Representations of Molecular Properties

Transcription

1 Representations of Molecular Properties We will routinely use sets of vectors located on each atom of a molecule to represent certain molecular properties of interest (e.g., orbitals, vibrations). The set of vectors for a property form the basis for a representation (or basis set). e.g. molecular coordinates. The mathematical description of the behavior of the vectors under the symmetry operations of the group forms a representation tti of thegroup. In most cases the representation for the property will be the sum of certain fundamental representations of the group, called irreducible representations. A representation is a set of symbols (called characters) that have the same combinational results as the elements of the group, as given in the group's multiplication table. i.e., a representation of a group must satisfy the multiplication table for that group. Typical characters in representations of point groups include the following: 0, ±1, ±2, ±3, ±i, ± є є = ± exp(2 /n)

2 Given the multiplication table: Irreducible Representations of C 2v The following set of substitutions has the same combinational relationships: The substitution of all 1's, which can be made for any point group, forms the totally symmetric representation, labeled A 1 in C 2v.

3 Irreducible Representations of C 2v (contd.) The following set of substitutions also satisfies the multiplication table: These characters form the A 2 irreducible representation of C 2v : 2 p 2v

4 Irreducible Representations of C 2v (contd.) Two other sets of substitutions also satisfy the multiplication table, forming the B 1 and B 2 irreducible representations of C 2v : Any other set of substitutions will not work: e.g.

5 Basic Character Table The collection of irreducible representations for a group is listed in a character table, with the totally symmetric representation listed first: The characters of the irreducible representations can describe the ways in which certain vector properties are transformed by the operations of the group.

6 Unit Vector Transformations Consider the effects of applying the operations of C 2v on a unit vector z. [Assume v = xz and v = vz ] The four 1 x 1 transformation matrices, taken as a set, are identical to the A 1 irreducible representation ofc C 2v.

7 Unit Vector Transformations "In C 2v the vector z transforms as A 1 representation (the totally symmetric representation). "In C 2v the vector z belongstothea 1 species (the totally symmetric species)." OR

8 Unit Vector Transformations Consider the effects of applying the operations of C 2v on a unit vector x. The four 1 x 1 transformation matrices, taken as a set, are identical to the B 1 irreducible representation of C 2v. "In C 2v the vector x transforms as B 1.

9 Unit Vector Transformations Consider the effects of applying the operations of C 2v on a unit vector y. The four 1 x 1 transformation matrices, taken as a set, are identical to the B 2 irreducible representation of C 2v. "In C 2v the vector y transforms as B 2.

10 Rotational Vector Transformations Consider the symmetry of a rotation about the z axis. The four 1 x 1 transformation matrices, taken as a set, are identical to the A 2 irreducible representation ofc C 2v. "In C 2v the vector R z transforms as A 2.

11 Character Table with Vector Transformations

12 Review of Matrices A matrix is a rectangular array of numbers that combines with other such arrays according to specific rules. The dimension of a matrix is give as rows vs. columns, i.e., m x n.

13 If two matrices are to be multiplied together they must be conformable; i.e., the number of columns in the first (left) matrix must be the same as the number of rows in the second (ih) (right) matrix. The product matrix has as many rows as the first matrix and as many columns as the second matrix. The elements of the product matrix, c ij, are the sums of the products a ik b kj for all values of k from 1 to m; i.e.,

14 Transformations of a General Vector in C 2v To see how the transformation properties of a general vectors relate to the irreducible representations of a group, consider the vector v. All symmetry operations for the C 2v point group can be described by the following equations: E C 2 v ( xz ) v ( yz )

15 A Representation with Matrices The operator matrices combine with each other in the same was as the operators do in the multiplication table, thus the character table can be re written to describe a reducible representation of the group, v v m m is described as a reducible representation of the C 2v point group as it can be broken down to a simpler form or reduced. The character ( ) of a matrix is the sum of the elements along the left to right diagonal of the matrix, i.e. the trace of the matrix. The matrix reducible representation m can be converted to the representation of characters v (also reducible) as follows: (E)=3 (C 2 )=1 ( v )=1 ( v )=1 v v v

16 Reduction of m by Block Diagonalization Breaking down m into its component irreducible representations is called reduction. The species into which m reduces are the those by which the vectors z, x, and y transform, respectively. v v m Each diagonal element, c ii,of each operator matrix expresses how one of the coordinates x, y, or z is transformed by the operation. Each c 11 element expresses the transformation of the x coordinate. Each c 22 element expresses the transformation of the y coordinate. Each c 33 element expresses the transformation of the z coordinate. The set of four c ii elements with the same i (across a row) is an irreducible representation. m = A 1 + B 1 + B 2

17 Following block diagonalization the m representation can be more conveniently displayed as v v v Summing the characters of the corresponding irreducible representations gives the same result as v, thus the three irreducible representations found by block diagonalization of m v = A 1 + B 1 + B 2 = m v v are the same as those found for v Relating this result to the vector v, which forms the basis for the reducible representation, we see that the symmetry of the vector is a composite of three irreducible representations (symmetry species), which represent the symmetry of more fundamental vectors the Cartesian axis, i.e., the three irreducible representations of which v is composed are the three symmetry species by which the unit vectors z (A 1 ), x (B 1 )andy (B 2 )transform. v

18 Dimensions of Representations In a representation of matrices, such as m, the dimension (d ) of the representation is the order of the square matrices of which it is composed. For a representation of characters, such as v, the dimension is the value of the character for the identity operation. v v v The dimension of the reducible representation (d r ) must equal the sum of the dimensions of all the irreducible ibl representations ti (d i ) of which h it iscomposed. v v v

19 More Complex Groups and Standard Character Tables The group C 3v has: Three classes of symmetry elements. Three irreducible representations. One irreducible representation has a dimension of d i = 2 (doubly degenerate).

20 Classes Geometrical Definition (Symmetry Groups): Operations in the same class can be converted into one another by changing the axis system through application of some symmetry operation of the group. Mathematical Definition (All Groups): The elements A and B belong to the same class if there is an element X within the group such that X 1 AX =B,whereX 1 is the inverse of X (i.e., XX 1 = X 1 X = E). If X 1 AX = B, wesaythatb is the similarity transform of A by X, orthata and B are conjugate to one another. The element X may in some cases be the same as either A or B.

21 Classes of C 3v by Similarity Transforms Take the similarity transforms on C 3 to find all members in its class: Take the similarity transforms on 1 to find all members in its class: only C 3 and C 3 2 only 1, 2 and 3

22 Transformations of a General Vector in C 3v The Need for a Doubly Degenerate Representation No operation of C 3v changes the z coordinate, thus every operation involves an equation of the form where we only need to describe any changes in the projection of v in the xy plane. The operator matrix for each operation is generally unique, but all operations in the same class have the same character ( ) from their operator matrices. Thus, we only need to examine the effect of one operation in each class. Transformation by E Transformation by 1 ( xz )

23 Transformation by C 3 From trigonometry: Thus the transformation matrix has non zero off diagonal elements.

24 Reduction by block diagonalization must result in blocks that are consistent across all matrices The presence of nonzero, off diagonal elements in the transformation matrix for C 3 restricts us to diagonalization into a 2 x 2 block and a 1 x 1 block. For all three matrices we must adopt a scheme of block diagonalization that yields one set of 2x2matricesandanothersetof1x1matrices. Converting to representations of characters gives a doubly degenerate irreducible representation and a non degenerate irreducible representation. Any property that transforms as E in C 3v will have a companion, with which it is degenerate, that will be symmetrically and energetically equivalent.

25 Direct Product Listings The last column of typical character tables gives the transformation properties of direct products of vectors. Among other things, these can be associated with the transformation properties of d orbitals in the point group. Correspond to d orbitals: z 2,x 2 y 2,xy,xz,yz,2z 2 x 2 y 2 Do not correspond to d orbitals: x 2,y 2,x 2 +y 2,x 2 +y 2 +z 2

26 Complex Conjugate Paired Irreducible Representations Some groups have irreducible representations with imaginary characters in complex conjugate pairs: C n (n >= 3), C nh (n >= 3), S 2n, T, T h The paired representations appear on successive lines in the character tables, joined by braces ({ }). Each pair is given the single Mulliken symbol of a doubly degenerate representation e.g., E, E 1, E 2, E', E", E g, E u. g Each of the paired complex conjugate representations is an irreducible representation in its own right.

27 Combining Complex Conjugate Paired Representations It is sometimes convenient to add the two complex conjugate representations to obtain a representation of real characters. When the pair has є and є* characters, where є = exp(2 i/n), the following identities are used in taking the sum: which combine to give є = exp(2 i/n) = cos2 /n + i sin2 /n є* = exp( 2 i/n) = cos2 /n sin2 /n є + є* = 2cos2 /n e.g., in C 3 є = exp(2 i/3)andє + є*=2cos2 /3 If complex conjugate paired representations are combined in this way, realize that the realnumber representation is a reducible representation.

28 In non linear groups: Mulliken Symbols Irreducible Representation Symbols A : non degenerate; symmetric to C n where (C n )>0. B : non degenerate; anti symmetric to C n where (C n ) < 0. E : doubly degenerate; (E)=2. T : triply degenerate; (T)=3. G : four fold degeneracy; (G)=4,observedinI and I h H : five fold degeneracy; (H)=5,observedinI and I h In linear groups C v and D h : A non degenerate; symmetric to C ; (C )=1. E doubly degenerate; (E)=2.

29 Mulliken Symbols (Modifying Symbols) With any degeneracyin any centrosymmetric groups: subscript g : gerade ; symmetric with respect to inversion ; i >0. subscript u : ungerade ;anti symmetric with respect to inversion ; i <0. With any degeneracy in non centrosymmetric non linear groups: prime ( ) : symmetric with respect to h ; ( h )>0. double prime ( ) : anti symmetric with respect to h ; ( h ) <0. With non degenerate representations in non linear groups: subscript 1 : symmetric with respect to C m (m < n)or v ; (C m ) > 0 or ( ( v ) > 0. subscript 2 : anti symmetric with respect to C m (m < n)or v ; (C m )<0or ( v )<0. With non degenerate representations in linear groups (C v and D h ): subscript + : symmetric with respect to C 2 or v ; ( C 2 2) ) =1or ( h h) ) =1. subscript : anti symmetric with respect to C 2 or v ; ( C 2 )= 1or ( h )= 1.

30 Relationships from the Great Orthogonality Theorem 1. The sum of the squares of the dimensions of all the irreducible representations is equal to the order of the group: 2. The number of irreducible representations of a group is equal to the number of classes. 3. In a given representation (irreducible or reducible) the characters for all operations belonging to g p ( ) p g g the same class are the same.

31 4. The sum of the squares of the characters in any irreducible representation equals the order of the group: 5. Any two different irreducible representations are orthogonal: (note: R = any operation of the group; R c = a class of operations; g c = order of the class)