量子化学与群论基础 9.ppt

* * 6.4 Subgroups, Homomorphisms and Direct Products 1. Subgroups Definition: A subset H of is a subgroup of G if it is itself a group under the same binary operation as G; if this is the case we write H ? G 2. Cyclic groups we concentrate on the subgroup of G generated by a single element. Proposition If g? G , the elements of gare just the powers gn of g for n ?Z. e.g. Cn={E, Cn, Cn2, Cn3… Cnn-1} 3 Homomorphisms Let G and H be groups. G ? H is called a homomorphism if  A homomorphism which is also a bijection is called an isomorphism. we say that G and H are isomorphic, and write G ? H C4={E, C4, C42, C43} U4={1, i, -1, -i} G = { ?,?,?,?} 4 Direct Products The focus of the previous section was on construction; given two groups, we saw how to construct a larger one from them. In this section we turn to the opposite problem of decomposition; given a group, we shall try to decide if it has been constructed out of two smaller ones in this fashion. If G is a group with normal subgroups G1 and G2 such that G=G1G2 and G1?G2={E}, we call G the direct product of G1 and G2, and write G=G1 ? G2 . 7 Point Group 7.1 Cn Cn={E, Cn, Cn2, Cn3… Cnn-1} C1={E} C2={E, C2} C2H2Cl2 H2O2 C3={E, C3, C32} E E, C2 E, C3,C32 E, C4, C2,C43 E, C5,C52, C53, C54 C1 C2 C3 C4 C5 Essential Symmetry Elements Cn 7.2 Cnv Cnv={E, Cn, Cn2, Cn3… Cnn-1, ?v(1), ?v(2), ?v(3), … ?v(n)} [?v(n)]2= E ?v(n) = [?v(n)]-1 C2v ={E, C2, ?v(1), ?v(2) } C1v ={E, ?v}=Cs C3v ={E, C3, C32, ?v(1), ?v(2) , ?v(3) } NH3 P4S3 C?v E, ?v E, C2 ,?v(1), ?v(2), E, C3,C32 , ?v(1), ?v(2) , ?v(3) E, C4, C2,C43 , ?v(1), ?v(2) , ?v(3) , ?v(4) E, C5,C52, C53, C54?v(1), ?v(2) , ?v(3) , ?v(4) ,?v(5) E, ? ? coincidental C, ? x ?v C1v = Cs C2v C3v C4v C5v C ? v Essential Symmetry Elements Cnv 7.3 Cnh Cnh =Cn?Cs= {E, Cn, Cn2, Cn3… Cnn-1} ?{E, ?h} C1h ={E, ?h}=Cs C2h ={E，C2， ?h ，i} C3h ={E, C3, C32, S3, S35, sh} E, ?v E，C2， ?h ，i, E, C3, C32, S3, S35, sh E, C4, C42, C43, S4, S43, i, sh E, C5, C52,C53, C54,S5,S57,S