Jonathan Tennyson and Ad Van Der Avoird- Effects of the Potential Anisotropy of the Calculated Fine-Structure Spectrum of O2He

8/3/2019 Jonathan Tennyson and Ad Van Der Avoird- Effects of the Potential Anisotropy of the Calculated Fine-Structure Spect

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Volume 105. number 1 CHEMICAL PHYSICS LETTERS 2 hiarch 19

EFFECTS OF THE POTENTIAL ANISOTROPYON THE CALCULATED FINE-STRUCTURE SPECTRUM OF 02HeJona than TENNYSONSERC, Daresbury Laboratory. Darerbury. IVam&yron, Cheshire WA4 4AD. UK

andAd VAN DER AVOIRDIllstirule of Theoretical Chemirt~. lhlive~i@ of Nijmegen. Toernooiveld, Nvmegen, The NetherlandsReceived 20 September 1983; in final form 19 December 1983

Ro-vibrational-spin calculatrons are performed on the OzHe van der Waak molecule using a recent empirical potentralenergy surface AU the bound states of the complex are presented, as are the fme-structure spectrum and weak-fiel d Zee-man splittings of the low-lying states The V , an&tropic term is sometimes found to have a signif imt (-10%) effectwhitst that of the hgher anisotropic terms is negligib le ( 2_ Faubel et al_ [5] recently published an empirical He-02 anisotropic interaction ptential which supplements the isotropic potential oBattagha et al. [6]_ This potential, which was obtained f rom a multiproperty fit to a variety of expemen& dara [3,61, contaiYns rerms up to A = $.

In this wark we apply the approach of Tennyson Mettes [l] to the fine-structure spectrum and weakfield Zeeman splittings of UzNe, pa+ng particularattention to the effect of truncating the potential epansion (1). In doing this we hope to assess the reh-ability of 02Ar calculations. based as they are, on atruncated potential This result is of practicalinterestti vrew of the problems of finding the lines in thebeam resonance experiments currently in progress [

For 02He the barrier to internal rotation is only12 cm-l , which, as we shall show, is less than thezerc+point energy of the ground state. Therefore, thww Qz Q$bz Js. zQC i d.!z&& ia Goz&&tit& for rnSt.anS~ uzA5 which is (probably) Ioc&sein the T-shaped potential well [S]. The model of va

0 009-2614/84/S 03.00 0 Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)


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Volume 105. number 1 CHEMICAL PHYSICS LE-ITERS 2 March 19der Avoird [4] i s, in principle, less appropriate for02He than for 02Ar, because of the lower rotationbamer. We shall demonstrate, however, that theresults of the present calculations on 02He can bewell understood on the basis of this model_

2. Calculations and resultsFor details of the method used the reader is refer-

red to the paper of Tennyson and Mettes [ 1). Thrsmethod employs a coupled spherical harmonic basisfor the angular coordinat= that can be writtenI((N.S)&C) JMJX INMp,r) represents the 02 rotations(by Bose-Emstein statistics N must be odd) andISI&) the angular momentum of the spin, for theground 3X; state of oxygen (S = 1). These couple toUMj), the total angular momentum quantum num-bers of pure 02 [VI. In the dimer I@_) couples wrthILML), the end-over-end rotations of the complex, togive IJMJX J, MJ and the parity of Iv + L are the onlygood quantum numbers of the complex

The calculations on He-02 used all angular func-tions withN G 7 which couple to a particular 1 MJ)_For the R stretching coordinate eight numerical f unc-tions were used These were the lowest f unctions generated by solving a model one-dimensional problembased on Vu [lo]- As Vu itself only supports onebound state, it was found necessary to lower Vu inthe model problem by 0.004 Eb for all R =Z13 4u,This;pproach was found to give only a small (O.OOScm ) error in the calcuked energy of the loweststate of Y~J, and proved a satisfactory method of gen-erating a basis set

Table 1 gives the energies of the 35 (36 when A =2) bound states of 02He obtained by these calcula-tions As the van der Waals minimum lies at 23.5cm: [5], all the bound states of the system lie abovehalf way to dissociation.

The levels calculated using the potential of Faubelet al. [S] truncated at A = 4 are very similar to thosecalculated with the full A = 8 potenti al_ However, theleve ls with A = 4 are shifted by up to 0.4 cm- (12GHz) in comparison with those computed with onlythe leading anisotropic term, A = 2.

The (relative) positions of 012 the bound-state levels,as listed in table 1, can be understood via van derAvoirds model [4] _This model is based on the obser-50

Table 1Calculated rotatiorrvibration-spin level s ) for On-He UIcm- as a function of the terms in the Legendre series 1)J Parity b) A=2 A=4 A=8

01

2

3

4501

2

3

4

56

ee

e

e

-6 160-9.621-6.708-4.529-3.086-7.948-5.719-3.897-2.081-7 465-6.265-4.147-1.860-4.667-2.468-4_064-1.163

-10.191-4.766-6.620-5.417-3.614-8.629-7.913-5.747-3.509-1.144-6.580-4347-1.720-5.986-3.962-1-952-2_233-0.095-1.689

-5.751-9.270-6 536-4.512-3.148-7.545-5.378-3.771-2.055-7.132-6.033-3.994-1.773-4.372-2.235-3.814-1.031-9.842-4.729-6 277-5.199-3.688-R-255-7.586-5.604-3.445-1.147-6.231-4.058-1624-5.730-3.780-1.786-1.997

-5.761-9.273-6.533-4.50R-3.143-7.549-5 380-3.766-2.051-7.135-6.033-3.994-1.771-4.374-2.236-3.874-1.030-9.8444.730-6.278-5.196-3.681-8.259-7 588-5.603-3 444-1.145-6.233-4.060-1.620-5-730-3.780-1.787-1-996

--1.553 -1.5529)All energies elative to dissociated He and O2 in its j = 1stateb)e=even o-odd.vation that the N= 1 rotational ground state of pur02 is split into ]MN] = 1 levels, which are lowered bthe anisotropic 02-X potential, and ]MN] = 0 levelwhich are raised (here MN refers to the component


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Volume 105. number 1 CHEMICAL PHYSICS LETTERS 2 March 19

N on the body-fiied L axis, lying along the vector R ).T h i s splitting destroys the 02 angular momentumj=N+ S as a constant of the motion. However, the zcomponent ofj (which rS the same as the z componeniof J =j + t) remains an approximately good quantumnumber (K). The 02X levels orlgmating from thelM~l= 1 states of 02 can be ordered in four distinctladders, each labelled by this quantum number K anda parity index p; the overall panty of the states isgiven by (-l)@/_ Those four ladders comprise 27of the 35 bound states of OzHe, while the remauung8 states are shown to form two additional ladders,with IKl= 0 and IKl= 1, onginating from the IMnl= 0 states of 02 (see fig. 1). According to the model[4], the positions of the rungs on the ladders are de-termined by the expressionBJ(J+ 1) where J 1s theoverall angular momentum quantum number (J ZIK I) and E is the end-over-end rotational constant of02X, . For 02He we find that B F= 7.5E km-1 Labeling .panly EIGHZI

0 c IL- I z-

6---7 t .- L_- !I_- I 3*_-I --z*2_ Ir___-

1.l*ijr- III=1

PR 2 , i s e a s i ly in c o rp o ra t e d inthese corrections; it does not occur in the ftrst-orderesults.

Table 2 presents fme-structure transition frequecies for the low-lying states of OzHe. Whilst the sptra obtained w i th A = 4 a n d A = 8 a re v e ry s imi la r ,th e t r a n s i t io n s c a lc u la t e d w i th A = 2 a re s h i f t e d . Ts izes of these shifts seem rather irregular: some frequencies are hardly altered by the terms with A > other f requencies are shifted by as much as 10%.Looking at the shifts of the levels m fig. 1, howevewe can observe the systematics. The (upward) shiftare largest for the IMNI = 1 ladders; some of thelM~l= 0 levels even shift d o w n w a rd s . Wi th in th ef i rs t group the shifts of the even p states in the IKIand IKI = 1 ladders are smaller. Moreover, they genally decrease for tie higher levels.

In a magnetic field the II + 1 degeneracy of eacstate is lif ted. To f irst order (or in a weak field), thsplittings between the levels is given by g,p~ CS,> &


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Volume 105. number 1 CHEMICALPHYSICS LETTERS 2 March 19Table 2Fxquency of the allowed (ai= 0, al, parity conserved) fu t~ st ru c tu re b a r &io n s in th e lo w -ly in g s t a t e s o f OlHe as a functionthe terms in the Legendre series (1). Energies of the lower state cakulated with A = 8 are given for comparison

J Parity - Transition frequency (GHz) a) Energy (cm-)

Ii> II ) n=2 A=4 A=82 .2 01 2 e2 1 02 3 02 2 01 i e2 1 01 0 e0 1 o1 2 e2 3 02 1 00 1 01 1 e2 2 00 0 01 2 eI 1 e0 1 0

21.47 20.0550-13 51-7260.24 59.3061-43 60-6986.40 794687-31 8215

- 96.30 91.63103.74 105.33107_06 106.88116.97 116.69128.37 125.82150.34 136.91132.12 139.20152.65 142.66153.49 144.19162 64 153-29171.60 164-M195.91 183.53197.15 184.49

20.10 -8.25951.76 -9.27359.37 -8.25960.71 -8-25979.61 -8.25982.15 -9-27391.82 -8.259

10529 -9.273106.92 -9A44116.70 P-273125.88 -8-259137.23 -8 259139.37 -9 844142.85 -9.273144.33 -8.259153.33 -9.844165.08 -9-273183.75 -9.273184.78 -9.844

a) cm- = 29.97925 GHZif tire 2 axis is taken as the direc tion of the magneticfield B. Table 3 gives cSr> values for the low-lyingstates of 02He. Again the splittings with A= 4 andA=8areverysimiiarandthosewithA=2~~woccasional marked differences: 3640% for the low-est two J = 1 (even) states.Table 3-Frrst-cnder weak-field) Zeeman split tings of the low-lyingstates of *He calculated as a function of the terms in theLegendre series (1). The energy of the unperturbwl stateWith A = 8 is given for c omparisonJ parity Energy G,@=rn--) n=2 _n=4 .A=81 e 9.273 0.07$8 0.0579 0.0584 -1 e 6533 0.14911 02618 0.25902 e 7549 0.4280 0_4275 0.4275-3 e 7:135 O-2737 0.2805 ..0.28031 0 6 278 0.4403 O-4176. 041872 o_ 8.259 O-2209 ~0.1959 0.19642 0 7.588 0.2786. 0.2975. 0.29713 -6 6.233 0.2726 O-2695 il.2696=52

3. ConclusionsThe fine-structure spectrum and weak-field Zeeman splittings of 02He have been calculated usingferent l evels of anisotropy in the Legendre series. I

has been shown that the calculated spectrum can binterpreted in terms of a simple model [4]. None othe results presented were sensitive to the h > 4 tein the Legendre expansion in line with the observathat the experimental data fitted by Faubel et al. are insensi tive to these higher anisotropic terms [ 1

Neglecting the h = 4 term made only a small (a%change to most of the calculated transition frequencies and Zeeman splittings However, some of thesresults proved to be more sensitive (~10%) to neglof this term. This suggests firstly that if observatioof fine structure of Zeeman spectra a re to be basedtiis sort bf ro-_tibrational-spin calculation, then theterm n e e d s to be included for complete confidencethe results and, secondIy, that experimental data ot h e bound states of 02He could also yield mfo rma-tion about &J4_ .Finally, we consider the accuracy of these cakul


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Volume 105. number 1 CHEMICAL PHYSICS LE-ITERS 2 March 19tions Errors can arise from three sources. from incom-pleteness of the variational basrs, from approximationsin the hamiltonian and irom inaccuracies m the poten-tial. Convergence tests on our basis showed that thebasis set used was well saturated so that this is unlikelyto be a significant source of error.The hamiltonian used contains two approximationsabove the usual Bomappenheimer approximation.Firstly the effects of the 02 vibrations were neglected,an approximation which has been shown to be verygood for the He-HF van der Waals molecule [ 12]_Secondly the fine-structure term was assumed to bethe same as in the 02 molecule [8] _This assumptionis physically reasonable because of the weakness ofthe van der Waals interactions_ It is expected to holdeven in solid 02, where the molecular fine-structureterm is responsible for the dominant magnetic aniso-tropy in the antiferromagnetlcally ordered a-phase[13]. The analogous approximation was used success-fully by Reuss and coworkers to fit the hyperfmelevels of van der Waals complexes of H2 [ 141. How-ever, final assessment of this assumption can only bemade by direct comparison with experiment

The potential-energy surface is usually the majorsource of error in m-vibrational calculations_ In thiswork we have used an empirical surface for which highaccuracy is claimed - the position of the mmunum towithin 2% and the anisotropy in the well region to

within 5% [5] - and it is thus hoped that the resupresented reflect this accuracy_

References111PI131[41[51161[71ISIPI

IlO1IllI1121[I31[I41

I. Tennyson and J. Mettes. Chem Phys 76 (1983) 1V. Ming&in and R-G_ Gordon, J. Chem Phys 70(1979) 3828.I-. Pirani and F. Vecchiocattiv& Chem. Phys 59 (198387_A. van der Avoird. J. Chem. Phys 79 (1983) 1170.M. Faubel, K.H. Kohl, J -P_ Toennies and F.A. G IXI-turco. J_ Chem. Phys 78 (1983) 5629.F. Battaglia, F.A. G ianturco, P. Casavecchia, F. Piraand F_ Vecchiocattrq Faraday Discussi ons Chem S73 (1982) 257.J. hfetteg D Lain 6 and J. Reuss, private communicatG. Henderson and G E. Ewing, J. Chem. Phys 59(1973) 2280.M. Mlzushima. The theory of rotating diatomic molcules (Wiley, New York, 1975) ch 9.G. Brocks and J. Tennyson, J_ hIoL Spectry_ 99 (198263.M. Faubel, private communi~rionJ Tennyson and B.T. Sutclif fc, J. Chem. Phys 79(1983) 43.G.C. DeFotis, Phyr Rev_ B23 (1981) 4714.J. Verberne and J. Reuss Chem Phyr 50 (1980) 1354 (1981) 189;hi. Wtijer, hl . Jacobs and J. Reuss, Chen Phys 63(1981) 247,257;hi. Waaijer and J_ Reusx, Chem. Phys 63 (1981) 263

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Jonathan Tennyson and Ad Van Der Avoird- Effects of the Potential Anisotropy of the Calculated Fine-Structure Spectrum of O2He