Laser Molecular SpectroscopyCHE466Fall 2009

David L. Cedeño, Ph.D.Illinois State University

Department of Chemistry

Elements of Symmetry

Utilizing Symmetry to Represent Molecular Motion

Molecules can be classified according to their symmetry. The classification is based on a mathematical categorization (using group theory) based on the so-called elements of symmetry which are defined via reference geometrical representations (point, axis, plane).

Consider the molecule H2O.

There are four elements of symmetry in this molecule: Identity (E), a principal axes of symmetry (twofold, C2) and 2 vertical planes (contain the symmetry axis, v(xz) and v(yz)). Any molecule that contains this and only this elements of symmetry belongs to the so called C2v group (Schonflies notation)

C2

z

yx

The Character Table and Molecular Symmetry

Character tables contain symmetry labels that represent the effects of operations around symmetry elements that make a group. The operations include rotation around symmetry axes, reflections thru a symmetry plane or inversions through a symmetry point.

The character table for C2v molecules is shown below

The symbols in the first column are called the Mulliken symbols and they represent the symmetry of the operation with respect to a given element. The characters represent the so-called irreducible representation of the group. The labels under each element column symbolize the irreducible character associated to the symmetry representation (notated via the Mulliken symbol) and the last two column are the so called transformation properties, which we will relate to motion and, of course, spectroscopy.

C2

z

yx

The Character Table and Molecular Symmetry

Symmetry Representations: Mulliken Symbols and their meaning

singly degenerate state which is symmetric with respect to rotation about the principal axis,

singly degenerate state which is antisymmetric with respect to rotation about the principal axis,

doubly degenerate,

triply degenerate,

(gerade, symmetric) the sign of the wavefunction does not change on inversion through the center of the atom,

(ungerade, antisymmetric) the sign of the wavefunction changes on inversion through the center of the atom,

(on or ) the sign of the wavefunction does not change upon rotation about the center of the atom,

(on or ) the sign of the wavefunction changes upon rotation about the center of the atom,

' = symmetric with respect to a horizontal symmetry plane ,

" = antisymmetric with respect to a horizontal symmetry plane .

Symmetry Classification StrategyThe following chart and link are used to determine the symmetry group to which a given molecule belongs:

Character table and interpretation of motion: Water

Translations

The symbols x, y, z in the colored column tell us the symmetry representation of translational motion along the x, y, and z coordinate axes. For instance, translation along the z axis is completely symmetrical respect to all elements of symmetry. This is because the motion vector along that axis is not changed if we operate around the symmetry element. On the other hand, translation along the x axis is antisymmetric with respect to rotation (180o) around the C2 axis and the reflection through the yz plane (walk yourself through transaltion along the y axis).The irreducible representation of rotations is: Гtrans = A1 + B1 + B2

z

yx

C2

rotation

C2

reflection

xz xz

Character table and interpretation of motion: Water

Rotations

The symbols Rx, Ry, Rz in the colored column tell us the symmetry representation of rotational motion around the x, y, and z coordinate axes. For example, rotational motion around the z axis is symmetrical respect to the C2 axis of symmetry, but antisymmetric with respect to reflection through the xz plane. Work yourself through the rotations around x and y axes.The irreducible representation of rotations is: Гrot = A2 + B1 + B2

z

yx

C2

rotation

C2

+ - + -

reflection

+ +- -

xz xz

Character table and interpretation of motion: Water

Vibrations

In order to represent vibrations, one needs the coordinates of motion of each nucleus, which is trivial for small molecules, but would need the assistance of computer software for large molecules. Still, the numer of vibrational motions in a molecule is (3N-6) for non-linear molecules and (3N-5) for linear moelcules, where N is the number of atoms, thus going individually for every motion is tedious. There is a way to know the irrreducible representation of vibrations by accounting for the irreducible representation of all motions (Г3N) and subtracting those of rotation and translation: Гvib = Г3N - Гtrans - Гrot

The following shows the three vibrational motions of water and their Mulliken symbolsThe vibrational representation is Гvib = 2A1 + B2

A1 B2 A1

Character table and interpretation of motion: Water

Finding the Reducible Representation of All Motions

The procedure involves finding the reducible representation for all motions. This one is obtained by looking at the number of atoms (n) that remain unchanged (i.e do not move) during a symmetry operation. This is then multiplied by the sum of the diagonal of the transformation matrix of the operation to yield the reducible representation. The table below summarizes the number of reducible representations as a function of symmetry operations (or elements)

SYMMETRY ELEMENT # OF REDUCIBLE REPRESENTATIONS

E 3n h,v n

i -3n Cn

k

nk2cos21 n

Snk

nk2cos21 n

n is the number of atoms that do not move during a symmetry operation around the symmetry element

Character table and interpretation of motion: Water

Finding the Reducible Representation of All Motions in Water

a ba b

E n = 3, Г(E) = 9

a b

C2

b a

n = 1, Г(C2) = -1

a b

yz

a b

n = 3, Г(yz) = 3

a b

xz

b a

n = 1, Г(xz) = 1

Reducing the reducible representation per symmetry representation:

h = total number of symmetry elementsГred,I = # of reducible representations per element (i)Г(ζ)irred = irreducible character per element corresponding to a given symmetry representationη = number of elements in the class#irred(ζ) = Symmetry representation

For water:

Character table and interpretation of motion: Water

3)(#

)113()111()111(11941)(#

1

1

A

A

irred

irred

))((h1)(# , irrediredirred

1)(#

)113()111()111(11941)(#

2

2

A

A

irred

irred

2)(#

)113()111()111(11941)(#

1

1

B

B

irred

irred 3)(#

)113()111()111(11941)(#

2

2

B

B

irred

irred

Reducing the reducible representation per symmetry representation:

For water the total reducible representation of all motions is:

Г3N = 3A1 + A2 + 2B1 + 3 B2 (a total of 9 motions)

Therefore the irreducible representation of vibrational motion is:

Гvib = Г3N - Гtrans - Гrot

Гvib = (3A1 + A2 + 2B1 + 3 B2) – (A1 + B1 + B2) – (A2 + B1 + B2)

Гvib = 2A1 + B2 (a total of 3 vibrational motions)

Character table and interpretation of motion: Water

A1 B2 A1

Ammonia belongs to the C3v symmetry group. It has a two equivalent (collinear) axes of rotation (C3 and C3

2) and 3 equivalent vertical planes.

The total reducible representation is given by :

Г(E) = 12, Г(C2) = 0, Г(v) = 2, which is reduced to: Г3N = 3A1 + A2 + 4E

From the character table: Гtrans = A1 + E Гrot = A2 + E

Thus, the vibrational representation is Гvib = 2 A1 + 2 E

There are two A1 modes (totally symmetric) and 2 pairs of doubly degenerate modes

Character table and interpretation of motion: Ammonia

C3v I 2C3 3v A1 1 1 1 Tz xx + yy,zz A2 1 1 -1 Rz E 2 -1 0 (Tx,Ty), (Rx, Ry) (xx - yy,xy),

(xz, yz)

It is possible to determine if a molecule has a permanent dipole moment by looking at its character table.A molecule has a permanent dipole moment if any of the translational motion of the molecule is totally symmetric.In other words if either Tx, Ty or Tz have an A1 representation, then the molecule has a permanent dipole moment.

Therefore molecules belonging to the following groups will have permanent dipole moments:C1, Cs, Cn, and Cnv (including C∞v).

Permanent Dipole Moment and Symmetry

C3v C2vC∞v

D6h

No dipole