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Molecular Symmetry Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . σ v(vertical): plane colinear with principal axis Reflection through a planes of symmetry (mirror plane) σ σ d(dihedral) Vertical, parallel to principal axis , parallel to C n and bisecting two C2' axes 4 Reflection through a planes of symmetry (mirror plane) σ h(horizontal): plane perpendicular to principal axis Inversion, Center of Inversion (i) A center of symmetry: A point at the center of the molecule. (x,y,z) (-x,-y,-z). It is not necessary to have an atom in the center (e.g. benzene). Tetrahedrons, triangles, and pentagons don't have a center of inversion symmetry. C2H6 Ru(CO) 6 C4H4Cl 2F2 5 Rotation-reflection, Improper rotation(S n) ° This is a compound operation combining a rotation 360 /n (C n) with a σ σ reflection through a plane perpendicular to the C n axis h.(C n followed by h) σ Cn=S n ° An improper rotation (or rotation–reflection), Sn, involves rotation about 360 /n followed by reflection through a plane that is perpendicular to the rotation axis. 6 Identity (E) Simplest symmetry operation. All molecules have this element. If the molecule does have no other elements, it is asymmetric. The identity operation amounts to doing nothing to a molecule and so leaves any molecule completely unchanged. CHFClBr SOFCl Successive Operations 7 Symmetry Point Groups •Symmetry of a molecule located on symmetry axes, cut by planes of symmetry, or centered at an inversion center is known as point symmetry . •Collections of symmetry operations constitute mathematical groups . •Each symmetry point group has a particular designation. Cn, C nh , C nv Dn, D nh , D nd S2n C∞v ,D∞h Ih, I Td , Th ,T Oh,O C1,C i, C s 8 HCl F2 C D ∞∞∞v ∞∞∞h Linear molecular species can be classified according to whether they possess a centre of symmetry (inversion centre) or not. 9 Tetrahedral Geometry P4 B4Cl 4 Octahedral Geometry Icosahedral Geometry 2−−− [W(CO) 6] [B 12 H12 ] 10 Identify the symmetry elements that are present in benzene. Scheme for assigning point groups of molecules and molecular ions 11 Cn Point Groups n = Cn E PBrClF C1 H2O2 C2 As(C 6H5)3 C3 M(NH 2CH 2CO 2)4 C4 Cnh Point Groups The direction of the C n axis is take as vertical, so a symmetry σ plane perpendicular to it is a horizontal plane, h. HOCl NH 2F BBrClF The point group is called C s (C 1h ) B(OH) N2F2 3 C C2h 3h 12 Cnv Point Groups If a mirror plane contains the rotational axis, the group is called a C nv group. H2O SF 4 C2v C2v NF SF Cl 3 CHCl 3 5 C3v C3v C4v Dn and Dnh Point Groups Adding a C 2 axis perpendicular to a C n axis generates one of the dihedral groups. There must be n C2 axes perpendicular to Cn Angle between rings not 0 ° or 90° D2 D3 σ Adding a h to a D n group generates a D nh group. 2- C2H4 [PtCl 4] D2h D4h 13 BF 3 D3h Dnd Point Groups Adding a vertical mirror plane to a Dn group in such a fashion to bisect adjacent C 2 axes generates a Dnd group. D2d Ferrocene Fe(C 5H5)2 D5d 14 Sn groups n σ For odd n, (S n) = h n For even n, (S n) = E The S n for odd n is the same as the C nh . 1,3,5,7-tetrafluorocyclooctatetraene Absence of mirror planes distinguish S n groups from D nd groups. Linear Groups H-Cl O=O C∞v D∞h σ Has a h High Symmetry Molecules CCl 4 SF 6 C60 Td O h Ih 15 The letter ‘A’. tetrachloroplatinate(II) GeH 3F HOCl SF 6 H O SeH 3F B(OH) 3 B H O O AsBr 5 (trigonal bipyramid) H H BH 3 B trans rotamer of Si H 2 6 H H Ni(en) 3 Crown-shaped S 8 16 Characteristic symmetry elements of some important classes of point groups. 17 Character Tables •Character tables contain, in a highly symbolic form, information about how something of interest (an bond, an orbital, etc.) is affected by the operations of a given point group . •Each point group has a unique character table, which is organized into a matrix. •Column headings are the symmetry operations , which are grouped into classes . •Horizontal rows are called irreducible representations of the point group. •The main body consists of characters (numbers), and a section on the right side of the table provides information about vectors and atomic orbitals . σ σ C2v EC2 v(xz) v’ (yz) 2 2 2 A1 1 1 1 1 z x , y , z A2 1 1 -1 –1 Rz xy B1 1 -1 1 –1 x, R y xz B2 1 –1 –1 1 y, R x yz σ C3v E 2C 3 3 v 2 2 2 A1 1 1 1 z x + y , z A2 1 1 -1 Rz 2 2 E 2 -1 0 (x,y), (R x, R y) (x -y , xy)(xz, yz) •Symmetry elements possessed by the point group are in the top row •Left hand column gives a list of symmetry labels •Gives information about degeneracies (A and B indicate non- degenerate, E refers to doubly degenerate, T means triply degenerate) •Main part of table contains characters (numbers) to label the symmetry properties (of MO’s or modes of molecular vibrations) 18 To obtain from this total set the representations for vibration only, it is necessary to subtract the representations for the other two forms of motion: rotation and translation . z z O y z x z y H y H y x x x Cartesian displacement vectors for a water molecule Translational Modes Rotational Modes A mode in which A mode in which all atoms are atoms move to moving in the rotate (change same direction, the orientation equivalent to of) the molecule. moving the There are 3 molecule. rotational modes for nonlinear molecules, and 2 rotational modes for linear molecules. 19 Selection Rules: Infrared and Raman Spectroscopy Infrared energy is absorbed for certain changes in vibrational energy levels of a molecule. -for a vibration to be infrared active , there must be a change in the molecular dipole moment vector associated with the vibration. For a vibration mode to be Raman active , there must be a change in the net polarizability tensor -polarizability is the ease in which the electron cloud associated with the molecule is distorted For centrosymmetric molecules, the rule of mutual exclusion states that vibrations that are IR active are Raman inactive, and vice versa The transition from the vibrational ground state to the first excited state is the fundamental transition . The two bending modes require the same amount of energy and are therefore degenerate. 20 σ σ C2v EC2 v(xz) v’ (yz) 2 2 2 A1 1 1 1 1 z x , y , z A2 1 1 -1 –1 Rz xy B1 1 -1 1 –1 x, R y xz B2 1 –1 –1 1 y, R x yz If the symmetry label (e.g. A 1, B 1, E) of a normal mode of vibration is associated with x, y, or z in the character table, then the mode is IR active . If the symmetry label (e.g. A 1, B 1, E) of a normal mode of vibration is associated with a product term ( x2, xy ) in the character table, then the mode is Raman active . σ σ C2v EC2 v(xz) v’ (yz) 2 2 2 A1 1 1 1 1 z x , y , z A2 1 1 -1 –1 Rz xy B1 1 -1 1 –1 x, R y xz B2 1 –1 –1 1 y, R x yz 21 σΞ 3σΞ D 3h E 2C 3 3C 2 hh 2S 3 3 v v ′ 2 2 2 A1 1 1 1 1 1 1 x + y , z ′ A 2 1 1 -1 1 1 -1 R z ′ E 2 -1 0 2 -1 0 (x , y ) (x 2 – y 2, 2 xy ) ′′ A 1 1 1 1 -1 -1 -1 ′′ A 2 1 1 -1 -1 -1 1 z ′′ E 2 -1 0 -2 1 0 (R x , R y )(xy , yz ) 22 The vibrational modes of CH 4 (Td), only two of which are IR active.
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