Predicted Landé g-factors for open shell diatomic molecules
aa r X i v : . [ phy s i c s . c h e m - ph ] J a n Predicted Land´e g -factors for open shell diatomicmolecules Mikhail Semenov, Sergei. N. Yurchenko, Jonathan Tennyson
Department of Physics and Astronomy, University College London, London, WC1E6BT, UK
Abstract
The program
Duo (Yurchenko et al. , Computer Phys. Comms., 202 (2016)262) provides direct solutions of the nuclear motion Schr¨odinger equationfor the (coupled) potential energy curves of open shell diatomic molecules.Wavefunctions from
Duo are used to compute Land´e g -factors valid for weakmagnetic fields; the results are compared with the idealized predictions ofboth Hund’s case (a) and Hund’s case (b) coupling schemes. Test calcula-tions are performed for AlO, NO, CrH and C . The computed g J ’s bothprovide a sensitive test of the underlying spectroscopic model used to rep-resent the system and an indication of whether states of the molecule arewell-represented by the either of the Hund’s cases considered. The compu-tation of Land´e g -factors is implemented as a standard option in the latestrelease of Duo . Keywords:
Diatomics, Zeeman, Lande factors, magnetic field, Hund’scases, ExoMol
Preprint submitted to Journal of Molecular Spectroscopy February 13, 2018 . Introduction
The lifting of the degeneracy of the energy levels in molecule by a mag-netic field is a well-known and well-studied phenomenon. Thus it has spawnedexperimental techniques such as laser magnetic resonance spectroscopy [1, 2],magnetic rotation spectroscopy [3], and optical Zeeman spectroscopy [4, 5].These techniques, for example, use the Zeeman effect to tune transitions inand out resonance by changing the applied magnetic field [6]. Zeeman effectscan also be probed directly using standard spectroscopic techniques to studymolecules in magnetic fields [7, 8, 9]. In a similar fashion, Zeeman effectsare increasingly being used to form, probe and trap molecules at ultra-coldtemperatures [10] for example by use of magnetically tunable Feshbach res-onances [11].Spectral shifts and splittings provide a remote sensing technique withwhich to study the Universe. Zeeman splitting of molecular spectra areactively being used to probe magnetic fields in a variety of astronomicalenvironments including sunspots [12, 13, 14], starspots [15], white dwarfs[16, 17] M-dwarfs [18] and potentially exoplanets [19].The Zeeman splitting patterns of the spectrum of an open shell diatomicmolecule can be calculated in a straightforward fashion provided that thequantum numbers characterizing states in question are known and are con-served. However, there are circumstances, such as resonance interactionsbetween nearby states via spin-orbit or other couplings where the quantumnumbers used to specify the electronic state associated with a given level arenot precisely conserved. In this case evaluation of Zeeman splitting as repre-sented by the Land´e g -factor is not straightforward and requires a numerical2reatment. It is such a treatment which is the focus of the present article.The important advantage of the Zeeman effect is that the associatedsplitting can be made large enough to separate otherwise degenerate spin-components (Λ-doublet). Moreover, the measurement of the g values can bemore accurate than the energy spacing [20].The Zeeman methodology for diatomics was introduced by Schade [21].Stolyarov et al. [22] investigated the perturbations in the calculation of theLand´e factors caused by interactions with other electronic states. In a veryrecent theoretical work Borkov et al. [23] presented a numerical model ofZeeman splitting based on the use of effective molecular Hamiltonians.Le Roy’s LEVEL [24] has become the program of choice for solving the di-atomic nuclear motion problem. However,
LEVEL can only treat open shellmolecules in limited circumstances [25, 26] and does not consider the couplingbetween states by spin-orbit and related effects which can have an importanteffect of the g -factors. For this reason we have written our own diatomicnuclear motion code Duo [27].
Duo explicitly treats open shell systems andcan allow for coupling between the various states involved. In this context,of particular interest to us are the many open-shell diatomic molecules whichare known to be present, or may be present, in hot astronomical atmospheressuch as those found in cool stars and exoplanets. Such species are being stud-ied as part of the ExoMol project [28]. Zeeman splittings in these moleculescan provide useful information on the magnetic fields present in these dis-tant bodies. So far ExoMol has created spectroscopic models for a numberof open shell diatomic species [26, 29, 30, 31, 32, 33, 34, 35, 36].In this paper we present extension to
Duo which allows Land´e g -factors to3e computed for individual states of open shell systems. As initial exampleswe focus on four systems studied by ExoMol, namely AlO [29], NO [37],CrH [33] and C [38, 39]. These systems were selected as ones of interest forExoMol and for which there are laboratory Zeeman spectra. These laboratorystudies are discussed below.
2. Theory
Discussions of the underlying theory and methodology used in
Duo isgiven elsewhere [27, 40, 41] so only key points are considered below.
Duo solves the diatomic nuclear motion problem using a Hund’s case (a) basis.This does not represent an approximation even for molecules poorly repre-sented by Hund’s case (a) since a complete set of angular momentum func-tions are used for a given total angular momentum, J . We note that thesame choice has been adopted by others [42, 43].The basis set used by Duo can be written as | n i = | state , J, Ω , Λ , S, Σ , v i = | state , Λ , S, Σ i| J, Ω , M i| state , v i , (1)where Λ is the projection of electron angular momentum on the molecu-lar axis; S is the electron spin quantum number with projection Σ along themolecular axis and Ω is corresponding projection of J . The vibrational quan-tum number is given by v and the label ‘state’ is used to denote the electronicstate which is required for both the state-dependent angualar momenta andthe vibrational state. M is the projection of the total angular momentumalong the laboratory axis Z and is therefore the magnetic quantum numberwhich quantizes the splitting of the levels in a weak magnetic field. Finally,4 n i is simply a compound index representing the various quantum numbers.These basis functions are symmetrized to give a definite parity, τ . Only J and τ are conserved quantum numbers with the addition of u/g for homonuclearmolecules. Duo obtains the wavefunctions for a given nuclear motion problem bydiagonalizing a coupled-states Hamiltonian. These wavefunctions, φ Jτλ , arethen given by φ Jτλ = X n C Jτλn | n i , (2)where λ denotes the electronic state.In the case of weak magnetic fields, the Zeeman splitting can be approx-imated by ∆ E B = g J M µ B, (3)where ∆ E B is the shift in energy of a state with total angular momentum J and projection of J along the field direction is M , g J is the Land´e factor, µ is the Bohr magneton, B is the magnetic field. Within a Hund’s case (a)representation, the Land´e g -factor is given by [20, 44]: g (a) J = ( g L Λ + g S Σ)Ω J ( J + 1) (4)where g S and g L are the standard electron spin and orbital g -factors respec-tively. If Λ and Σ are conserved quantities for a given rovibronic state thenthis expression is analytic; below this will be known as the QN(a) approxi-mation meaning good quantum numbers in Hund’s case (a).The good-quantum number approximation has been also introduced andused in the case of the NiH spectroscopy by Gray et al [20]. Here we used g J
5s a total Land´e factor which includes all other contributions from the elec-tron spin and orbital angular momenta; this is different from the definitionconventionally used in Zeeman experimental studies, see, for example, Gray et al [20].The corresponding expression for the Hund’s case (b) Land´e factor canbe approximated by Berdyugina and Solanki [44] g (b) J = g L J ( J + 1) (cid:26) Λ [ J ( J + 1) + N ( N + 1) − S ( S + 1)] N ( N + 1) (cid:27) + g S J ( J + 1) [ J ( J + 1) − N ( N + 1) + S ( S + 1)] , (5)where N is the rotational quantum number. If Λ and N are conserved quan-tities for a given rovibronic state then this expression is analytic; below thiswill be known as the QN(b) approximation meaning good quantum numbersin Hund’s case (b).The intermediate (and more general) case can be modeled using the G matrix with the following matrix elements [44]: G Σ , Σ = ( g L Λ + g S Σ)Ω J ( J + 1) , (6) G Σ , Σ ± = g S p S ( S + 1) − Σ(Σ ± p J ( J + 1) − Ω(Ω ± J ( J + 1) δ v,v ′ δ Λ , Λ ′ δ S,S ′ . (7)In practice Λ and Σ are not generally conserved when spin-orbit and othercurve coupling effects are taken into account. In this case one can use the Duo wavefunctions to compute g J for a given rovibronic state by averagingover the corresponding wavefunction as given by G DuoΣ , Σ = X n | C Jτλn | ( g L Λ n + g S Σ n )Ω n J ( J + 1) , (8)6 DuoΣ , Σ ± = X n X n ′ C Jτλn ∗ C Jτλn ′ δ v,v ′ δ Λ , Λ ′ δ S,S ′ × g S p S n ( S n + 1) − Σ n (Σ n ± p J ( J + 1) − Ω n (Ω n ± J ( J + 1) , (9)where S n , Λ n and Σ n are the values of S , Λ and Σ taken in basis function | n i and Ω n = Λ n + Σ n . case (a). In the following we also assume g L = 1 and g S = 2 . g J ’s eval-uated using Duo wavefunctions and using the QN approximation as givenby. ∆ g (x) J = g Duo J − g (x) J . (10)where (x) is (a) or (b) as appropriate.
3. Results
As an initial test case Land´e g -factors were computed for aluminium oxide(AlO). The spectroscopic model used for AlO is due to Patrascu et al. [29, 40]which is based on ab initio curves tuned to reproduce the extensive set ofexperimental spectra. The model comprises three electronic states: X Σ + ,A Π and B Σ + . The latter two states lie, respectively, 5406 and 20688cm − above the ground states. The closeness of the X and A states leads tosignificant mixing. A recent study of radiative lifetime [45] using this modelshowed strong effects due to X – A mixing but that the B state appearedlargely unperturbed.Figure 1 shows the difference between g J ’s evaluated using Duo wave-functions and using the QN approximation These results show systematic7ffects. Firstly, it would appear that the g J -factors for the X Σ + and B Σ + are significantly better represented in the Hund’s case (b) than case (a). Sec-ondly, the A Π appears closer to Hund’s case (a), although in this case thedifferences are smaller. Finally, there are a number perturbations caused bya coupling between levels in the X and A states. Such interactions have beennoted before [45].Changes due to the X – A state coupling appearing as well pronouncedstructures in Fig. 1 suggest that the B state g -factors are largely unchangedfrom the idealised values. In order to illustrate how the coupling betweendifferent states affect the values of the Land´e factors in Fig. 2 we show energycrossings between the X ( v = 8) and A ( v = 3, Σ = 1 / / J progressions of theLand´e factors (lower display) appear at the same J values (13.5 and 21.5) asthe two crossings. This is where the wavefuncitons are extremely mixed andthe quantum number approximation, Eq. (4), becomes very poor. The neteffect from the Duo model even in case of strongest couplings of AlO is stillrelatively small, of the order of 10 − for J = 50 see Fig. 3.Figure 3 shows the g J values computed using the three methods: Duowavefunctions, Hund’s case (a) and Hund’s case (b) approximations for theX ( v = 0) state. The character of g J changes from (b) to (a) as energyincreases, illustrating importance of the proper modelling of the Zeemaneffect.We should note the study by Gilka et al. [46], where the effects of cou-plings between orbital- and spin-angular momenta of the X and A states onthe g S values were also studied in their ab initio calculations of the g -tensor8f AlO. Figure 1: Difference between Land´e g -factors obtained for AlO using Duo wavefunctionsand the QN approximations.
The nitric oxide (NO) molecule provides a rather simple test of ourmethodology. McConkey et al. [37] recently constructed a spectroscopicmodel and generated the associated line list for NO considering only theX Π electronic ground state. McConkey et al. consider all 6 major isotopo-logues of the system; here we restrict ourselves to N O for which thereare some limited, experimental studies on the behavior of its ground statero-vibrational transitions in a magnetic field, albeit a relatively strong one[47]. These observations have been subject of recent models [23] which showthat for most of the field strengths considered it was necessary to move be-yond the linear Zeeman effect considered here. We note the Zeeman effectin NO has also been used to ascertain the distribution of NO in the Martianatmosphere [48].The spectroscopic model of McConkey et al. [37] made extensive use ofexperimental data in determining both the shape of the X Π potential energy9 X , v = , f X , v = , e A , v = , E - . J ( J + ) / c m - A, v=3,
A, v=3, X, v=8, e X, v=8, f A, v=3, g ( a ) J J Figure 2: Reduced energy term values of AlO in the region of the crossing between X, v = 8 and A, v = 3 (upper display) and difference between Land´e g -factors obtained forAlO using Duo wavefunctions and the QN Hund’s case (a) approximation.
20 40 60 80 100 120 140 160 180 200-0.4-0.20.00.20.40.60.81.0
Hund’s case (a) Hund’s case (b)Duo intermediate g J Energy (cm -1 ) Figure 3: Land´e g -factors of AlO obtained using Duo wavefunctions with Eqs. (8,9) andthe QN approximation, Hund’s cases (a) and (b). curve and the various coupling terms. However even with a high qualityfit their model does not predict transition frequencies with the accuracyrequired for studying the relatively small Zeeman splittings. We thereforefollow McConkey et al. and adopt as our zero-field energy levels the empiricalvalues they determine.The experimental study of Ionin et al. [47] only considered in detail thesplitting of the Π / Q(2.5) fundamental transition at 1875.7228 cm − as afunction of magnetic field. There strategy was to use the magnetic field totune the transition into resonance with CO laser lines. Ionin et al. observed3 components of this transition, namely the ( M ′ , M ”) ones associated with(1 / , − / / , /
2) and (5 / , /
2) which they observed using the CO laserline 9 → − . Bringing the lines into resonancerequired magnetic fields of approximately 3.8, 4.2 and 5.6 T respectively.11trong fields are required since the g factors for the ground and first excitedvibrational states are fairly similar so the transition frequency only dependsweakly on B .If the Zeeman splittings were linear then the three lines considered wouldall lie at the same frequency. In practice this is only true up to about 2T and for fields above this value the quadratic Zeeman effects become in-creasingly important [47, 23]. The previous studies suggest that only the( M ′ , M ”) = (1 / , − /
2) transition frequency varies approximately linearlywith field strength in the 2 – 6 T region. Our calculations place this transitionat 1876.29 cm − for a field of 3.8 T, in excellent agreement with the obser-vations. This values are obtained from our calculated g J factors of 0.316625and 0.316857 for the Π / J = 2 . v = 0 and v = 1 respectively. The carbon dimer is a very well studied system [38, 49] whose spectrumis widely used for studying astronomical and terrestrial plasmas. The manysystems of C electronic bands are well-known to have many perturbationsdue to couplings between states, something that should be reflected in theLand´e g -factors.The ExoMol model for C considers the eight lowest electronic states ofC : X Σ +g , A Π u , B ∆ g , B ′ Σ +g , a Π u , b Σ − g , c Σ +u , and d Π g . Dueto strong interactions between rovibronic states in this system especiallyat high rotational excitations, the quantum numbers becomes meaningfuland therefore very difficult to correlate between Hund’s cases. Therefore werestrict ourselves to states with J ≤
35 and energies up to 35,000 cm − .Figure 4 gives an overview of our computed Land´e g -factors for C . It is12 igure 4: Difference between Land´e g -factors obtained for C using Duo wavefunctionsand the QN approximation. clear that for this system the changes caused by coupling between states arelarge.Apart from the characteristic spikes as in the case of AlO, these Land´efactors show that these deviated values also build well defined horizontalpatterns. These patterns should indicate a deviation of the C spectra fromHund’s case (a). The transition from Hund’s cases (a) and (b) for different J is a well-known issue in the analysis of the rovibronic spectra of C , see,for example, [50].
4. CrH
As a third example we consider the CrH molecule, another astronomicallyimportant species. The ExoMol model [33] for this system considers thelowest 8 electronic states: X Σ + , a Σ + , A Σ + , B Π, b Π, C ∆, c ∆and the lowest dissociative Σ + state. Here we consider states with J ≤
35 and energies up to 35,000 cm − . As the model used does not considerelectronic states with thresholds above 20,000 cm − , where new states become13ncreasingly dense, it is likely that our calculations will underestimate theperturbation of the g -factors in this region. However even below 10,000 cm − ,where the states all belong to X Σ + electronic state the perturbations arefairly large.Figure 5 gives an overview of our computed Land´e g -factors for the low-est two sextet states of CrH compared to the QN approximations. Clearlythere are lots of structures. For the X Σ + state, QN (a) appears a poorapproximation in all cases; QN (b) does better but still shows pronouncedstructures starting at about 4000 cm − . Above about 14000 cm − all X state g J factors appear highly perturbed. This is also true for the singlet A state.Chen et al. [6] measured effective g values for a few levels in CrH; Table 1compares these measurements (A Σ + , v = 1) with our results. This tablealso shows that the Hund’s case (b) approximation to g J is more appropriatethan case (a). Chen et al. [6] also reported an averaged g value for theA Σ + , v = 0 state over the measured values for four states J = 3 / , N = 1, J = 5 / , N = 0, J = 5 / , N = 1, J = 7 / , N = 1 of 2.0081(20). Using theDuo g J values to produce the same averaging we obtained 1.9976, which isin very close agreement to the experiment. The agreement for other levels isnot as good, suggesting that our model for CrH needs further improvement.
5. Conclusion
We have developed a numerical procedure for evaluating Land´e g -factorsfor diatomic molecules without making any assumptions about conservedquantum numbers. This method is tested for four molecules AlO, NO, C and CrH. It would seem that besides making predictions for g -factors, the14 -1 )-1012 ∆ g J ( x ) A Σ + case (a)A Σ + case (b)X Σ + case (a)X Σ + case (b) Figure 5: Difference between Land´e g -factors obtained for CrH using Duo wavefunctionsand the QN approximation.Table 1: Comparison between measured values of g J for CrH due to Chen et al . [6] andour calculated values using the Duo wavefunction approach, Hund’s cases (a) and (b)approximations. state [6] Duo QN(a) QN(b) N = 0 , J = 5 / N = 1 , J = 3 / N = 1 , J = 5 / N = 1 , J = 7 / v=3v=2v=0 g (a)J g (b)J g ( x ) J Energy (cm -1 )v=1 Figure 6: Difference between Land´e g -factors obtained for CrH using Duo wavefunctions,the QN case (a) and (b) approximations. comparison between our computed value and the value predicted under theassumption of particular Hund’s case gives a clear means of distinguishingthose levels which are best represented by Hund’s case (a) from those whichare approximately Hund’s case (b)-like.The accuracy of our predicted g J factors depend on a number of factors:(a) the accuracy of the underlying spectroscopic model used and, in particu-lar, its ability to reproduce coupling between different electronic states, (b)our ability to solve this model by, for instance, converging the basis set repre-sentation and (c) any assumptions made about angular momentum couplingswithin the system. Although our procedure is based on a Hund’s case (a)coupling scheme, our general formulation means that no actual approxima-tions are made by adopting a (complete) basis formulated within this scheme.Similarly it is relatively easy, and computationally cheap, to use large vibra-16ional basis sets when converging the problem. This means that the choiceof spectroscopic model is likely to be the major source of uncertainty in ourcalculations or, conversely, that available measurements of Land´e g -factorshave the potential to be used to improve the spectroscopic model. In additionwe note that our formulation is only appropriate when the changes dependlinearly with the magnetic field. Inclusion of non-linear effects require a moresophisticated treatment which we plan to study in future in work.It is our plan to routinely compute Land´e g -factors for all open shelldiatomic species studied as part of the ExoMol project from now on. Tothis end, the new ExoMol data format [51] has been adjusted to include thecomputed values of g J for each state as part of the states file made availablefor each isotopologue studied. Acknowledgements
We thank Patrick Crozet for drawing our attention to mistakes in thepublished version of this article. This work was supported by the EuropeanResearch Council under Advanced Investigator Project 267219 and the COSTaction MOLIM (CM1405).
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