The molecule ethylene has D2h symmetry, with the 8 symmetry operations {e,c2=c2z,c2=c2y,c2=c2x,i,h=xy,d=xz,v=yz} (See D2hEthyleneN.pdb in ExplorePointGroups...nb) a. How many classes are there in D2h ? Do all symmetry operations commute? b. Construct a 4-dimensional representation 4d for the basis {H1s,H4s,H3s,H4s} where Hks is a s-orbital centered at the position of Hydrogen atom k. (This is the same as the permutation representation for the 4 hydrogen atoms). c. Find the characters of this representation. How could you have obtained them without explicitly writing down the 44 representation matrices? d. Given that the decomposition of a representation is unique, use the character table below to find out which of the irreducible representations Ag,B1g,,B3u occurs how many times in the reduction of 4d. e. For one-dimensional representations the characters are also irreducible representation 'matrices' p(T). Start with any Hks orbital to construct the corresponding irreducible basis functions by the projection method: k=TD2hp(T)P^(T)Hks