Lecture 4 Character Table Symmetry operations Point group

Lecture 4

Character Table Symmetry operations Point group Mülliken symbols Characters +1 symmetric behavior -1 antisymmetric Each row is an irreducible representation

Classes of operations dxy, dxz, dyz, as xy, xz, yz dx 2 - y 2 behaves as x 2 – y 2 dz 2 behaves as 2 z 2 - (x 2 + y 2) px, py, pz behave as x, y, z s behaves as x 2 + y 2 + z 2 x, y, z Symmetry of translations (p orbitals) Rx, Ry, Rz: rotations

Symmetry of Atomic Orbitals

Effect of the 4 operations in the point group C 2 v on a translation in the x direction

Naming of Irreducible representations • One dimensional (non degenerate) representations are designated A or B. • Two-dimensional (doubly degenerate) are designated E. • Three-dimensional (triply degenerate) are designated T. • Any 1 -D representation symmetric with respect to Cn is designated A; antisymmétric ones are designated B • Subscripts 1 or 2 (applied to A or B refer) to symmetric and antisymmetric representations with respect to C 2 Cn or (if no C 2) to sv respectively • Superscripts ‘ and ‘’ indicate symmetric and antisymmetric operations with respect to sh, respectively • In groups having a center of inversion, subscripts g (gerade) and u (ungerade) indicate symmetric and antisymmetric representations with respect to i

Character Tables • Irreducible representations are the generalized analogues of s or p symmetry in diatomic molecules. • Characters in rows designated A, B, . . . , and in columns other than E indicate the behavior of an orbital or group of orbitals under the corresponding operations (+1 = orbital does not change; -1 = orbital changes sign; anything else = more complex change) • Characters in the column of operation E indicate the degeneracy of orbitals • Symmetry classes are represented by CAPITAL LETTERS (A, B, E, T, . . . ) whereas orbitals are represented in lowercase (a, b, e, t, . . . ) • The identity of orbitals which a row represents is found at the extreme right of the row • Pairs in brackets refer to groups of degenerate orbitals and, in those cases, the characters refer to the properties of the set

Definition of a Group • A group is a set, G, together with a binary operation : such that the product of any two members of the group is a member of the group, usually denoted by a*b, such that the following properties are satisfied : – (Associativity) (a*b)*c = a*(b*c) for all a, b, c belonging to G. – (Identity) There exists e belonging to G, such that e*g = g*e for all g belonging to G. – (Inverse) For each g belonging to G, there exists the inverse of g, g-1, such that g-1*g = g*g-1 = e. • If commutativity is satisfied, i. e. a*b = b*a for all a, b belonging to G, then G is called an abelian group.

Examples • The set of integers Z, is an abelian group under addition. What is the element e, identity, such that a*e = a? 0 What is the inverse of the a element? -a

As applied to our symmetry operators. For the C 3 v point group What is the inverse of each operator? A * A-1 = E E C 3(120) C 3(240) sv (1) sv (2) sv (3) E C 3(240) C 3(120) sv (1) sv (2) sv (3)

Examine the matrix represetation of the C 2 v point group C 2 E - sv(xz) s’v(yz)

Multiplying two matrices (a reminder) Most of the transformation matrices we use have the form

E sv(xz) C 2 What is the inverse of C 2? What is the inverse of sv? C 2 sv s’v(yz) = =

What of the products of operations? C 2 E E * C 2 = ? sv * C 2 = ? C 2 sv(xz) s’v(yz) = s’v =

Classes Two members, c 1 and c 2, of a group belong to the same class if there is a member, g, of the group such that g*c 1*g-1 = c 2

Properties of Characters of Irreducible Representations in Point Groups • Total number of symmetry operations in the group is called the order of the group (h). For C 3 v, for example, it is 6. 1 + 2 + 3 = 6 • Symmetry operations are arranged in classes. Operations in a class are grouped together as they have identical characters. Elements in a class are related. This column represents three symmetry operations having identical characters.

Properties of Characters of Irreducible Representations in Point Groups - 2 The number of irreducible reps equals the number of classes. The character table is square. 1 + 2 + 3 = 6 3 by 3 1 1 22 6 The sum of the squares of the dimensions of the each irreducible rep equals the order of the group, h.

Properties of Characters of Irreducible Representations in Point Groups - 3 For any irreducible rep the squares of the characters summed over the symmetry operations equals the order of the group, h. A 1: 12 + (12 + 12 ) + = 6 A 2: 12 + (12 + 12 ) + ((-1)2 + (-1)2 ) = 6 E: 22 + (-1)2 = 6

Properties of Characters of Irreducible Representations in Point Groups - 4 Irreducible reps are orthogonal. The sum of the products of the characters for each symmetry operation is zero. For A 1 and E: 1 * 2 + (1 *(-1) + 1 *(-1)) + (1 * 0 + 1 * 0) = 0

Properties of Characters of Irreducible Representations in Point Groups - 5 Each group has a totally symmetric irreducible rep having all characters equal to 1

Reduction of a Reducible Representation Irreducible reps may be regarded as orthogonal vectors. The magnitude of the vector is h-1/2 Any representation may be regarded as a vector which is a linear combination of the irreducible representations. Reducible Rep = S (ai * Irreducible. Repi) The Irreducible reps are orthogonal. Hence Sym ops S(character of Reducible Rep)(character of Irreducible Repi) = ai * h Or ai = Sym ops S(character of Reducible Rep)(character of Irreducible Repi) / h

These are block-diagonalized matrices (x, y, z coordinates are independent of each other) Irreducible representations Reducible Rep

C 2 v Character Table to be used for water Symmetry operations Point group Mülliken symbols Characters +1 symmetric behavior -1 antisymmetric Each row is an irreducible representation

Let’s use character tables! Symmetry and molecular vibrations # of atoms degrees of freedom Translational modes Rotational modes Vibrational modes N (linear) 3 x 2 3 N-5 = 1 Example 3 (HCN) 9 3 2 4 N (nonlinear) 3 N 3 3 3 N-6 Example 3 (H 2 O) 9 3 3 3

Symmetry and molecular vibrations A molecular vibration is IR active only if it results in a change in the dipole moment of the molecule A molecular vibration is Raman active only if it results in a change in the polarizability of the molecule In group theory terms: A vibrational motion is IR active if it corresponds to an irreducible representation with the same symmetry as an x, y, z coordinate (or function) and it is Raman active if the symmetry is the same as x 2, y 2, z 2, or one of the rotational functions Rx, Ry, Rz

How many vibrational modes belong to each irreducible representation? You need the molecular geometry (point group) and the character table Use the translation vectors of the atoms as the basis of a reducible representation. Since you only need the trace recognize that only the vectors that are either unchanged or have become the negatives of themselves by a symmetry operation contribute to the character.

A shorter method can be devised. Recognize that a vector is unchanged or becomes the negative of itself if the atom does not move. A reflection will leave two vectors unchanged and multiply the other by -1 contributing +1. For a rotation leaving the position of an atom unchanged will invert the direction of two vectors, leaving the third unchanged. Etc. Apply each symmetry operation in that point group to the molecule and determine how many atoms are not moved by the symmetry operation. Multiply that number by the character contribution of that operation: E=3 s=1 C 2 = -1 i = -3 C 3 = 0 That will give you the reducible representation

Finding the reducible representation E=3 s=1 C 2 = -1 i = -3 C 3 = 0 G 3 x 3 9 1 x-1 -1 3 x 1 3 1 x 1 (# atoms not moving x char. contrib. ) 1

Now separate the reducible representation into irreducible ones to see how many there are of each type S G 9 -1 3 1 A 1 = 1/4 (1 x 9 x 1 + 1 x(-1)x 1 + 1 x 3 x 1 + 1 x 1 x 1) = 3 A 2 = 1/4 (1 x 9 x 1 + 1 x(-1)x 1 + 1 x 3 x(-1) + 1 x 1 x(-1)) = 1

Symmetry of molecular movements of water Vibrational modes

Which of these vibrations having A 1 and B 1 symmetry are IR or Raman active? Raman active IR active

Often you analyze selected vibrational modes Example: C-O stretch in C 2 v complex. n(CO) 2 x 1 2 0 x 1 0 Find: # vectors remaining unchanged after operation. G

A 1 is IR active B 1 is IR and Raman active G 2 0 = A 1 + B 1 A 1 = 1/4 (1 x 2 x 1 + 1 x 0 x 1 + 1 x 2 x 1 + 1 x 0 x 1) = 1 A 2 = 1/4 (1 x 2 x 1 + 1 x 0 x 1 + 1 x 2 x-1 + 1 x 0 x-1) = 0 B 1 = 1/4 (1 x 2 x 1 + 1 x 0 x 1 + 1 x 2 x 1 + 1 x 0 x 1) = 1 B 2 = 1/4 (1 x 2 x 1 + 1 x 0 x 1 + 1 x 2 x-1 + 1 x 0 x 1) = 0

What about the trans isomer? Only one IR active band no Raman active bands Remember cis isomer had two IR active bands and one Raman active

Symmetry and NMR spectroscopy The # of signals in the spectrum corresponds to the # of types of nuclei not related by symmetry The symmetry of a molecule may be determined From the # of signals, or vice-versa

Molecular Orbitals

Atomic orbitals interact to form molecular orbitals Electrons are placed in molecular orbitals following the same rules as for atomic orbitals In terms of approximate solutions to the Scrödinger equation Molecular Orbitals are linear combinations of atomic orbitals (LCAO) Y = caya + cbyb (for diatomic molecules) Interactions depend on the symmetry properties and the relative energies of the atomic orbitals

As the distance between atoms decreases Atomic orbitals overlap Bonding takes place if: the orbital symmetry must be such that regions of the same sign overlap the energy of the orbitals must be similar the interatomic distance must be short enough but not too short If the total energy of the electrons in the molecular orbitals is less than in the atomic orbitals, the molecule is stable compared with the atoms

Combinations of two s orbitals (e. g. H 2) Antibonding Bonding More generally: Y = N[ca. Y(1 sa) ± cb. Y (1 sb)] n A. O. ’s n M. O. ’s

Electrons in antibonding orbitals cause mutual repulsion between the atoms (total energy is raised) Electrons in bonding orbitals concentrate between the nuclei and hold the nuclei together (total energy is lowered)

Both s (and s*) notation means symmetric/antisymmetric with respect to rotation s s* s* Not s

Combinations of two p orbitals (e. g. H 2) s (and s*) notation means no change of sign upon rotation p (and p*) notation means change of sign upon C 2 rotation

Combinations of two p orbitals

Combinations of two sets of p orbitals

Combinations of s and p orbitals

Combinations of d orbitals No interaction – different symmetry d means change of sign upon C 4

Is there a net interaction? NO YES NO

Relative energies of interacting orbitals must be similar Strong interaction Weak interaction

Molecular orbitals for diatomic molecules From H 2 to Ne 2 Electrons are placed in molecular orbitals following the same rules as for atomic orbitals: Fill from lowest to highest Maximum spin multiplicity Electrons have different quantum numbers including spin (+ ½, ½)

O 2 (2 x 8 e) 1/2 (10 - 6) = 2 A double bond Or counting only valence electrons: 1/2 (8 - 4) = 2 Note subscripts g and u symmetric/antisymmetric upon i

Place labels g or u in this diagram s*u p*g pu sg

s*u g or u? sg p*g pu d*u dg

Orbital mixing Same symmetry and similar energies ! shouldn’t they interact?

s orbital mixing When two MO’s of the same symmetry mix the one with higher energy moves higher and the one with lower energy moves lower

Molecular orbitals for diatomic molecules From H 2 to Ne 2 H 2 sg 2 (single bond) He 2 sg 2 s*u 2 (no bond)

E (Z*) Paramagnetic due to mixing DE s > DE p C 2 pu 2 (double bond) C 22 - pu 2 sg 2(triple bond) O 2 pu 2 p*g 1 (double bond) paramagnetic O 22 - pu 2 p*g 2 (single bond) diamagnetic

Bond lengths in diatomic molecules Filling bonding orbitals Filling antibonding orbitals

Photoelectron Spectroscopy

O 2 N 2 sg (2 p) pu (2 p) s*u (2 s) p*u (2 p) sg (2 p) s*u (2 s) pu (2 p) Very involved in bonding (vibrational fine structure) (Energy required to remove electron, lower energy for higher orbitals)