Vibrations. Matti Hotokka Department of Physical Chemistry Åbo Akademi University

Transcription

1 Vibratins Matti Htkka Department f Physical Chemistry Åb Akademi University

2 Harmnic scillatr V(r) Schrödinger s equatin Define q = r - r e V ( q) = 1 2 fq 2 α = f hν r e r 2 2 h d + V ( q) Ψ( q) = EΨ( q) 2 2 8π m dq 1 E = hν ( v + ) Slutin: Quantum number v = 0, 1, 2,... v e 2 1/ 2α q Ψ v ( q) = N v H v ( q) e 2 2 ν e f = 1 2π µ

3 Harmnic scillatr Zer-pint energy E = ν h e Same distance between energy levels E( v + 1 v) = hν e Istpe effect: E.g. e(h 2 ) = 4395 cm -1 e(d 2 ) = 3118 cm -1

4 Istpe effect 100 Example: Water H 2 O T % Wavenumber (1/cm) 100 D 2 O T % Wavenumber (1/cm)

5 Anharmnic scillatr V(r) ( T + V ) Ψ = EΨ Realistic frm f ptential energy curve. Many pssible functins: Mst ppular are plynmial and the Mrse ptential. ( αr 1 ) V ( r) = D e e If V(r) is Mrse ptential the Schrödinger equatin can be slved analytically. 2 Slutins: quantum number v = 0,1,2, E = hν ( v + ) hν x ( v + ) v e 2 e e 2 2 r

6 Anharmnic scillatr Distance between energy levels varies E( v + 1 v) = hν 2hν x ( v + 1) e e e Almst the same transitin energy fr 1 0but large discrepances fr higher levels Harmnic apprximatin is quite gd fr the lwest transitin and much simpler s harmnic apprximatn is used.

7 Degrees f freedm Ttally 3N degrees f freedm. Translatins f the whle mlecule require 3 degrees f freedm. Rtatins require further 3 degrees f freedm Internal mtins, 3N-6 degrees f freedm, 3N- 5 in linear mlecules

8 Nrmal vibratins Independent f each ther Cmmn starting level, all unexcited

9 Nrmal vibratins Independent f each ther Cmmn starting level, all unexcited Symmetry adapted

10 Nrmal vibratins Independentfeachther Symmetryadapted Invlvethewhlemlecule

11 Zer-pint energy Fr quantum chemists: the zer-pint energy may be quite large E.g., CO 2 : nrmal frequencies 1340, 667 and 2349 cm-1. Each cntributes t the zer-pint energy by ½ v ~ e. Thus we btain = 2177 cm -1 Spectrscpy measures distances between energy levels, nt abslute energies, s the zer-pint energy is irrelevant.

12 Nrmal crdinate analysis Given the stiffnesses (frce cnstants) f individual bnds (valence crdinates, really) and pssible interrelatins,calculate hw the atms mve and what are the harmnic energy levels.

13 Nrmal crdinate analysis Mdern methd Use mlecular mechanics calculatins Interactins Bnd distances Bnd angles Trsin angles 1-4 interactins Crss-terms Out cme harmnic (even anharmnic) energy levels and shapes f the mtins

14 Nrmal crdinates Quantum chemical methds Use gd basis sets (TZP, 6-311G(d,p) r such) There is a systematic errr, apply a crrectin factr t vibratinal energies Only harmnic frequencies are available (anharmnic are very expensive t calculate)

15 Nrmal crdinates The traditinal methd Wilsn s GF matrix methd Requires a gd frce cnstant matrix Requires carefully defined valence crdinates

16 Frce cnstant matrix Example: Kim Palmö, HU, 1987 H1 H4 H2 C C H3 CC CH1 CH2 CH3 CH4 H1CH2 H3CH4 CC CH CH CH CH H1CH H3CH CH1, CH3, CH1,2 CH3,4

17 Nrmal crdinates Wilsn s methd Out cme Exact mtins Crrect symmetries Vibratin energies (The selectin rules were analyzed separately) A g A g A g A u R 1623 R 3019 R 1342 B 1g B 1g B 1u B 2g R 3272 R 1050 IR 949 R 943 B 2u B 2u B 3u B 3u IR 3105 IR 995 IR 1443 IR 2989

18 Grup vibratins Basically all vibratins invlve all atms in the mlecule Hwever, sme vibratins are mstly cncentrated in a certain functinal grup f the mlecule => grup frequencies

19 Nrmal vibratins Full stp. Here cmes an animatin using a separate prgram.

20 Symmetry f the vibratins Cpy frm character tables the characters f the xyz translatins, xyz. Fr each class f symmetry peratins, cunt the statinary atms, N stat. Multiply the lines xyz and N stat => tt. Subtract the translatins f the whle mlecule, Subtract the rtatins f the whle mlecule, Reduce the resulting representatin with characters rt. xyz. vib.

21 Symmetry f the vibratins Example: equilaterial triangle D 3h E 2C 3 h 2S 3 3C 2 3 v A 1 ' A 2 ' R z E (x,y) A 1 " A 2 " z E (R x, R y ) rt xyz N stat tt xyz rt vib a A1' = 1/12( )=1 a A2' = 1/12( )=0 a E' = 1/12( )=1 a A1" = 1/12( )=0 a A2" = 1/12( )=0 a E = 1/12( )=0 vib = A 1 '+ E

22 Vibratinal states N vibratin is excited: ttally symmetrical One vibratin excited: the vibratinal state has the symmetry f the excited state Generally: Let the vibratins have the symmetries 1, 2,... M. Let vibratin n be excited t state v n. The the ttal vibratinal state has the symmetry v1 v2 Γ = ( Γ ) ( Γ ) ( Γ ) v M state 1 2 M

23 Vibratinal states Example: Equilateral triangle Vibratinal mtins belng t A 1 'and E Grund state, nne excited: state = A 1 ' Fundamental excitatin, A 1 'excited 1 0: state = A 1 ' Fundamental excitatin, E'excited 1 0: state = E' Overtne, E'excited 2 0: state = E' E = A 1 '+[A 2 ']+E = A 1 '+E (N.B. Special rules fr pwers f E!) Cmbinatin band, bth A 1 'and E'excited 1 A 1 ' E'= E 0: state =

24 Pssible transitins Carbn dixide Fundamental: 2349 cm -1 Overtne: 2001 cm -1 Cmbinatin band: 2007 cm -1 v=0 ν 1 ν 2 ν 3 R1340 cm -1 IR667 cm -1 IR2349 cm -1

25 !!! Selectin rules Vibratinal transitin take place between vibratinal states. Usually fundamental transitin: initial state is the ttally symmetrical grund state and final state has the same symmetry as the vibratin Tw measuring techniques Infrared spectrscpy Absrptin; use the diple peratr Raman spectrscpy Use the plarizability peratr

26 $ $ " % ' & & % % & & % & & % % & & % Selectin rules Example: the equilateral triangle Evaluate the symmetry f integral D 3h E 2C 3 h 2S 3 3C 2 3 v A 1 ' X 2 +y 2, z 2 Ψf O Ψi Γf Γ Γ O i A 2 ' R z E (x,y) (X 2 -y 2, xy) Operatr O is fr infrared A 1 " spectrscpy and fr Raman A 2 " z spectrscpy. E (R x, R y ) (xz, yz) If there is a ttally symmetrical cmpnent in the prduct, the D 3h A 1 A 2 E transitin is allwed. A 1 A 1 A 2 E Infrared A 2 A 1 E Vibratin A 1 ': Grund state A 1 ', excited state A 1 ' E A 1 +[A 2 ]+E x: A 1 ' E A 1 '=E ; nt allwed y: the same z: A 1 ' A 2 " A 1 '=E ; nt allwed Vibratin E': Grund state A 1 ', excited state E x: A 1 ' E E =A 1 '+A 2 '+E ; ALLOWED! y: the same z: A 1 ' A 2 " E'=E ; nt allwed #

27 ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ( Selectin rules Example: the equilateral triangle Raman Vibratin A 1 ': Grund state A 1 ', excited state A 1 ' x2+y2: A 1 ' A 1 ' A 1 '=A 1 '; ALLOWED! z2: the same x2-y2: A 1 ' E A 1 '=E ; nt allwed xy: the same xz: A 1 ' E A 1 '=E ; nt allwed yz: the same Vibratin E': Grund state A 1 ', excited state E x2+y2: E A 1 ' A 1 '=E '; nt allwed z2: the same x2-y2: E E A 1 '=A 1 '+A 2 '+E ; ALLOWED! xy: the same xz: E E A 1 '=A 1 "+A 2 "+E ; nt allwed yz: the same SUMMARY: Excitatin f vibratin A 1 ' - nt allwed in infrared spectrscpy - allwed in Raman spectrscpy Excitatin f vibratin E - allwed in infrared spectrscpy - allwed in Raman spectrscpy

28 Selectin rules Example: Water

29 -, * + *. Fermi resnance Example: Carbn dixide v=0 ν 1 ν 2 ν 3 R1340 cm -1 IR667 cm -1 IR2349 cm -1 First vertne 1334 cm -1 has tw cmpnents, u u= g+ + g. The first interacts with the 1 line because the symmetry is the same.

30 Fermi resnance Example: Carbn dixide Raman spectra. N line at 1340 cm -1. Bth lines have changed psitin. Bth lines have apprx the same intensity s the vertne has brrwed intensity frm the allwed fundamental. G. W. Bndarenk, J. Appl. Spectrsc., 45 (1986) 1573.