Spectroscopy and Characterisation. Dr. P. Hunt Rm 167 (Chemistry) web-site:

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1 Spectroscop and Characterisation Dr. P. unt Rm 167 (Chemistr) web-site:

2 Resources Web copies course notes download slides model answers links to good web-sites Reading Recomended Texts Kieran Mollo, Group Theor for Chemists, 2004, arwood Publishing, Chichester. PW Atkins and RS Friedman, Molecular Quantum Mechanics, xford Universit Press, xford Secondar Text Alan Vincent, Molecular Smmetr and Group Theor, Second edition, 2001, John Wile & Sons Ltd, Chichester. elective reading background material that supports lectures ptional specialist material that explains difficult concepts in more detail if ou are interested in a wider perspective

3 Molecular Spectra light -> molecule -> spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules

4 Molecular Spectra light -> molecule -> spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules Ke relationship wavelength / wavenumber hc λ = E a E b f= c /λ (frequenc) ν=1/λ (wavenumber) ΔE=hc/λ =hf =hcν

5 light -> molecule -> spectra Molecular Spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules Ke relationship incident light is: hc λ = E a E b

6 light -> molecule -> spectra Molecular Spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules Ke relationship incident light is: hc λ = E a E b absorbed (rotational and vibrational spectra and UV-vis spectra) transmitted (objects colour) absorbed transmitted

7 light -> molecule -> spectra Molecular Spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules Ke relationship incident light is: absorbed (rotational and vibrational spectra and UV-vis spectra) transmitted (objects colour) reflected (reflectance spectra) hc λ = E a E b reflected

8 light -> molecule -> spectra Molecular Spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules Ke relationship incident light is: absorbed (rotational and vibrational spectra and UV-vis spectra) transmitted (objects colour) reflected (reflectance spectra) scattered (Raman spectra) hc λ = E a E b scattered

9 light -> molecule -> spectra Molecular Spectra light incident on a sample interacts with molecules various spectra are obtained which contain information about the molecules Ke relationship incident light is: hc λ = E a E b absorbed (rotational and vibrational spectra and UV-vis spectra) transmitted (objects colour) reflected (reflectance spectra) scattered (Raman spectra) emitted later (fluorescence spectra) eject an electron (photo-electron spectra) e- eject electron fluorescence

10 Molecular Spectra large range of energ cannot be spanned b single source or detector different interactions in different (reduced) energ ranges different scales of energ quantisation ν(cm -1 ) E(kJ mol -1 ) rotational levels (microwave) ν 10 cm -1 E 0.1 kj/mol vibrational levels (IR) ν 1000 cm -1 E 1 kj/mol electronic levels (UV-vis) ν 10, 000 cm -1 E 100 kj/mol Fig 2

11 This Course stud light interacting with molecules vibrational and electronic spectroscop certain relationships must hold: Selection Rules these are determined b smmetr! smmetr is important more generall chemistr, phsics, mathematics ver important in Quantum Mechanics matrix representation of smmetr operators perturbation theor smmetr is important in determining vibrations of a molecule selection rules for vibrational spectroscop (IR and Raman) nature of the state of a molecule electronic spectrum of a molecule electronic spectroscop (colour and UV-vis)

12 Example produced Pd(N3)2Cl2 but which isomer is present cis or trans? look at the IR spectrum Pd-Cl stretching mode 350cm -1 cis-isomer C2v smmetr and will have 2-modes trans-isomer D2h smmetr and will have 1-mode Cl Cl Pd cis- N 3 N 3 3 N Cl Cl Pd N 3 trans- IR Raman trans D2h b 3u a g cis C2v a 1, b 2 a 1, b 2 spectrum from Nakamoto Infrared and Raman Specta of Inorganic and Coordination Compounds, 5 th Edition (1997), John Wile & Sons, New York, PartB, p10, Fig III-5. Fig 3

13 utline Revision on Character Tables Matrix representation of Smmetr perators Matrix mechanics

14 Character Tables point group class of smmetr operations k=number of operations characters D 3h E 2C 3 3C 2 σ h 2S 3 3σ v smmetr labels A 1 ' A 2 ' E' R (T x, T ) x 2 + 2, 2 smmetr of quadratic functions ie das (x 2-2, x) A 1 " A 2 " T smmetr of cartesian axes E" (R x, R ) (, x) irreducible representation Fig 4

15 Character Tables point group class of smmetr operations k=number of operations characters D 3h E 2C 3 3C 2 σ h 2S 3 3σ v smmetr labels A 1 ' A 2 ' E' R (T x, T ) x 2 + 2, 2 smmetr of quadratic functions ie das (x 2-2, x) A 1 " A 2 " T smmetr of cartesian axes E" (R x, R ) (, x) irreducible representation

16 Character Tables point group class of smmetr operations k=number of operations characters D 3h E 2C 3 3C 2 σ h 2S 3 3σ v smmetr labels A 1 ' A 2 ' E' R (T x, T ) x 2 + 2, 2 smmetr of quadratic functions ie das (x 2-2, x) A 1 " Mulliken Notation extra notes added on m web-site if ou want to know what the letters and subscripts mean A 2 " E" irreducible representation T (R x, R ) (, x) smmetr of cartesian axes

17 Character Tables point group class of smmetr operations k=number of operations characters D 3h E 2C 3 3C 2 σ h 2S 3 3σ v smmetr labels A 1 ' A 2 ' E' R (T x, T ) x 2 + 2, 2 smmetr of quadratic functions ie das (x 2-2, x) A 1 " A 2 " T smmetr of cartesian axes E" (R x, R ) (, x) irreducible representation

18 Character Tables point group class of smmetr operations k=number of operations characters D 3h E 2C 3 3C 2 σ h 2S 3 3σ v smmetr labels A 1 ' A 2 ' E' R (T x, T ) x 2 + 2, 2 smmetr of quadratic functions ie das (x 2-2, x) A 1 " A 2 " E" T Translation smmetr of cartesian axes (R x, R ) (, x) irreducible representation

19 Character Tables point group class of smmetr operations k=number of operations characters D 3h E 2C 3 3C 2 σ h 2S 3 3σ v smmetr labels A 1 ' A 2 ' E' x 2 + 2, 2 Rotation smmetr of R quadratic functions ie das (T x, T ) (x 2-2, x) A 1 " A 2 " T smmetr of cartesian axes E" (R x, R ) (, x) irreducible representation

20 Character Tables point group smmetr labels D 3h E 2C 3 3C 2 σ h 2S 3 3σ v A 1 ' A 2 ' E' class of smmetr operations k=number of operations characters R (T x, T ) Quadratic functions x 2 + 2, 2 smmetr of quadratic functions ie das (x 2-2, x) A 1 " A 2 " T smmetr of cartesian axes E" (R x, R ) (, x) irreducible representation

21 orbital: rbitals: determine smmetr x smmetr elements of the molecule: C 2 () σ v () σ v (x) under a smmetr operation orbital maps onto itself => +1 or if inverted => -1 then compare to the character table to determine smmetr label hence this orbital b1 C 2v point group determine the orbitals smmetr: E C 2 σ v () σ v (x) C 2 σ σ v v Γ C 2v E C 2 σ v (x) σ v '() which matches the B 1 irreducible representation: C 2v A 1 E C 2 σ v (x) σ v '() h=4 A B 1 B x Fig 5

22 Translation: x C 2 () σ v () σ v (x) Use the same process for the translation and rotation of the whole molecule Important! C 2v E C 2 σ v (x) σ v '() Γ T B 2 b 2 Fig 6

23 Rotation: C 2 () C 2v E C 2 σ v (x) σ v '() Γ R A 2 b 2 σ v () Fig 7 looking down -axis σ v (x)

24 CoM motion omework to confirm for ourself: Use the same process for the translation and rotation of the whole molecule Important! x T x B 1 A 1 T x x B 2 B 1 R x R Fig 8

25 Binar Functions Use the same process for the translation and rotation of the whole molecule Important! C 2v E C 2 σ v (x) σ v '() Γ d bb 2 2 Fig 9

26 Smmetr perators smmetr operations smmetr elements smmetr operator operator C 2 acts on wavefunction mathematical representation of the action (phsical operation) operator acts on something (hence the brackets) molecule wavefunction banana vibration! allows us to write an equation for the smmetr operation C 2 C 2 ψ 2 = ψ 2 operator C 2 acts on molecule Eq. 1

27 Matrix Representation we can represent a smmetr operator as a matrix notation D(R) D stands for Darstellung=representation in German R stands for the operator determine D(R) b examining the effect of an operation on the quantit under consideration (the basis) ou have alread meet several matrix representations of smmetr operators! go through a number of examples now

28 Matrix Representation the rotation matrix from maths lectures D(θ) v(x,) v'(x',') x' ' = ' cos(θ) sin(θ) 0 sin(θ) cos(θ) x ' v'(x',') θ v(x,) x' x θ takes on specific values depending on the tpe of axis θ=0º and 180º for the C2 θ=0º, 120º, 240º for the C3 θ=0º, 90º, 180º, 270º for the C4 Quantised! D(C 2 ()) = -axis comes out of the page C 2 () operation: cos(π ) sin(π ) 0 sin(π ) cos(π ) 0 = Fig Eq. 3

29 Flight Dnamics Fig R(x) = 0 cos(θ) sin(θ) Roll 0 sin(θ) cos(θ) cos(θ) 0 sin(θ) R() = Pitch sin(θ) 0 cos(θ) R() = Yaw cos(θ) sin(θ) 0 sin(θ) cos(θ) Image from:

30 Matrix Representation eas to determine the matrix representation work out what happens to unit vectors placed on the center of an atom! Important! under E the x,, and vectors do not move E -vector -vector x-vector x' ' = ' x = 0 0 1!#" # $ D(E) 1x x = 0x x Fig 12 x

31 Matrix Representation under C2 vector is unchanged x and vectors are rotated and become negative C 2 () x x' ' = ' x = 0 0 1!## "## $ D(C 2 ) 1x x = 0x x Fig 13

32 In Class Activit determine the matrix representation for σv() σ v () x' ' = ' x =!## "## $ D(σ v () = x Fig 14

33 In Class Activit determine the matrix representation for σv() σ v () x x' ' = ' x = 0 0 1!#"# $ D(σ v () 1x x = 0x x x Fig 14

34 Matrix Representation each smmetr operation can be represented b a matrix eisenberg VERY powerful technique allows us to use matrix algebra and matrix mechanics on smmetr operators Werner eisenberg developed matrix mechanics as well as the eisenberg uncertaint principle Nobel prie in phsics in 1932 when he was 31 for the creation of quantum mechanics Fig 15 photo from the wikipedia web-site: en.wikipedia.org/wiki/werner_karl_eisenberg

35 Matrix Representation Dirac reformulate Schrödinger s equation in a matrix notation MUC more powerful form of the equation Paul Dirac noticed connections between eisenberg s matrices and Schrödinger s wave mechanics also introduced bra-ket notation and the delta function shared the Nobel Prie in phsics with Schrödinger in 1933 for the discover of new productive forms of atomic theor Fig 16 photo from the wikipedia web-site: en.wikipedia.org/wiki/paul_dirac Important! smmetr and Schrödinger equation now on equal footing: both in matrix formulation

36 The Basis previous basis: As/Ms or Cartesian vectors other basis sets are possible, determine the matrix: C 2 (b) C 3 () work out what happens to the basis item under each smmetr operation Important! C 2 (c) b c B a C 2 (a) Fig 17 a range of examples using D3 components of D3h point group for B3 B B(s) B B(p x,p ) the Boron atomic orbitals s and p individual atoms single bonding M b ψ=s a +s b +s c a c Fig 18

37 The Basis comparing the D(R) of D3 components of D3h point group extra notes on the web-site if ou cannot see how these matrices are generated D 3 E C 1 3 () C 1 3 () C 1 2 (a) C 1 2 (b) C 1 2 (c) B(s) B(p ) B(p x, p ) ( a, b, c ) (s a + s b + s c )

38 The Basis comparing the D(R) of D3 components of D3h point group single sa has a 1X1 matrix p stas alone and has a 1X1 matrix px and p are degenerate 2X2 matrix individual atoms 3X3 matrix single M 1X1 matrix extra notes on the web-site if ou cannot see how these matrices are generated D 3 E C 1 3 () C 1 3 () C 1 2 (a) C 1 2 (b) C 1 2 (c) B(s) B(p ) B(p x, p ) ( a, b, c ) (s a + s b + s c )

39 The Basis comparing the D(R) of D3 components of D3h point group the matrix changes with the basis the matrix has the same dimension as the number of basis elements D 3 E C 1 3 () C 1 3 () C 1 2 (a) C 1 2 (b) C 1 2 (c) B(s) B(p ) B(p x, p ) ( a, b, c ) (s a + s b + s c )

40 Combining Smmetr perations example: C 2 ()σ v (x){ 2 } =σ v (){ 2 } show using diagrams for the RS: σ v () x σ v x Fig 20

41 Combining Smmetr perations example: C 2 ()σ v (x){ 2 } =σ v (){ 2 } show using diagrams for the LS: operate FIRST with σv TEN C2 for operators alwas work from the inside out not all operators commute ou cannot change the order Important! σ v (x) C 2 () x σ v x x C 2 x Fig 21

42 Combining Smmetr perations final structures are the same RS=LS C 2 ()σ v (x){ 2 } =σ v (){ 2 } x x

43 Using Matrices example: C 2 ()σ v (x){ 2 } =σ v (){ 2 } show using diagrams for the LS: x' ' ' = !# #" ### $!##" ## $ D(C 2 ) D(σ v (x) x x' ' ' = !# #" ### $ D(C 2 ) x = x Eqn 4

44 Using Matrices example: C 2 ()σ v (x){ 2 } =σ v (){ 2 } show using diagrams for the RS: x' ' ' = !##"## $ D(σ v () x = x Eqn 5 final vectors are the same!

45 Combining Smmetr perations final vectors are the same RS=LS C 2 ()σ v (x){ 2 } =σ v (){ 2 } x' ' ' !### "### $!##"## $ = D(C 2 ) D(σ υ (x)) x = x x' ' '!##"## $ = D(σ υ ()) x = x x and match the results found using diagrams

46 Combining Smmetr perations using matrix multiplication: C 2 ()σ v (x){ 2 } =σ v (){ 2 } !##" ## $ D(C 2 )! " ## ## $ D(σ v (x)) = !##" ## $ D(σ v ()) algebra on operations parallels algebra on matrices! Important!

47 In Class Acitivit: show using diagrams and then matrices C 2 ()σ v (){ 2 } =σ v (x){ 2 } !## "## $ D(E)! " ## ## $ D(C 2 ) ## ## $ D(σ v ())! " ## ## $ D(σ v (x))! "

48 Ke Points be able to list the processes light incident on a sample undergoes and identif the related spectroscopic techniques be able to draw a schematic representation of the energ levels associated with various transitions, and identif the relevant wavelengths of light be able to define the components of a character table be able to draw clear diagrams showing smmetr operations be able to generate the matrix representation of a smmetr operator for a given basis be able to discuss how matrix representations change for different basis be able to use diagrams and matrix mechanics to form smmetr operation and smmetr matrix products be able to discuss the important relationship between phsical operations and mathematical operators

49 Resources Web copies course notes download slides model answers links to good web-sites Reading Recomended Texts Kieran Mollo, Group Theor for Chemists, 2004, arwood Publishing, Chichester. PW Atkins and RS Friedman, Molecular Quantum Mechanics, xford Universit Press, xford Secondar Text Alan Vincent, Molecular Smmetr and Group Theor, Second edition, 2001, John Wile & Sons Ltd, Chichester. elective reading background material that supports lectures ptional specialist material that explains difficult concepts in more detail if ou are interested in a wider perspective