1、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 theor
2、y 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 c
3、entral 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 hel
4、d 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 equivalenc
5、e, 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 mathema
6、tical 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 p
7、oint) 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
8、) (-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, CCn 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
9、= 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 pri
10、nciple 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 (
11、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
12、,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,Abst
13、ract 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.,Examples The integers and any of the binary operations of arithmetic:=+: 1+5=6 (1)=- : 1 -5= -4 5 -1 (2)(12-
14、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.,Closure
15、 Require 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.,Associativity Require that if F,G,H G,
16、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 element Require that in G there is an element , the identity, such that E G =G E=G . For the integers, the i
17、dentity 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.,Inverse For every element G G there exists an element denoted G-1 s
18、uch that G-1 G =G .G -1 =E. For the integers, the inverse of k is -k . There is no inverse under multiplication in general. Under division every element may 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 translatio
19、n. For a rotation the inverse is the same rotation in the opposite sense (matrix inverse).,Commutativity If 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 integer multiplication; integer sub
20、traction is not. Translations are commutative, and so are successive rotations around the same axis.,GroupsThe 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).,Fo
21、r each GG there exists an element G -1G such thatG -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 themselves only, forming a subgroup.,The order of a subgroup must be a divisor of the orde
22、r of the group (Lagrange).,Subgroups,If HG, and GG but GH, G H is a left coset and HG is a right coset.,Cosets,Classes If there is at least one X G such thatH =XGX-1, G,H GH is conjugate to G . Clearly, if H is conjugate to G , G is conjugate to H : they are mutually conjugate. A subset of the elements of G in which all the elements are mutually conjugate is called a conjugacy class, or simply class.,
