Chem 59-651 Symmetry and Introduction to Group Theory Symmetry is all around us and is a fundamental property of nature. Chem 59-651 Symmetry and Introduction to Group Theory The term symmetry is derived from the Greek word “symmetria” which means “measured together”. An object is symmetric if one part (e.g. one side) of it is the same* as all of the other parts. You know intuitively if something is symmetric but we require a precise method to describe how an object or molecule is symmetric. Group theory is a very powerful mathematical tool that allows us to rationalize and simplify many problems in Chemistry. A group consists of a set of symmetry elements (and associated symmetry operations) that completely describe the symmetry of an object. We will use some aspects of group theory to help us understand the bonding and spectroscopic features of molecules. Chem 59-651 We need to be able to specify the symmetry of molecules clearly. F H No symmetry – CHFClBr Br Cl F H Some symmetry – CHFCl2 H H Cl Cl More symmetry – CH2Cl2 Cl Cl H Cl More symmetry ? – CHCl3 Cl Cl What about ? Point groups provide us with a way to indicate the symmetry unambiguously. Chem 59-651 Symmetry and Point Groups Point groups have symmetry about a single point at the center of mass of the system. Symmetry elements are geometric entities about which a symmetry operation can be performed. In a point group, all symmetry elements must pass through the center of mass (the point). A symmetry operation is the action that produces an object identical to the initial object. The symmetry elements and related operations that we will find in molecules are: Element Operation Rotation axis, Cn n-fold rotation Improper rotation axis, Sn n-fold improper rotation Plane of symmetry, σ Reflection Center of symmetry, i Inversion Identity, E The Identity operation does nothing to the object – it is necessary for mathematical completeness, as we will see later. Chem 59-651 ∞ n-fold rotation - a rotation of 360°/n about the Cn axis (n = 1 to ) O(1) 180° O(1) H(2) H(3) H(3) H(2) In water there is a C2 axis so we can perform a 2-fold (180°) rotation to get the identical arrangement of atoms. H(3) H(4) H(2) 120° 120° N(1) N(1) N(1) H(4) H(2) H(4) H(3) H(2) H(3) In ammonia there is a C3 axis so we can perform 3-fold (120°) rotations to get identical arrangement of atoms. Chem 59-651 Notes about rotation operations: - Rotations are considered positive in the counter-clockwise direction. - Each possible rotation operation is assigned using a superscript integer m m of the form Cn . n - The rotation Cn is equivalent to the identity operation (nothing is moved). H(3) H(2) H(4) 1 C 2 C3 3 N(1) N(1) N(1) H(2) H(4) H(4) H(3) H(3) H(2) H(2) 3 C3 = E N(1) H(4) H(3) Chem 59-651 m Notes about rotation operations, Cn : - If n/m is an integer, then that rotation operation is equivalent to an n/m - fold rotation. 2 1 2 1 3 1 e.g. C4 = C2 , C6 = C3 , C6 = C2 , etc. (identical to simplifying fractions) Cl(5) Cl(2) Cl(3) 1 2 1 C4 C4 = C2 Cl(2) Ni(1) Cl(3) Cl(4) Ni(1) Cl(5) Cl(5) Ni(1) Cl(4) Cl(4) Cl(3) Cl(2) 3 C4 Cl(4) Cl(3) Ni(1) Cl(2) Cl(5) Chem 59-651 m Notes about rotation operations, Cn : - Linear molecules have an infinite number of rotation axes C∞ because any rotation on the molecular axis will give the same arrangement. C(1) O(2) O(2)C(1) O(3) C(1) O(2) N(1)N(2) N(1) N(2) Chem 59-651 The Principal axis in an object is the highest order rotation axis. It is usually easy to identify the principle axis and this is typically assigned to the z-axis if we are using Cartesian coordinates. Ethane, C2H6 Benzene, C6H6 The principal axis is the three-fold The principal axis is the six-fold axis axis containing the C-C bond. through the center of the ring. The principal axis in a tetrahedron is a three-fold axis going through one vertex and the center of the object. Chem 59-651 Reflection across a plane of symmetry, σ (mirror plane) σ O(1) v O(1) H(2) H(3) H(3) H(2) These mirror planes are called “vertical” mirror σ planes, v, because they contain the principal axis. σ O(1) v O(1) The reflection illustrated in the top diagram is through a mirror plane H(2) H(3) H(2) H(3) perpendicular to the plane of the water molecule. The plane shown on the bottom is in the same plane as the Handedness is changed by reflection! water molecule. Chem 59-651 Notes about reflection operations: - A reflection operation exchanges one half of the object with the reflection of the other half. σ - Reflection planes may be vertical, horizontal or dihedral (more on d later). - Two successive reflections are equivalent to the identity operation (nothing is moved). σ σ h A “horizontal” mirror plane, h, is perpendicular to the principal axis. This must be the xy-plane if the z- axis is the principal axis. σ σ σ In benzene, the h is in the plane d d of the molecule – it “reflects” each atom onto itself. σ h Vertical and dihedral mirror σ σ planes of geometric shapes. v v Chem 59-651 Inversion and centers of symmetry, i (inversion centers) In this operation, every part of the object is reflected through the inversion center, which must be at the center of mass of the object. 1 F Cl 22F Cl 1 2 Br i Br 1 1 221 1Br 2 Br 1 Cl F2 2 Cl F1 i [x, y, z] [-x, -y, -z] We will not consider the matrix approach to each of the symmetry operations in this course but it is particularly helpful for understanding what the inversion operation does. The inversion operation takes a point or object at [x, y, z] to [-x, -y, -z]. Chem 59-651 m n-fold improper rotation, Sn (associated with an improper rotation axis or a rotation-reflection axis) This operation involves a rotation of 360°/n followed by a reflection perpendicular to the axis. It is a single operation and is labeled in the same manner as “proper” rotations. F1 F2 H1 F F4 1 1 S4 H4 C H2 H3 F F 2 F 3 1 2 S4 F3 F4 F1 H2 σ 90° h H C H 1 1 3 C2 F3 H4 2 F4 S4 H3 H2 C H4 H1 σ Note that: S1 = , S2 = i, and sometimes S2n = Cn (e.g. in box) this makes more sense when you examine the matrices that describe the operations. Chem 59-651 Identifying point groups We can use a flow chart such as this one to determine the point group of any object. The steps in this process are: 1. Determine the symmetry is special (e.g. octahedral). 2. Determine if there is a principal rotation axis. 3. Determine if there are rotation axes perpendicular to the principal axis. 4. Determine if there are mirror planes. 5. Assign point group. Chem 59-651 Identifying point groups Chem 59-651 Identifying point groups Special cases: Perfect tetrahedral (Td) e.g. P4, CH4 -2 Perfect octahedral (Oh) e.g. SF6, [B6H6] -2 Perfect icosahedral (Ih) e.g. [B12H12] , C60 Chem 59-651 Identifying point groups Low symmetry groups: Only* an improper axis (Sn) e.g. 1,3,5,7-tetrafluoroCOT, S4 F1 F4 F2 F3 Only a mirror plane (Cs) e.g. CHFCl2 F H Cl Cl Chem 59-651 Identifying point groups Low symmetry groups: Only an inversion center (Ci) e.g. (conformation is important !) F Cl Br Br Cl F No symmetry (C1) e.g. CHFClBr F H Br Cl Chem 59-651 Identifying point groups Cn type groups: σ A Cn axis and a h (Cnh) e.g. B(OH)3 (C3h, conformation is important !) H O B H O O H e.g. H2O2 (C2h, conformation is important !) H O O H Note: molecule does not have to be planar e.g. B(NH2)3 (C3h, conformation is important !) Chem 59-651 Identifying point groups Cn type groups: Only a Cn axis (Cn) e.g. B(NH2)3 (C3, conformation is important !) H H N B H N N H H H e.g. H2O2 (C2, conformation is important !) H O O H Chem 59-651 Identifying point groups Cn type groups: σ A Cn axis and a v (Cnv) e.g. NH3 (C3v) H N H H e.g. H2O2 (C2v, conformation is important !) O O H H Chem 59-651 Identifying point groups Cn type groups: σ A Cn axis and a v (Cnv) e.g. NH3 (C3v, conformation is important !) H H e.g. carbon monoxide, CO (C∞v) H There are an infinite number of possible N C axes and σ mirror planes. H H n v H H H H C O - e.g. trans-[SbF4ClBr] (C4v) Cl F F F Sb OC F SbBrCl F F F Br F Chem 59-651 Identifying point groups Dn type groups: A Cn axis, n perpendicular C2 axes σ and a h (Dnh) e.g.