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L10_Group_TheoryII - Group theory II: exploita1on of...

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Unformatted text preview: Group theory II : exploita1on of symmetry Character Table Point group Symmetry opera1ons Mülliken symbols Characters +1 symmetric behavior  ­1 an1symmetric Each row is an irreducible representa1on Naming of Irreducible representa5ons •  •  •  •  •  •  •  One dimensional (non degenerate) representa1ons are designated A or B. Two ­dimensional (doubly degenerate) are designated E. Three ­dimensional (triply degenerate) are designated T. Any 1 ­D representa1on symmetric with respect to Cn is designated A; an1symmetric ones are designated B Subscripts 1 or 2 (applied to A or B refer) to symmetric and an1symmetric representa1ons with respect to C2 or (if no C2) to σv respec1vely Superscripts ‘ and ‘’ indicate symmetric and an1symmetric opera1ons with respect to σh, respec1vely In groups having a center of inversion, subscripts g (gerade) and u (ungerade) indicate symmetric and an1symmetric representa1ons with respect to i Point Group with 2D Irr. Repr. Symmetry opera1on of NH3 Ammonia has E, 2C3 (C3 and C23) and 3σv Point group = C3v Proper1es of C3v (any group) a.  Must contain E b.  Each opera1on must have an inverse; doing both gives E (right to leV) c.  Any product equals another group member d.  Associa1ve property The threefold symmetry of NH3 makes for complex transforma1on matrices The C3V point group 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 Characters of Irreducible Representa1ons in Point Groups •  Total number of symmetry opera5ons in the group is the order of the group (h) For C3v, it is 6. 1 + 2 + 3 = 6 •  Symmetry opera1ons are arranged in classes. Opera1ons in a class are grouped together as they have iden1cal characters. Elements in a class are related. This column represents three symmetry opera1ons having iden1cal characters. Characters of Irreducible Representa1ons in Point Groups 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. Characters of Irreducible Representa1ons in Point Groups For any irreducible rep the squares of the characters summed over the symmetry opera1ons 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 Proper1es of Characters of Irreducible Representa1ons in Point Groups  ­ 4 Irreducible reps are orthogonal. The sum of the products of the characters for each symmetry opera1on is zero. For A1 and E: 1 * 2 + (1 *( ­1) + 1 *( ­1)) + (1 * 0 + 1 * 0 + 1 * 0) = 0 Classes of operations x, y, z Symmetry of transla1ons (p orbitals) Rx, Ry, Rz: rota1ons 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 Character Tables •  •  Irreducible representa1ons are the generalized analogues of σ or π 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 opera1ons (+1 = orbital does not change;  ­1 = orbital changes sign; anything else = more complex change) Characters in the column of opera1on 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 iden1ty 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 proper1es of the set •  •  •  •  ...
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