Representations of Molecular Properties

• We will routinely use sets of vectors located on each atom of a molecule to represent certainmolecular properties of interest (e.g., orbitals, vibrations).

The set of vectors for a property form the basis for a representation (or basis set) 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 symmetryti f th f t ti f thoperations of the group forms a representation of the group.

In most cases the representation for the property will be the sum of certain fundamentalrepresentations of the group, called irreducible representations.

• A representation is a set of symbols (called characters) that have the same combinationalresults 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)є = ± exp(2/n)

Irreducible Representations of C2v• Given the multiplication table:

• 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 totallysymmetric representation, labeled A1 in C2v .

Irreducible Representations of C2v (contd.)

• The following set of substitutions also satisfies the multiplication table:

• These characters form the A2 irreducible representation of C2v :2 p 2v

Irreducible Representations of C2v (contd.)• Two other sets of substitutions also satisfy the multiplication table, forming the B1 and B2

irreducible representations of C2v :

• Any other set of substitutions will not work: e.g.

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 certainvector properties are transformed by the operations of the group.

Unit Vector Transformations• Consider the effects of applying the operations of C2v 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 A1 irreduciblerepresentation of Crepresentation of C2v .

Unit Vector Transformations• "In C2v the vector z transforms as A1 representation (the totally symmetric representation).”

OR

• "In C2v the vector z belongs to the A1 species (the totally symmetric species)."2v g 1 p ( y y p )

Unit Vector Transformations• Consider the effects of applying the operations of C2v on a unit vector x.

The four 1 x 1 transformation matrices, taken as a set, are identical to the B1 irreduciblerepresentation of C2v .

"In C2v the vector x transforms as B1 .”

Unit Vector Transformations• Consider the effects of applying the operations of C2v on a unit vector y.

The four 1 x 1 transformation matrices, taken as a set, are identical to the B2 irreduciblerepresentation of C2v .

"In C2v the vector y transforms as B2 .”

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 A2 irreduciblerepresentation of Crepresentation of C2v .

"In C2v the vector Rz transforms as A2 .”

Character Table with Vector Transformations

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• The dimension of a matrix is give as rows vs. columns,

i.e., m x n.

• If two matrices are to be multiplied together they must be conformable; i.e., the number ofcolumns in the first (left) matrix must be the same as the number of rows in the second( i h ) i(right) matrix.

• The product matrix has as many rows as the first matrix and as many columns as the secondmatrix.

• The elements of the product matrix, cij , are the sums of the products aikbkj for all values of kfrom 1 to m; i efrom 1 to m; i.e.,

Transformations of a General Vector in C2v• 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 C2v point group can be described by the following equations:

EE

C2 v (xz ) v’ (yz )

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 reduciblerepresentation of the group,

v v’

m

• m is described as a reducible representation of the C2v point group as it can be broken down

m

m is described as a reducible representation of the C2v point group as it can be broken downto a simpler form or reduced.

• The character () of a matrix is the sum of the elements along the left‐to‐right diagonal ofthe matrix, i.e. the trace of the matrix. The matrix reducible representation m can beconverted to the representation of characters v (also reducible) as follows:

(E) = 3 (C2) = 1 (v) = 1 (v’ ) = 1

’v v

v

• Breaking down into its component irreducible representations is called reduction.

Reduction of m by Block DiagonalizationBreaking 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

E h di l l t f h t t i h f th di t• Each diagonal element, cii, of each operator matrix expresses how one of the coordinates x,y, or z is transformed by the operation.

• Each c11 element expresses the transformation of the x coordinate.

• Each c22 element expresses the transformation of the y coordinate.

• Each c33 element expresses the transformation of the z coordinate.

• The set of four cii elements with the same i (across a row) is an irreducible representation.ii ( ) p

m =  A1 +  B1 +  B2

• 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 sameresult as v , thus the three irreducible representations found by block diagonalization of mare the same as those found for v

’v =  A1 +  B1 +  B2 =  m

• Relating this res lt to the ector v hich forms the basis for the red cible representation

v v’

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 – theCartesian axis,

i.e., the three irreducible representations of which v is composed are the three symmetryspecies by which the unit vectors z (A1 ), x (B1 ) and y (B2 ) transform.

• In a representation of matrices, such as , the dimension (d ) of the representation is the

Dimensions of RepresentationsIn a representation of matrices, such as m , the dimension (d ) of the representation is theorder of the square matrices of which it is composed.

• For a representation of characters, such as v , the dimension is the value of the characterfor the identity operation.

v v’

v

• The dimension of the reducible representation (dr ) must equal the sum of the dimensions ofll th i d ibl t ti (d ) f hi h it i d

v

all the irreducible representations (di ) of which it is composed.

v v’

v

More Complex Groups and Standard Character Tables

• The group C3v has:

Three classes of symmetry elements Three classes of symmetry elements.

Three irreducible representations.

One irreducible representation has a dimension of di = 2 (doubly degenerate).

Classes

• Geometrical Definition (Symmetry Groups): Operations in the same class can be convertedinto one another by changing the axis system through application of some symmetryoperation of the group.

• Mathematical Definition (All Groups): The elements A and B belong to the same class ifthere is an element X within the group such that X‐1AX = B, where X‐1 is the inverse of X (i.e.,XX‐1 = X‐1X = E).

If X‐1AX = B, we say that B is the similarity transform of A by X, or that A and B areconjugate to one another.

The element Xmay in some cases be the same as either A or B.

Classes of C3v by Similarity Transforms

• Take the similarity transforms on C3 tofind all members in its class:

• Take the similarity transforms on 1 tofind all members in its class:f f

only C3 and C32 only 1 , 2 and 3

Transformations of a General Vector in C3vThe Need for a Doubly Degenerate Representation

• No operation of C3v changes the z coordinate, thus every operation involves an equation ofthe 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 sameclass 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)

• Transformation by C3

• From trigonometry:

• Thus the transformation matrix has non‐zero off diagonal elements.

• 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 C3 restrictsThe presence of nonzero, off diagonal elements in the transformation matrix for C3 restrictsus 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 of2 x 2 matrices and another set of 1 x 1 matrices.

• Converting to representations of characters gives a doubly‐degenerate irreduciblerepresentation and a non‐degenerate irreducible representation.

• Any property that transforms as E in C3v will have a companion, with which it is degenerate,that will be symmetrically and energetically equivalent.

Direct Product Listings

• The last column of typical character tables gives the transformation properties of directproducts of vectors.

• Among other things these can be associated with the transformation properties of d orbitals• Among other things, these can be associated with the transformation properties of d orbitalsin the point group.

Correspond to d orbitals: z2, x2– y2, xy, xz, yz, 2z2 – x2 – y2

2 2 2 2 2 2 2 Do not correspond to d orbitals: x2, y2, x2+ y2, x2+ y2 + z2

Complex‐Conjugate Paired Irreducible Representations

• Some groups have irreducible representations with imaginary characters in complexconjugate pairs:

C (n >= 3) C (n >= 3) S T TCn (n >= 3), Cnh (n >= 3), S2n, T, Th

• The paired representations appear on successive lines in the character tables, joined byb ({ })braces ({ }).

• Each pair is given the single Mulliken symbol of a doubly degenerate representation

e.g., E, E1, E2, E', E", Eg, Eu.g

• Each of the paired complex‐conjugate representations is an irreducible representation in itsown right.

Combining Complex‐Conjugate Paired Representations

• It is sometimes convenient to add the two complex‐conjugate representations to obtain arepresentation of real characters.

• When the pair has є and є* characters, where є = exp(2 i/n), the following identities areused in taking the sum:

є = exp(2i/n) = cos2/n + i sin2/n

є* = exp(–2i/n) = cos2/n – sin2/np( )

which combine to give

є + є* = 2cos2/n

e.g., in C3 є = exp(2 i/3) and є + є* = 2cos2/3

• If complex‐conjugate paired representations are combined in this way, realize that the real‐number representation is a reducible representation.

Mulliken Symbols‐ Irreducible Representation Symbols

• In non‐linear groups:

A : non‐degenerate; symmetric to Cn where (Cn) > 0.

B : non degenerate; anti symmetric to C where (C ) < 0B : non‐degenerate; anti‐symmetric to Cn where (Cn) < 0.

E : doubly‐degenerate; (E) = 2.

T : triply‐degenerate; (T) = 3.

G : four‐fold degeneracy; (G) = 4, observed in I and Ih

H : five‐fold degeneracy; (H) = 5, observed in I and Ih

• In linear groups C∞v and D∞h :

A non‐degenerate; symmetric to C∞ ; (C∞) = 1.

E doubly‐degenerate; (E) = 2.

• With any degeneracy in any centrosymmetric groups:

Mulliken Symbols (Modifying Symbols)With any degeneracy in 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.p ( ) y p h ( h)

• With non‐degenerate representations in non‐linear groups:

subscript 1 : symmetric with respect to Cm (m < n) or v ;

(C ) > 0 ( ) > 0(Cm) > 0 or (v) > 0.

subscript 2 : anti‐symmetric with respect to Cm (m < n) or v ;

(Cm) < 0 or (v) < 0.

• With non‐degenerate representations in linear groups (C∞v and D∞h ):

subscript + : symmetric with respect to∞C2 or∞v ;

(∞C2) = 1 or (∞h) = 1.( 2) ( h)

subscript : anti‐symmetric with respect to∞C2or∞v ;

(∞C2) = ‐1 or (∞h) = ‐1.

Relationships from theGreat Orthogonality Theorem

1. The sum of the squares of the dimensions of all the irreducible representations is equal to theorder 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 tog p ( ) p g gthe same class are the same.

4. The sum of the squares of the characters in any irreducible representation equals the order ofthe group:

5. Any two different irreducible representations are orthogonal:

(note: R = any operation of the group; Rc = a class of operations; gc = order of the class)