Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $
III.
A.RESULTSOFCALCULATIONMatrixelements
Thesimple,thoughprovidingtherequiredaccuracyexpressionsforthemultipolema-trixelements(16),(17),and(18)areobtainedusing
thefollowingreliableapproximations.Firstly,thematrixelementsarecompletelydeterminedbytheBOpotentialforthehydrogenmoleculeV(ρ)whichisfairlywellknownfromthecalculations[6,7,14,15].As(dtµ)Xeeisproducedinlow-energytµ+DXcollisions,onlythelowestvibrationalstatesshouldbetakenintoaccount.ForthesestatesarelocalizedneartheminimumofV(ρ)attheequilibriuminternucleardistancea≈1.4a.u.,itisnaturaltousetheharmonicapproximation
Vh(ρ)=1
2µ2ω2(ρ a)2[1 αM(ρ a)]+V0.(20)
whichtakesintoaccountthenexttermoftheexpansioninρ a.Theapproximation(20)accuratelyreproducestheexactenergiesofthelowestvibrationalstatescalculatedin[6,7].
1)/2µ2a2≈10 4istwoordersofmagnitudesmallerthanthevibrationalenergyω≈10 2.Secondly,therotationalenergyin(3)forthehydrogen-likemoleculel(l+1)/2µ2ρ2≈l(l+Therefore,underausualapproximation,thecentrifugaltermistreatedperturbatively,i.e.,theeigenenergiesaregivenby
Enl=En0+vrl(l+1)(21)
andthewavefunctionsΦnl(ρ)willbetakenindependentoflinthesameapproximation.Indeed,therotationalspectrumcalculatedin[6,7]isingoodagreementwiththeaboveexpression(21)withvr≈1/2µ2a2 10 4.Thus,undertheaboveapproximations,theradialwavefunctionΦnl(ρ)inthepotential(19)coincideswiththeharmonic-oscillatorwavefunctionandthemultipolematrixelements(16),(17),(18)arereducedtol-independentexpressions
DUnν= 2 √n+1δn+1,ν,
MUnn=µ2ω2,QUnn=2µ2ω2.(22)