量子化学与群论基础.ppt

6 Group theory,6.1 Introduction,Group Theory is one of the most powerful mathematical tools used in Quantum Chemistry and Spectroscopy.It allows the user to predict,interpret,rationalize,and often simplify complex theory and data.,Group theory can be considered the study of symmetry.,Group theory is a basic structure of modern algebra,consisting of a set of elements and an operation.,Group theory is the subject of intense study within mathematics,and is used in many scientific fields.e.g.,groups are used in chemistry to describe the symmetries of molecules,and the Lorentz group is a central part of special relativity.Also,the theory of groups plays a central role in particle physics,where it has led to the discovery of new elementary particles.,1985,Fullerenes,1990,Kratcshmer,The involvement of symmetry in chemistry has a long history;in 540 BC the society of Pythagoras held that the earth had been produced from the cube,fire from the tetrahedron,air from the octahedron,water from the icosahedron,and the heavenly sphere from the regular dodecahedron.,Symmetry exists all around us and many people see it as being a thing of beauty.,Symmetry,is related to equivalence,mutually corresponding arrangement of various parts of a body,producing a proportionate,balanced form.,At its heart is the fact that the Set of Operations associated with the Symmetry Elements of a molecule constitute a mathematical set called a Group.This allows the application of the mathematical theorems associated with such groups to the Symmetry Operations.,6.2 Symmetry elements and operations,Symmetry operations A symmetry operation is defined as:movement of a molecule to a new orientation in which every point in the molecule is coincident with an equivalent point(or the same point)of the molecule in its original orientation.,Symmetry Elements A symmetry element is a geometrical entity(a line,plane or point)with respect to which one or more symmetry operations may be carried out.,Symmetry elements and operations,1.Types of symmetry operation,(a)Inversion,i(x,y,z)-(-x,-y,-z)in(x,y,z)-(-1)n x,(-1)n y,(-1)n z),Ni(CN)42-,C2H4,benzene,Matrix representation of a inversion:,(c)Proper rotations,C Cn is a rotation about the axis by 2/n Thus,C2 is a rotation by 180,while C3 is a rotation by 120.,(b)Identity,E,no change at all,i2n=E,n=integer in=i for odd n,Principle axis is always defined as the axis with the highest order.,Matrix representation of a proper rotation:,Cnm is a rotation about the axis by m 2/n Note:Cnn=E=Cn2n=Cn3n Cn axis generates n operations:Cn,Cn2,Cn3 Cnn,(d)Reflections,v:in a plane which contains the principle axis(suffix v for“vertical”).h:in a plane principle axis(suffix h for“horizontal”).d:in a plane containing principle axis and bisecting lower order axes(suffix d for“dihedral”or“diagonal”).,(xy):(x,y,z)-(x,y,-z),(e)Improper rotations,S Sn=Cn h,N3S2PCl4O2,Sn=h Cn=Cn h(Cn and h always commute).(Note that in general,R1R2 does not equal R2R1),2.Operator multiplication,As was implicit above,the consecutive application of two symmetry operations may be represented algebraically by the product of the individual operations.,The product of two operators is defined by,The identity operator does nothing(or multiplies by E),The associative law holds for operators,The commutative law does not generally hold for operators.In general,e.g.order,C2 C2=C22=E,v(yz)v(yz)=E v(xz)v(xz)=E,v(xz)C2=v(yz),Multiplication table,Order:,6.3 Mathematical groups,Abstract Group Theory,Consider a set of objects Gand a product rule denoted that allows us to combine them.Denoted F G,where F,GG.G can be objects such as numbers or variables,or operators.,ExamplesThe integers and any of the binary operations of arithmetic:=+:1+5=6(1)=-:1-5=-4 5-1(2)(12-3)-7=3 12-(3-7)=16(3)=:12 3=4 3 12(not even an integer)(4)Note that so far there are no requirements that should obey certain rules,such as commutativity or closure.,Translations or rotations of a physical object in two or three dimensions.Here denotes successive transformations.,ClosureRequire that if F,G G,then F G G and G F G.Note that this does not imply FG=GF.For example,the integers are closed under addition,multiplication,and subtraction,but not under division.Successive rotations and translations in M dimensions are closed.,AssociativityRequire that if F,G,H G,we have(F G)H=F(G H).For example,the addition and multiplication of integers is associative,whereas subtraction is not.Successive translations and rotations are associative.,Identity elementRequire that in G there is an element,the identity,such that E G=G E=G.For the integers,the identity for addition is 0,for multiplication it is 1;there is no identity for division.For translations the identity is the null operation,for rotations it is the identity rotation which is given in matrix form by a unitmatrix.,InverseFor every element G G there exists an elementdenoted G-1 such that G-1 G=G.G-1=E.For the integers,the inverse of k is-k.There is no inverseunder multiplication in general.Under division every elementmay appear to be its own inverse,but this is not so,since 1 is not the identity.For a translation the inverse is-1 times the original translation.For a rotation the inverse is the same rotation in the opposite sense(matrix inverse).,CommutativityIf the set G has the property that for any two elements F,G G we have F G-G F=0,then the elements of G commute.Integer addition is commutative,and so is integermultiplication;integer subtraction is not.Translations are commutative,and so are successive rotations around the same axis.,Groups The elements of a set G together with a product rule form a group G if:,G,H G,GHG(closure).,F,G,H G,F(GH)=(FG)H(associativity).,An element EG exists such that EG=GE=GGG(identity).,For each GG there exists an element G-1G such that G-1 G=GG-1=E(inverse).,If in addition GH-HG G,H G,G is Abelian.,Multiplication table,G=E,C2,v(xz),v(yz),We can see that some elements multiply among themselvesonly,forming a subgroup.,The order of a subgroup must be a divisor of the order of thegroup(Lagrange).,Subgroups,If HG,and GG but GH,G H is a left coset and HG isa right coset.,Cosets,ClassesIf there is at least one X G such that H=XGX-1,G,H G H is conjugate to G.Clearly,if H is conjugate to G,G is conjugate to H:they aremutually conjugate.A subset of the elements of G in which all the elements aremutually conjugate is called a conjugacy class,or simply class.,