Multiplication Tables, Rearrangement Theorem Each row and each column in the group multiplication table lists each of the group elements once and only once. (Why must this be true?) From this, it follows that no two rows may be identical. Thus each row and each column is a rearranged list of the group elements. Subgroups A subgroup is a group within another group - a subset of group elements. A supergroup is a group obtained by adding new elements to a group (to give a larger group). The order of any subgroup g of a group of order h must be a divisor of h: h g k where k is an integer 2 2 Similarity Transforms, Classes Classes, cont. A is said to be conjugate with, if there exists any element of the group, X, such that A X X (i) Every element is conjugate with itself. (ii) If A is conjugate with, then is conjugate with A. (iii) If A is conjugate with and C, then and C are conjugate with each other. 3 A complete set of elements that are conjugate to one another within a group is called a class of the group. The number of elements in a class is called its order. The orders of all classes must be integral factors of the order of the group. 4 3 4
The Symmetry Operations Reflections in a plane: σ Reflection (in a plane) Inversion (through a point) Rotation (about a proper axis) - through an angle 2π/n Improper Rotation (about an improper axis) Identity (do nothing) σ i Cn Sn E Examples: Cl Cl S O N R 5 6 Examples r Cl Cl r Inversion: i Cl PMe 3 Me 3 P Cl Cl Nb Nb Cl Cl PMe 3 PMe 3 Cl Cat s Eye Nebula (3-year old supernova) Inversion 7 8
Rotations about an axis: C n Improper Rotations: S n Examples O C 2 O Y X X Y X P C 3 (coming out of the paper ) Example S 4 3 C C 3 C 3 C 3 Note: S σ S 2 i 2 S 2n C n Y 9 S, S 2 are just σ and i S 4 in action Axis bisects angles Rotate 9 Reflect through mirror Mirror perpendicular to axis 2
Groups with more Symmetry i Octahedra Cnv groups C 4 C 3, S 6 σv3 σ v2 σ v C 5 σ d σ d2 2 C 4, C 2 ( C 4 ), S 4 σ d σ h σv4 σ v5 σv C 2 Ni(CO)C 5 5 σv2 WOCl4 3 4 Tetrahedra Icosahedral Molecules C 3 C 2, S 4 σ d 2 inside C 6 [ 2 2 ]2-5 6
C v and D h Start Point Group Flow Chart Step Step 2 Special Groups (a) Linear? C v, D h? (b) Multiple high-order axes? T, T h, T d, O, O h, I, I h? Low Symmetry (no axes): C, C s, C i Step 3 Only S n (n even) axis: S 4, S 6, S 8,..., C n axis (not simple consequence of S 2n ) Step 4 Step 5 No C 2 s axes to C n n C 2 s axes to C n Is the molecule linear? If so, does it have inversion symmetry? σ h n σ v s no σ s σ h n σ d s no σ s C nh C nv C n D nh D nd D n 7 8 Examples Enneper s minimal surface A unopened can of soup, unlabeled; A opened, empty soup can? [N5] + (isolated by Christie) Ethane, staggered SF4, PF5, rf5 Enneper s minimal surface: [Co(en)3] 3+, [Co(en)2Cl2 ] + ignore ring conformations, -atoms [22] 2-, [C2] -,,2-C22 x y z u u 3 3 + uv2 v v3 3 + vu2 u 2 v 2 9 2
More Examples C [ 2 2 ]2- [C 2 ] C C,2-C 2 2 N N N r C C C r 2 22 uilding up Groups from Generators The philosophy of the ermann-mauguin (international) notation is to supply the minimal number of group operations in the group symbol (and those operations generate the rest ). Point group examples: C 2v mm ; D 2h mmm ; C 4v 4mm ; 4/mmm? ; 422? ; 42m? ; 3m? 32 Crystallographic Point Groups Schoenflies C,C i,c 2,C s,c 2h,C 2v, D 2, D 2h,C 4,S 4,C 4h,C 4v, D 2d, D 4, D 4h,C 3,S 6,C 3v, D 3, D 3d, C 6,C 3h,C 6h, D 3h,C 6v, D 6, D 6h,T,T h,t d,o,o h ermann-mauguin,,2,m,2 m, mm,222, mmm, 4, 4,4 m, 4mm, 42m, 422,4 m mm, 3, 3, 3m, 32, 3m, 6, 6,6 m,6m2,6mm,622, 6 m mm,23,m3, 43m,432, m3m These are all the point symmetries compatible with translational symmetry of crystals. 23 24
Vectors quickie review Dot (Scalar) product b c bccosθ where b b and c c In any set of orthogonal coordinates, b c b i c i b b 2! b n b n i 25 b θ c c c 2 " c n c n Vectors quickie review Cross product (3-D only) b c b c bcsinθ where b b and c c To compute cross products: ˆx ŷ ẑ b x b y b z ( b y c z b z c y ) ˆx ( b x c z b z c x )ŷ + ( b x c y b y c x )ẑ c x c y c z 26 a θ c b Cartesian Vectors All vectors in normal 3-D space are built up with Cartesian unit vectors: x ˆ, y ˆ, ˆ z (calculus books use! i,! j, k! ) x! y! z! Any general vector, r, is written as: x x r y + y + x + y + z xx! + yy! + zz! z z 27 Scalar Vector Vector b c b b 2 b 3 b 4 b 5 b b 2 b 3 b 4 b 5 c c 2 c 3 c 4 c 5 b c, b 2 c 2, etc. 28
Scalar Matrix Matrix Matrix Vector Vector C b b 2 b 3 b 4 b 5 b 2 b 22 b 23 b 24 b 25 b 3 b 32 b 33 b 34 b 35 b 4 b 42 b 43 b 44 b 45 b 5 b 52 b 53 b 54 b 55 b b 2 b 2 b 3 b 4 b 5 b 22 b 32 b 42 b 52 b 3 b 23 b 33 b 43 b 53 b 4 b 24 b 34 b 44 b 54 b 5 b 25 b 35 b 45 b 55 a a 2 a 3 a 2 a 22 a 32 a 3 a 23 a 33 a 4 a 24 a 34 a 5 a 25 a 35 Ab c b b 2 b 3 c c 2 c 3 c c 2 c 3 c 4 c 2 c 22 c 32 c 42 c 3 c 23 c 33 c 43 c 4 c 24 c 34 c 44 c 5 c 25 c 35 c 45 c, c 2, etc. b b 2 a 4 a 5 a 42 a 52 a 43 a 53 a 44 a 54 a 45 a 55 a 4 b + a 42 b 2 + a 43 b 3 + a 44 b 4 + a 45 b 5 c 4 b 4 b 5 c 4 c 5 c 5 c 52 c 53 c 54 c 55 29 3 Matrix Multiplication Summation Notation a a 2 a 3 a 4 a 5 b b 2 b 3 b 4 A C b 5 c c 2 c 3 c 4 c 5 A C a 2 a 3 a 4 a 5 a 22 a 32 a 42 a 52 a 23 a 33 a 43 a 53 a 24 a 34 a 44 a 54 a 25 a 35 a 45 a 55 b 2 b 3 b 4 b 5 b 22 b 32 b 42 b 52 b 23 b 33 b 43 b 53 b 24 b 34 b 44 b 54 b 25 b 35 b 45 b 55 c 2 c 3 c 4 c 5 c 22 c 32 c 42 c 52 c 23 c 33 c 43 c 53 c 24 c 34 c 44 c 54 c 25 c 35 c 45 c 55 k a ik b kj c ij a4 b2 + a42 b22 + a43 b32 + a44 b42 + a45 b52 c42 row index column index 3 32
lock Factored Matrices Matrices Definitions, Properties a a2 a 2 a22 b2 a 33 a 43 a 53 a 34 a 44 a 54 a 35 a 45 a 55 a a 2 b b 2 c c 2 a 2 a 22 b 2 b 22 c 2 c 22 b b 2 b22 b 33 b 43 b 53 a 33 a 43 a 53 b 34 b 35 b 44 b 45 b 54 b 55 a 34 a 44 a 54 a 35 a 45 a 55 c c2 c 2 c22 b 33 b 34 b 35 b 43 b 44 b 45 b 53 b 54 b 55 c 33 c 43 c 53 c 33 c 43 c 53 c 34 c 44 c 54 c 34 c 44 c 54 c 35 c 45 c 55 c 35 c 45 c 55 Character, χ (often called the Trace) of a Matrix if C A and D A, then χ C χ D. Conjugate matrices have equal characters Transpose (A T ): (A T ) ij (A) ji just transpose the rows and columns Matrices and Geometrical Transformations Orthogonal Matrices inverse matrix is just the transpose 33 34 Why are the matrices of symmetry operations always orthogonal matrices? Symmetry and Physical Properties Assume an orthogonal set of unit column vectors, ˆx! ; ˆx 2! ; ˆx 3! ; " ˆx n!, note that ˆx i T i ˆx j δ ij The action of symmetry operation R on these unit vectors will map them R on to a new set of vectors, { ˆx i,i,!,n} { ŷ i,i,!,n}. ut by the rules of matrix multiplication, the columns of must just be the ŷ i vectors: R ŷ ŷ 2 ŷ 3! ŷ n. Since symmetry operations are just rotations or reflections, the set of vectors, { ŷ i,i,!,n}, are mutually orthogonal. Therefore, the columns of R are orthogonal. Dipole moments moment must be invariant under all symmetry operations Chirality 2nd-rank tensors (e.g., polarizability tensor, hyperfine and g-tensors in EPR; in the solid state: conductivity, susceptibility, & stress tensors) - skip this. 35 36
Chirality - Symmetry requirement If no S n axis - molecule is Chiral Λ-enantiomer -enantiomer Λ-enantiomer -enantiomer If a molecule is not superimposable on its mirror image, it said to be chiral. 37 The act of superimposing molecules after reflecting is equivalent to executing a rotation. 38