5.04 Principles of Inorganic Chemistry II

Transcription

1 MIT OpenourseWare Principles of Inorganic hemistry II Fall 8 For information about citing these materials or our Terms of Use, visit:

2 5.4, Principles of Inorganic hemistry II Prof. Daniel G. Nocera Lecture 3: Irreducible Representations and haracter Tables Similarity transformations yield irreducible representations, Γ i, which lead to the useful tool in group theory the character table. The general strategy for determining Γ i is as follows: A, B and are matrix representations of symmetry operations of an arbitrary basis set (i.e., elements on which symmetry operations are performed). There is some similarity transform operator v such that A = v A v B = v B v = v v where v uniquely produces block-diagonalized matrices, which are matrices possessing square arrays along the diagonal and zeros outside the blocks A B A = A B = B = A 3 B 3 3 Matrices A, B, and are reducible. Sub-matrices A i, B i and i obey the same multiplication properties as A, B and. If application of the similarity transform does not further block-diagonalize A, B and, then the blocks are irreducible representations. The character is the sum of the diagonal elements of Γ i. As an example, let s continue with our exemplary group: E, 3, 3, σ v, σ v, σ v by defining an arbitrary basis a triangle v v '' A v ' B The basis set is described by the triangles vertices, points A, B and. The transformation properties of these points under the symmetry operations of the group are: 5.4, Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page of 7

3 A A E B = B = A B σ V A A = = B B A B A B B = = 3 A A B σ V A B B = A = A B A 3 B = A = B A B σ V A B = B = A A B These matrices are not block-diagonalized, however a suitable similarity transformation will accomplish the task, ; v = v = Applying the similarity transformation with 3 as the example, v v = = = 3 * , Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page of 7

4 if v 3 * v is applied again, the matrix is not block diagonalized any further. The same diagonal sum is obtained *though off-diagonal elements may change). In this case, 3 * is an irreducible representation, Γ i. The similarity transformation applied to other reducible representations yields: v E v = E* = v 3 v = 3 * = 3 3 v σ v v = σ v * = v σ v v = σ v * = 3 3 v σ v v = σ v * = 3 3 As above, the block-diagonalized matrices do not further reduce under reapplication of the similarity transform. All are Γ irr s. Thus a 3 3 reducible representation, Γ red, has been decomposed under a similarity transformation into a ( ) and ( ) block-diagonalized irreducible representations, Γ i. The traces (i.e. sum of diagonal matrix elements) of the Γ i s under each operation yield the characters (indicated by χ) of the representation. Taking the traces of each of the blocks: E 3 3 σ v σ v σ v E 3 3σ v Γ Γ Γ Γ Note: characters of operators in the same class are identical 5.4, Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page 3 of 7

5 This collection of characters for a given irreducible representation, under the operations of a group is called a character table. As this example shows, from a completely arbitrary basis and a similarity transform, a character table is born. The triangular basis set does not uncover all Γ irr of the group defined by {E, 3, 3, σ v, σ v, σ v }. A triangle represents artesian coordinate space (x,y,z) for which the Γ i s were determined. May choose other basis functions in an attempt to uncover other Γ i s. For instance, consider a rotation about the z-axis, The transformation properties of this basis function, R z, under the operations of the group (will choose only operation from each class, since characters of operators in a class are identical): E: R z R z 3 : R z R z σ v (xy): R z R z Note, these transformation properties give rise to a Γ i that is not contained in a triangular basis. A new ( x ) basis is obtained, Γ 3, which describes the transform properties for R z. A summary of the Γ i for the group defined by E, 3, 3, σ v, σ v, σ v is: E 3 3σ v Γ Γ from triangular basis, i.e. (x, y, z) Γ 3 from R z Is this character table complete? Irreducible representations and their characters obey certain algebraic relationships. From these 5 rules, we can ascertain whether this is a complete character table for these 6 symmetry operations. 5.4, Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page 4 of 7

6 Five important rules govern irreducible representations and their characters: Rule The sum of the squares of the dimensions, l, of irreducible representation Γ i is equal to the order, h, of the group, l i = l + l + l = h i order of matrix representation of Γ i (e.g. l = for a ) Since the character under the identity operation is equal to the dimension of Γ i (since E is always the unit matrix), the rule can be reformulated as, [ x (E)] i i = h character under E Rule The sum of squares of the characters of irreducible representation Γ i equals h [ x (R)] R i = h character of Γ i under operation R Rule 3 Vectors whose components are characters of two different irreducible representations are orthogonal [x i (R)] [x j (R)] = R for i j Rule 4 For a given representation, characters of all matrices belonging to operations in the same class are identical Rule 5 The number of Γ i s of a group is equal to the number of classes in a group. 5.4, Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page 5 of 7

7 With these rules one can algebraically construct a character table. Returning to our example, let s construct the character table in the absence of an arbitrary basis: Rule 5: E ( 3, 3 ) (σ v, σ v, σ v ) 3 classes 3 Γ i s Rule : l + l + l 3 = 6 l = l =, l = Rule : All character tables have a totally symmetric representation. Thus one of the irreducible representations, Γ i, possesses the character set χ (E) =, χ ( 3, 3 ) =, χ (σ v, σ v, σ v ) =. Applying Rule, we find for the other irreducible representation of dimension, χ (E) χ (E) + χ ( 3 ) χ ( 3 ) + 3 χ (σ v ) χ (σ v ) = Since χ (E) =, χ (E) + χ ( 3 ) + 3 χ (σ v ) = consequence of Rule 4 + χ ( 3 ) + 3 χ (σ v ) = χ ( 3 ) =, χ (σ v ) = For the case of Γ 3 (l 3 = ) there is not a unique solution to Rule + χ 3 ( 3 ) + 3 χ 3 (σ v ) = However, application of Rule to Γ 3 gives us one equation for two unknowns. Have several options to obtain a second independent equation: Rule : + [χ 3 ( 3 )] + 3[χ 3 (σ v )] = 6 Rule 3: + χ 3 ( 3 ) + 3 χ 3 (σ v ) = or + χ 3 ( 3 ) + 3 ( ) χ 3 (σ v ) = Solving simultaneously yields χ 3 ( 3 ) =, χ 3 (σ v ) = Thus the same result shown on pg 4 is obtained: E 3 3σ v Γ Γ Γ 3 5.4, Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page 6 of 7

8 Note, the derivation of the character table in this section is based solely on the properties of characters; the table was derived algebraically. The derivation on pg 4 was accomplished from first principles. The complete character table is: operations 3v Schoenflies symbol for A point group A E E 3 3σ v z x + y, z R z (x,y)(r x,r y ) (x y, xy) (xz,yz) Mulliken characters basis functions symbols for the Γ i Γ i s of: l = A or B A is symmetric (+) with respect to n B is antisymmetric ( ) with respect to n l = l = 3 E T subscripts and designate Γ i s that are symmetric and antisymmetric, respectively to s; if s do not exist, then with respect to σ v primes ( ) and double primes ( ) attached to Γ i s that are symmetric and antisymmetric, respectively, to σ h for groups containing i, g subscript attached to Γ i s that are symmetric to i whereas u subscript designates Γ i s that are antisymmetic to i 5.4, Principles of Inorganic hemistry II Lecture 3 Prof. Daniel G. Nocera Page 7 of 7