james r. henderson et al- calculated ro-vibrationai spectrum of ^7-li3^+ and ^7-li2^6li^+

8/3/2019 James R. Henderson et al- Calculated ro-vibrationai spectrum of ^7-Li3^+ and ^7-Li2^6Li^+

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Spectrochimica Acta, Vol. 44A, No. 12 , pp . 1287 1290, 1988. 0584-8539/88 $3.00+ 0.00Pr in ted in G r ea t B r i ta in . Per g a mo n P r es s p lc

Calculated ro-vibrationai sp ectrum of VL i~-and 7 L i e 6 L i +JAMES R. HENDERSON, STEVEN MILLER and JONATHAN TENNYSON*

Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, U.K.(Received 23 M ay 1988; n f inal form 21 J u n e 1988;accepted 24 J u n e 1988)

Abstract--Calculated ro-vibrational energy levels J ~ ~ m + n + ~ , (1)for a given value of N. In Eqn (1) m, n and j are thenumbe r of quan ta in the basis function representingthe r, R and 0 coordinates respectively. Note that ineach expansion the first function has qu ant um num berzero. Tables 1 and 2 demonstrate the convergence ofthe 7Li~-and 7Li26Li + ban d or igins as a function of N.In both cases the fundamentals were converged to~.01 cm-1 for N=9, an accuracy very much higherthan one can expect for the potential.The first step of the rotat ional ly excited calculationsinvolves diagonalising a series of vibrat ional calcu-lations in which it is assumed that the projection of thetota lang ular mo mentu m onto the body-fixed z-axis, k,

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1288 JAMES R. HENDERSON et al .Table 1. Calculated* vibrational band origins for Liiin cm-

v,vJ Symmetry Parity? N=6 N=l N=8 N=9 SDF[2]011 E e 226.27 226.04 226.01 225.99 225.98

0 225.94 225.89 225.11 225.98100 A, e 298.92 298.79 298.78 298.77 298.76020 A, e 449.01 448.35 447.93 447.80 447.76022 E e 453.86 451.83 451.32 451.10 451.050 451.95 451.15 450.97 451.07111 E e 522.58 521.00 520.66 520.57 520.540 521.16 520.52 520.37 520.55200 A, e 596.80 595.61 595.53 595.48 595.47

*Parameters used for Morse oscillator-like functions-r coordinate: r, = 3.96 A,D,=10100cm-1,o,=285cm-1;Rcoordinate:r,=2.65i(,D,=8800cm-,w,=285rm-.t Parity of Legendre functions used in the basis+ = even; o = odd.

Table 2. Calculated vibrational band origins for Liz6Li+ incm-vlvz3 Symmetry N=7 N=8 N=9

010 A, 231.44 231.41 231.40001 B2 231.93 231.93 231.93100 AI 307.31 307.30 307.30020 4, 459.16 458.88 458.80002 AI 462.35 462.11 462.06011 B2 462.92 462.83 462.81101 R, 534.34 534.28 534.28110 4, 535.17 534.92 534.86200 A, 612.50 612.36 612.34

is a good quantum nut&x. These results are thenenergy ordered and the lowest I eigenvectors used as abasis for the fully-coupled problem [S]. It was foundthat the rotational calculations converged rapidlywith I. For all the results presented here the first stepwas solved with N= 8 and Z taken as 25 x (J+ 1-p),where p=O or 1 and the total parity of the state isgiven by (-1) JcP This gave rotational levels con-verged to better than 0.01 cn- 1relative to the appro-priate band origin.The calculations used the programs SELECT,TRIATOM and ROTLEV [13] which have beenupdated to drive a new program DIPOLE [ 141. Thisallows for the calculation of transition dipole mo-ments, the theory of which can be found elsewhere [ 15,16). All calculations were performed on the Cray-1Scomputers at the University of London ComputerCentre.

RESULTS AND DISCUSSIONTable 1 presents results for the vibrational band

origins of Li: as a function of N. The slight splittingbetween theeven and odd components of the degener-ate E states is caused by the failure of our method toallow for the full DJk symmetry of the system. Thissplitting can provide a useful indication of conver-gence. Our results are well converged for N = 8 andagree closely with those of SDF, which are presentedfor comparison.

Table 2 presents the analogous results for 7Li,6Lif.Again the results are well converged at the N = 8 level.We note that the splitting between v2 and vJ caused bylowering the symmetry of the system by isotopicsubstitution is only 0.5 cn- i.Rotational calculations were performed for valuesof the total angular momentum, 5~4. Rather thanpresenting the individual levels, we give the results ofleast-squares fits to the more compact parameterisedHamiltonians due to WATSONet al. [17,18-j. For a fulldiscussion of assigning and parameterising the levelsof X, systems see WATSON [19]. We note howeverthat K in Tables 3 and 5 refers to the projection of .Zalong the C, symmetry axis of Li$. It was found thatincluding up to quartic terms gave a very goodrepresentation of our data as can be judged by thestandard deviations given in Tables 3 and 4.Liz6Li+ is an asymmetric top with ~-0.62. Onefeature of the results for Liz6Li+ presented in Table 4is the very strong Coriolis interaction between thenearby vI and vJ fundamentals. Because the rotationalmanifolds of the two states not only overlap but arealso strongly coupled, they were fitted simultaneouslywith no attempt at vibrational pre-assignment. In-stead, the vibrational states were assigned by analysisof the eigenvectors produced by the fitting procedure.

Table 3. Rotational constants [19] for Li: in cm-ParameterBC10SDJ105DJK105DKKCz105$10s+102q105q10SqKNo.*103ot

Ground state0.53190.26410.9-1.40.6

160.01

v1 v20.5260 0.53210.2603 0.2629-9.9 2.62.6 1.4-0.7 -0.7-0.2588-4.42.5-1.0071.0

-16.516 324.1 1.2*Number of levels fitted.tstandard deviation on fits.


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Spectra of Lii and Liz6Lif

Table 4. Rotational constants [I83 for Liz6Li+ in cm-

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Parameter Ground state 1 2 3A 0.5908B 0.5320C 0.2779105AJ, 0.5105AJK -0.1105AKK 0.81036J103V

__

0.5867 0.6030 0.57770.5293 0.5206 0.54060.2763 0.2173 0.3343

-0.1 - 185.1 224.21.3 443.7 - 876.60.2 - 182.0 708.6

1.10 -0.91-1.88 1.78

1031J 2.61103iK -3.16103% - 8.09No.* 24 24 48lo%? 0.0 0.0 6.9

*Number of levels fitted.tStandard deviation on fits.

Table 5. Dipole allowed ro-vibrational transitions for Li:. All transitionsare for the v2 fundamental. The energy levels are given relative to the J = 0

state of the vibrational ground state. (Powers of ten in brackets)E I? wi/ S(f-i)J G -J K (cm-) (cm- ) (cm _ ,) (Debye)

Ol,, 11 225.986 0.796 225.190 0.661(-3)10~, 10 227.29 1 1.064 226.227 0.195(-2)II,, 11 227.053 0.796 226.256 0.977( 3)lo_, 00 227.312 0.000 227.3 12 0.1301-2)

Even for J = 1 there was up to 20% mixing betweenthe levels. As in the equivalent bands of H,D+ [20],any experimental analysis of these fundamentals willundoubtedly prove difficult.

Tables 5 and 6 present dipole allowed transitions forLi: and Liz6Li+. Only transitions involving thevibrational ground state where J equals 0 or 1 foreither level are given. These give a representativesample of the transition intensities to be expected forthis system. Intensities for other transitions are avail-able from the authors on request.

For 7Lilthe only anbwed transitions involve excita-tion of the v2 fundamental. For the v2 state the 1doubling introduces the extra quantum numbers G(= K + U) and U (= f 1) [ 193. For this band there isthe additional selection rule G = K.

In contrast, for Liz6Li+ all three fundamentals arei.r. active and the ion also has a pure rotationalspectrum as a result of its permanent dipole along the6 axis. If Li: was a rigid system, the transition dipolefor the 1, ,-O,, transition would be equal to thepermanent dipole of 7Li26Li+. In fact this transition

Table 6. Dipole allowed rotational and ro-vibrational transitions from the vibrational groundstate of Liz6Li+. The energy levels are given relative to the J=O state of the vibrational groundstate. (Powers of ten in brackets)

v Jk,K -JE- KI4I Type (cm-) (cm-) i ,(cm ) (D&y&00223223331111

bbbb;:baa;:bbb

1.123 0.810 0.313 0.26 l(0)0.869 0.0 0.869 0.174(O)231.400 0.869 230.531 0.151(-3)231.888 1.123 230.765 0.165(-3)231.925 0.810 231.115 0.169(-2)232.527 0.810 231.717 0.231(-3)231.917 0.0 231.917 0.152(-3)233.105 1.123 231.982 0.259 (- 2)233.047 0.869 232.178 0.25 1 ( - 2)233.100 0.0 233.100 0.168(-2)307.299 0.869 306.430 0.126(-3)308.106 1.123 306.983 0.187(-3)308.416 0.810 307.606 0.186(-3)308.163 0.0 308.163 0.122(-3)


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1290 JAMESR. HENDERSON t al.

gives a value of p of 0.4172 D in very close agreementwith the value calculated for the equilibrium geometryat the centre of mass of 0.4157 D. For the ro-vibra-tional transitions, the vg a-type transitions are roughlyan order of magnitude stronger than the vi and v2 b-type lines.In conclusion, we have presented results for thosero-vibrational transitions of Liz which would appearto be most promising for a laboratory observation ofthe system. As our ro-vibrational calculations are ofproven accuracy it is necessary to concentrate on thepotential energy surface in estimating. any errors inour results. Error estimates. for the potential aredifficult. However, we note that our rotational con-stants for the vibrational ground state of Li: agreenot only with SDFs estimates using the same poten-tial, but also those of MARTIN et al. [21] obtainedfrom. a pseudo-potential calculation.Acknowledgements-We would like to thank Dr B. T.SUTCLIFFE or drawing this problem to our attention and DrE. I. VON NAGY-FELSOFJU~CIor supplying results piior topublication.

REFERENCES[l] S. J. DUNNE, D. J. SEARLES and E. I. VO NNAGY-FELSOBUKI,Spectrochim. Acta 4U, 699 (1987).[Z] D. J. SEARLES, S. J . DUNNE and E. I. VONNAGY-FELSOBUKI,Spectrochim. Acta 44A, 505 (1988).

[3] D. J. SEARLES, S. J. DUNNE and E. I. VONNAGY-FELSOBUKI, pectrochim. Act! MA, 985 (1988).[4] J. TENNYSON and B. T. SUTCLIFFE, J. them. Sk.Faraday Trans. 2 82, 1151 (1986).[S] B. T. S ~TCLIFFE, S. MILLER and J. TENNYSON,Comput.Phys. Commun., in press.[6] S. MILLER and J. T ENNYSON, . molec. Spectrosc. 128,183 (19871.

[7j S. MILLET and J. TENNYSON, . molec. Spectrosc.530 (1988).PIc91WICl11WI[ii;Cl51Cl61T171

WI

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1%S. M;LLizRand J. TENNYSON,Chem. Phys. L&t. 145,117(1988).S. MI LLER and J. TENNYSON,Astrophys. Jl., in press.J. TENNYSON, . MILLER and B. T . S UTCLIFFE,J. them.Sot . Faraday Trans. 2, in press.J. TENNYSON nd B. T. SUTCLIFFE,J. &em. Phys. 77,4061 (1982); ib id., 79, 43 (1983).J. TENNYSON.Commct. Phvs. Ren. 4. 1 (1986).J. TENNYSON;Cornput. Ph;s. Co&&n. 42, 257 (1986).J. TENNYSON nd S. MILLER, Comput. Phys. Commun.,to be submitted.G. BROCKS, . TENNVSON nd A. VANDER AVOIRD,J.them. Phys. SO, 3223 (1984).S. MILLER, J. TENNYSON nd B. T. SUTCLIFFE,Molec.Phys., to be submitted.J. K. G. WATSON, . C. FOSTER,A. R. W. MCKELLAR,P.BERNATH, . AIUIANO,. S. PAN, M. W. CROFTON,R. S.ALTMAN nd T. OKA$ Can. J. Phys. 62, 1886 (1984).S. C. FOSTER, A. R. W. MCKELLAR, I. R . PETERKIN, .K. G. WATSON, F. S. PAN, M. W. CROFTON, . S.ALTMAN and . OKA, . them. Phys. 84,91 (1985).J. K. G. WATSON, . rnol ec Spectrosc. 103, 350 (1984).J.-T. SHY, FARLEY and W. H. WING, Phys. Rev. A 24,1146 (1981).J. L. MARTIN,R. CAR and J. BUTTET,J. them. Phys. 78,5646 (1983).

james r. henderson et al- calculated ro-vibrationai spectrum of ^7-li3^+ and ^7-li2^6li^+

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