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 $
tions[9,10,18]wherethel-independentenergyshiftwasobtainedbyscalingtheresultfortheatom-likefour-bodysystem(dtµ)e.Aspointedoutinthispaper,themonopolecon-tributioncalculatedinRef.[8]dependsonthechoiceofthecoordinatesystemthatdoesnotallowacomparison.Forthisreason,onlythequadrupolecontributiontothe rst-orderPTenergyshiftsofRef.[8]willbecomparedwiththepresentresults.ThequadrupolecontributionofRef.[8]isgiveninTableIIofthatpaper,whileinthepresentapproach E(1)
Q=(10/3)v0AL2(l1l1),asfollowsfromEqs.(11),(22),and(27).Thebothresultsareinexcellentagreementwitheachother,asshowninTableI.Notethatinthepresentapproach
lLAL2(l1l1) E(1)(1)
Q EQ[8] v0
TABLEI:Quadrupolecontributions E(1)
Q(inmeV)tothe rst-orderPTenergyshiftsofthe
presentcalculationandthosefromRef.[8]fordi erentlandL.AlsopresentedaretheangularintegralsAL2(l1l1)andtheparameter v0correspondingtotheenergyshiftsofRef.[8].thedependenceonangularmomentaiscompletelydeterminedbythefactorAL2(l1l1)whichisalsopresentedinTableI.Toagoodaccuracy,theresultsofRef.[8]revealthesamedependenceonangularmomentawhichapprovesthedescriptionofenergyshiftsbyasingleparameterv0.Toemphasizethisfact,thequadrupolecorrectioncalculatedinRef.[8]isexpressedintheform E(1)
Q=(10/3) v0AL2(l1l1)withthevariable v0presentedinTableI.
Indeed,v 0ispracticallyindependentoflandLandagreeswithv0=1.81meV.Agreementbetweenthepresentone-parameterresultforthequadrupolecorrectionandtheelaboratesix-bodycalculation[8]isagoodargumentforthevalidityofthepresentapproach.Atlast,oneshouldobtainWlln
1(34),whichrequiresevaluationofJλ 1+Eν
E