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 $
viathedimensionlessintegrals
J0(z)=∞ 0[sinδ0(k) (k2+3)(k/2)cosδ0(k)]dk2
(k2+1)4(k2+z)
B.Shiftandsplittingofenergylevels.(30)
Energyshiftsareobtainedbysolvingthesecularequation(6)whichisreduced,dueto
theselectionrulesforangularmomentalandl1,toa2×2matrixequationforl,l1=L±1(L=0)andascalarequationforl=l1=L=0.TheenergyshiftswithrespecttotheunperturbedrovibrationalenergiesEnl+ε11aredenotedas 0(n)and ±(nl)forl=Landl=L±1,respectively.NotethatthestatewithL=0andl=l1=1isuncoupled;however,itsenergyshift +(n1)willbedeterminedinthesamemannerasfortheotherL=0states.
nThe rst-orderPTmatrixelementsVllinEq.(6)arecalculatedbysubstitutingthe1
M,QradialintegralsUnn(22),thequadrupolemomentumQ(27),andtheangularintegrals
nAL2(l1l11)(A4)inEq.(11).NotethatVll1appearstobeindependentofthevibrational
quantumnumbernandthisindexwillbeomittedinwhatfollows.Thematrixelementsarescaledbyasingledimensionalparameter
v0=2m1m3ω2Ca
2L+1 1
L(L+1)2 1 inwhichthe rstrowandcolumncorrespondtol,l1=L 1andthesecondonesto
l,l1=L+1.
(33)