Vibrational Spectroscopy 1

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Curtis Banks
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1 Applied Spectroscopy Vibratioal Spectroscopy Recommeded Readig: Bawell ad McCash Chapter 3 Atkis Physical Chemistry Chapter 6

2 Itroductio What is it? Vibratioal spectroscopy detects trasitios betwee the quatised vibratioal eergy levels associated with bod stretchig ad/or bod agle bedig i molecules. How do we do it? Trasitios are observed by measurig the amout of ifrared radiatio that is absorbed or emitted by vibratig molecules i solid, liquid or gas phases. Why do we do it? A kowledge of the vibratioal level spacigs gives us the value of the stretchig (or bedig) force costats which characterise the stifess of a bod, allows us to estimate the bod dissociatio eergy, ad gives us a meas of idetifyig characteristic fuctioal groups of atoms withi large molecules.

3 Vibratioal Spectroscopy Chemical bods are ot rigid, the atoms i a molecule vibrate about a equilibrium positio. The force required to stretch or compress the bod legth to r is proportioal to (r-r eq ) The eergy required to stretch the bod is E k r eq ( ) r kx where x (r-r eq ) m m r eq k force costat of bod, uits N.m - Just like a sprig which obeys Hooke s Law. r

4 Potetial Eergy Curve Coectio betwee shape of potetial eergy curve ad stregth of bod k. dv dx ( ) ( ) V + x + x +... V x d V V dx ( ) x x kx d V dx Potetial eergy curve is approximately parabolic ad the force costat k is related to the potetial by d V k dx d V Big è steep potetial è large k dx Small d V è shallow potetial è small k dx CO, k 9 Nm -, è strog. HCl, k 56 Nm -, è weak.

5 Schrodiger Equatio for Simple Harmoic Oscillator Quatum Approach µ ψ x ( x) + kx ψ ( x) Eψ( x) Where μ is the effective (reduced) mass of the molecule mm µ m + m From Quatum II otes we kow that the wavefuctios are give by ψ N y ( y) e where H (y) are the Hermite Polyomials, ad The eergy levels are quatised ad are give by E H 4 x y h with α α + ω ω 4π where is the vibratioal quatum umber,,,, 3, k μ mk

6 Hermite Polyomials ad SHO wavefuctios

7 Wavefuctios First four wavefuctios ad correspodig probability distributios for the SHO potetial.

8 Selectio Rules Dipole momet of a vibratig polar diatomic molecule which is fixed ad aliged i space

9 Selectio Rules Molecules may iteract with a electromagetic field through their electric dipole momets. If they possess a oscillatig dipole momet the they ca absorb or emit photos. A oscillatio is ifra-red active if it chages i magitude or directio whe the atoms are displaced relative to each other Selectio Rules:. Dipole momet must chage as molecule oscillates.. Δ ± + è absorptio of photo Examples: - è emissio of photo HCl, HI, H O, O X, H X,... δ _ δ + Time Dipole momet

10 Justificatio for Selectio Rules I φf µ ˆ φ Selectio Rules arise from the trasitio dipole momet i Cosider a oe dimesioal oscillator (e.g. diatomic molecule). The dipole momet arises from two partial charges ±δq separated by a distace R R e + x. The µ ˆ Rˆ δq Reδq + xˆ δq µ ˆ + xˆ δq Where μ is the dipole momet operator whe the atoms are at their equilibrium separatio. The, assumig that the iitial ad fial states are differet (i f), φ f µ ˆ φi μ φf φi + δq φf xˆ φi First term is zero because the oscillator states are orthogoal. So the trasitio momet is φ f µ ˆ φi δq φf xˆ φi ad because δq dμ dx we ca write the trasitio momet more geerally as : φ f µ ˆ φ i φ f xˆ φ i dµ dx the right had side is zero uless there is a varyig dipole momet. So there is o absorptio uless the molecule has a chagig dipole momet.

11 Justificatio for Selectio Rules II The specific selectio rule is determied by cosiderig the value of the matrix elemet φ f xˆ φ i We write out the matrix elemet explicitly i terms of the Hermite polyomials ad evaluate the itegral φ f xˆ φ i N H f f xˆn H exp i i ( ) ( ) y dy α N H ŷn H exp y To evaluate the itegral we use the recursio relatio f f i i dy This turs the matrix elemet to yh H + H+ H f H i H f H i α N N f i ( ) ( ) y dy + H H exp y H H exp f i f i+ i dy

12 Now usig the orthogoality coditio H f H i+ exp ( ) y dy π i! i if if f f i i We see that the first itegral is zero uless f i - ad the secod itegral is zero uless f i +. So the trasitio dipole momet is zero uless Δ ±. A Third Selectio Rule At this poit we ote that there is o agular mometum associated with the radial vibratioal motio of the molecule. Sice agular mometum must be coserved whe a photo is emitted or absorbed this implies that all vibratioal trasitios must be accompaied by a chage i the rotatioal quatum umber J. So i reality all vibratioal spectra are due to vibratioal-rotatioal trasitios i the molecule with a simultaeous chage i the J quatum umber. We will come back to this poit later!!!

13 E Eergy Levels i Harmoic Oscillator Vibratioal eergy is quatised + ω o ω o Vibratioal terms of a molecule G () eergies of vibratioal states expressed i waveumbers. E G hc where k μ ω hc ( ) o + + νo ν k o πc µ Uits: cm- E ω o ( ) G νo Zero poit eergy Example: H 35 Cl has a force costat of 59 cm -. Fid its oscillatio frequecy ν µ amu.6 kg + 35 ν o ω π π 9. 3 Hz 56 N. m.6 7kg Ad therefore c 3 8 m.s λ ν 3 9. sec m 3.3µ m ν 99 cm We ca excite the H 35 Cl vibratio with radiatio of this wavelegth è ifra-red

Vibratioal Spectrum (Diatomic) From the selectio rules we see that the waveumbers of the allowed trasitios are give by 3 G( + ) G( ) + + ν ν All allowed trasitios occur at the same frequecy υ oly oe

14 Vibratioal Spectrum (Diatomic) From the selectio rules we see that the waveumbers of the allowed trasitios are give by 3 G( + ) G( ) + + ν ν All allowed trasitios occur at the same frequecy υ oly oe lie i the spectrum. Positio of lie depeds o ν ad therefore k ad μ. At room temperature kt cm -. For most molecules ν cm > therefore at room temp oly the lowest vibratioal state will be occupied. è domiat spectral trasitio will be è. Fudametal trasitio ν G(4) G(3) G() G() G() cm -

15 Populatio of Eergy Levels I diatomic molecules the vibratioal trasitios typically have waveumbers i the rage 5 to cm - (.5 to.5 ev). The populatios of the vibratioal eergy levels are give by the Boltzma distributio. N N j i ΔE exp kt (Note that vibratioal levels are o-degeerate so there is o degeeracy factor). At room temperature (3), kt /4 ev è kt /hc cm - this is less tha the typical separatio of the eergy levels So at room temperature almost all of the molecules will be i the groud ( ) state ad the predomiat trasitio is adsorptio from the to state.

Example HCl Example: H 35 Cl has a fudametal vibratio at ν 99cm fid the relative populatio of the vibratioal state at room temperature (3K) ad at (5K) At room temperature kt/hc cm -.

16 Example HCl Example: H 35 Cl has a fudametal vibratio at ν 99cm fid the relative populatio of the vibratioal state at room temperature (3K) ad at (5K) At room temperature kt/hc cm -. ν At 5K kt/hc 3475 cm -. kt From Boltzma distributio N N T 3K ΔE hc exp exp ν kt kt 99cm exp 3. 7 cm At 3K oly 3 i millio molecules are i the vib. state the rest are i the groud ( ) state. N N T 5 K ΔE exp kt 99cm exp 3475cm hc exp ν kt.43 At 5 K about 43 % of the molecules are i the state.

The Aharmoic Oscillator Real molecules do ot behave like harmoic oscillators. The Harmoic Oscillator does ot dissociate; it ca have but (r-r eq ), does ot make physical sese.

17 The Aharmoic Oscillator Real molecules do ot behave like harmoic oscillators. The Harmoic Oscillator does ot dissociate; it ca have but (r-r eq ), does ot make physical sese. I real molecules the vibratioal eergy levels get closer together as E icreases. I real molecules the harmoic oscillator approximatio breaks dow. Must cosider additioal terms i the Potetial Eergy. V(x) d V x dx + 3 d V x 6 3 dx 3 + Iclude Aharmoic terms i potetial.

18 The Morse Potetial Use a empirical fuctio that is a good represetatio of the true potetial curve. The Morse potetial is oe such fuctio. Where V a r hcdeq e a µω hcd eq ( r ) 8π eq µν hcd eq D eq is the depth of the Morse potetial Note that the Dissociatio Eergy D is give by ω hcν D Deq Deq Uits of D ad D eq are eergy, J or ev Note eergy levels get closer together as icreases

19 The Morse Potetial A way of seeig the physics behid the Morse potetial is to expad it a( rr ) a( rr ) a( r r ) V hc + eq eq eq Deq e Deq e e Short rage repulsio term a( r ).5 e r eq log rage attractio term asymptote.5 D eq e a r ( ) r eq

20 Usig the fact that e x + The Morse Potetial ( x)! x + x 3 x 6 4 x expadig the expoetial term i the Morse potetial ad collectig terms gives Vr ( ) D ar r ar r ar r hc eq eq eq eq For the Harmoic Oscillator potetial V(r) /k(r-r eq ). So we ote that for small values of r-r eq, the cubic ad higher terms will be small, the first term will domiate ad we retur to the harmoic oscillator with Vr ( ) D ar r hc eq eq Put rhs ½ k Morse x kmorse Deqa Morse potetial Harmoic Oscillator potetial with cubic ad higher aharmoic terms added to make the shape more realistic.

21 The Morse Potetial We ca solve the Schrodiger equatio with the Morse potetial to fid the allowed eergy levels (eigevalues). Difficult but ca be doe aalytically. Eergy levels give by o e o E ω χ ω + + Aharmoic correctio μ k ω o with The vibratioal terms are give by ( ) hc E G e e χ ν ν χ ν o o o χ e the Aharmoicity costat, depeds o molecule, small correctio factor. a eq e 4D µω ν χ The umber of vibratioal states is ow fiite,,,,. max. Above max the molecule dissociates (breaks apart)

Selectio Rules for Aharmoic Oscillator Selectio Rules:. Dipole momet must chage as molecule oscillates.. Δ ±, ±, ±3, all trasitios allowed but oly the ± ± trasitio is strog. 3.

22 Selectio Rules for Aharmoic Oscillator Selectio Rules:. Dipole momet must chage as molecule oscillates.. Δ ±, ±, ±3, all trasitios allowed but oly the ± ± trasitio is strog. 3. Coservatio of agular mometum, so really we have vibratioal-rotatioal trasitios. Fudametal Trasitio ν G() G ν χ e ( ) ( ) First OvertoeTrasitio ν G() G ν 3 χ e ( ) ( ) Secod OvertoeTrasitio ν 3 G(3) G( ) 3 ν( 4 χ e ) Ca calculate eergies of higher overtoe trasitios, fud itesity decreases rapidly for st higher overtoes. d cm -

23 Maximum vibratioal quatum umber I geeral the trasitio eergies are give by ( ) ν ν ε ε ε e + + χ Δ This shows that as the vibratioal quatum umber icreases the eergy levels get systematically closer together ad go to zero at the dissociatio limit. Aother way of seeig this is to recogise that the eergy fuctio is a parabolic fuctio of whose limit lies at D eq. Takig the derivative with respect to ad settig it equal to zero gives o e o ν ε ν χ + + ν o χ e ν o d dε + solvig gives the vibratioal idex associated with the dissociatio limit o e o D ν χ ν roud dow to the earest iteger to fid max the maximum vibratioal quatum umber.

24 Example: Overtoe bads i HCl The ifrared spectrum of HCl has vibratioal lies at 886cm - (strog, fudametal), 5668 cm - (weak, st overtoe ) ad 8347 cm - (very weak, secod overtoe). Fid χ e ad ν From data give we have Fudametal G() G ν χ e 886 cm st Overtoe G() G ν 3χ e 5668 cm d Overtoe G(3) G 3 ν 4χ e 8347cm Three equatios, two ukows, ca solve equatios to fid ν 99 cm χ e.74 ν vibratioal frequecy of simple harmoic oscillator Note that χ e is dimesioless

25 Hot Bads At low temperature oly the state has a sigificat populatio, so oly trasitios from occur. However as temperature icreases the some of the molecules will be i the state ad absorptio trasitios out of the state will be observed. These are called Hot Bads ad their itesities deped o the temperature of the gas. First Hot BadTrasitio ν G() G ν 4 χ e ( ) ( ) Secod Hot BadTrasitio ν 3 G(3) G ν 5 χ e ( ) ( ) st Hot Bad fud st d cm -

26 The Birge-Spoer Plot Whe several vibratioal trasitios are detectable i a spectrum the a graphical techique called a Birge-Spoer plot may be used to determie the dissociatio eergy of the molecule. The basis of the Birge-Spoer plot is that the sum of successive itervals ΔG from the zero-poit eergy to the dissociatio limit is the Dissociatio eergy. D ΔG + ΔG ΔG+ The area uder the plot of ΔG +/ agaist +/ is equal to this sum ad therefore to D (see plot o ext page). The plot decreases liearly whe oly the Χ e term is take ito accout ad the Do ca be foud by extrapolatio. I real systems the plots are oliear ad therefore overestimate D.

27 The Birge-Spoer Plot

28 Example usig the Birge-Spoer plot The observed vibratioal trasitios i H + are observed at the followig values: 9 cm - 46 cm cm cm cm cm cm cm cm cm - 33 cm - 98 cm cm cm cm cm - Determie the dissociatio eergy of the H + molecule. Birge-Spoer Method: Plot the lie separatios agaist + /, extrapolate liearly to the horizotal axis the measure the area uder the curve (use formula for triagle or cout the squares). (See Birge-Spoer plot for this problem o ext page).

29 Dissociatio eergy of H + Birge-Spoer Method: The total area uder the curve is 4 squares. Each square correspods to cm -, so the dissociatio eergy is,4 cm -.

Chapter 5 Vibrational Motion

Chapter 5 Vibrational Motion Fall 4 Chapter 5 Vibratioal Motio... 65 Potetial Eergy Surfaces, Rotatios ad Vibratios... 65 Harmoic Oscillator... 67 Geeral Solutio for H.O.: Operator Techique... 68 Vibratioal Selectio Rules... 7 Polyatomic