Symmetry. Using Symmetry in 323

Transcription

1 Symmetry Powerful mathematical tool for understanding structures and properties Use symmetry to help us with: Detecting optical activity and dipole moments Forming MO s Predicting and understanding spectroscopy of inorganic compounds Infrared, Raman and UV-visible Using Symmetry in 323 Symmetry tools and language Sec Flow chart p. 22 Vibrational spectroscopy Sec 4.8

2 Symmetry and the Exam Recognize symmetry elements Identify the important elements present in a molecule Assign the point group of an object or molecule Read a character table Calculate vibrational normal modes Symmetry Elements A symmetry element is present if the operation is performed and the object is indistinguishable from its original state Element C n σ i S n E Name n-fold rotation mirror plane Center of inversion Improper rotation axis Identity Operation rotate by 360 /n Reflection through a plane Inversion through the center of the object Rotation as C n followed by reflection in perpendicular mirror plane Do nothing 2

3 Rotation Axis: C n Rotation of the object around an axis by 360/n degrees Higher symmetry object may have will have multiple C n Identify principal rotation axis as the one with the highest n Center of Inversion: I Inverts all atoms through the centre of the object 3

4 Inversion vs Rotation (C 2 ) Mirror Planes: σ Reflection of object through a mirror plane Objects in the plane are reflected onto themselves, objects on either side of the plane are reflected to the other side Three types σ h : Horizontal, perpendicular to principal axis σ v : Vertical, parallel to principal axis σ d : Dihedral, same plane as σ v related by half a rotation of the principal axis 4

5 Dihedral vs. Vertical Typically σ d and σ v are related by rotation of 80/n Labelling: Exception: when n=2, label is σ v not σ d Rule of Thumb: dihedral planes pass through fewer atoms (i.e. is dihedral to the angle of the bonds) Improper Rotation Axis: S n Rotate 360/n followed by reflection in mirror plane perpendicular to axis of rotation All planar molecules have an S n 5

6 Special Cases: S and S 2 S = σ h and S 2 = i Comparing Symmetry Compare NH 3 and BF 3 6

7 Point Groups collections of symmetry elements are summarized into Point Groups these are groups as strictly defined by mathematical group theory short form method for identifying all of the symmetry elements present in a molecule T d E, 8C 3, 3C 2, 6S 4, 6σ d 7

8 Identifying Molecular Symmetry Requires knowing molecular geometry First draw a Lewis structure Use VSEPR to predict molecular geometry Use VSEPR geometry to identify symmetry elements present Classify molecule according to its point group Lewis Structures. Count valence electrons available, include net charges. 2. Write skeleton structure, drawing bonds between atoms using up two valence electrons for each bond. 3. Distribute remaining electrons to most electronegative species first to fill electron shells. 4. Satisfy unfilled octets where possible by drawing multiple bonds 8

9 Resonance Structure and Formal Charge Resonance allows for non-integer bond order and delocalized electron distribution Formal Charge Predicts which of multiple possible structures are more favourable FC = group # bonded electrons unshared 2 electrons VSEPR Relies on the electron distribution around the central atom Bonded pairs Typically ignore bond order (,2 etc) Ignore what atom it is bound to Occupy less volume than lone pair Lone pairs (on the central atom) Bulkier than bonds Have largest impact on molecular geometry 9

10 0

11 Examples SO 4 2- PCl 5 Character Tables Character tables are tell-all manuals of symmetry, tabulated in your text Using group theory, lists of behaviour under the symmetry elements in a point group are tabulated, these are called irreducible representations and they are said to span the group Therefore, a portion of a molecule can be described by some linear combination of these irreducible representations.

12 Reading the Character Tables C 3v E 2C 3 3σ v A x 2 +y 2 +z 2 z A 2 - R z E 2-0 (x,y), (R x,r y ) (x 2 -y 2,xy) Character values: means no change - means change of sign 2, 0 sum of multicomponent behaviour Mulliken Labels A,B,E,T indicating degeneracy A vs B : symmetric or antisymmetric wrt highest order rotation axis,2 : symmetric or antisymmetric wrt C2 axis or σv g,u : symmetric or antisymmetric wrt i, : symmetric or antisymmetric wrt σv 2

13 D 3h Character Table D 3h E 2C 3 3C 2 σ h 2S 3 3σ v A X 2 +y 2, z 2 A R z E (x,y) (x 2 -y 2, xy) A A Z E (R x, R y ) (zx, yz) Vibrational Spectroscopy IR and Raman spectroscopy demonstrate changes in the vibrational energy state of a molecule Vibrations: atomic displacements within a molecule that leave the molecule centre of mass and orientation unchanged Larger or smaller in energy than electronic transitions? Larger or smaller in energy than rotational transitions? 3

14 Normal Modes Observable vibration is seemingly chaotic But vibration of a molecule can be broken down into a sum of vibrational modes known as normal modes normal modes must be consistent with the symmetry of the molecule i.e. they transform as one of the species in the character table of the point group Normal Modes vs Motion Consider some atomic displacements of a linear molecule: translation rotation vibration 4

15 How Many Normal Modes Mathematically: each atomic displacement is described by 3 vectors (x,y,z) Therefore there are 3 x N possible displacement vectors for the whole molecule However translation and rotation are not vibrations and must be subtracted from the sum of atomic displacements Anything left over is a vibrational normal mode Translation and Rotation Translation: all atomic displacements point the same way (centre of mass of molecule moves) 3 possible (x,y,z) Rotation: orientation of the molecule changes about some axis 3 possible around each axis Unless molecule is linear (2 in this case) 5

16 Normal Modes normal modes are motions which do not stimulate other normal modes (i.e. they are orthogonal to each other) typically represented using internal coordinates, not Cartesian coordinates 3N 6 possible normal modes H 2 O N = 3 so 3x3-6 normal modes 3 normal modes expected Normal Modes of BF 3 How many normal modes should BF 3 Have? BF 3 has 4 atoms Apply 3N-6 formula: 3x4 6 = 6 normal modes 6

17 Symmetry of Normal Modes As with any other portion of a molecule, use character tables of the point group of the whole molecule to assign symmetry species of a normal mode i.e. the normal mode transforms as one of the irreducible representations in the character table Symmetry of the Normal Modes of Water H 2 O has C 2v symmetry C 2v E C 2 σ v σ v A z X2, y2, z2 A R z xy B - - x, R y zx B y, R x yz 7

18 Example: BF 3 Assign the symmetry of these normal modes of BF 3 IR and Raman Activity Can assign the symmetry but don t know if they actually show up (i.e. the activity) Raman and IR have different activity rules IR: motion must change the dipole moment of the molecule Raman: motion must alter the polarizability of the molecule From the character tables: Use the basis functions (right most column) A mode is IR active if it transforms as x,y or z A mode is Raman active if it transforms as 2 nd order basis 8

19 Activity of BF 3 Normal Modes a Use the basis functions in the character table to identify activity a 2 Vibrations & Symmetry II Symmetry can also be used with atomic displacements to predict symmetry and number of modes from scratch Doesn t build normal modes for us Just tells us the number and symmetry of each mode Uses method of unshifted atoms and decomposition formula 9

20 Unshifted Atoms Method. Determine degrees of freedom (3xN) 2. Set up cartesian coordinate system on all atoms 3. Assign molecule s point group 4. Find reducible representation of symmetry behaviour of the axis on each atom. If atom moves all axes contribute 0 2. If axis is unaffected, it contributes +, if reversed, otherwise 0 5. Factor into irreducible representation (using decomposition formula) 6. Remove Translation and Rotational Modes 7. Assign IR or Raman Activity to remaining modes Decomposition Formula ai = gχi ( R) χt ( R) h R a i :number of times the irreducible representation χ i appears h :number of symmetry operations in the point group (order of the group) g :number of symmetry elements in the class (the number in front of the symmetry element in the point group) χ I (R) : character of the irreducible representation for the R th symmetry element χ T (R) : character of the reducible representation (the number of shifted axes from the atoms) for the R th symmetry element 20

21 An Example: F 2 O Steps:. N = 3 therefore 9 modes total including rotation and translation 2. Axes as shown: 3. Point group: C 2v 4. Calculate χ T (R) Calculate χ T (R) C 2v E C 2 σ v σ v A z X2, y2, z2 A R z xy B - - x, R y zx B y, R x yz χ T (R) = 9-3 2

22 C 2v B 2 χ T Decomposition Now for each symmetry element, do decomposition formula E C 2 σ v ai = gχi ( R) χ h A σ v A B R ( R) a A =¼ (xx9 + xx - + xx3 + xx) a A = ¼(2) =3 Therefore 3a modes Repeat for each symmetry element T Translation and Rotation Decomposed reducible representation (χ T ) is χ T = 3a +a 2 +3b +2b 2 Need to remove translation and rotation Basis functions show us how: Translation = a + b + b 2 Rotation = a 2 + b + b 2 So left over is vibration χ vib : χ vib = 2a +b 22

23 Basis Functions Translation: x, y,z Rotations: R x, R y, R z C 2v E C 2 σ v σ v A z x2, y2, z2 A R z xy B - - x, R y zx B y, R x yz Raman and IR Activity Basis functions can also tell us activity (IR or Raman) IR activity is indicated by first order functions (x,y,z) Raman activity is indicated by second order functions (x 2, y 2, z 2, xy, xz, yz) So: Raman: a b χ vib = 2a +b IR: a b 23

24 A Note from Reality We have only calculated fundamental modes Might see more or fewer bands Ignores overtones, combinations and crystal splitting Spectral windows Sensitivity issues Overlap (ability to resolve two bands of similar energy) Symmetry doesn t say anything about intensity! Vibrational Spectroscopy Infrared and Raman spectroscopy probe the relative motions of atoms within molecules, i.e. vibrational spectroscopy these molecular motions are often indicative of the the properties that materials researchers are interested in VS is one technique which fills the gap between elemental analysis and mechanical testing 24

25 IR Absorbance: What is Happening? Molecular Schematic Vibrational Energy Level Diagram Infrared Spectroscopy Almost all IR experiments share these processes:. Expose sample to IR light from glowbar source 2. Sample absorbs some but not all IR, remainder is transmitted or reflected 3. Detect remaining IR 4. Determine amount of absorbed IR Note: Experiments generally require incomplete absorbance (i.e. the detector needs to see some response so it knows how much was absorbed), therefore strong absorbers and scattering surfaces are difficult to analyze by most IR techniques. 25

26 IR: What does it look like? IR spectra of a fluorinated resin draw-down over time no catalyst, xylene solvent solvent evaporation is observed absence of catalyst almost stops crosslinking as indicated by residence time of the isocyanate peak (2050 cm - ) Raman: What is Happening? B A 2 3. Laser (A) shines on the sample. 2. Some of the molecules are excited. 3. The molecules relax but might return to a different vibrational level than in step. Relaxation emits light (B), the wavelength of light emitted differs from that of the laser light by the difference of the vibrational energy levels. 26

27 Raman Units Raman: What does it look like? Raman investigation of TPOZ modified with increasing amounts of a :5 :4 :3 :2 : carboxylic acid ratio of TPOZ:Acetic Acid as shown changes in molecular structure indicated by new Raman bands appearing Wavenumber (cm - ) Infrared versus Raman Infrared amount of IR absorbed depends on the strength of the dipole of the atoms involved in the motion strong dipole molecules such as H 2 O and CO 2 are very strong absorbers, i.e. problems in open air sensitive, large number of sampling techniques available to handle different types of samples Raman does not depend on the dipole, can be used in open air emission based so it requires very sensitive detectors less sensitive than IR, few techniques, multiple excitation sources available for handling different materials 27

28 Fourier Transform Spectroscopy an interference pattern can be transformed (decoded) into the different frequencies that make up the pattern similar to hearing a musical chord, the sound you hear is the sum of the individual notes played together Process:.measure all frequencies at once, record detector signal as a function of time 2.Fourier transform into sum of discrete frequencies 3.display results intensity versus frequency Symmetry and Vibrational Spectroscopy Symmetry Symmetry Elements Point Groups Character Tables Vibrational Spectroscopy Symmetry of Normal Modes Reducible Representations IR and Raman Spectroscopy 28