The hyperfine energy levels of alkali metal dimers: ground-state homonuclear molecules in magnetic fields
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t The hyperfine energy levels of alkali metal dimers:ground-state homonuclear molecules in magnetic fields
J. Aldegunde ∗ and Jeremy M. Hutson † Department of Chemistry, Durham University, South Road, DH1 3LE, United Kingdom (Dated: November 2, 2018)We investigate the hyperfine energy levels and Zeeman splittings for homonuclear alkali-metaldimers in low-lying rotational and vibrational states, which are important for experiments designedto produce quantum gases of deeply bound molecules. We carry out density-functional theory(DFT) calculations of the nuclear hyperfine coupling constants. For nonrotating states, the zero-field splittings are determined almost entirely by the scalar nuclear spin-spin coupling constant. Bycontrast with the heteronuclear case, the total nuclear spin remains a good quantum number in amagnetic field. We also investigate levels with rotational quantum number N = 1, which have long-range anisotropic quadrupole-quadrupole interactions and may be collisionally stable. For thesestates the splitting is dominated by nuclear quadrupole coupling for most of the alkali-metal dimersand the Zeeman splittings are considerably more complicated. PACS numbers:
I. INTRODUCTION
There have been enormous advances over the last yearin experimental methods to produce ultracold moleculesin their rovibrational ground state at microkelvin tem-peratures. Ospelkaus et al. [1] produced KRb mole-cules in high-lying states by magnetoassociation (Fes-hbach resonance tuning) and then transferred them bystimulated Raman adiabatic passage to levels of the Σ + ground state bound by more than 10 GHz. This wasthen extended by Ni et al. [2] to produce molecules in( v, N ) = (0 , v and N are the quantum numbersfor molecular vibration and mechanical rotation. Danzl et al. [3, 4] have carried out analogous experiments onCs dimers, while Lang et al. [5, 6] have produced Rb molecules in the lowest rovibrational level of the lowesttriplet state. There have also been considerable successesin direct photoassociation to produce low-lying states[7, 8, 9, 10].A major goal of the experimental work is to producea stable molecular quantum gas. However, such a gascan form only if (i) a large number of molecules are inthe same hyperfine state and (ii) the molecules are stableto collisions that occur in the gas. In particular, inelas-tic collisions that transfer internal energy into relativetranslational energy cause heating and/or trap loss. It isthus very important to understand the hyperfine struc-ture of the low-lying levels and its dependence on ap-plied electric and magnetic fields. In a previous paper,we explored the hyperfine levels of heteronuclear alkalimetal dimers in rotationless levels [11]. The purpose ofthe present paper is to extend this work to homonuclearmolecules, which have important special features. Wealso explore N = 1 levels, which may be collisionally sta- ∗ Electronic address: E-mail: [email protected] † Electronic address: E-mail: [email protected] ble for homonuclear molecules and which interact withlonger-range forces than N = 0 levels. II. MOLECULAR HAMILTONIAN
The Hamiltonian of a diatomic molecule in the pres-ence of an external magnetic field can be decomposed intofive different contributions: the electronic, vibrational,rotational, hyperfine and Zeeman terms. For Σ mole-cules in a fixed vibrational level, the first two terms takea constant value and the rotational, hyperfine and Zee-man parts of the Hamiltonian may be written [12, 13, 14] H = H rot + H hf + H Z , (1)where H rot = B v N − D v N · N ; (2) H hf = H Q + H IN + H t + H sc = X i =1 V i : Q i + X i =1 c i N · I i + c I · T · I + c I · I ; (3) H Z = − g r µ N N · B − X i =1 g i µ N I i · B (1 − σ i ) . (4)where the index i refers to each of the nuclei in themolecule. N , I and I represent the operators for me-chanical rotation and for the spins of nuclei 1 and 2. Therotational and centrifugal constants of the molecule aregiven by B v and D v (but centrifugal distortion is ne-glected in the present work). We use N rather than J for mechanical rotation because we wish to reserve J forthe angular momentum including electron spin for futurework on triplet states.The hyperfine Hamiltonian of equation 3 consistsof four different contributions. The first is the elec-tric quadrupole interaction H Q , with coupling constants( eqQ ) and ( eqQ ) . It represents the interaction of thenuclear quadrupoles ( eQ i ) with the electric field gra-dients q i created by the electrons at the nuclear posi-tions. The second is the spin-rotation term H IN , whichdescribes the interaction of the nuclear magnetic mo-ments with the magnetic moment created by the rota-tion of the molecule. Its coupling constants are c and c . For a homonuclear molecule with identical nuclei,( eqQ ) = ( eqQ ) and c = c . The last two terms repre-sent the interaction between the two nuclear spins; thereis both a tensor component H t , with coupling constant c , and a scalar component H sc , with coupling constant c . The second-rank tensor T represents the angular partof a dipole-dipole interaction.The Zeeman Hamiltonian H Z has both rotational andnuclear Zeeman contributions characterized by g -factors g r , g and g . For homonuclear molecules g = g . Thenuclear shielding tensor σ i is approximated here by itsisotropic part σ i ; terms involving the anisotropy of σ i areextremely small for the states considered here.The nuclear g -factors and the quadrupole moments ofthe nuclei are experimentally known [15].For homonuclear molecules we neglect the effect ofelectric fields, though in principle there are small ef-fects due to anisotropic polarizabilities and the molecularquadrupole moments can interact with the gradients ofinhomogeneous fields. III. EVALUATION OF THE COUPLINGCONSTANTS
The rotational g -factors are known experimentally forall the homonuclear alkali metal dimers [16]. However,the only such species for which the nuclear hyperfine cou-pling constants have been determined accurately is Na [17]. We have therefore evaluated the remaining couplingconstants using density-functional theory (DFT) calcula-tions performed with the Amsterdam density functional(ADF) package [18, 19] with all-electron basis sets andincluding relativistic corrections. A full description ofthe basis sets, functionals, etc. used in the calculationshas been given in our previous paper on heteronuclearsystems [11]. In the present work, the calculations werecarried out at the equilibrium geometries ( R e = 2 .
67 ˚Afor Li [20], 3.08 ˚A for Na [21], 3.92 ˚A for K [22], 4.21˚A for Rb [23] and 4.65 ˚A for Cs [24]). This give resultsthat are approximately valid not only for v = 0 statesbut also for other low-lying vibrational states.The values for the coupling constants are given in ta-ble I. It may be seen that the DFT results for Na arewithin about 30% of the experimental values, and similaraccuracy was obtained for other test cases in our previ-ous work [11]. The accuracy is likely to be comparablefor the other cases studied here. This level of accuracyis adequate for the purpose of the present paper, whichaims to explore the qualitative nature of the Zeeman pat-terns. Most of our conclusions are insensitive to the exact magnitudes of the coupling constants. IV. HYPERFINE ENERGY LEVELS
Our previous work [11] showed that the zero-field split-ting for heteronuclear diatomic molecules in N = 0 statesis determined almost entirely by the scalar nuclear spin-spin interaction. This remains true for homonuclear mol-ecules in N = 0 states. We show below that for N > and Li , with smallerbut significant contributions from the remaining couplingconstants.For all systems except Li , the scalar spin-spin cou-pling is considerably stronger than the spin-rotation andtensor spin-spin couplings. Knowledge of the nuclearquadrupole coupling constant eQq and the scalar spin-spin coupling constant c is therefore sufficient to un-derstand the hyperfine splitting patterns. We will focushere on Rb and Rb , which form a convenient pairthat approximately cover the range of values of the ratio | c / ( eqQ ) | . Lang et al. [6] have produced Rb in thelowest rovibrational level of the lowest triplet state, butas far as we are aware not yet in the singlet state.The hyperfine energy levels are obtained by diagonal-izing the matrix representation of the Hamiltonian (1)in a basis set of angular momentum functions. In orderto facilitate the assignment of quantum numbers to theenergy levels, two different basis sets are employed, | ( I I ) IM I N M N i (spin-coupled basis); (5) | ( I I ) IN F M F i (fully coupled basis) . (6)where I and F are the total nuclear spin and total angularmomentum quantum numbers and M I and M F are theirprojections onto the Z axis defined by the external field.The matrix elements of the different terms in the Hamil-tonian in each of the basis sets are calculated throughstandard angular momentum techniques [25]. Explicitexpressions are given in the Appendix.For homonuclear molecules, nuclear exchange symme-try dictates that not all possible values of the total nu-clear spin I can exist for each rotational level. For mol-ecules in Σ + states, only even I values can exist foreven N and only odd I for odd N . This is true for ei-ther fermionic or bosonic nuclei but is reversed for Σ + states. Table II summarizes the I - N pairs compatiblewith the antisymmetry of the wave function under nu-clear exchange for the Rb isotopomers. A. Zeeman splitting for N = 0 homonuclear alkalidimers The Zeeman splittings for the N = 0 hyperfine levelsof Rb and Rb are shown in figure 1. The zero-fieldsplittings are in most respects similar to those found for TABLE I: Nuclear quadrupole moment ( Q ), electric quadrupole coupling constant ( eqQ ), nuclear g -factor ( g ), spin-rotationcoupling constant ( c ), tensor spin-spin coupling constant ( c ), scalar spin-spin coupling constant ( c ), absolute value of the c / ( eqQ ) ratio, isotropic part of the nuclear shielding ( σ ) and rotational g -factor ( g r ) for the homonuclear alkali dimers. Allthe quantities except the nuclear quadrupole moments, the nuclear g -factors and the rotational g -factors were evaluated usingDFT calculations (see section III). Both experimental [17] and theoretical results are presented for Na . Q (fm ) eqQ (MHz) g c (Hz) c (Hz) c (Hz) | c / ( eqQ ) | σ (ppm) g r6 Li -0.082 0.00123 0.822 161 137 32 0.026 102 0.1259 Li -4.06 0.0608 2.171 365 955 226 0.0037 102 0.1080 Na (Exp.) 10.45 -0.459 1.479 243 303 1067 0.0023 — 0.0386 Na (DFT) — -0.456 — 299 298 1358 0.0030 613 — K K -7.3 0.362 -0.324 -42 8 163 0.00045 1313 0.0207 K Rb Rb Cs -0.355 0.0486 0.738 96 119 12993 0.27 6461 0.0054TABLE II: Values of I permitted by the nuclear exchangesymmetry for even and odd rotational levels of Rb and Rb . N I even 0, 2, 4 Rb ( I Rb = 5 /
2) odd 1, 3, 5even 0, 2 Rb ( I Rb = 3 /
2) odd 1, 3 heteronuclear molecules in the ground rotational state[11]. The similarities can be summarized as follows: • The scalar nuclear spin-spin interaction and the nu-clear Zeeman effect are the only two terms in themolecular Hamiltonian with nonzero diagonal ele-ments for N = 0. • The electric quadrupole and the tensor nuclearspin-spin interactions are not diagonal in N , cou-pling the N , N + 2 and N − eQq and c are very much smaller thanthe rotational spacings, so that in practice it is ad-equate to include one excited rotational level. Con-vergence for N = 0 is reached with N max = 2 andconvergence for N = 1 is reached with N max = 3. • The scalar spin-spin interaction is diagonal in boththe spin-coupled and fully coupled basis sets, which for N = 0 are identical, h ( N = 0) IM I | c I · I | ( N = 0) IM I i = 12 c [ I ( I + 1) − I Rb ( I Rb + 1)] . (7)Except for a very small contribution coming fromthe coupling with N = 2 levels, these diagonal ele-ments determine the zero-field splitting.Despite the similarity of the zero-field levels, there areimportant differences between the Zeeman splittings forheteronuclear and homonuclear molecules. For heteronu-clear dimers [11], levels with the same M I but different I exhibit avoided crossings as a function of magnetic field.Because of this, I is no longer a good quantum numberat high field but the individual nuclear spin projections M I and M I become nearly conserved. For homonucleardimers, however, different energy levels that correspondto the same value of M I are parallel, so that no avoidedcrossings appear as a function of the field. Both I and M I remain good quantum numbers regardless of the value ofthe magnetic field but M I and M I are not individuallyconserved. This is illustrated in figure 1. It arises becausethe nuclear Zeeman term, which is the only nondiagonalterm for N = 0 in the heteronuclear case, is diagonal forhomonuclear molecules. Its nonzero elements are givenby equation A.11 of the Appendix, h ( N = 0) IM I | H IZ | ( N = 0) IM I i = − g Rb µ N B Z (1 − σ Rb ) M I . (8)The nuclear Zeeman term is diagonal because the g -factors of the two nuclei are equal and not because ofnuclear exchange symmetry. The N = 0 block of themolecular Hamiltonian for a heteronuclear dimer withtwo identical nuclear g -factors would also be diagonal.The conservation of the total nuclear spin I and non-conservation of M I and M I at high fields may have I I M I Magnetic Field (G) E n e r g y ( k H z ) Rb c c I I M I Magnetic Field (G) E n e r g y ( k H z ) Rb c FIG. 1: Zeeman splitting of hyperfine levels for N = 0 statesof Rb (upper panel) and Rb (lower panel). important consequences for the selection rules in spec-troscopic transitions used to produce ultracold moleculesand for the collisional stability of molecules in excitedhyperfine states. B. Zeeman splitting for N = 1 homonuclear alkalidimers Ultracold homonuclear molecules in N = 1 states areparticularly interesting because they are likely to be sta-ble with respect to inelastic collisions to produce N = 0,at least for collisions with non-magnetic species suchas other molecules in Σ states. Such collisions can-not change the nuclear spin symmetry and thus can-not change N from odd to even. Inelastic collisionsmay well be stronger for collisions of molecules in tripletstates, because of magnetic interactions between electronand nuclear spins. Transitions between odd and evenrotational levels are permitted in atom-exchange colli-sions, such as occur in collisions with alkali metal atoms[26, 27, 28, 29, 30, 31].Homonuclear molecules do not possess electric dipolemoments but do have quadrupole moments. The quadrupole-quadrupole interaction is anisotropic and isproportional to R − , so is longer-range than the R − dis-persion interaction that acts between neutral atoms andmolecules. The quadrupole-quadrupole interaction aver-ages to zero for rotationless states ( N = 0), but not for N >
0. Quantum gases of rotating homonuclear mole-cules may thus exhibit anisotropic effects.For
N >
0, all the terms in the Hamiltonian (1) havematrix elements diagonal in N . Some of these are nondi-agonal in hyperfine quantum numbers, so the energy levelpatterns are much more intricate. The zero-field splittingis dominated in most cases by the electric quadrupoleinteraction and the scalar nuclear spin-spin term. Theremaining constants ( c and c ) make much smaller con-tributions except for the two Li isotopomers. For Li ,all the terms in the hyperfine hamiltonian contribute sig-nificantly. For Li , the splitting is dominated by theelectric quadrupole interaction but contributions from allthe remaining terms are significant.Figure 2 shows the “building-up” of the zero-field N = 1 hyperfine energy levels for Rb and Rb inthree steps: first, only the rotational and the electricquadrupole terms are considered; secondly, the scalarspin-spin interaction is included; and thirdly, the spin-rotation and the tensor spin-spin interaction terms areadded to complete the hyperfine Hamiltonian. For Rb ,the electric quadrupole term alone determines the energylevel pattern, while for Rb there is a significant addi-tional contribution from the scalar spin-spin interaction,attributable to the relatively large value of c for thismolecule (see table I).The quantum numbers that label the zero-field energylevels are included in figure 2. The total angular mo-mentum quantum number F is always a good quantumnumber at zero field. In some cases, when there is onlyone pair of values I and N that can couple to give the re-sultant F , I is also a good quantum number. Otherwise, I is mixed and the values given in figure 2 are ordered ac-cording to their contribution to the eigenstate: the firstquantum number listed identifies the largest contribu-tion.The Zeeman splittings for N = 1 states of Rb and Rb for different ranges of magnetic fields are shown infigures 3 and 4. Each zero-field level splits into 2 F + 1states with different projection quantum numbers M F .Although in principle both the nuclear ( H IZ ) and the ro-tational ( H NZ ) Zeeman terms contribute to the splitting, g Rb ≫ g r so that the rotational Zeeman term contributesonly about 1% for Rb and less than 0.5% for Rb .In contrast with the N = 0 case, the Hamiltonian for N = 1 is not diagonal and energy levels correspondingto the same M F value display avoided crossings. Themagnetic field values at which the avoided crossings arefound, between 0 and 2000 G for Rb (lower panel offigure 4) and between 0 and 200 G for Rb (lower panelof figure 3), scale with the ratio between the electricquadrupole constant and the nuclear g -factor.For larger magnetic fields, M I and M N become indi- E n e r g y ( k H z ) Rb H rot + H Q H rot + H Q + H sc H rot + H hf F I E n e r g y ( k H z ) Rb H rot + H Q H rot + H Q + H sc H rot + H hf F I FIG. 2: Zero-field hyperfine splitting for N = 1 states of Rb (upper panel) and Rb (lower panel). For eachspecies, the hyperfine energy levels obtained when only the ro-tational and electric quadrupole terms are included are showin the left column. The effect of adding the scalar spin-spininteraction is displayed in the central column and, finally, theright column shows the splitting when the whole hyperfineHamiltonian is considered. All the energies are referred tothe lowest hyperfine level for the complete Hamiltonian. vidually good quantum numbers and the energy levelscorresponding to the same value of M I gather together.Both features are illustrated in figure 4 where, for thesake of clarity, the values of M N are included only in thelower panel. Equation A.11 shows that the matrix repre-sentation of the nuclear Zeeman term in the spin-coupledbasis is diagonal with nonzero elements proportional to M I and independent of any other quantum number. Asthe magnetic field increases the nuclear Zeeman termsbecomes dominant and the slope of the energy levels isdetermined by M I .The results in figure 4 neglect the diamagnetic Zee-man interaction, which is not completely negligible atthe highest fields considered (up to 4000 G). The justifi-cation for this is as follows. The Hamiltonian for the dia-magnetic Zeeman interaction [12] consists of two termsproportional to the square of the magnetic field: one de-pending on the trace of the magnetizability tensor andthe other is proportional to its anisotropic part. The first F Magnetic Field (G) E n e r g y ( k H z ) Rb F Magnetic Field (G) E n e r g y ( k H z ) Rb M F = 0 F = 4 F = 1 FIG. 3: Zeeman splitting for N = 1 states of Rb (upperpanel) and Rb (lower panel). The inset for Rb showsthe avoided crossing of M F = 0 states. term has a value around 200 kHz at 4000 G for Rb .Although this quantity is not negligible, it has not beenincluded because it simply shifts all the energy levels bythe same amount and has no effect on splittings. Thesecond term is diagonal in the spin-coupled basis set andits nonzero elements depend on N and M N . For Rb at 4000 G it would shift the energy levels by about 15kHz. It is therefore very small compared to the nuclearZeeman effect. V. CONCLUSIONS
We have explored the hyperfine energy levels and Zee-man splitting patterns for low-lying rovibrational statesof homonuclear alkali-metal dimers in Σ states. We havecalculated the nuclear hyperfine coupling constants for allcommon isotopic species of the homonuclear dimers fromLi to Cs and explored the energy level patterns in detail -8000-4000040008000 Magnetic Field (G) E n e r g y ( k H z ) Rb M I -5-4-3-2-1012345 Magnetic Field (G) E n e r g y ( k H z ) Rb ( M F = 0) M I M N -1 10 01 -1 FIG. 4: Zeeman splitting for N = 1 states of Rb . Theavoided crossings for M F = 0 are shown in the lower panel. for Rb and Rb .For rotationless molecules ( N = 0 states), the zero-field splitting arises almost entirely from the scalar nu-clear spin-spin coupling. The levels are characterized bya total nuclear spin quantum number I and states withdifferent values of I are separated by amounts between 90 Hz for K and 160 kHz for Cs . When a magneticfield is applied, each level splits into 2 I + 1 componentsbut all the levels with a particular value of M I are paral-lel. This is different from the heteronuclear case, and forhomonuclear molecules I remains a good quantum num-ber in a magnetic field. However, the projection quantumnumbers M I and M I for individual nuclei do not be-come nearly good quantum numbers at high fields forhomonuclear molecules. These differences in quantumnumbers may have important consequences for spectro-scopic selection rules and for the collisional stability ofmolecules in excited hyperfine states.Molecules in excited rotational states are also of con-siderable interest. In particular, molecules in N = 1states may be collisionally stable because transitions be-tween even and odd rotational levels require a changein nuclear exchange symmetry. Molecules in excited ro-tational states have anisotropic quadrupole-quadrupoleinteractions that are longer-range than dispersion inter-actions. The hyperfine energy level patterns are consid-erably more complicated for N = 1 states than for N = 0states and I is not in general a good quantum numbereven at zero field.The results of the present paper will be importantin studies that produce ultracold molecules in low-lyingrovibrational levels, where it is important to understandand control the population of molecules in different hy-perfine states. Acknowledgments
The authors are grateful to EPSRC for funding of thecollaborative project QuDipMol under the ESF EURO-CORES Programme EuroQUAM and to the UK NationalCentre for Computational Chemistry Software for com-puter facilities.
APPENDIX: A
Explicit expressions for the matrix elements of the molecular Hamiltonian terms are now provided. The equationsare valid for homonuclear molecules.The matrix elements for the rotational term ( H rot ) are given by h N M N ( I I ) IM I | H rot | N ′ M N ′ ( I I ) I ′ M I ′ i = δ NN ′ δ M N M N ′ δ II ′ δ M I M I ′ B N ( N + 1) (A.1) h N ( I I ) IF M F | H sc | N ′ ( I I ) I ′ F ′ M F ′ i = δ NN ′ δ II ′ δ F F ′ δ M F M F ′ B N ( N + 1) . (A.2)The matrix elements for the electric quadrupole interaction ( H Q ) are given by h N M N ( I I ) IM I | H Q | N ′ M N ′ ( I I ) I ′ M I ′ i = ( eqQ ) − I { N + 1)(2 N ′ + 1)(2 I + 1)(2 I ′ + 1) } / × N N ′ ! I I − I I ! − ( I I I I ′ I ) × X F M F " ( − F +2 M F (2 F + 1) N I FM N M I − M F ! × N ′ I ′ FM N ′ M I ′ − M F ! ( I N FN ′ I ′ ) (A.3) h N ( I I ) IF M F | H Q | N ′ ( I I ) I ′ F ′ M F ′ i = δ F F ′ δ M F M F ′ ( eqQ ) − N + N ′ + F +2 I { N + 1)(2 N ′ + 1) } / × { (2 I + 1)(2 I ′ + 1) } / N N ′ ! I I − I I ! − ( I N FN ′ I ′ ) ( I I I I ′ I ) (A.4)The matrix elements for the spin-rotation interaction ( H IJ ) are given by h N M N ( I I ) IM I | H IJ | N ′ M N ′ ( I I ) I ′ M I ′ i = δ NN ′ δ II ′ c ( − N − M N − M I +2 I +1 I + 1) × { (2 N + 1) N ( N + 1)(2 I + 1) I ( I + 1) } / ( I I I I I ) × X p " ( − p N N − M N p M N ′ ! I I − M I − p M I ′ ! (A.5) h N ( I I ) IF M F | H IJ | N ′ ( I I ) I ′ F ′ M F ′ i = − δ NN ′ δ II ′ δ F F ′ δ M F M F ′ c ( − F +2 I − M F − I (2 F + 1) × (2 I + 1) { (2 N + 1) N ( N + 1)(2 I + 1) I ( I + 1) } / ( I I I I I ) × X F ′′ ,M F ′′ " ( − F ′′ + M F ′′ (2 F ′′ + 1) ( N F IF ′′ N ) × X p (cid:20) ( − p F F ′′ − M F p M F ′′ ! (cid:21) ( I F ′′ NF I ) (A.6)The matrix elements for the scalar nuclear spin-spin interaction ( H sc ) are given by h N M N ( I I ) IM I | H sc | N ′ M N ′ ( I I ) I ′ M I ′ i = δ NN ′ δ M N M N ′ δ II ′ δ M I M I ′ c [ I ( I + 1) − I ( I + 1)] (A.7) h N ( I I ) IF M F | H sc | N ′ ( I I ) I ′ F ′ M F ′ i = δ NN ′ δ II ′ δ F F ′ δ M F M F ′ c [ I ( I + 1) − I ( I + 1)] (A.8)The matrix elements for the tensor nuclear spin-spin interaction ( H t ) are given by h N M N ( I I ) IM I | H t | N ′ M N ′ ( I I ) I ′ M I ′ i = − c √ − I − M I − M N I ( I + 1)(2 I + 1) { (2 N + 1)(2 N ′ + 1) } / × { (2 I + 1)(2 I ′ + 1) } / N N ′ ! I I I I I I ′ × X p " ( − p N N ′ − M N p M N ′ ! I I ′ − M I − p M I ′ ! (A.9) h N ( I I ) IF M F | H t | N ′ ( I I ) I ′ F ′ M F ′ i = − δ F F ′ δ M F M F ′ c √ − N ′ + N + I + F I ( I + 1)(2 I + 1) × { (2 N + 1)(2 N ′ + 1)(2 I + 1)(2 I ′ + 1) } / N N ′ ! ( N I FI ′ N ′ ) I I I I I I ′ (A.10)The matrix elements for the nuclear Zeeman term ( H IZ ) are given by h N M N ( I I ) IM I | H IZ | N ′ M N ′ ( I I ) I ′ M I ′ i = − δ NN ′ δ M N M N ′ δ II ′ δ M I M I ′ g µ N B Z (1 − σ ) M I (A.11) h N ( I I ) IF M F | H IZ | N ′ ( I I ) I ′ F ′ M F ′ i = − δ NN ′ δ II ′ δ M F M F ′ g µ N B Z (1 − σ )( − F − M F × ( − N +2 I (2 I + 1) { F ′ + 1)(2 F + 1)(2 I + 1) I ( I + 1) } / × F F ′ − M F M F ! ( I F NF ′ I ) ( I I I I I ) (A.12)The matrix elements for the rotational Zeeman effect ( H NZ ) are given by h N M N ( I I ) IM I | H NZ | N ′ M N ′ ( I I ) I ′ M I ′ i = − δ NN ′ δ M N M N ′ δ II ′ δ M I M I ′ g r µ N B Z M N (A.13) h N ( I I ) IF M F | H NZ | N ′ ( I I ) I ′ F ′ M F ′ i = − δ NN ′ δ II ′ δ M F M F ′ g r µ N B Z ( − N + I + F + F ′ − M F +1 × { (2 F ′ + 1)(2 F + 1)(2 N + 1) N ( N + 1) } / F F ′ − M F M F ! ( N F IF ′ N ) (A.14) [1] S. Ospelkaus, A. Pe’er, K.K. Ni, J.J. Zirbel, B. Neyen-huis, S. Kotochigova, P.S. Julienne, J. 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