Molecular Symmetry and Group Theory - Physical Chemistry | CHEM 3510, Study notes of Physical Chemistry

Download Molecular Symmetry and Group Theory - Physical Chemistry | CHEM 3510 and more Study notes Physical Chemistry in PDF only on Docsity! CHEM 3510 Fall 2007 149 Unit X Molecular Symmetry and Group Theory A. Introduction to Molecular Symmetry 1. The molecular symmetry properties can be used to: a. Reduce the high-order secular determinants in Hückel method b. Determine the IR or Raman activity of vibrational normal modes c. Label and designate molecular orbitals d. Derive selections rules for spectroscopic transitions 2. Symmetry elements and symmetry operations a. The symmetry of a molecule is described by its symmetry elements. b. Each symmetry element has (one or more) symmetry operations associated with them. Symmetry elements Symmetry operations Examples Description Symbol Symbol Description Identity E Ê No change n-Fold axis of symmetry nC nĈ Rotation about the axis by 360/n degrees C2, C3 C4, C6 Plane of symmetry (mirror plane) ⎪⎩ ⎪ ⎨ ⎧ d h v σ σ σ σ ⎪⎩ ⎪ ⎨ ⎧ d h v σ σ σ σ ˆ ˆ ˆ ˆ Reflection through a plane Center of symmetry i î Reflection through the center of symmetry n-Fold rotation reflection axis of symmetry (improper rotation) nS nŜ Rotation about the axis by 360/n degrees followed by reflection through a plane perpendicular to the axis CHEM 3510 Fall 2007 150 c. The axis with the highest value of n is called the principal axis. d. The planes of symmetry can be: □ σv: the plane of symmetry is parallel to a unique axis or to a principal axis □ σh: the plane of symmetry is perpendicular to a unique axis or to a principal axis □ σd: the plane of symmetry bisects the angle between C2 axes that are perpendicular to a principal axis ○ σd is a special type of a σv plane. e. A symmetry element may have more than one symmetry operation associated with it. □ The 3-fold axis of symmetry (C3) has two symmetry operations associated with it. ○ 3Ĉ (rotation with 1203 360 = degrees) ○ 33 2 3 ˆˆˆ CCC = (rotation with 240 degrees) □ The 4-fold axis of symmetry (C4) has three symmetry operations: ○ 4Ĉ (rotation with 904 360 = degrees) ○ 44 2 4 ˆˆˆ CCC = (rotation with 180 degrees) ○ 444 3 4 ˆˆˆˆ CCCC = (rotation with 270 degrees) 3. Point groups a. A group (or set) of symmetry operations constitutes a point group. b. Each point group consists of a number of symmetry elements. c. The total number of symmetry operations is called the order of the point group. □ The total number of symmetry operations can be greater than the total number of symmetry elements. CHEM 3510 Fall 2007 153 g. Molecule examples: identify all the symmetry elements and the point group that each molecule belongs to. H2O XeF4 C6H5Cl H2C=C=CH2 Xe F FF F Cl E,C2,2σv E,C4,4C2,i,S4,σh,2σv,2σd E,C2,2σv E,S4,3C2,2σd C2v D4h C2v D2d SO3 CH2Cl2 C6H6 trans-ClHC=CHCl S O O O C H H Cl Cl E,C3,3C2,σh,S3,3σv E,C2,2σv E,C6,3C2,i,S6,σh,3σv,3σd E,C2,i,σh D3h C2v D6h C2h C2H4 BF3 CH3Cl CH4 Cl HH H E,3C2,i,3σv E,C3,3C2,σh,S3,3σv E,C3,3σv E,4C3,3C2,3S4,6σd D2h D3h C3v Td cis-ClHC=CHCl NH3 naphthalene meta-C6H4Cl2 C2v C3v D2h C2v CHEM 3510 Fall 2007 154 h. A flow diagram for determining the point group of a molecule. □ Start at the top and answer the question posed in each diamond (Y = yes, N = no). □ Example: benzene CHEM 3510 Fall 2007 155 B. Group Theory 1. Introduction a. The set of symmetry operations of a molecule form a point group. b. A group is a set of entities (A, B, C…) that satisfy certain requirements: □ Combining (i.e., multiplying) any 2 members of the group gives a member of the group. ○ “A group must be closed under multiplication.” □ The multiplication must be associative: A (B C) = (A B) C □ The set of entities (i.e., the members of the group) contains an identity element E such that: E A = A E (and E B = B E) □ For every entity in the group (for example A) there is an inverse (A–1) that is also a member of the group so that: A A–1 = A–1 A = E (and B B–1 = B–1 B = E) 2. Group multiplication table a. Example of a group of symmetry operations: the case of water □ There are four symmetry operations: Ê , 2Ĉ , vσ̂ , 'ˆvσ □ These four symmetry operations form the C2v point group. □ Consider an arbitrary vector (u) and investigate how each combination of symmetry operations change the vector. First Operation Second Operation Ê 2Ĉ vσ̂ 'ˆ vσ Ê Ê 2Ĉ vσ̂ 'ˆ vσ 2Ĉ 2Ĉ Ê 'ˆ vσ vσ̂ vσ̂ vσ̂ 'ˆ vσ Ê 2Ĉ 'ˆ vσ 'ˆ vσ vσ̂ 2Ĉ Ê b. The table above is called the group multiplication table of the C2v point group. O HH u CHEM 3510 Fall 2007 158 f. The case of C3v point group □ The irreducible representations of the C3v point group: Ê 3Ĉ 2 3Ĉ vσ̂ 'ˆ vσ ''ˆ vσ A1 (1) (1) (1) (1) (1) (1) A2 (1) (1) (1) (–1) (–1) (–1) E ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 10 01 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −− 2 1 2 3 2 3 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− − 2 1 2 3 2 3 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −10 01 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− 2 1 2 3 2 3 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −− 2 1 2 3 2 3 2 1 □ Two-dimensional irreducible representations are designated by E (not the same as the symmetry operation E). ○ It can be obtained by applying the symmetry operations to a vector in x-y plane. ○ Because x and y transform together, the result of a given operation is written as a linear combination of x and y. □ The x and y are said to form a basis for E or to belong to E. g. Three-dimensional irreducible representations are designated by T. h. For almost all applications of group theory one do not uses the complete matrices, only the sum of the diagonal elements called its trace or, in group theory, its character. i. The characters of the irreducible representations of a point group are displayed in a table called character table. j. Certain symmetry operations (for example 3Ĉ and 2 3Ĉ or vσ̂ , 'ˆvσ , and ''ˆvσ in the C3v point group) are essentially equivalent (have the same characters) and are said to belong to the same class. k. The number of classes is equal to the number of irreducible representations (⇒ the character tables are squared). l. For the point groups that has a center of symmetry i, the irreducible representations are labeled as g or u to describe whether they are symmetric or antisymmetric under the inversion. CHEM 3510 Fall 2007 159 m. Examples of character table for some useful point groups. v3C Ê 2 3Ĉ 3 vσ̂ 1A 1 1 1 z 22 yx + , 2z 2A 1 1 –1 zR E 2 –1 0 ),( yx ),( yx RR ),( 22 xyyx − ),( yzxz v2C Ê 2Ĉ vσ̂ vσ ′ˆ 1A 1 1 1 1 z 2x , 2y , 2z 2A 1 1 –1 –1 zR xy 1B 1 –1 1 –1 x, yR xz 2B 1 –1 –1 1 y, xR yz h2C Ê 2Ĉ î hσ̂ gA 1 1 1 1 zR 2x , 2y , 2z , xy gB 1 –1 1 –1 xR , yR xz , yz uA 1 1 –1 –1 z uB 1 –1 –1 1 x, y h3D Ê 2 3Ĉ 3 2Ĉ hσ̂ 2 3Ŝ 3 vσ̂ 1A′ 1 1 1 1 1 1 22 yx + , 2z 2A′ 1 1 –1 1 1 –1 zR E ′ 2 –1 0 2 –1 0 ),( yx ),( 22 xyyx − 1A ′′ 1 1 1 –1 –1 –1 2A ′′ 1 1 –1 –1 –1 1 z E ′′ 2 –1 0 –2 1 0 ),( yx RR ),( yzxz CHEM 3510 Fall 2007 160 n. Description of character tables □ The second to last column of the character table lists how the three axis (x, y, and z) (or the translation along the three axis) and how the rotation along the three axis (Rx, Ry, and Rz) transform in that particular point group. ○ It can also be said that x (or y or z) for a basis of a certain irreducible representation. ○ For the C2v point group, x forms a basis for the B1 representation, y forms a basis for the B2 representation, and z forms a basis for the A1 representation. ○ For the C3v point group, x and y form jointly a basis for the two dimensional representation E. ○ Example of rotation around the z axis in the C3v point group: – Depict the rotation around an axis by vectors. top view – The rotation along the z axis transforms as the A2 representation. □ The last column lists how combinations of the axis (also components of the molecular polarizability) (x2, y2, xy, etc) transform in that particular point group. ○ When there is no axis that transform as a two-dimensional irreducible representation, the combination of the axis is just the product between the particular axis. zzv zzv zzv zz zz zz RR RR RR RRC RRC RRE −=′′ −=′ −= = = = )(ˆ )(ˆ )(ˆ )(ˆ )(ˆ )(ˆ 2 3 3 σ σ σ CHEM 3510 Fall 2007 163 □ Find the coefficients aj, multiply by )ˆ(Riχ and sum over R̂ : 44 344 21 ji j R i j j R i RRaRR = ≠ ∑∑∑ = ifonly 0 ˆˆ )ˆ()ˆ()ˆ()ˆ( χχχχ and considering ijj R i hRR δχχ =∑ )ˆ()ˆ( ˆ ∑=⇒ R ii RRha ˆ )ˆ()ˆ( χχ ∑∑ ==⇒ classesˆ )ˆ()ˆ()ˆ(1)ˆ()ˆ(1 RRRn h RR h a i R ii χχχχ i. Example: Reduce the reducible representation Γ= 4 2 0 2 as a sum of irreducible representations belonging to C2v group. 4)ˆ( =Eχ ; 2)ˆ( 2 =Cχ ; 0)ˆ( =vσχ ; 2)ˆ( =′vσχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 212101214 4 1 1A =×+×+×+×=a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 112101214 4 1 2A =−×+−×+×+×=a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 012101214 4 1 1B =−×+×+−×+×=a ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 112101214 4 1 2B =×+−×+−×+×=a 2212 BAA ++=Γ⇒ □ Verification: 2A1 2×1 2×1 2×1 2×1 A2 1 1 –1 –1 B 2 1 –1 –1 1 Γ 4 2 0 2 CHEM 3510 Fall 2007 164 C. Applications of Molecular Symmetry 1. Hückel theory for benzene a. When using the pz orbitals on various C atoms, Hückel theory leads to: 0 10001 11000 01100 00110 00011 10001 = x x x x x x b. Instead of using atomic orbitals, use “symmetry orbitals” that are obtained as a linear combination of the pz orbitals: )( 6 1 6543211 ψψψψψψφ +++++= )( 6 1 6543212 ψψψψψψφ −+−+−= )22( 12 1 6543213 ψψψψψψφ +−−−+= )22( 12 1 6543214 ψψψψψψφ −−−++= )22( 12 1 6543215 ψψψψψψφ −−+−−= )22( 12 1 6543216 ψψψψψψφ −+−−+−= c. Using these orbitals leads to the following determinant: 0 1 2 10000 2 110000 001 2 100 00 2 1100 000020 000002 = − − − − + + + + − + xx xx xx xx x x CHEM 3510 Fall 2007 165 □ This determinant is easier to solve and the solutions are 2,1,1 ±±±=x . □ The construction of those “symmetry orbitals” takes advantage of the symmetry of the system. 2. The use of symmetry to predict the elements of the secular determinant that are zero. a. Look at ∫= τφφ dHH jiij ˆ * and ∫= τφφ dS jiij * b. These integrals should be independent on the molecule orientation (the value should not change upon applying a symmetry operation): ijjiij SdRRSR ==⇒ ∫ τφφ ˆˆˆ * ijjiij HdRHRRHR ==⇒ ∫ τφφ )(ˆ)ˆ(ˆ)(ˆˆ * c. The overlap integrals □ Consider φi* is a basis for one irreducible representation Γa. □ Consider φj is a basis for one irreducible representation Γb. ⇒ ** )ˆ(ˆ iai RR φχφ = and jbj RR φχφ )ˆ(ˆ = ∫ ==⇒ ijbajibaij SRRdRRS )ˆ()ˆ()ˆ()ˆ( * χχτφφχχ ○ So the )ˆ()ˆ( RR ba χχ product should be equal to 1 for every symmetry operation. If the )ˆ()ˆ( RR ba χχ is equal to –1 ⇒ Sij should be zero for the equation above to be true. □ Sij ≠ 0 only if φi and φj are bases of the same irreducible representation. □ Sij = 0 only if φi and φj are bases of different irreducible representations. d. The exchange integrals □ The molecular Hamiltonian operator is symmetric under all symmetry operations. ijijbAajiij HHRRRdRHRRHR ===⇒ ∫ )ˆ()ˆ()ˆ()(ˆ)ˆ(ˆ)(ˆˆ 1 * χχχτφφ