Fundamentals - Symmetry and Chemical Application - Lecture Slides, Slides of Applied Chemistry

Download Fundamentals - Symmetry and Chemical Application - Lecture Slides and more Slides Applied Chemistry in PDF only on Docsity! 1 Fundamentals of Group Theory 1 C734b: Symmetry and : Symmetry and Chemical Applications Part I: Fundamentals of Group Theory Fundamentals of Group Theory 2 The Rules of the Game Group: a collection of objects called elements which obey certain rules which interrelate them: Rule 1: The product of any 2 elements in the group and the square of each element must be an element in the group. Let the set of elements = {gk} When we say multiplication → gigj ≡ “carry out operation implied by gj and then that implied by gi”. This is a right-to-left convention ∴ Rule 1 implies “closure” for all gi, gj ε {gk}, gigj = gℓ where gℓ is a member of {gk} In group theory multiplication is not necessarily commutative; that is, gigj ≠ gjgi However, if they do, the groups are called Abelian groups docsity.com 2 Fundamentals of Group Theory 3 Rule 2: One element in the group must commute with all others and leave them unchanged. ≡ identity element (designated by E) { }kiiii gggEgEg ∈∀==⇒ Rule 3: The associative law of multiplication holds: ( ) ( ) kjikji gggggg =⇒ This property holds for any continued product For example: ( )( )( )( ) ( )( )( ) ( ) ( ) ( ) etc. HGFEDCBA HGFEDCBA HGFEDCBA gggggggg gggggggg gggggggg = = Fundamentals of Group Theory 4 Rule 4: Every element gi must have an inverse or reciprocal, gi-1 which is also an element of the group { } { }kikiiiii ggggEgggg ∈∀∈== −−− 111 ; Group Multiplication Tables The number of elements g in a group, G, is called the order of the group, say “h”. )(hGG ≡∴ This means there are h x h = h2 possible products to completely and uniquely define a group, G (abstractly) These can be presented in a group multiplication table. Consists of h rows and h columns. Each row and column is labelled by a group element. docsity.com 6 Fundamentals of Group Theory 11 Example: Permutation Group S(3) A permutation operator P rearranges a set of objects. If for example P{a, b, c, …} = {b, a, c,…} This means that P ≡ operator which interchanges a and b. Important: Pij means “interchange objects CURRENTLY at the locations ORGINALLY occupied by objects i and j. Means can consider the original configuration as objects allocated to certain boxes (like electrons in orbitals). ∴ Pij means “interchange the contents of the ith and jth box, whatever they currently happen to be”. Fundamentals of Group Theory 12 Consider 3 objects. The number of permutations is 3! = 3x2x1 = 6. P0 ≡ E ∴ if 1 2 3 P0 1 2 3 ≡ 1 2 3 = original configuration Let P1 and P2 correspond to the two cyclic permutations: P1 1 2 3 = 2 3 1 P2 1 2 3 = 3 1 2 docsity.com 7 Fundamentals of Group Theory 13 Let P3, P4 and P5 correspond to the 3 binary interchanges: P3 1 2 3 = 1 3 2 P4 1 2 3 = 3 2 1 P5 1 2 3 = 2 1 3 {P0, P1, P2, P3, P4, P5} constitute a group: S(3) If so all binary products will be elements of S(3) Fundamentals of Group Theory 14 Example: Binary products with P1: P0 1 2 3 P0P1 2 3 1 P1P1 3 1 2 P2P1 1 2 3 2 1 3 1 3 2 3 2 1 P3P1 P4P1 P5P1 = P1 = P2 = P0 = P5 = P3 = P4 = one row or one column as required by Rearrangement Theorem. 2nd operation 1st operation In this way one can construct the entire multiplication table. docsity.com 8 Fundamentals of Group Theory 15 Multiplication Table for the S(3) permutation group 0214355 1023544 2105433 3541022 4350211 5432100 543210)3( PPPPPPP PPPPPPP PPPPPPP PPPPPPP PPPPPPP PPPPPPP PPPPPPS Fundamentals of Group Theory 16 Conjugate Elements and Classes Elements can be separated into smaller sets called classes using a similarity transformation If Gggg kji ∈,, and kjii gggg = −1 then gk is the transform of gj and gj and gk are conjugate elements The complete set of elements conjugate to gi form a class. The number of elements in a class is called the order of the class (≡ integral factor of h) i) Every element is conjugate with itself. True if there is at least one element X such that: { }kiii ggXgXg ∈= − anyfor1 Works if X = E docsity.com 11 Fundamentals of Group Theory 21 Subgroups A subset H of G contained within G that is itself a group with the same laws of binary composition is a subgroup of G Note: in S(3), {P0, P1, P2} satisfies closure and is therefore a subgroup. E is always a trivial subgroup of order 1. Some groups have no subgroups other than E; some have more than one. Restriction: The order of any subgroup h, of a group of order g must be a divisor (factor) of g that is, g/h = k where k is an integer. Fundamentals of Group Theory 22 Proof: Let sub group = {A1, A2, A3, …, Ah} (order = h). Take an element B which is a member of G but not in {A1, A2, A3, …, Ah} Form h products of B with the subgroup elements. = {BA1, BA2, BA3, …, Bah} These products are not in the subgroup For example: if BA2 = A4 and A5 = A4-1 ⇒ BA2A5 = A4A5 = BE =B This is impossible since B is not a member of the subgroup. docsity.com 12 Fundamentals of Group Theory 23 Therefore, {A1, A2, …, Ah} and {BA1, BA2, …, BAh} form a larger group of at least 2h members. If g > 2h choose a different element C which is a member of G but not {A1, A2, …, Ah} or {BA1, BA2, …, BAh} ⇒ g must be ≥ 3h Repeat this k times until there are no more elements which are different from {Ai}, {BAi}, {CAi} etc. Then g = kh where k is an integer ∴ ∴ g/h = k However, it does not follow that for a given group that there are subgroups of all orders which are divisors of g. Furthermore there can more than one subgroup of a given order. Fundamentals of Group Theory 24 Question: Can groups as a whole be multiplied? Answer is yes. Direct Products Suppose A = {ai} and B = {bj} are two groups of order a and b, respectively. If BbAaabba jiijji ∈∀∈∀= , the direct product BAG ⊗= is a also a group of order ab with elements aibj = bjai, i =1, …, a; j = 1, …, b Example: A = {a1, a2} B = {b1, b2, b3} BAG ⊗= = {a1B, a2B} or {Ba1, Ba2} = {a1b1, a1b2, a1b3, a2b1, a2b2, a2b3} Order = 2x3 =6 More on direct products later. They’re important! docsity.com 13 C734b Fundamentals of Group Theory 25 Two important terms in group theory are isomorphic and homomorphic Two groups are isomorphic if there is a one-to-one correspondence between the elements of the two groups Isomorphic mapping: A B C A’ B’ C’ Isomorphic implies if AB = C then A’B’ = C’ Both groups have the same multiplication table except perhaps for a change in symbols or in the meaning of the operations Fundamentals of Group Theory 26 Two groups are homomorphic if there is a many-to-one relationship between some of the elements of the group {a1, a2, a3} {b1, b2} a’ b’ The structure of the two homomorphic groups are no longer identical but multiplication rules are preserved This will be seen when discussing Character Tables. docsity.com