量子化学与群论基础

《量子化学与群论基础》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、6.4Subgroups,HomomorphismsandDirectProducts1.SubgroupsDefinition:AsubsetHofisasubgroupofGifitisitselfagroupunderthesamebinaryoperationasG;ifthisisthecasewewriteHG2.CyclicgroupsweconcentrateonthesubgroupofGgeneratedbyasingleelement.PropositionIfgG,theelementsofarejustthepowersgnofgfornZ.

2、e.g.Cn={E,Cn,Cn2,Cn3…Cnn-1}3HomomorphismsLetGandHbegroups.GHiscalledahomomorphismifAhomomorphismwhichisalsoabijectioniscalledanisomorphism.wesaythatGandHareisomorphic,andwriteGHC4={E,C4,C42,C43}U4={1,i,-1,-i}G={,,,}4DirectProductsThefocusoftheprevioussectionwasonconstruction;giventwogro

3、ups,wesawhowtoconstructalargeronefromthem.Inthissectionweturntotheoppositeproblemofdecomposition;givenagroup,weshalltrytodecideifithasbeenconstructedoutoftwosmalleronesinthisfashion.IfGisagroupwithnormalsubgroupsG1andG2suchthatG=G1G2andG1G2={E},wecallGthedirectproductofG1andG2,andwriteG=G1G

4、2.7PointGroup7.1CnCn={E,Cn,Cn2,Cn3…Cnn-1}C1={E}C2={E,C2}C2H2Cl2H2O2C3={E,C3,C32}CnEssentialSymmetryElementsC1C2C3C4C5EE,C2E,C3,C32E,C4,C2,C43E,C5,C52,C53,C547.2CnvCnv={E,Cn,Cn2,Cn3…Cnn-1,v(1),v(2),v(3),…v(n)}[v(n)]2=Ev(n)=[v(n)]-1C2v={E,C2,v(1),v(2)}C1v={E,v}=CsC3v={E,C3,C32,v(1),

5、v(2),v(3)}NH3P4S3CvCnvEssentialSymmetryElementsC1v=CsC2vC3vC4vC5vCvE,vE,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,coincidentalC,xv7.3CnhCnh=CnCs={E,Cn,Cn2,Cn3…Cnn-1}{E,h}C1h={E,h}=CsC2h={E，C2，h，i}C3h

6、={E,C3,C32,S3,S35,sh}CnhEssentialSymmetryElementsC1h=CsC2hC3hC4hC5hE,vE，C2，h，i,E,C3,C32,S3,S35,shE,C4,C42,C43,S4,S43,i,shE,C5,C52,C53,C54,S5,S57,S53,S59,sh7.4DnDn={E,Cn,Cn2,Cn3…Cnn-1,C2(1),C2(2),C2(3),…C2(n)}D2={E,C2,C2(1),C2(2)}D3={E,C3,C32,C2(1),C2(2),C2(3)}DnEssentialSymmetryElementsD2D3

7、D4D5E，C2，C2(1),C2(2)E,C3,C32,C2(1),C2(2),C2(3)E,C4,C42,C43,C2(1),C2(2),C2(3),C2(4)E,C5,C52,C53,C54,C2(1),C2(2),C2(3),C2(4)C2(5)7.5DnhDnh=DnCs={E,Cn,Cn2,Cn3…Cnn-1,C2(1),C2(2),C2(3),…C2(n)}{E,h}=：{E，Cn，Cn2，Cn3…Cnn-1；C1(1)，C2(2)…C2(n)；σh，Sn1，