1. In elass we derived selection mulss for transitions between electronic states of the bydrogen atom. In doing so, we used a little known known rule for the value of an integral containing three eplerrical harmonic fusetiene. Hene we will show that the rules fow directly frum the properties of v() and vie) that combise to form the spherical harmonics. The important poest that we will deal with in detail in Chem 113C, is for all zllowed trimsitions the tratsition dipole mement integral mast be none-zero. That is: re=r=r(r)r(r)dr=0 Where H=ef=e(x+y+2)=rsin()cos()g=rsin()sin()]m=1,2,z=recs()m=0 We noted that the integral over r is always none-xero, so the selection rulcs dopind on the 0 and o inlegrals. There ars two quantum numbers we are soneerned with: 1 Which will depend on the 0 integral m. Which will depend on the op inegral a) We will start with the / quntum number ()bes=NbeePllnis()=cos() The legendre polynomials P1iem lave a recurien relationship cos(ee)Pili=21+11[(lm+1)Pi+1m+(I+m)Pi1l1] Use this relationsip and the fact the wavefunctioes genented from these polynomials are onthogonal to show that =1 b) The firs few lependre polynomials are P00=11;P10=cos();P11=sin(); P20= 1/2(3cos2()1):P21=(121)11(cos()sin():P22=3sin2() Show that the P6b,P11,P11 functions obcy the recursion relationship. c) Now les use n(p) to get the selection rules for the wa quantus aumber: m()=(21)1/2einpm=0,1,2,+1 i) use the x-dipole to shore Aw=0 ii) in class we showed that xiy=etivsin() Use the o part to show Aw=1