only isotropic interactions. This has been experimentally observed by Tscherbul *et al.* (2009) for the ground state Ga and In atoms. An atom in the ^{2}P_{1/2} fine-structure level interacts with He isotropically, whereas the ^{2}P_{3/2} level splits out in two components with the body-fixed projections of **J** |*Ω*|=1/2 or 3/2, making such interaction anisotropic. The difference in the Zeeman relaxation rates for two levels grows up with the magnitude of SO coupling and approaches six orders of magnitude.

Of interest here is a quite remarkable opposite example. While the ground state nitrogen atoms were successfully trapped (Hummon *et al.* 2008), attempts to trap the heaviest member of the same group, , failed (Maxwell *et al.* 2008). Accurate *ab initio* calculations pointed out the second-order (inter-multiplet) SO coupling as the prime reason. It induces the splitting of the |*Ω*|=1/2 and 3/2 components in the Bi–He complex, thus making it anisotropic.

For an atom, the dipole polarizability is the simplest parameter that reflects the interaction anisotropy. An atom in a homogeneous electric field has the same axial symmetry as the atom–atom collision complex. Moreover, polarizability directly or indirectly (through dispersion coefficients) determines the lowest order long-range interaction with an atomic ion or another atom. No wonder that the results of trapping experiments mentioned above have been successfully interpreted in terms of the static dipole polarizability anisotropy for lanthanides (Rinkleff & Thorn 1994*a*,*b*; Buchachenko *et al.* 2006*a*, 2007; Chu *et al.* 2007) and for transition metals (Chu *et al.* 2005; Klos 2005). Example of the ^{2}P multiplet is trivial: the dipole polarizability of the ^{2}P_{1/2} level is isotropic. (We neglect here weak hyperfine interactions; see, however, §5.)

It is instructive to explore the same analogy for the S-state atoms bearing the spin greater than 1/2. In addition to the fifth-group atoms from N to Bi mentioned above, the ground state polarizability anisotropy induced by the SO coupling may appear in two groups of the transition metals (Cr, Mo and Mn, Tc, Re), as well as in two f-elements (Eu and Am). To our knowledge, it was measured by Stark spectroscopy only for the Eu atom (Martin *et al.* 1968), whereas no theoretical calculations have been performed so far.

This work, therefore, represents the first systematic study of the anisotropic polarizabilities induced by the SO coupling. In order to get a uniform picture of the effect for atoms with different electronic structures, we used the ‘conventional’ techniques of the *ab initio* quantum chemistry combined with the finite-field approach, in which the polarizability is obtained from the dependence of energy on the external electric field strength (Cohen & Roothaan 1965; Pople *et al.* 1968). At the first stage, the benchmark calculations on the scalar polarizabilities were performed within the scalar-relativistic (SR) approximation. At the second stage, the effect of the vectorial SO interaction was taken into account through the state-interacting SO configuration interaction (SI-SOCI) method by Berning *et al.*(2000). In this approach, the SO operator matrix is evaluated on the basis of the SR wave functions that describe a preselected set of atomic states. Though it has already been successfully tested in the finite-field atomic polarizability calculations (Buchachenko 2010), a word of caution is in order when considering its results as quantitative. Rigorously, the method requires convergence with respect to the set of atomic states involved in the SO treatment. (This convergence issue should not be confused with the convergence with respect to the dipole-allowed transitions to excited states pertinent to the polarizability expansion through the oscillator strengths.) Expansion of this set usually presents difficulties for the SR stage associated to the loss of accuracy or even to hardly surmountable convergence problems. For this reason, the tensor ground state polarizability values presented here cannot be regarded as precise (except, perhaps, those for N, P and As atoms), but still provide new information for useful qualitative analysis. Moreover, they can be used for testing the methods of the atomic structure theory, for which the calculation of the small polarizability values may also be challenging.

The results for atoms belonging to the distinct groups are arranged below in three consecutive sections and discussed, as a whole, in §5. Concluding remarks follow.

## 2. The *n*s^{2}*n*p^{3} atoms from N to Bi

### (a) Ground state polarizabilities

All *ab initio* calculations were performed using the molpro program package (Werner *et al.* 2009) within the *C*_{2v} point group to model the axial symmetry of an atom plus field system. The dipole field was applied along the space-fixed *z* axis and the calculations of the electronic energies were accomplished at 13 field strengths *F* from 0 to 0.05 atomic units. Parallel polarizabilities *α*_{∥}, simply *α* throughout the paper, were extracted by the polynomial fit to the low-field portion of this dependence (seven points up to 0.00075 atomic units). The high-field portion was used only for very small anisotropic polarizability values, see below. The energy threshold was always set to 10^{−12} atomic units. All the polarizabilities are given below in atomic units.

At the first stage, we calibrated the basis sets and obtained the benchmark SR polarizability values in the uncoupled |*LM*_{L}*SM*_{S}〉 representation, where *M*_{L} and *M*_{S} are the projections of the orbital and spin electronic momenta onto the field axis, respectively. Standard restricted Hartree–Fock (RHF) calculations were followed by the restricted coupled cluster calculations with single, double and non-iterative triple excitations, CCSD(T), which correlated all the electrons considered explicitly. Extensive preliminary testing revealed that the standard augmented correlation-consistent polarized valence basis sets of quintuple quality (aug-cc-pV5Z) provide good accuracy for the polarizability values. For N, P and As, the second-order Douglas–Kroll (DK) SR correction (Reiher & Wolf 2004) was taken into account and the bases were used with the specially adapted contraction by de Jong *et al.* (2001). For As, Sb and Bi, the relativistic effective core potentials (ECPs) ECP10MDF (Stoll *et al.* 2002), ECP28MDF (Metz *et al.* 2000) and ECP60MDF (Metz *et al.* 2000), respectively, were employed with the corresponding aug-cc-pV5Z basis sets (Peterson 2003). The number in the ECP acronyms specifies the number of electrons absorbed.

So obtained polarizabilities denoted as are collected in the upper part of table 1 together with the selected literature values (Alpher & White 1959; Zeiss & Meath 1977; Flambaum & Sushkov 1978; Sadlej 1991, 1992; Kellö & Sadlej 1992; Stiehler & Hinze 1995; Das & Thakkar 1998; Miller 1998; Hohm *et al.* 2000; Chu & Dalgarno 2004; Roos *et al.* 2004; Lupinetti & Thakkar 2005; Maroulis 2007). Our recommended values are given in boldface. Benchmark non-relativistic (NR) numerical RHF data by Stiehler & Hinze (1995) and results by Thakkar and co-workers (Das & Thakkar 1998; Lupinetti & Thakkar 2005) perfectly validate the present approach for the lightest N and P atoms, when the SR correction is negligible. For heavier atoms, present results are superior over the previous ones (we made a preference to our estimate for Sb since the calculations by Maroulis (2007) disregarded the SR effects). The results for As indicate that the all-electron DK and ECP calculations give very close results, so all the subsequent calculations were made with the more economic ECP option.

At the second stage, we considered the vectorial SO interaction within the multi-reference methods. The calculations used the same basis sets and SR corrections as described above and were performed for all the states arising from the lowest *n*s^{2}*n*p^{3} electronic configuration, namely, ^{4}S^{°}, ^{2}D^{°} and ^{2}P^{°}. Starting orbitals were generated by the complete active space multi-configurational self-consistent field (CASSCF) method, with the averaging over all the components of these electronic states classified within the *C*_{2v} symmetry. The average values of the squared *M*_{L} projections were strictly controlled. Following (Anderson & Sadlej 1992), we used *n*s, *n*p, *n*d,(*n*+1)s,(*n*+1)p active atomic orbitals (AOs) to distribute five valence electrons. The multi-reference single and double configuration interaction (MRCI) calculations followed, with the *n*s, *n*p, *n*d active orbitals and no core AOs for N, 1s^{2},2s^{2} core for P and (*n*−2)s^{2},(*n*−2)p^{2} core for As, Sb and Bi atoms. Davidson’s correction (Langhoff & Davidson 1974) was included.

The MRCI results for the polarizabilities are listed in table 1. In general, they agree well with the more accurate CCSD(T) values, though the deviation grows up with the atomic number approaching 0.7 atomic units (1.5%) for Bi. Agreement with less accurate but systematic second-order complete active space perturbation theory (CASPT2) calculations by Roos *et al.* (2004) is also quite satisfactory.

Finally, the SI-SOCI calculations were performed for the SO matrix spanned by all the components of the SR states ^{4}S^{°}, ^{2}D^{°} and ^{2}P^{°}. Note that only the ^{2}P^{°} state is coupled directly with the ground ^{4}S^{°} one by the SO interaction. The full Breit–Pauli SO operator was used for the internal part of the configuration interaction space and the mean-field approximation—for the external part (Berning *et al.* 2000).

The SO-coupled eigenvectors were assigned within the coupled angular momentum |*JM*〉 representation and the respective polarizabilities were obtained from the field dependences. They were used to evaluate the scalar and tensor polarizability components (Angel & Sandars 1968) as
2.1
(The same equations hold for and polarizabilities to be considered below for excited states.) For the particular case of *J*=3/2, one has
2.2
Special care was taken to extract the tensor component. We first built up the corresponding combinations of energies and then fitted them to the polynomials of various orders. Initial part of the field dependences, where the energy differences are below or close to the convergence threshold, was excluded from the fit and some points at the field strengths above 0.00075 atomic units were added.

The calculated and polarizabilities are given in the lower part of table 1. The scalar component is contrasted to a few *J*-resolved values obtained by the oscillator strength summation and with the recommended data. More useful is the comparison to the MRCI values, which reflects the influence of the SO interaction. The tensor component, entirely owing to the SO effect, is always negative and increases by six orders of magnitude from N to Bi. The absolute value of the tensor component is very close to the SO effect on the scalar component when the latter is visible (As, Sb and Bi).

### (b) Excited states

Table 2 presents the computed energies of the fine-structure levels and their deviations from experimental data (Ralchenko *et al.* 2007). For the multiplet, the agreement is always good. The results for the multiplet are subjected to inaccuracies for the lightest N atom (inversion of the quasi-degenerate levels) and the heaviest Bi atom (large error for *J*=3/2 level). The latter problem originates from the intrusion of the ^{2}P state from the excited 6p^{2}7s configuration (Ralchenko *et al.* 2007) not included in the calculations. The same state probably affects the splitting in the Sb atom.

Excited-state polarizabilities , (MRCI) and , (SI-SOCI) are collected in table 3. For the lightest N and P atoms, literature estimates are available (Stevens & Billingsley 1973; Anderson & Sadlej 1992; Stiehler & Hinze 1995). The only correlated CASPT2 calculations by Anderson & Sadlej (1992) gave , and , for N; , and , for P, in reasonable agreement with the more accurate present data. It can be seen that all the multiplets arising from the *n*s^{2}*n*p^{3} electronic configuration have similar (within 15%) scalar polarizabilities.

The influence of the SO coupling can be addressed using the well-known formulae connecting *α*^{L} and *α*^{J} values for an isolated multiplet under the mean-field approximation (Angel & Sandars 1968)
2.3
where {:::} is the 6-*j* symbol. In our particular case, tensor polarizabilities are related as
2.4

In figure 1, the non-vanishing SO-coupled tensor polarizabilities are plotted against the respective SR values. Strong deviations from the lines representing the above mean-field equalities are evident for the pair of *J*=3/2 levels in atoms heavier than P. They reflect very large SO interaction between the ^{2}D^{°} and ^{2}P^{°} multiplets: according to the hyperfine structure analysis for the Bi atom by Landman & Lurio (1970), it is twice as large as the Coulomb energy separating two states. The same trend can be seen for the scalar polarizabilities.

## 3. The *n*d^{5}(*n*+1)s atoms Cr and Mo

For the Cr and Mo atoms, we concentrated on the ground ^{7}S and the first excited ^{5}S states, both arising from the *n*d^{5}(*n*+1)s configuration with the different filling of the *n*d shell. The methods employed were similar to those described in §2, but the polarizabilities were determined from the shorter grid of the field strengths (six to eight values up to 0.001 atomic units).

Preliminary tests indicated the convergence of the ground state scalar polarizability for the augmented correlation-consistent polarized weighted core-valence sets of quintuple-zeta quality, aug-cc-pwCV5Z (Balabanov & Peterson 2005; Figgen *et al.* 2009). The NR RHF value obtained with this set for Cr coincides with the numerical RHF result by Stiehler & Hinze (1995), whereas the effect of the SR DK correction on it is similar to that reported by Baranowska *et al.* (2007) (table 4). All the calculations presented below accounted for this correction and used the aug-cc-pwCV5Z basis contracted accordingly (Balabanov & Peterson 2005). For the Mo atom, the basis set of the same quality was used with the ECP28MDF ECP (Figgen *et al.* 2009). The most accurate values were obtained by the CCSD(T) method, with the correlation of all electrons treated explicitly. For the ground states, they agree with the previous theoretical data within 6 per cent (table 4). Scalar polarizability only slightly increases from Cr to Mo and from the ground ^{7}S to the excited ^{5}S state.

The choice of atomic states to be included in the SI-SOCI calculations for the transition metals is less straightforward than for the main-group atoms. We used the maximum set of the states for which we were able to converge the state-averaged CASSCF solutions with the proper *M*_{L} projections within the *n*d,(*n*+1)sp active space. It included ^{7}S, ^{5}S, *n*d^{4}(*n*+1)s^{2} ^{5}D, *n*d^{5}(^{4}G)(*n*+1)s ^{5}G and *n*d^{5}(^{4}P)(*n*+1)s ^{5}P multiplets. According to NIST Atomic Spectra Database (Ralchenko *et al.* 2007), the latter state is the only one that couples with the lowest S states by the SO interaction in the Mo atom. In Cr, however, 3d^{4}4s^{2} ^{3}P state lying at *ca* 23 500 cm^{−1} can also affect the ^{5}S state, but we failed to include it in the calculations. The MRCI calculations were performed for each *M*_{L} component of the ^{7}S, ^{5}S, ^{5}D, ^{5}G and ^{5}P multiplets using the same choice of active AOs and keeping in core 1s^{2},2s^{2},2p^{6} shells for Cr and the 4s^{2} shell for Mo (according to the CCSD(T) results, associated error does not exceed 0.5%). So, obtained MRCI wave functions were used to build up the SO Hamiltonian matrix in the SI-SOCI method.

Table 4 indicates that the MRCI method can quantitatively reproduce the value only for the ground state Cr. Its error with respect to the CCSD(T) method is remarkable in other circumstances.

The SO-coupled polarizabilities were obtained using equation (2.1)
3.1
for *J*=3 (^{7}S) and
3.2
for *J*=2 (^{5}S) cases. Effect of the SO coupling on the scalar polarizability, taken as the difference between the MRCI and the SI-SOCI values, is very small, except for the ^{5}S state of the Mo atom (table 4). The latter case, however, may reflect the deficiency of the MRCI treatment. The tensor polarizabilities are all negative and small, though they increase by an order of magnitude from the ^{7}S_{3} to the ^{5}S_{2} level.

For excited non-S states, the convergence issue of the SI-SOCI method is much more crucial. For this reason, we presented for them, in table 5, only the SR MRCI results that can be considered as the qualitative estimates. Even at this level of theory, the *M*_{L} components of the polarizabilities of the D and G multiplets behave irregularly, preventing reliable estimation of the tensor components. Agreement between the calculated zero-field energies and the centres of fine-structure multiplets obtained from the experimental data (Ralchenko *et al.* 2007) is generally acceptable, but remarkably worse than that attained for the fifth-group atoms in §2.

## 4. The *n*d^{5}(*n*+1)s^{2} atoms Mn, Tc and Re

The calculations on the *n*d^{5}(*n*+1)s^{2} ^{6}S atoms Mn, Tc and Re were performed using the similar approaches. The aug-cc-pwCV5Z basis sets were employed with the SR DK correction (Balabanov & Peterson 2005) for Mn and with ECP28MDF (Peterson *et al.* 2007) and ECP60MDF (Figgen *et al.* 2009) for Tc and Re, respectively. The ground state SR polarizabilities obtained by the RHF and the CCSD(T) methods with the correlation of all electrons treated explicitly are presented in the table 6 together with a few literature values. The NR RHF results attest the quality of the basis set, whereas the CASPT2 (Roos *et al.* 2005) and recommended (Miller 1998) results confirm the trend observed in the CCSD(T) calculations.

According to the NIST Atomic Spectra Database (Ralchenko *et al.* 2007) and references cited therein, none of the low-lying states of the Mn and Tc atoms can couple with the ground state by the SO interaction. It implies that the present SI-SOCI method is not applicable for their ground state polarizability anisotropies. We can only state that the values for Mn and Tc atoms should be significantly smaller than those for Cr and Mo, respectively, as estimated in §3.

The Re atom has a different electronic spectrum (Ralchenko *et al.* 2007). Though its lowest electronic energy levels are highly mixed, the analysis by Wyart (1978) showed the appearance of the low-energy 5d^{5}6s^{2} ^{4}P state that can interact with the ground one. We, therefore, performed the MRCI calculations with the 5d,6s active AOs, 5p AOs correlated as doubly occupied and 5s AO included in the core for the 5d^{5}6s^{2} ^{6}S, ^{4}P, ^{4}G and 5d^{6}6s ^{6}D states. The reference CASSCF wave functions were obtained for the same orbital active space and with the averaging over the same set of the electronic states. The SR ground state value computed in this way agrees very well with the benchmark CCSD(T) result (table 6).

The SI-SOCI calculations followed. The polarizabilities of the ^{6}S_{5/2} level presented in table 6 were calculated according to equation (2.1),
4.1

The MRCI results for excited states are given in table 5. The large error in the ^{4}P energy with respect to the centre of the ^{4}P_{J} multiplet (Wyart 1978); Ralchenko *et al.* (2007) casts some doubts in the accuracy of our estimation.

## 5. Discussion

In order to bring the results for all atoms studied to an equal footing, we explored the correlation of the ground state tensor polarizability with the simplest estimate of the SO coupling strength that scales, for a hydrogen-like atom, as a fourth power of the nuclear charge. Figure 2 shows quite good correlation. We also included in it the measured results for the atom, atomic units, obtained by Martin *et al.* (1968). This state is primarily coupled by the SO interaction with the very distant 4f^{7}5d^{2} e^{6}P^{°} state (Wyart 1985; Ralchenko *et al.* 2007) that cannot be included in the SI-SOCI calculations. The Eu tensor polarizability is lower than the general trend prediction, as should also happen for the Mn and Tc atoms according to our speculations above. Note that these exceptions occur for atoms with extremely stable d^{5}s^{2} and f^{7}s^{2} configurations.

The correlation with the nuclear charge also helps to compare the typical scales for polarizability anisotropies induced by different interactions. Comparison of the *n*s^{2}*n*p^{3} atoms Al and Tl with P and Bi, respectively, is especially relevant for this purpose. The tensor polarizability of the Al(^{2}P^{°}) owing to the Coulomb interaction is equal to −8.4 atomic units (Martin *et al.* 1968), the value almost five orders of magnitude larger than that of the atom. The SI-SOCI calculations (Buchachenko 2010) did not reveal any prominent effect of the SO interaction on the polarizabilities of the Al atom. The ground fine-structure level of Al exhibits the polarizability anisotropy owing to the hypefine interaction. According to the measurements by Angel *et al.* (1974), the tensor component amounts to (8.1±0.7)×10^{−4} atomic units, i.e. it is even greater than those induced by the SO coupling in P atom. For Tl atom, the SO effect on the tensor polarizability was calculated as −4 atomic units (Buchachenko 2010), in close agreement to that found here for Bi. The ratio of the values for (Petersen *et al.* 1968) and atoms does not exceed 7. The tensor polarizability of the , induced by the hyperfine interaction measured by Gould (1976) as −1.5×10^{−4} atomic units, is more than four orders of magnitude smaller than the tensor polarizability of Bi owing to the SO coupling.

In general, the *Z*^{4} scaling may serve for the order-of-magnitude guess for the SO effect on the atomic polarizabilities. There is not much data clearly isolating this effect beyond the mean-field approximation, because its estimation comes as the tiny difference between two relatively large SO-coupled and SR polarizabilities, either scalar or tensor. By contrast, the tensor polarizabilities of the S-state atoms provide the well-defined estimate, as it comes as the finite value with the vanishing SR counterpart.

Deeper analysis is possible within a two-state model that has the simplest form for the fifth-group atoms. In the absence of an electric field, the ^{4}S^{°} state is connected with the ^{2}P^{°} one by the SO matrix element *ζ*. In the |*LM*_{L}*SM*_{S}〉 representation, the SO Hamiltonian matrix reduces to two blocks,
if *M*=±1/2 and
if *M*=±3/2, where *E*_{P} denotes the energy of the ^{2}P^{°} state. Taking into account that the energy of the ground state varies with the field as , the energy of the excited state varies as depending on *M*_{L}, and assuming that *ζ* does not depend on the field, one can apply the second-order perturbation theory for corrections to the ground state energy in both blocks. The coefficients at the terms quadratic in *F* defines and , respectively. Using equation (2.2), one gets
5.1
and
5.2
Equation (5.1) says that the effect of the SO interaction on the scalar polarizability is proportional to the scalar polarizability difference for excited and ground states, whereas equation (5.2) says that the tensor polarizability of the ground state mirrors the tensor polarizability of the excited state. Both components scale quadratically as the SO coupling increases and the excitation energy decreases.

The proportionality coefficients at *f*_{0} and *f*_{2} in equations (5.1) and (5.2) are three and two, respectively. However, preliminary analysis indicated that they do not reproduce our numerical results well. It is more instructive to use equation (5.2) as the correlation rule. In figure 3, we plot the absolute values of the polarizabilities against the *f*_{2} values as calculated from the MRCI and SI-SOCI parameters for the ground and the lowest P states (^{2}P^{°} for N–Bi, ^{5}P for Cr, Mo and ^{4}P for Re atoms, see above). The correlation (5.2) holds reasonably well with the proportionality factor 10.3±0.2.

Finally, we wish to return to the question that motivated this study, namely, the efficiency of the Zeeman relaxation in the magnetic traps. Successful trapping has been reported so far for N (Hummon *et al.* 2008); Cr (Weinstein *et al.* 1998*b*), Mn (Nguyen *et al.* 2005; Connolly *et al.* 2010), Mo (Hancox *et al.* 2004*b*) and Eu (Kim *et al.* 1997) atoms, whereas unsuccessful attempts were made on Re and Bi atoms (Maxwell *et al.* 2008). Among the trappable atoms, Mo exhibits the largest polarizability anisotropy, 0.1 atomic units, whereas among the non-trappable atoms, Re has the smallest, 0.9 atomic units (both taken by the absolute value). Since the trapping efficiency is often characterized by the ratio of the elastic to the inelastic collision rates (Ketterle & Van Druten 1996), it is instructive to consider the ratios of the scalar to tensor polarizabilities, 880 and 70 for Mo and Re, respectively. From the same ratios, we can conclude that the P and As atoms should be well trappable, while the trapping of the Sb atoms looks questionable. Of course, these simple considerations disregard any difference in the conditions of the particular trapping experiment.

## 6. Concluding remarks

We presented here the first systematic study of the anisotropic static dipole polarizabilities induced by the SO coupling in the ^{2S+1}S, *S*>1/2, atoms. We found that the associated tensor polarizability component scales with the nuclear charge as *Z*^{4}, much faster than that induced by the Coulomb or hyperfine interactions. We hope that our calculations will be of use for estimating the second-order SO effect on the polarizabilities of other atoms and for assessing the methods of atomic structure theory. Our auxiliary results for the scalar polarizability components are the most accurate to date.

## Acknowledgements

This work was supported by the Russian Foundation for Basic Research (project 08-03-00414) and by the Moscow State University.

- Received August 21, 2010.
- Accepted October 18, 2010.

- This journal is © 2010 The Royal Society