Symmetry and Introductionto Group Theory Disclaimer: Some lecture note slides are adopted from CHEM 59- 250 - Originally by Dr. Samuel Johnson Power point slides from Inorganic Chemistry 4th edition by Gary L. Miessler and Donald A. Tarr
Symmetry and Point GroupsI. Introduction A. Symmetry is present in nature and in human culture
B. Using Symmetry in Chemistry 1. Understand what orbitals are used in bonding 2. Predict IR spectra or Interpret UV-Vis spectra 3. Predict optical activity of a moleculeII. Symmetry Elements and Operations A. Definitions 1. Symmetry Element = geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations can be carried out 2. Symmetry Operation = a movement of a body such that the appearance after the operation is indistinguishable from the original appearance (if you can tell the difference, it wasn’t a symmetry operation) B. The Symmetry Operations 1. E (Identity Operation) = no change in the object a. Needed for mathematical completeness b. Every molecule has at least this symmetry operation
2. Cn (Rotation Operation) = rotation of the object 360/n degrees about an axis a. The symmetry element is a line b. Counterclockwise rotation is taken as positive c. Principle axis = axis with the largest possible n value d. Examples:C23 = two C3’sC33 = E C17 axis
3. s (Reflection Operation) = exchange of points through a plane to an opposite and equidistant point a. Symmetry element is a plane b. Human Body has an approximate s operation c. Linear objects have infinite s‘s d. s h = plane perpendicular to principle axis e. s v = plane includes the principle axis f. s d = plane includes the principle axis, but not the outer atoms sd O C O O H H sh sv
4. i (Inversion Operation) = each point moves through a common central point to a position opposite and equidistant a. Symmetry element is a point b. Sometimes difficult to see, sometimes not present when you think you see it c. Ethane has i, methane does not d. Tetrahedra, triangles, pentagons do not have i e. Squares, parallelograms, rectangular solids, octahedra do
5. Sn (Improper Rotation Operation) = rotation about 360/n axis followed by reflection through a plane perpendicular to axis of rotation a. Methane has 3 S4 operations (90 degree rotation, then reflection) b. 2 Sn operations = Cn/2 (S24 = C2) c. nSn = E, S2 = i, S1 = s d. Snowflake has S2, S3, S6 axes
C2 sdC. Examples: 1. H2O: E, C2, 2s O H H sv 2. p-dichlorobenzene: E, 3s, 3C2, i Cl Cl 3. Ethane (staggered): E, 3s, C3, 3C2, i, S6 H H H C C H H H 4. Try Ex. 4-1, 4-2
III. Point Groups A. Definitions: 1. Point Group = the set of symmetry operations for a molecule 2. Group Theory = mathematical treatment of the properties of the group which can be used to find properties of the molecule B. Assigning the Point Group of a Molecule 1. Determine if the molecule is of high or low symmetry by inspection a. Low Symmetry Groups
2. If not, find the principle axis3. If there are C2 axes perpendicularto Cn the molecule is in DIf not, the molecule will be in C or Sa. If sh perpendicular to Cn then Dnh or Cnh If not, go to the next stepb. If s contains Cn then Cnv or Dnd If not, Dn or Cn or S2nc. If S2n along Cn then S2n If not Cn
C. Examples: Assign point groups of molecules in Fig 4.8
Vertical or Dihedral Mirror Planes and S2n AxesExamples: XeF4, SF4, IOF3, Table 4-4, Exercise 4-3
D. Properties of Point Groups 1. Symmetry operation of NH3 a. Ammonia has E, 2C3 (C3 and C23) and 3sv b. Point group = C3v 2. Properties of C3v (any group) a. Must contain E b. Each operation must have an inverse; doing both gives E (right to left) c. Any product equals another group member d. Associative property
We need to be able to specify the symmetry of molecules clearly. F HNo 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.
Symmetry and Point GroupsPoint groups have symmetry about a single point at the center of mass ofthe system.Symmetry elements are geometric entities about which a symmetryoperation can be performed. In a point group, all symmetry elements mustpass through the center of mass (the point). A symmetry operation is theaction that produces an object identical to the initial object.The symmetry elements and related operations that we will find inmolecules are:The Identity operation does nothing to the object – it is necessary formathematical completeness, as we will see later. Element Operation Rotation axis, Cn n-fold rotation Improper rotation axis, Sn n-fold improper rotation Plane of symmetry, s Reflection Center of symmetry, i Inversion Identity, E
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 getthe identical arrangement of atoms. H(3) H(4) H(2) 120° 120° N(1) N(1) N(1) H(2) H(4) H(3) H(2)H(4) H(3) In ammonia there is a C3 axis so we can perform 3-fold (120°) rotations to get identical arrangement of atoms.
Notes about rotation operations:- Rotations are considered positive in the counter-clockwise direction.- Each possible rotation operation is assigned using a superscript integer m of the form Cnm.- The rotation Cnn is equivalent to the identity operation (nothing is moved). H(3) H(2) H(4) C31 C32 N(1) N(1) N(1)H(2) H(4) H(4) H(3) H(2) H(3) H(2) C33 = E N(1) H(4) H(3)
Notes about rotation operations, Cnm: - If n/m is an integer, then that rotation operation is equivalent to an n/m - fold rotation. e.g. C42 = C21, C62 = C31, C63 = C21, etc. (identical to simplifying fractions) Cl (5) Cl (2) Cl (3) C41 C42 = C21Cl (2) Ni (1) Cl (3) Cl (4) Ni (1) Cl (5) Cl (5) Ni (1) Cl (4) Cl (4) Cl (3) Cl (2) C43 Cl (4) Cl (3) Ni (1) Cl (2) Cl (5)
Notes about rotation operations, Cnm:- Linear molecules have an infinite number of rotation axes C because anyrotation on the molecular axis will give the same arrangement. C(1) O(2) O(2) C(1) O(3) C(1) O(2) N(2) N(1) N(1) N(2)
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 axis The principal axis is the six-fold 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.
Reflection across a plane of symmetry, s (mirror plane) O(1) sv O(1)H(2) H(3) H(3) H(2) These mirror planes are Handedness is changed by reflection! called “vertical” mirror planes, sv, because they contain the principal axis. O(1) sv O(1) The reflection illustrated in the top diagram is through a H(3) H(3) mirror plane perpendicular H(2) H(2) to the plane of the water molecule. The plane shown on the bottom is in the same plane as the water molecule.
Notes about reflection operations:- A reflection operation exchanges one half of the object with the reflection ofthe other half.- Reflection planes may be vertical, horizontal or dihedral (more on sd later).- Two successive reflections are equivalent to the identity operation (nothing ismoved). A “horizontal” mirror plane, sh, is sh perpendicular to the principal axis. This must be the xy-plane if the z- axis is the principal axis. In benzene, the sh is in the plane sd sd of the molecule – it “reflects” each atom onto itself. sh Vertical and dihedral mirror sv planes of geometric shapes. sv
Inversion and centers of symmetry, i (inversion centers)In this operation, every part of the object is reflected through the inversioncenter, which must be at the center of mass of the object. F Cl 1 F Cl 2 2 1 Br Br 2 i 1 1 2 Br 2 1 Br1 2 Cl F Cl F 1 2 2 1 i [x, y, z] [-x, -y, -z]We will not consider the matrix approach to each of the symmetry operations in thiscourse but it is particularly helpful for understanding what the inversion operation does.
n-fold improper rotation, Snm (associated with an improper rotation axis ora rotation-reflection axis) This operation involves a rotation of 360°/nfollowed by a reflection perpendicular to the axis. It is a single operation andis labeled in the same manner as “proper” rotations. F1 F2 H1F4 S4 1 F1 H4 C H2 H3 F2 F3 S41 F2 H2 F3 F4 F1 H1 C H3 90° sh H4 C21 F3 S42 F4 H3 H2 C H4 H1 Note that: S1 = s, S2 = i, and sometimes S2n = Cn (e.g. in box) this makes more sense if you examine the final result of each of the operations.
Identifying point groupsWe can use a flow chart such as thisone to determine the point group ofany object. The steps in this processare:1. Determine the symmetry is special(e.g. octahedral).2. Determine if there is a principalrotation axis.3. Determine if there are rotation axesperpendicular to the principal axis.4. Determine if there are mirror planes.5. Assign point group.
Identifying point groups Special cases:Perfect tetrahedral (Td) e.g. P4, CH4Perfect octahedral (Oh) e.g. SF6, [B6H6]-2 Perfect icosahedral (Ih) e.g. [B12H12]-2, C60
Identifying point groups Low symmetry groups: Only* an improper axis (Sn) e.g. 1,3,5,7-tetrafluoroCOT, S4 F1 F4 F2 F3Only a mirror plane (Cs)e.g. CHFCl2 F H Cl Cl
Identifying point groupsLow symmetry groups:Only an inversion center (Ci)e.g. (conformation is important !) F Cl Br Br Cl FNo symmetry (C1)e.g. CHFClBr F H Br Cl
Identifying point groups Cn type groups:A Cn axis and a sh (Cnh)e.g. B(OH)3 (C3h, conformation is important !) H O H B O O H O O H He.g. H2O2 (C2h, conformation is important !) Note: molecule does not have to be planar e.g. B(NH2)3 (C3h, conformation is important !)
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 He.g. H2O2 (C2, conformation is important !) H O O H
Identifying point groups Cn type groups:A Cn axis and a sv (Cnv)e.g. NH3 (C3v) H N H He.g. H2O2 (C2v, conformation is important !) O O H H
Identifying point groups Cn type groups:A Cn axis and a sv (Cnv)e.g. NH3 (C3v, conformation is important !) H H e.g. carbon monoxide, CO (Cv) H There are an infinite number of possible H N H Cn axes and sv mirror planes. H H H H C Oe.g. trans-[SbF4ClBr]- (C4v) Cl F F F Sb O C F Sb Br Cl F F F Br F
Identifying point groups Dn type groups:A Cn axis, n perpendicular C2 axesand a sh (Dnh)e.g. BH3 (D3h) H H B H H B H He.g. NiCl4 (D4h) Cl (2) lC lC iN lC lC Cl (4) Ni (1) Cl (5) lC lC iN lC lC Cl (3)
Identifying point groups Dn type groups: e.g. pentagonal prism (D5h)A Cn axis, n perpendicular C2 axesand a sh (Dnh)e.g. Mg(5-Cp)2 (D5h in the eclipsed conformation) Mg Mg View down the C5 axis e.g. square prism (D4h) e.g. carbon dioxide, CO2 or N2 (Dh) There are an infinite number of possible Cn axes and sv mirror planes in addition to the sh. O O O C
Identifying point groups Dn type groups:A Cn axis, n perpendicular C2 axesand no mirror planes (Dn)-propellor shapese.g. Ni(CH2)4 (D4) H H H H Ni H H H H H HH H NiH H H H H H H H Ni H H H H
e.g. (SCH2CH2)3 (D3 conformation is important!) e.g. propellor (D3)e.g. Ni(en)3 (D3 conformation is important!) en = H2NCH2CH2NH2
Identifying point groups Dn type groups:A Cn axis, n perpendicular C2 axesand a sd (Dnd)e.g. ethane, H3C-CH3(D3d in the staggered conformation) H H H H H H H H H H H H dihedral means between sides or planes – this is where you find the C2 axes
e.g. Mg(5-Cp)2 and other metallocenes in the staggered conformation (D5d) Fe Mg Al M View down the C5 axis These are pentagonal antiprismse.g. triagular e.g. square e.g. allene or a tennis ball (D2d)antiprism (D3d) antiprism (D4d)
Summary of point group identification1. Determine the symmetry is special(e.g. octahedral).2. Determine if there is a principalrotation axis.3. Determine if there are rotation axesperpendicular to the principal axis.4. Determine if there are mirror planes.5. Assign point group.