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Mulliken Labels for Representations

Character Tables . Operations in the same class are grouped togetherMost symmetric representation is written first (usually the totally symmetric one)Numbers under the operations are characters of matrices representing the effect of symmetry operationsThe Character Tables have information on symm

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Mulliken Labels for Representations

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    1. Mulliken Labels for Representations ? Non-Degenerate Representations = A or B Character under identity E = 1; (symmetric) Character under Principal axis = 1 then Symbol = A Character under identity E = 1; (symmetric) Character under Principal axis = -1 (anti-symmetric) Symbol = B When symmetric to inversion (i): add g = gerade (e.g. A1g) When anti-symmetric to inversion: add u = ungerade (A1u) ? Degenerate Representations: Doublet: Symbol =E ( Example: Eg in Oh point group) Triplet: Symbol = T (Example: T2g in Oh point group)

    2. Character Tables Operations in the same class are grouped together Most symmetric representation is written first (usually the totally symmetric one) Numbers under the operations are characters of matrices representing the effect of symmetry operations The Character Tables have information on symmetry characteristics of orbitals, directions, rotations, vibrations, functions etc.. (x, y) = degenerate; x, y = sharing a representation but not degenerate Similar representations are differentiated by subscripts or superscripts e.g. A1 and A2 or E’ and E’’

    3. fgb

    6. Reducible Representations

    7. Application of Group Theory and Symmetry to Vibrational Spectroscopy Find symmetry of all possible molecular motion [Cu(N)4(O)] (C4v) -Set up Cartesian coordinate system on each atom and allow each atom freedom to move along x, y, and x directions – 6 atoms hence 18 motions in total) -Determine reducible representation for members of the C4v point group on all 18 motions and reduce it using the formula Allow for rotational and translational motion Determine symmetry of vibrational motion Determine IR-active vibrations Determine Raman-active Vibrations

    9. Set up and Reduce a Reducible Representation for all of the possible motions

    10. Representations of the C4V Point Group

    11. Selection Rules for Vibrational Spectroscopy Find symmetry of all possible molecular motion G motion = 4A1 + A2 +2B1 + B2 + 5E Allow for rotational and translational motion Rotation – transform as (Rx + Ry) (E) , Rz (A2) Translation – transform as (x, y) (E) and z (A1) Determine symmetry of vibrational motions G vib = 3A1+ 2B1 + B2 + 3E Determine IR-active vibrations (Transforming as components of dipole moments, x, y, z) : 3A1 + 3E Determine Raman-active Vibrations (Transforming as components of polarization x2, y2, z2, xy, xz,, yz etc..): 3A1+ 2B1 + B2 + 3E Determine coincidences (vibrations which are both IR and Raman active): 3A1 + 3E

    12. Water Vibrations

    13. Symmetry & Group Theory Applications to d-Bonding Questions to be answered: 1)What are Irreducible Representations of d-Molecular Orbitals? 2) What linear combinations of ligand atomic orbitals result in Ligand Group Orbitals (LGOs) suitable for d-bonding? 3)What are the rough shapes of the LGOs? 4) What central atom orbitals are suitable for bonding with LGOs?

    14. d-Bonding in M-C bonds of [M(CN)5]2- in D3h

    15. Projection Operator Method

    16. For A1’: multiply characters of A1’ by the result indicating effect of D3h on members a and b (see Table slide 13). Simplify and Normalize: Thus for a & d: Projection yields: 6a + 6d which simplifies to: a + d ?LGO(A1’) = (?a + ?d)/(v2) For b, c, e: 4b + 4c + 4e ; i.e. b + c + e ?LGO(A1’) = (?b + ?c + ?e )/(v3) For A2’’: ?LGO(A2”) = (?a - ?d)/(v2) For the Degenerate Pair of LGOs E’: ?LGO(E’) = (2?b - ?c - ?e )/(v6) , and an orthogonal LGO:(?c - ?e)/(v2)

    17. Which M3+ (3d- element) orbitals are suitable for M-C in [M(CN)5]2- d-Bonding? From D3h Character Tables bonding and non-bonding orbitals are: For Bonding Orbitals 2A1’ : 3d - z2 and 4s A2”: 4pz E’: 3d – (x2-y2, xy) and 4p(x, y) (which has higher energy hence not preferred) Non-bonding: 3d: (xz, yz) –E2” 4p: (x, y) –E’

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