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Part 2.2: Symmetry and Point Groups 1

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VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline 2

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3 Valence shell electron pair repulsion (VSEPR) theory VSEPR

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You know intuitively if something is symmetric but we require a precise method to describe how an object or molecule is symmetric. Definitions: – The quality of something that has two sides or halves that are the same or very close in size, shape, and position. (Webster) – When one shape becomes exactly like another if you flip, slide or turn it. (mathisfun.com) – The correspondence in size, form and arrangement of parts on opposites sides of a plane, line or point. (dictionary.com) Symmetry Elements Symmetry Operations Symmetry 4

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Symmetry Elements – a point, line or plane with respect to which the symmetry operation is performed – is a point of reference about which symmetry operations can take place Symmetry Operations – a process that when complete leaves the object appearing as if nothing has changed (even through portions of the object may have moved). Symmetry Do Something! If A, B and C are equivalent. 5

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Symmetry Element: Inversion Center Each atom in the molecule is moved along a straight line through the inversion center to a point an equal distance from the inversion center. For any part of an object there exists an identical part diametrically opposite this center an equal distance from it. 6

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Symmetry Element: Axis A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis. An axis around which a rotation by 360°/n results in an object indistinguishable from the original. 7

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Symmetry Element: Plane A molecule can have more than one plane of symmetry. A plane of reflection through which an identical copy of the original is given. 8

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Symmetry Elements Point = Center of Symmetry = 2 dimensions Line = Axis of Symmetry Plane = Plane of Symmetry = 0 dimension = 1 dimension Volume of symmetry? 9

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Symmetry elements – Axis – Mirror plane – Center of inversion Symmetry Operations – Identity (E) – Proper Rotation (C n ) – Reflection ( ) – Inversion (i) – Improper Rotation (S n ) Elements and Operations 10

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Leaves the entire molecule unchanged. All objects have identity. Abbreviated E, from the German 'Einheit' meaning unity. E Identity (E) ~ doing nothing! 11

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Proper Rotation (C n ) Rotation about an axis by 360°/n C 2 rotates 180°, C6 rotates 60°. Each rotation brings is indistinguishable from the original. C5C5 C2C2 C n m : apply a C n rotation m times 12

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Rotations (C n ) C n m : apply a C n rotation m times 13 360°/6 = 60°

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More than 1 C n BH 3 Highest n = principle axis (z-direction) benzene C 6, C 3, C 2 6 x ⊥ C 2 14 C 3 3 x ⊥ C 2

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Reflection ( ) Take each atom in the molecule and move it toward the reflection plane along a line perpendicular to that plane. Continue moving the atom through the plane to a point equidistant from the plane on the opposite side of the plane 15

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v = through atom vertical plane d = between atoms/bonds dihedral planes h = perpendicular to the primary rotation axis Reflection ( ) Multiple Reflection Planes benzene 3 x v 3 x d 1 x h 16

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Reflection ( ) Multiple Reflection Planes v or xz C2C2 1 2 v ’ or yz 17

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Inversion (i) Each atom in the molecule is moved along a straight line through the inversion center to a point an equal distance from the inversion center. 18

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Inversion (i) Does not require an atom at the center of inversion. C60 Benzene 19

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Rotation about an axis by 360°/n followed by reflection through the plane perpendicular to the rotation axis. Labeled as an S n axis. Improper Rotation (S n ) S n m : apply a C n rotation then reflection m times 20

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Improper Rotation (S n ) S n m : apply a C n rotation then reflection m times Allene B2H4B2H4 21

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Improper Rotation (S n ) S n m : apply a C n rotation then reflection m times n = even 22

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Improper Rotation (S n ) S n m : apply a C n rotation then reflection m times n = odd 23

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– Identity (E) – Proper Rotation (C n ) – Reflection ( ) – Inversion (i) – Improper Rotation (S n ) Symmetry Operations 24

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Side note: Crystals Molecular Symmetry Crystallographic Symmetry Center of Mass Unchanged Identity Rotation Reflection Inversion Additional Symmetry Operations 25

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Screw Symmetry Glide-reflection Symmetry The combination of reflection and translation. The combination of rotation and translation. Crystallographic Symmetry Operations 26

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Groups of Symmetry Operations Many molecules have 2 or more symmetry operations/elements. Not all symmetry operations occur independently. Molecules can be classified and grouped based on their symmetry. Molecules with similar symmetry elements and operations belong to the same point group. Called point group because symmetry operations are related to a fixed point in the molecule (the center of mass). 27

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Point Groups Infinite number of molecules, finite number of point groups. 28

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Nonaxial (no rotation) - C 1, C s, C i Cyclic (rotational) -C n, C nv, C nh, S n Dihedral ( ⊥ C 2 ) - D n, D nd, D nh Polyhedral - T, T h, T d, O, O h, I, I h Linear - C ∞v, D ∞h Types of Point Groups http://symmetry.jacobs-university.de/ 29

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There are no symmetry elements except E Nonaxial: C 1 Many molecules are C 1 30

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Symmetry elements: E and Nonaxial: C s 31

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Symmetry elements: E and i Nonaxial: C i 32

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Cyclic: C n Symmetry elements: E and C n 33 C3C3 C2C2 C6C6 [6]-rotane

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Cyclic: C nv Symmetry elements: E, v / d and C n 34 C 2v C 2 principle axis v contains C 2 axis d contains C 2 axis C 6v C 6 principle axis 3 x v contains C 6 axis 3 x d contains C 6 axis gallium( 6 -C 6 Me 6 )

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Cyclic: C nh Symmetry elements: E, h, S n and C n 35 C 3h C 3 principle axis h ⊥ to C 3 axis S 3 axis C 6h C 6 principle axis h ⊥ to C 6 axis S 6 axis hexakis(Me 2 N)benzene

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Cyclic: C nh Symmetry elements: E, h, S n and C n 36

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Cyclic: S n Symmetry elements: E and S n 37 S4S4 S 4, C 2 principle axis S6S6 S 6, C 3 principle axis tetrabromopentane 18-crown-6 Only S 4, S 6, S 8, etc.

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38 Dihedral Point Groups CnCn DnDn D nd D nh C 2 perpendicular to C n C2C2 CnCn C2C2 CnCn C2C2 dd hh

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39 Dihedral: D n D3D3 Symmetry elements: E, C n and ⊥ C 2 D2D2 (SCH 2 CH 2 ) 3 D3D3

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40 Dihedral: D n Symmetry elements: E, C n and ⊥ C 2 Ni(en) 3 en = H 2 NCH 2 CH 2 NH 2 1 x C 3 3 x ⊥ C 2

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41 Dihedral: D nd Symmetry elements: E, C n, ⊥ C 2 and d D 2d Allene

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42 Dihedra: D nd Symmetry elements: E, C n, ⊥ C 2 and d D 2d Staggard Sandwich complexes (D 5d ) View down the C 5 axis

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43 Dihedra: D nh Symmetry elements: E, C n, ⊥ C 2 and h Benzene D 6h

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44 Dihedra: D nh Symmetry elements: E, C n, ⊥ C 2 and h D 2h D 3h D 5h D 4h D 6h

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45 Dihedra: D nh Symmetry elements: E, C n, ⊥ C 2 and h X X X X Y Y trans-MY 2 X 4 OsCl 2 (CO) 4 PtCl 4

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46 Dihedral Point Groups D 5d D 5h Ferrocene (Fc) D5D5

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Polyhedral Point Groups More than two high-order axes Sometimes referred to as the Platonic solids. Earth Air Water Fire

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Polyhedral Point Groups More than two high-order axes I, O and T rarely found No mirror planes! Point Groups Tetrahedral - T, T h, T d Octahedral - O, O h Icosahedral - I, I h

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Polyhedral: Tetrahedron Point Groups Has d No Has h

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Polyhedral: T No Only Rotational Symmetry! [Ca(THF) 6 ] 2+ Very rare!

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Polyhedral: T d Rotational + d Most common tetrahedral point group

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Polyhedral: T h Rotational + h Has h

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Polyhedral: Octahedron Point Groups Has h No

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Polyhedral: Octahedron No V 6 P 8 O 24 -core Dodeka(ethylene)octamine Very rare!

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Polyhedral: Octahedron

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Polyhedral: Icosahedral Point Groups Has h No I snub dodecahedron IhIh Icosahedron E, 15C 5, 12C 5 2, 20C 3, 15C 2 E, 15C 5, 12C 5 2, 20C 3, 15C 2, i, 12S 10, 12S 10 3, 20S 6, 15σ I couldn’t find a molecule

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Polyhedral: Icosahedral Has h IhIh Icosahedron E, 15C 5, 12C 5 2, 20C 3, 15C 2, i, 12S 10, 12S 10 3, 20S 6, 15σ B 12 H 12

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Polyhedral Point Groups More than two high-order axes I, O and T rarely found No mirror planes! Point Groups Tetrahedral - T, T h, T d Octahedral - O, O h Icosahedral - I, I h

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60 High Symmetry BxHxBxHx

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Linear Point Groups Point Groups Has h No C ∞v D ∞h E, C ∞, ∞σ v E, C ∞, ∞σ v, ∞C 2, I HCl, CO, NC NCS, HCN, HCCH H 2, N 2, O 2, F 2, Cl 2 CO 2, BeH 2, N 3

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Nonaxial (no rotation) - C 1, C s, C i Cyclic (rotational) -C n, C nv, C nh, S n Dihedral ( ⊥ C 2 ) - D n, D nd, D nh Polyhedral - T, T h, T d, O, O h, I, I h Linear - C ∞v, D ∞h Types of Point Groups http://symmetry.jacobs-university.de/ 62

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Memorize – C s : E and S – O h : E, 3S 4, 3C 4, 6C 2, 4S 6, 4C 3, 3σ h, 6σ d, i Flow chart 63 Assigning point groups

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67 CO 2

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∞∞ 68

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70 Cotton’s “five-step” Procedure

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71 General Assignments VSEPR GeometryPoint group LinearD∞h Bent or V-shape C 2v Trigonal planar D 3h Trigonal pyramidal C 3v Trigonal bipyramidal D 5h Tetrahedral Td Td Sawhorse or see-saw C 2v T-shape C 2v Octahedral Oh Square pyramidalC 4v Square planar D 4h Pentagonal bipyramidal D 5h For MX n M is a central atom X is a ligand

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http://symmetry.otterbein.edu/gallery/index.html 72 Webpage

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2-D vs 3-D Global vs. Local Dynamic Molecules 73 Pitfalls in Assigning Symmetry

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74 2-D vs 3-D Molecules D 3h C3C3

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75 Global vs. Local Symmetry vs. C 2v CsCs h + 2 + 2 cycloaddition

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76 Global vs. Local Symmetry D3D3 OhOh A 1 = z 2 E = x 2 -y 2, xy, xz, yz E g = z 2, x 2 -y 2 T 2g = xy, xz, yz

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77 Global vs. Local Symmetry hemoglobin C2C2 heme CsCs porphyrin D 4h

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78 Global vs. Local Symmetry No translation Quasicrystals Dan Shechtman 2011 Nobel Prize in Chemistry Crystals

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79 Dynamic Molecules Amine Inversion 10 ^10 per second Symmetry Through the Eyes of a Chemist Changes in: Symmetry Dipole moment Chirality Vibrational motion

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80 Dynamic Molecules

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81 a)NH 3 b)SF 6 c)CCl 4 d)H 2 C=CH 2 e)H 2 C=CF 2 CO 2 OCS Assign the point group

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82 Stereographic Projections two-fold rotational axis three-fold rotational axis Solid line = v Dashed line = d Objection Position = in plane x = below plane o = above plane Dashed circle = no d Solid circle = h

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83 Stereographic Projections S 4 axis S 4 and C 2 C 3, 3C 2, S 6, and 3 d

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84 Stereographic Projections What about I h ?

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85 Stereographic Projections 6C 5, 10C 3, 15C 2, 15σ

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86 What Point Group? D 4h C 6h D3D3

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Symmetry Operation Math Easy Alternative: Represent symmetry operations with matrices. 87 xz O (x o, y o, z o ) O (x o, -y o, z o ) H1 (x h1, y h1, z h1 ) H1 (x h1, -y h1, z h1 ) H2 (x h2, y h2, z h2 ) H2 (x h2, -y h2, z h2 ) C5C5 C1 (x c1, y c1, z c1 ) C60 (x c60, y c60, z c60 )... Object after xz = Matrix for xz

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Matrix Math Outline 88 Matrices Trace/character Matrix algebra Inverse Matrix Block Diagonal Direct Sum Direct Product Symmetry operation matrix

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A matrix is a rectangular array of numbers, or symbols for numbers. These elements are put between square brackets. Generally a matrix has m rows and n columns: m Rows n columns Square matrix, m=n 89 Matrices

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Method to simplify a matrix. The character of a matrix is the sum of its diagonal elements. square matrix 90 Trace/Character

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Unit Matrix/Identity: All elements in the diagonal are 1 and the rest are zero. Column Matrix: used to represent a point or a vector. The Identity Matrix (E) 91 Special Matricies

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Addition: Subtraction: 92 Matrix Algebra For two matricies with equal dimensions: Multiply by a constant:

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93 Matrix Multiplication If two matrices are to be multiplied together they must be conformable; i.e., the number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix. To get the first member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding elements of the first column of the second matrix and the results are added. To get the second member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding members of the second column of the second matrix and the results are added, and so on.

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94 Matrix Multiplication If two matrices are to be multiplied together they must be conformable; i.e., the number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix. To get the first member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding elements of the first column of the second matrix and the results are added. To get the second member of the first row, all elements of the first row of the first matrix are multiplied by the corresponding members of the second column of the second matrix and the results are added, and so on.

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95 Matrix Multiplication

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Non-square matrix- The number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix. 96 Matrix Multiplication 2 columns 2 Rows columns of a = rows of b 3x2 times 2x3 = 3x3

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Non-square matrix- The number of columns in the first (left) matrix must be the same as the number of rows in the second (right) matrix. 97 Matrix Multiplication 2 columns 2 Rows columns of a = rows of b 3x2 times 2x3 = 3x3

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98 Matrix Multiplication: Vectors Identity Matrix (E) vector

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99 Inverse Matrices If two square matrices A and B multiply together, and A B = E then B is said to the inverse of A (written A -1 ). B is also the inverse of A. A B E aka A -1 Rules that govern groups: 4)For every element in the group there exists an inverse.

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A block-diagonal matrix has nonzero values only in square blocks along the diagonal from the top left to the bottom right 100 Block-diagonal Matrix

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Block Diagonal Matrix Row 1 and so on… Full Matrix MultiplicationBlock Multiplication Important in reducing matrix representations. 101

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102 Direct Sum The direct sum is very different from ordinary matrix addition since it produces a matrix of higher dimensionality. A direct sum of two matrices of orders n and m is performed by placing the matrices to be summed along the diagonal of a matrix of order n+m and filling in the remaining elements with zeroes = ⊕ ⊕ Γ (5) (g) = Γ (2) (g) ⊕ Γ (2) (g) ⊕ Γ (1) (g)

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103 Direct Product The direct product of two matrices (given the symbol ⊗ ) is a special type of matrix product that generates a matrix of higher dimensionality. 2 x 2 ⊗ 2 x 2 = 4 x 4

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Matrix Math in Group Theory Easy Alternative: Represent symmetry operations with matrices. 104 Object after X = Matrix for X X = Symmetry Operations – Identity (E) – Proper Rotation (C n ) – Reflection ( ) – Inversion (i) – Improper Rotation (S n )

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x,y or z stay the same = 1 on the diagonal x,y or z shift negative = -1 on the diagonal If a coordinate is transformed into another coordinate into the intersection position, respectively. These intersection positions will be off the matrix diagonal. A symmetry operation can be represented by a matrix. 105 Symmetry Operation Matrix

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Reflection ( ) 106

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Inversion (i) 107

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Rotation (C n ) x and y are dependent 1 0 0 0 0 z does not change Trigonometry! 108

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Rotation (C n ) sin = O/H cos = A/H x = O y = A l = H x 1 = l cos y 1 = l sin x 2 = l cos ( ) y 2 = l sin ( ) x 1, y 1 x 2, y 2 ? 109

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Rotation (C n ) General Matrix for C n C2C2 C3C3 For C 3 θ = 120° cos θ = -1/2 and sin θ = √3/2 For C 2 θ = 180° cos θ = -1 and sin θ = 0 110

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Rotation (C n ) General Matrix for C n 111

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Improper Rotation (S n ) h C n = S n = = 112

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C 2h Matrix Math Symmetry Operations: E, C 2(z), i, h E C 2(z) hh i 113

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C 3v Matrix Math Symmetry Operations: E, C 3, C 3 2, v’,v”,v’’’ E C31C31 v’v’ C32C32 v ’’ v ’’’ v’v’ v ’’ x y 114

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Summary 115

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VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Review 116

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