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 $
I.INTRODUCTION
Itisknownthatonestoppedmuoninadeuterium-tritiummixtureyieldsmorethan100nuclearfusionreactions.Theprocessofmuon-catalyzedfusionhasbeenintensivelystudiedandadetaileddescriptionexistsintheliterature,e.g.,inreviewarticles[1,2,3,4].Oneofthekeyprocessesinthemuon-catalyzedfusioncycleistheformationofthehydrogen-likeexoticmolecule(dtµ)Xee(forthesakeofgeneralityXstandsforeitherisotoped,t,orp),inwhichadtµmesicmoleculesubstitutesforoneofthenucleiinthehydrogenmolecule.ItiswidelyacceptedthattheresonancemechanismproposedbyVesman[5]isresponsibleforthehighrateofthedtµformation.Duetothismechanism,adtµmesicmoleculeinalooselyboundexcitedstatetobeproducedbylow-energycollisionsoftµmesicatomsandDXmoleculesinaresonanceprocess
tµ+DX →(dtµ)Xee
followedbydtµtransitiontothetµgroundstate.Therateoftheresonanceprocessissensitivetothepreciseresonancepositionandanaccuracybettera1meVisnecessarytoobtainreasonabletheoreticalestimatesoftheformationrate[1,2,3,4].
Resonanceformationcantakeplaceiftheenergyreleasedindtµbindingistransferredtotherovibrationalexcitationoftheexoticmolecule(dtµ)Xee.Thisisactuallythecaseasdtµhasalooselyboundexcitedstatewithanangularmomentumλ=1andbindingenergywhichiscomparabletovibrationalquantumofthe(dtµ)Xeemolecule.Inanon-relativisticapproximation,di erentcalculationsdeterminewithagoodaccuracythebindingenergyoftheisolateddtµmesicmolecule[1,2,3,4].Toobtaintheprecisevalueofthebindingenergyonehastocorrectthenon-relativisticenergyforrelativistice ects,hyper nee ects, nitenuclearsize,vacuumpolarization,andothers.Theresonancepositionisdetermined,besidesthebindingenergyofisolateddtµ,bytheenergyoftherovibrationalexcitationofthehydrogen-likemolecule(dtµ)XeewithonenucleusbeingtheparticleXandtheothertheexciteddtµmesicmolecule.Asthe”size”oftheexciteddtµmesicmoleculewithλ=1isoftheorderof0.05a.u.[8],whichismuchsmallerthantheinternucleardistanceinthewholemolecule,therovibrationalspectrumof(dtµ)Xeecanbecalculatedtoagoodapproximationbytreatingdtµasapoint-likechargedparticle[6,7].Nevertheless,toreachanaccuracyoftheorderofatenthofameVoneshouldtakeintoaccounttheenergyshiftwhicharisesduetotheinternalstructureandmotionofadtµmesicmolecule.