Laser Molecular Spectroscopy CHE466 Fall 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 H 2 O. There are four elements of symmetry in this molecule: Identity (E), a principal axes of symmetry (twofold, C 2 ) 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 C 2v group (Schonflies notation) C2C2 z y x

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. C2C2 z y x

The Character Table and Molecular Symmetry Symmetry Representations: Mulliken Symbols and their meaning

Symmetry Classification Strategy The 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: Г trans = A 1 + B 1 + B 2 z y x C2C2 rotation C2C2 reflection  xz

Character table and interpretation of motion: Water Rotations The symbols R x, R y, R z 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 C 2 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: Г rot = A 2 + B 1 + B 2 z y x C2C2 rotation C2C reflection  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 (Г 3N ) and subtracting those of rotation and translation: Г vib = Г 3N - Г trans - Г rot The following shows the three vibrational motions of water and their Mulliken symbols The vibrational representation is Г vib = 2A 1 + B 2 A1A1 B2B2 A1A1

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)

Character table and interpretation of motion: Water Finding the Reducible Representation of All Motions in Water ab ab E n = 3, Г(E) = 9 ab C2C2 ba n = 1, Г(C 2 ) = -1 ab  yz ab n = 3, Г(  yz ) = 3 ab  xz ba n = 1, Г(  xz ) = 1

Reducing the reducible representation per symmetry representation: h = total number of symmetry elements Г red,I = # of reducible representations per element (i) Г(ζ) irred = irreducible character per element corresponding to a given symmetry representation η = number of elements in the class # irred (ζ) = Symmetry representation For water: Character table and interpretation of motion: Water

Reducing the reducible representation per symmetry representation: For water the total reducible representation of all motions is: Г 3N = 3A 1 + A 2 + 2B B 2 (a total of 9 motions) Therefore the irreducible representation of vibrational motion is: Г vib = Г 3N - Г trans - Г rot Г vib = (3A 1 + A 2 + 2B B 2 ) – (A 1 + B 1 + B 2 ) – (A 2 + B 1 + B 2 ) Г vib = 2A 1 + B 2 (a total of 3 vibrational motions) Character table and interpretation of motion: Water A1A1 B2B2 A1A1

Ammonia belongs to the C 3v symmetry group. It has a two equivalent (collinear) axes of rotation (C 3 and C 3 2 ) and 3 equivalent vertical planes. The total reducible representation is given by : Г(E) = 12, Г(C 2 ) = 0, Г(  v ) = 2, which is reduced to: Г 3N = 3A 1 + A 2 + 4E From the character table: Г trans = A 1 + E Г rot = A 2 + E Thus, the vibrational representation is Г vib = 2 A E There are two A 1 modes (totally symmetric) and 2 pairs of doubly degenerate modes Character table and interpretation of motion: Ammonia

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 T x, T y or T z have an A 1 representation, then the molecule has a permanent dipole moment. Therefore molecules belonging to the following groups will have permanent dipole moments: C 1, C s, C n, and C nv (including C ∞v ). Permanent Dipole Moment and Symmetry C 3v C 2v C∞vC∞v D 6h No dipole