Lecture 6.

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Presentation transcript:

Lecture 6

Character Tables Group theory makes use of the properties of matrices Idea: When an operation, O, (proper rotation, improper rotation, reflection, inversion) is done on the function, f(x,y,z) O f(x,y,z) the result is a value taken on by the original function at some other point (x’,y’,z’). Or O f(x,y,z) = f (x’,y’,z’) We know the function f(x,y,z) the problem is to determine the values (x’,y’,z’). The nature of the operation tells us where to look. 2

Character Tables Group theory makes use of the properties of matrices Each operation may be expressed as a transformation matrix: [New coordinates] = [transformation matrix][old coordinates] Example: in Cartesian coordinate system, reflection in x = 0 plane Changes the value of x to –x (multiplies it by -1) Leaves y unchanged (multiplies it by 1) Leaves z unchanged (multiplies it by 1) To see the result of the operation at (x, y, z) look at the original object at (x’, y’, z’). = Results of transformation. Transformation matrix Original coordinates

Recall Technique of Matrix multiplication = V’ M V To get an element of the product vector a row in the operation square matrix is multiplied by the original vector matrix. For example V’2 = y’ = M2,1 * V1 + M2,2 * V2 + M2,3 * V3 y’ = 0 * x + 1 * y + 0 * z = y

Character Tables - 2 The matrix representation of the symmetry operations of a point group is the set of matrices corresponding to all the symmetry operations in that group. The matrices record how the x,y,z coordinates are modified as a result of an operation. For example, the C2v point group consists of the following operations E: do nothing. Unchanged. C2: rotate 180 degrees about the z axis: x becomes –x; y becomes –y and z unchanged. sv (xz): y becomes –y sv’ (yz): x becomes -x E C2 sv (xz): sv’ (yz):

Operations Applied to Functions - 1 Transform the coordinates. Consider f(x) = x2 sv’ (f(x)) = sv(x2) = (-x)2 = x2 = f(x) or sv’ (f(x)) = 1 * f(x) f(x) is an eigenfunction of this reflection operator with an eigenvalue of +1. This is called a symmetric eigenfunction. Similarly f(x) = x3 sv’ (f(x)) = -1 * f(x) f(x) is an eigenfunction of this reflection operator with an eigenvalue of -1. This is called a antisymmetric eigenfunction.

Plots of Functions, x2 Reflection yields. Here f(x) is x2. It can be seen to be a symmetric function for reflection at x = 0 because of mirror plane. The reflection carries out the mapping shown with the red arrows. x2 is an eigenfunction of s with eigenvalue 1

Plots of Functions, x3 Reflection yields. Here f(x) is x3. It can be seen to be a antisymmetric function for reflection at x = 0. The reflection carries out the mapping shown with the red arrows. x3 is an eigenfunction of s with eigenvalue -1

Plots of Functions - 2 Reflection yields. Here f(x) is x3. It can be seen to be a antisymmetric function for reflection at x = 0. The reflection carries out the mapping shown with the red arrows.

Operations Applied to Functions - 2 Now consider f(x) = (x-2)2 = x2 – 4x + 4 sv’ (f(x)) = sv(x-2)2 = (-x-2)2 = x2 + 4x + 4 f(x) = (x-2)2 is not an eigenfunction of this reflection operator because it does not return a constant times f(x). Reflection yields this function, not an eigenfunction. Neither symmetric nor antisymmetric for reflection thru x = 0.

Let’s look at Atomic Orbitals Reflection Get the same orbital back, multiplied by +1, an eigenfunction of the reflection, symmetric with respect to the reflection. The s orbital forms the basis of an irreducible representation of the operation s orbital z

Atomic Orbitals s Reflection Get the same orbital back, multiplied by -1, an eigenfunction of the reflection, antisymmetric with respect to the reflection. The p orbital behaves differently from the s orbital and forms the basis of a different irreducible representation of the operation p orbital z

Simplest ways that objects can behave for a group consisting of E and sh , the reflection plane. Irreducible Representations. Basis of the Irreducible Reps. Cs E sh A’ A” 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz s orbital is spherical behaves as x2 + y2 + z2. s orbital is A’. The s orbital is an eigenfunction of both E and sh. pz orbital has a multiplicative factor of z times a spherical factor. Behaves as A”. pz is an eigenfunction of both E and sh.

sp Hybrids Reflection Do not get the same hybrid back multiplied by +1 or -1 or some other constant. Not an eigenfunction. hybrid z The two hybrids form the basis of a reducible representation of the operation Recall: the hybrid can be expressed as the sum of an s orbital and a p orbital. = + Reduction: expressing a reducible representation as a combination of irreducible representations.

Reducible Representations Use the two sp hybrids as the basis of a representation h1 h2 sh operation. E operation. h1 becomes h1; h2 becomes h2. h1 becomes h2; h2 becomes h1. = = The reflection operation interchanges the two hybrids. The hybrids are unaffected by the E operation. Proceed using the trace of the matrix representation. 0 + 0 = 0 1 + 1 = 2

Our Irreducible Representations Cs E sh A’ A” 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz The reducible representation derived from the two hybrids can be attached to the table. G 2 0 (h1, h2) Note that G = A’ + A”

Return to polynomials: f(x) = (x-2)2 = x2 – 4x + 4 sv (f(x)) = sv(x-2)2 = (-x-2)2 = x2 + 4x + 4 =g(x) Neither f nor g is an eigenfunction of s but, taken together, they do form an reducible representation since they show what the s operator does. Approaching the problem in the same way as we did for hybrids we can carry out the reduction this way u(x) = ½ (f(x) + g(x)) = ½ (f(x) + s f(x)) = x2 + 4 symmetric, unchanged by the s operator. Behaves as A’ v(x) = ½ (f(x) - g(x)) = ½ (f(x) - s f(x)) = -4x, antisymmetric, multiplied by -1 by the s operator. Behaves as A’’

Character Table Each row is an irreducible representation Symmetry operations, Classes Point group x, y, z Symmetry of translations (p orbitals) Rx, Ry, Rz: rotations Characters +1 symmetric behavior -1 antisymmetric Mülliken symbols Each row is an irreducible representation

Character Tables - 3 Irreducible representations are not linear combinations of other representation (Reducible representations are) # of irreducible representations = # of classes of symmetry operations Instead of the matrices, the characters are used (traces of matrices) A character Table is the complete set of irreducible representations of a point group

Operation E C2 sv sv’ Transformation 1 -1 Effect of the 4 operations in the point group C2v on a translation in the x direction. The translation is simply multiplied by 1 or -1. It forms a basis to show what the operators do to an object. Operation E C2 sv sv’ Transformation 1 -1

Character Table Verify this character. It represents how a function that behaves as x, Ry, or xz behaves for C2. 21

Symmetry of translations (p orbitals) Another point group, C3v. Classes of operations x, y, z Symmetry of translations (p orbitals) Rx, Ry, Rz: rotations dxy, dxz, dyz, as xy, xz, yz dx2- y2 behaves as x2 – y2 dz2 behaves as 2z2 - (x2 + y2) px, py, pz behave as x, y, z s behaves as x2 + y2 + z2

Symmetry of Atomic Orbitals

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 C2  Cn or (if no C2) 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 = 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? What is the inverse of the a element? -a

As applied to our symmetry operators. For the C3v point group What is the inverse of each operator? A * A-1 = E E C3(120) C3(240) sv (1) sv (2) sv (3) E C3(240) C3(120) sv (1) sv (2) sv (3)

Examine the matrix represetation of the C2v point group - s’v(yz) sv(xz)

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

C2 sv(xz) s’v(yz) E What is the inverse of C2? C2 = What is the inverse of sv? sv =

C2 sv(xz) s’v(yz) E What of the products of operations? C2 E * C2 = ? sv * C2 = ? s’v =

Classes Two members, c1 and c2, of a group belong to the same class if there is a member, g, of the group such that g*c1*g-1 = c2

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 C3v, 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 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. A1: 12 + (12 + 12 ) + = 6 A2: 12 + (12 + 12 ) + ((-1)2 + (-1)2 + (-1)2 ) = 6 E: 22 + (-1)2 + (-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 A1 and E: 1 * 2 + (1 *(-1) + 1 *(-1)) + (1 * 0 + 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 * IrreducibleRepi) The Irreducible reps are orthogonal. Hence S(character of Reducible Rep)(character of Irreducible Repi) = ai * h Or ai = S(character of Reducible Rep)(character of Irreducible Repi) / h Sym ops Sym ops

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

C2v Character Table to be used for water Symmetry operations Point group Characters +1 symmetric behavior -1 antisymmetric Mülliken symbols 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 2 3N-5 = 1 Example 3 (HCN) 9 4 N (non- linear) 3N 3N-6 3 (H2O)

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 mode is IR active if it corresponds to an irreducible representation with the same symmetry of a x, y, z coordinate (or function) and it is Raman active if the symmetry is the same as A quadratic function x2, y2, z2, xy, xz, yz, x2-y2 If the molecule has a center of inversion, no vibration can be both IR & Raman active

You need the molecular geometry (point group) and the character table 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.

Apply each symmetry operation in that point group to the molecule 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 C2 = -1 i = -3 C3 = 0 That will give you the reducible representation

Finding the reducible representation 3x3 9 1x-1 -1 3x1 3 1x1 1 (# atoms not moving x char. contrib.) G

Now separate the reducible representation into irreducible ones to see how many there are of each type S 9 -1 3 1 G A1 = 1/4 (1x9x1 + 1x(-1)x1 + 1x3x1 + 1x1x1) = 3 A2 = 1/4 (1x9x1 + 1x(-1)x1 + 1x3x(-1) + 1x1x(-1)) = 1

Symmetry of molecular movements of water Vibrational modes

Which of these vibrations having A1 and B1 symmetry are IR or Raman active? IR active

Often you analyze selected vibrational modes Example: C-O stretch in C2v complex. n(CO) 2 x 1 2 0 x 1 2 x 1 2 0 x 1 G Find: # vectors remaining unchanged after operation.

Both A1 and B1 are IR and Raman active 2 G G = A1 + B1 A1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1 A2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x-1) = 0 B1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1 B2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x1) = 0

What about the trans isomer? Only one IR active band and 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

If the total energy of the electrons in the molecular 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

Y = N[caY(1sa) ± cbY (1sb)] Combinations of two s orbitals (e.g. H2) Antibonding Bonding More generally: Y = N[caY(1sa) ± cbY (1sb)] n A.O.’s n M.O.’s

(total energy is raised) 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 Not s

Combinations of two p orbitals (e.g. H2) s (and s*) notation means no change of sign upon rotation p (and p*) notation means change of sign upon C2 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 C4

Is there a net interaction? NO NO YES

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

for diatomic molecules From H2 to Ne2 Molecular orbitals for diatomic molecules From H2 to Ne2 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)

symmetric/antisymmetric O2 (2 x 8e) 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 sg g or u? p*g pu d*u dg

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

When two MO’s of the same symmetry mix 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

for diatomic molecules Molecular orbitals for diatomic molecules From H2 to Ne2 H2 sg2 (single bond) He2 sg2 s*u2 (no bond)

E (Z*) DE s > DE p C2 pu2 pu2 (double bond) Paramagnetic due to mixing C2 pu2 pu2 (double bond) C22- pu2 pu2 sg2(triple bond) O2 pu2 pu2 p*g1 p*g1 (double bond) paramagnetic O22- pu2 pu2 p*g2 p*g2 (single bond) diamagnetic

Bond lengths in diatomic molecules Filling bonding orbitals Filling antibonding orbitals

Photoelectron Spectroscopy

O2 N2 sg (2p) p*u (2p) pu (2p) sg (2p) pu (2p) s*u (2s) s*u (2s) Very involved in bonding (vibrational fine structure) s*u (2s) (Energy required to remove electron, lower energy for higher orbitals)