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Lecture 7

Lecture 7. Basis of a Representation: A set of objects capable of demonstrating the effects of the all the symmetry operations in a Point Group. A set of functions, atomic orbitals on a central atom or ligands, or common objects.

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Lecture 7

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  1. Lecture 7

  2. Basis of a Representation: A set of objects capable of demonstrating the effects of the all the symmetry operations in a Point Group. A set of functions, atomic orbitals on a central atom or ligands, or common objects. Representation: How the operations affect the basis objects by transforming them into themselves or each other by the operations. The representation is an array showing how the basis objects transform. Ususally we do not need the full matrices but only the trace of the matrices. These values are called the characters. The collected set of character represntations is called the character table. • Irreducible representations: A representation using a minimal set of basis objects. A single atomic orbital (2s, 2pz), a linear combination of atomic orbitals or hybrids (h1 + h2), etc. An irreducible representation may be minimally based on • A single object, one dimensional, (A or B) • Two objects, two dimensional, (E) • Three objects, three dimensional, (T) Reducible representation: How the basis objects transform. By adding and subtracting the basis objects the reducible representation can be reduced to a combination irreducible representations. # of irreducible representations = # of classes of symmetry operations

  3. 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 1 dimensional basis to show what the operators do to an object.

  4. Symmetry of Atomic Orbitals

  5. Naming of Irreducible representations • One dimensional (non degenerate) representations are designated A or B. (A basis object is only changed into itself or the negative of itself by the symmetry operations) • Two-dimensional (doubly degenerate) are designated E. (Two basis object are required to repesent the effect of the operations for an E representation. In planar PtCl42- the px and py orbitals of the Pt, an E representation, are transformed into each other by the C4 rotation, for instance.) • Three-dimensional (triply degenerate) are designated T. (Three objects are interconverted by the symmetry operations for the T representations. In tetrahedral methane, Td, all three p orbitals are symmetry equivalent and interchanged by symmetry operations) • 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  svrespectively • Superscripts ‘ and ‘’ indicate symmetric and antisymmetric behavior respectively with respect to sh. • In groups having a center of inversion, subscripts g (gerade) and u (ungerade) indicate symmetric and antisymmetric representations with respect to i But note that while this rationalizes the naming, the behavior with respect to each operation is provided in the character table.

  6. Character Tables • You have been exposed to symmetry considerations for diatomic molecules: s or p bonding. • Characters 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 E column indicate the dimension of the irreducible representation (of degenerate orbitals having same energy) • Irrecible representations are represented by CAPITAL LETTERS (A, B, E, T,...) whereas orbitals of that symmetry behavior 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

  7. 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 ( a*b = b*a) for all a, b belonging to G, then G is called an Abelian group. The symmetry operations of a Point Group comprise a “group”.

  8. Example Consider the set of all integers and the operation of addition (“*” = +) Is this set of objects (all integers) associative under the operation? (a*b)*c = a*(b*c) Is there an identity element, e? a*e = a Yes, (3 + 4) + 5 = 3 + (4+5) Yes, 0 For each element is there an inverse element, a-1? a-1 * a = e Yes, 4 + (-4) = 0 We have a group. Abelian? Is commutativity satisfied for each element? a * b = b * a Yes. 3 + (-5) = (-5) + 3

  9. 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)

  10. Examine the matrix representation of the elements of the C2v point group - E C2 s’v(yz) sv(xz)

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

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

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

  14. C2 C2 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 Consider PtCl4 C2(x) C2(y) So these operations belong to the same class?

  15. C4 C2 C2 C2(y) = C2(x) = C4 = Since C4 moves C2(x) on top of C2(y) it is an obvious choice for g C43 =

  16. C4 C2(y) C43 C2(x) = Belong to same class! How about the other two C2 elements?

  17. 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.

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

  19. 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 + 2 (12) + 3 (12) = 6 = # of sym operations = 1+2+3 A2: 12 + 2 (12) + 3((-1)2) = 6 E: 22 + 2 (-1)2 + 3 (0)2 = 6

  20. Properties of Characters of Irreducible Representations in Point Groups - 4 Irreducible reps are orthogonal. The sum over the symmetry operations of the products of the characters for two different irreducible reps is zero. For A1 and E: 1 * 2 + 2 (1 *(-1)) + 3 (1 * 0) = 0 Note that for any single irreducible rep the sum is h, the order of the group.

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

  22. Reduction of a Reducible Representation. Given a Reducible Rep how do we find what Irreducible reps it contains? 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 for the reducible rep and a particular irreducible rep i 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

  23. Reducible Representations in Cs = E and sh 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

  24. Let’s observe one helpful thing here. Only the objects (hybrids) that remain themselves, appear on the diagonal of the transformation of the symmetry operation, contribute to the trace. They commonly contribute +1 or -1 to the trace depending whether or not they are multiplied by -1. h1 h2; do not become themselves, interchange h1 , h1; become themselves = = 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

  25. The Irreducible Representations for Cs. The reducible representation derived from the two hybrids can be attached to the table. Note that G = A’ + A”

  26. The Irreducible Representations for Cs. The reducible representation derived from the two hybrids can be attached to the table. Let’s verify some things. Order of the group = # sym operations = 2 A’ and A’’ are orthogonal: 1*1 + 1+(-1) = 0 Sum of the squares over sym operations = order of group = h The magnitude of the A’ and A’’ vectors are each (2) 1/2: magnitude2 = ( 12 + (+/- 1)2)

  27. Now let’s do the reduction. We assume that the reducible rep G can be expressed as a linear combination of A’ and A’’ G = aA’ A’ + aA’’ A’’; our task is to find out the coefficients aA’ and aA’’ aA’ = (1 * 2 + 1 *0)/2 = 1 aA’’ = (1 * 2 + 1 *0)/2 = 1 Or again G = 1*A’ + 1*A’’. Note that this holds for any reducible rep G as above and not limited to the hybrids in any way.

  28. Water is C2v. Let’s use the Character Table Symmetry operations Point group Characters +1 symmetric behavior -1 antisymmetric Mülliken symbols Each row is an irreducible representation

  29. Let’s determine how many independent vibrations a molecule can have. It depends on how many atoms, N, and whether the molecule is linear or non-linear.

  30. 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

  31. 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.

  32. 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 atomsare 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

  33. Finding the reducible representation E = 3 s = 1 C2 = -1 i = -3 C3 = 0 3x3 9 1x-1 -1 3x1 3 1x1 1 (# atoms not moving x char. contrib.) G

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

  35. Symmetry of molecular movements of water Vibrational modes

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

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

  38. G 2 0 2 0 Both A1 and B1 are IR andRamanactive = 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

  39. 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

  40. 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

  41. Molecular Orbitals

  42. 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

  43. 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

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

  45. 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)

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

  47. 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

  48. Combinations of two p orbitals

  49. Combinations of two sets of p orbitals

  50. Combinations of s and p orbitals

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