Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy). Predict IR spectra or Interpret UV-Vis spectra Predict optical activity of a molecule 1 2 Molecular Symmetry Symmetry impacts Physical properties Reactions Molecular orbitals Electronic structure Molecular vibrations Group theory Behavior of molecule based on symmetry Symmetry analysis Application of symmetry Orbital symmetry Vibrational symmetry 3 Symmetry is all around us and is a fundamental property of nature. 4 5 6 1
Facial symmetry Invariance to transformation as an indicator of facial symmetry: No! Yes 7 Mirror image 8 What is symmetry? invariance* to transformation in space nature of the transformation determines the type of symmetry determines crystal packing, orbital overlap, spectroscopic properties Symmetry operations and elements *an invariance (meaning in mathematics and theoretical physics) is a property of a system which remains unchanged under some transformation. 9 10 Symmetry operations and elements Group Theory: mathematical treatment of symmetry. Symmetry Operation = an operation performed on an object which leaves it in a configuration that is indistinguishable from the original appearance (or action which molecular symmetry unchanged). e.g. rotation through an angle, reflection. Symmetry Elements = the points, lines, or planed to which a symmetry operation is carried out. Symmetry operations and elements Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane Inversion center Inversion of a point x, y, z to x,-y,-z Proper axis Rotation by (360/n) C n Improper axis 1. Rotation by (360/n) 2. Reflection in plane perpendicular to rotation axis i S n 11 12 2
1. Identity operation (E) The identity operation, E, states that the object exists Also denoted as C 1 or C n n Its existence is demanded by the math of group theory, and common sense 2. Planes and Reflection (σ) or Mirror plane (m) CHFClBr D-Glucose and L-Phenylalanine 13 14 Mirror plane symmetry element is a plane all of the points of a molecule are passed through the plane hands Mirror plane H 2 O Eyes glasses chair 15 16 Reflection across a plane of symmetry, (mirror plane) 2 = E Examples: Mirror plane v v Handedness is changed by reflection! 17 18 3
3. Inversion, Center of Inversion (i) Center of Inversion (i) The inversion operation takes a points through the center of symmetry of the molecule to an equal distance on the other side. A point at the center of the molecule (x, y, z) ---> (-x, -y, -z) 19 20 Examples: Center of Inversion (i) Examples: Center of Inversion (i) 21 22 4. Proper axes of rotation (Cn) Rotational symmetry rotation by an angle, such that n = 360 n is the order of the rotation C n the symmetry element is a line, about which the rotation takes place a C n axis generates n operations, which form a cyclic group or subgroup n = 2 n = 5 n = 6 360/2 360/5 360/6 180 o 72 o 60 o i. e. C 4 generates C 41, C 42 =C 2, C 43, C 44 =E 23 24 4
Examples: Proper axes of rotation (Cn) 25 26 C nm = E 180 120 120 180 27 28 29 30 5
Note: C 42 = C 21, C 62 = C 31, C 63 = C 2 1 C 4 1 C 42 = C 2 1 5. Rotation-reflection, Improper axis (Sn) S n - where n indicates the order of the rotationcomposed of two successive geometry transformations: first, a rotation through 360 /n about the axis of that rotation (C n ), and second, reflection through a plane perpendicular to Cn ( h). symmetry element is both a line and a plane C 4 3 31 32 Improper axis (Sn) Improper axis (Sn) CH 4 1) 90 rotation 2) reflection 33 34 CH 4 35 36 6
n-fold improper rotation, S n m rotation axes and mirror plane molecules: (H 2 O) S 4 1 S 4 1 90 h C 2 1 S 4 2 Note that: S 1 =, S 2 = i, and sometimes S 2n = C n (e.g. in box) this makes more sense if you examine the final result of each of the operations. 37 38 rotation axes and mirror plane molecules: (C 6 H 6 ) rotation axes and mirror plane molecules: (BF 3 ) 39 40 rotation axes and mirror plane molecules Examples for the different basic symmetry operations and symmetry elements 41 42 7
43 44 Point Groups Point Groups Point Group = the set of symmetry operations for a molecule. Group Theory = the mathematical treatment of the properties of groups, can be used to determine the molecular orbitals, vibrations, and other properties of the molecules. 45 46 Non rotation group (Low Symmetry) Group Symmetry Examples C1 E CHFClBr Single axis group Group Symmetry Examples C n E, C n H 2 O 2 Cs E, h H2C=CClBr C nv E, C n, nσ v H 2 O C n subgroup + n v Ci E, i HClBrC=CHClBr C nh E, C n, σ h, S n, i B(OH) 3 Cn subgroup + h 47 48 8
C n Point Groups C nv Point Groups 49 50 C nh Point Groups Single axis group Group Symmetry Examples S 2n E, S 2n 1,3,5,7-tetrafluoracyclooctatetrane C v E, C, σ v HCl 51 52 S n Point Groups Dihedral groups Group Symmetry Examples D n E, C n, nc2 NiN 6 D nd E, C n, nc 2, S 2n S 8 D nh E, C n, nc 2,σ h, nσ v D h E, i, C v, σ v 53 54 9
Dihedral groups D n and D nh Point Groups adding a C 2 perpendicular to the C n requires that there must be n C 2 axes perpendicular to the C n D n : C n + n C 2 axes (molecules in this group must have a zero dipole moment and be optically active D nh : D n + h D nd : D n + v. The v operations will bisect the adjacent C 2 axes 55 56 D nd Point Groups h v 57 58 Cubic groups Group Symmetry Examples T d E, C 3, C 2, S 4, σ d CCl 4 O h E, C 3, C 2, C 4, i, S 4, S 6, σ d,σ h I h E, C 3, C 2, C 5, i, S 10, S 6,σ 59 60 10
High Symmetry molecules Platonic solids 61 62 Point Groups p-dichlorobenzene: E, 3, 3C 2, i or E, C 2, 2 v Ethane (staggered): E, 3, C 3, 3C 2, i, S 6 63 64 Perpendicular C 2 axes Horizontal Mirror Planes 65 66 11
Vertical or Dihedral Mirror Planes and S 2n Axes 67 68 Application of Symmetry Construction and labeling of molecular orbitals Molecular properties Polarity Chirality 69 70 Polar Molecule Chiral Molecules 71 72 12
Representation of Point Groups Properties and Representations of Groups Matrices Why Matrices? The matrix representations of the point group s operations will generate a character table. We can use this table to predict properties. C ij = A ik x B kj C ij = product matrix; i rows and j columns A ik = initial matrix; i rows and k columns B kj = initial matrix; k rows and j columns 73 74 1. Matrix Representations of C 2v 1) Choose set of x,y,z axes - z is usually the C n axis - xz plane is usually the plane of the molecule 2) Examine what happens after the molecule undergoes each symmetry operation in the point group (E, C 2, 2 ) E Transformation Matrix x,y,z x,y,z What matrix times x,y,z doesn t change anything? E Transformation Matrix 75 76 C 2 Transformation Matrix x,y,z -x, -y, z Correct matrix is: v (xz) Transformation Matrix x,y,z x,-y,z Correct matrix is: v (yz) Transformation Matrix x,y,z -x,y,z Correct matrix is: These 4 matrices are the Matrix Representation of the C 2v point group All point group properties transfer to the matrices as well Example: E v (xz) = v (xz) 77 78 13
2. Reducible and Irreducible Representations 2). Reducible and Irreducible Representations 1). Character = sum of diagonal from upper left to lower right (only defined for square matrices) The set of characters = a reducible representation ( ) or shorthand version of the matrix representation For C 2v Point Group: E C 2 v (xz) v (yz) E C 2 v (xz) v (yz) 3-1 1 1 79 80 Irreducible Representations Reducible Repr. Axis used E C 2 v (xz) v (yz) x 1 1 1 1 y 1 1 1 1 z 1 1 1 1 3 1 1 1 81 Totally symmetric representatio n of the group Point group Classes of symmetry operations Symmetry or Mulliken labels, each corresponding to a different irreducible representation Character Tables C 4v E 2C 4 (z) C 2 2s v 2s d A 1 1 1 1 1 1 z x 2 +y 2, z 2 z 3, z(x 2 +y 2 ) A 2 1 1 1-1 -1 R z - - B 1 1-1 1 1-1 - x -y z(x -y ) 2 2 2 2 B 2 1-1 1-1 1 - xy xyz E 2 0-2 0 0 (x, y) (R x, R y ) (xz, yz) (xz 2, yz 2 ) (xy 2, x 2 y) (x 3, y 3 ) Characters (of the IRs of the group) 1 indicate that operation leave the function unchange -1 indicate that operation reverses the function Basis functions having the same symmetry as the IR cubic functions quadratic functions linear functions translations along specified axis R, rotation about specified axis Symmetries of the s, p, d, and f orbitals can be found here (by their labels). Ex: the d xy orbital shares the same symmetry as the B 2 IR. The s orbital always belongs to the totally symmetric representation (the first listed IR of any point group). 82 Irreducible Representation Labels Character Tables R = any symmetry operation = character (#) i,j = different representations (A1, B2, etc ) h = order of the group (4 total operations in the C 2v case 83 84 14
85 86 87 88 89 90 15
The C 3v character table Symmetry operations The complete C 4v character table NH 3 Irreducible representations The order h is 6 There are 3 classes A 1 transforms like z. E does nothing, C 4 rotates 90 o about the z-axis, C 2 rotates 180 o about the z-axis, v reflects in vertical plane and d in a diagonal plane. 91 92 A 2 transforms like a rotation around z. E +R z C 4 +R z C 2 +R z v -R z d -R z 93 94 95 96 16
97 98 Molecular Vibrations Molecular Vibrations 99 100 101 Theory Electron repulsion Bond breaking Vibrational modes Depends upon number of atoms and degrees of freedom (*Degrees of Freedom = possible atomic movements in the molecule) - 3N degrees of freedom for a molecule of N atoms Constraints due to Translational and rotational motion of molecule Motion of atoms relative to each other - Linear molecules Only 2 rotations change the molecule 3N 5 vibrations - Nonlinear molecules 3 translations (along x, y, z) 3 rotations (around x, y, z) 3N 6 vibrations 102 17
Selection Rules: Infrared and Raman Spectroscopy Water molecule (C 2V ) 103 104 Reducible Representations 105 106 Reducible to Irreducible representation The other entries for can also be found without the matrices E: all 9 vectors are unchanged--> 9 C 2 : H atoms change position in C 2 rotation, so all vectors have zero contribution to the character. O atom vectors in x and y are reversed, each contributing -1 and in z direction is the same, contributing 1. --> (C 2 ) = (-1)+(-1)+1 = -1 v (xz): reflection in the plane of the molecule changes the direction of all the y vector, the x and z are unchanged. ---> 3-3+3 = 3. v (yz): reflection perpendicular to the plane of the molecule changes the position of H atoms so their contribution is zero, the x vector on O changes direction, the y and z are unchanged. ---> -1+1+1 = 1 9x9 vector 107 Reducible representation of H 2 O 108 18
Reducible to Irreducible representation Reducible to Irreducible representation H 2 O a p = n R (R) p (R) a p = n R (R) p (R) 109 irrep. B1 B2 3A1 + A2 + 3B1 + 2B2 110 111 112 IR and Raman Active B2 113 114 19
A1 B1 B2 B2 1 115 116 117 118 119 120 20
Vibrational modes of SO 2 (C 2v ) 121 122 BCl 3 123 124 125 21