Chapter 5 Vibrational Motion

Transcription

1 Fall 4 Chapter 5 Vibratioal Motio Potetial Eergy Surfaces, Rotatios ad Vibratios Harmoic Oscillator Geeral Solutio for H.O.: Operator Techique Vibratioal Selectio Rules... 7 Polyatomic Molecules Beyod the harmoic oscillator approximatio Chapter 5 Vibratioal Motio Potetial Eergy Surfaces, Rotatios ad Vibratios Suppose we assume the uclei of a molecule are fixed, the we ca solve the Schrodiger equatio for the electros H ˆ ( r, r,... r ) = E ( r, r,... r ) N N This is a complicated problem, that will be discussed later (Chapter 7 ad beyod) We would get the (groud state) eergy at a particular uclear cofiguratio { R } Hece, assume we ca solve this We ca fit a curve through the poits V({ R }) This V({ R }) is called the potetial eergy surface (PES) ad is a crucial cocept i chemistry, eg. Chapter 5 Vibratioal Motio 65

2 Fall 4 Miima o the PES we associate with differet isomers, for example with the reactio A+ B C + D. Saddle poits o the PES we associate with trasitio states Obtaiig the miima (+ curvature) we ca get thermochemical iformatio. From TS we get ito rates of reactios (Chem54, Chem 35) Ideally, oce we have obtaied PES, we should solve for the Quatum eergies ivolvig the uclei. This is very complicated, ad today ca oly be doe for small molecules (up to 5 atoms). What ca be doe is solvig for molecular rotatios + vibratios i the harmoic oscillator + Rigid Rotor approximatio The crux is to make a quadratic approximatio to the PES aroud each miimum ( = molecular species) V (! R) V (R e ) + αβ k αβ ) α ) β Chapter 5 Vibratioal Motio 66

3 Fall 4 A molecule with N atoms has 3 N degrees of freedom 3 overall traslatios 3 overall rotatios or rotatios for liear molecules (3N 6) Vibratios, [3N 5 for liear molecule] k αβ : α,β =...3N All of the iformatio V (! R e ),! R e ad k αβ (! R e ) is obtaied from a electroic structure program (like Gaussia, Gamess, Turbomole ) Oce we have these, oe ca use exact calculatios to solve for Harmoic oscillator + Rigid Rotor eergies. The harmoic (quadratic) potetial is a approximatio, but ofte works well, certaily for stiff molecules/ degrees of freedom. What we will do ext is: discuss harmoic oscillator for diatomic molecules Groud state of harmoic oscillator All excited states usig operator techique Geeralize to polyatomic molecules More accurate solutio for diatomic (Beyod H.O.) Selectio rules, vibratioal spectroscopy Harmoic Oscillator I the followig chapter we will go o to discuss rotatios For a geeral polyatomic molecule we ca defie H.O. Hamiltoia as H = α For a diatomic this ca be reduced to! P M α + α a,b=...3n Ĥ =! d µ + kx k ab ) a ) b Here µ = M M is the [kg] reduced mass M + M x = ( R R e ) is deviatio [m] from equilibrium k is the force costat [Nm ] Chapter 5 Vibratioal Motio 67

4 Fall 4 I McQuarrie this is derived usig classical equatios of motio. I will post o the Website a geeral derivatio to get the H.O. form for a geeral polyatomic, but will ot discuss i class. Here I will simply use the form for H. The groud state solutio has the form o I will discuss the full solutio. e α x /. Let us determie the costat α, ad the eergy. Later! d e α x / αx / = α xe d x / x / x / e α e α x e α = α + α d µ + kx e α x / =! µ α! µ α x + kx e α x / = E e α x / E =! µ α k! α µ = α =! µk E =! µ µk! =! k µ =!ω ω = k µ ; α = µ! ω Geeral Solutio for H.O.: Operator Techique (see appedix 5 i McQuarrie) Defie q = x α e α x / Solutio Ĥ =! m d + kx e α x / α = µω /! = µk /! e q / q =! µk x x =! µk q Chapter 5 Vibratioal Motio 68

5 Fall 4 Ĥ =! µ x = q x µk! q =! µk q q + k! µk q =! k µ q + q =!ω d dq + q q are called dimesioless coordiates (Check that q =! µk x is ideed dimesioless!) Commutatio Relatio: d q, = dq d d q q f( q) = f( q) dq dq Defie ew operators: ˆ d b + = q dq The!ωb + b =!ω =!ω ˆ d b = q+ dq q d dq q + d dq q d dq + q, d dq = Ĥ!ω Hˆ ˆ ω b + = h b+ Commutatio Relatios, betwee b operators: bb ˆ, ˆ = b ˆ +, bˆ + = ˆˆ+ d d bb, = q+, q dq dq d d d d = [ qq, ] +, q q,, dq dq dq dq Chapter 5 Vibratioal Motio 69

6 Fall 4 Hece = ( + ( ) ) = bb ˆ, ˆ + = ˆ+ b, bˆ = Usig this form of the Hamiltoia ad the commutatio relatios we ca derive the eigevalues ad eigefuctios of H.O.!! a) If ( q ) is eigefuctio with eigevalue E the Proof: i) ˆ + b ( q) is eigefuctio with E + = E +!ω ii) bˆ ( q) is eigefuctio with E = E!ω i) Ĥ ˆb + ii) ( ) =!ω ˆb + ˆb + ˆb + =!ω ˆb + ˆb, ˆb+ + ˆb + ˆb+ ˆb + = ˆb ( +!ω + Ĥ ) = (E +!ω ) ˆb + ( ) =!ω ˆb + ˆb + Ĥ ˆb ˆb =!ω ˆb +, ˆb + ˆb ˆb + + ˆb =!ω ˆb + ˆb ˆb + ˆb + = ˆb (!ω + Ĥ ) = E!ω ( ) ˆb ˆb + ad ˆb are called the raisig ad lowerig operators, or ladder operators Chapter 5 Vibratioal Motio 7

7 Fall 4 What about the groud state? ˆb = E!ω ( ) ˆb Still true, but E!ω < E!! Oly way out: b ˆ ( q ) = d q+ ( q) dq = Differetial equatio with solutio ( ) q q = e d q q q+ e = ( q q) e = dq Puttig it all together 4 q ( q) = e ormalized π ( q ) ( ˆ ) ( )! b + = q ormalized ˆ d b + = q dq E = + h ω,,,3... = First couple of uormalized fuctios i terms of q : Chapter 5 Vibratioal Motio 7

8 Fall 4 ( ) q q = e ( ) d q q q e = qe dq q / ( ) d q / / ( q q q qe = q ) e dq ( ) d 3 ( q q q q ) qe dq = ( q 6 q) e Etc. We ca obtai all eigefuctios by differetiatio / 3 q / ( x ) substitute x = q α + ormalize The wave fuctios take the form H ( q) e q H ( ) q are Hermite polyomials, they are either odd or eve fuctios. (Each polyomial cotais oly odd or oly eve terms) Vibratioal Selectio Rules Later we will discuss more geerally the radiatio process. For ow the trasitio dipole momet is used to defie the stregth of a spectroscopic traslatio *( x ) µ ( x ) m( x ) m d µ µ ( x) = µ ( x) + x+... x µ is the dipole at equilibrium distace, d µ is the chage i dipole with chagig x (iterucleardistace) Eg. N : d µ = HF: d CO: d *( x) µ ( x) ( x) µ large µ small m = µ *( x) m( x) (if m) Chapter 5 Vibratioal Motio 7

9 Fall 4 d µ + *( xx ) m( x ) d µ α *( qq ) m( qdq ) dµ α *( )( ˆ+ ˆ q b b) m( q) dq + bˆ( m )~ ˆ + m b ( m )~ m+ For Harmoic Oscillator we ca oly get allowed trasitios if Δ = + or Δ = ΔE = ±!ω Polyatomic Molecules! Ĥ = + α µ α α,β k αβ ) α ) β By some maipulatio (see otes o webpage) this ca be writte as a sum of H.O. Hamiltoias.tedious derivatios Ĥ =! ω i d i dq + i q i + =! ω ˆbi ˆbi i + i The coordiates q : are liear combiatios of atomic displacemet vectors 3 N coordiate 3 traslatio, 3 rotatio, (3 N 6) vibratio For liear molecule, rotatio aroud axis is ot a displacemet rotatios, ad (3 N 5) vibratios Eg. Water Normal modes : display symmetry of molecule Chapter 5 Vibratioal Motio 73

10 Fall 4 The eigefuctios are simply products of dimesioal H.O. fuctios ( q ) ( q ) ( q ) k l m 3 E k,l,m = k +!ω + l +!ω + m +!ω 3 Very simple solutio. Oe eeds to diagoalize the mass weighted hessia k αβ M α M β 3N 3N Matrix Reasoable approximatios to all vibratioal levels Statistical Mechaics (recall Chem 35) Beyod the harmoic oscillator approximatio The true potetial is ot quadratic. For large molecules this is ot so easy to correct (people would use low order perturbatio theory, based o a quartic force field) For smalls molecules, i particular diatomics, oe ca solve for the full vibratioal problem The exact eergies are ot equidistat. A better approximatio is E() =!ω e +!ω x + e e The eergy levels are usually called g(v), v =,vibratioal quatum umber There are a fiite umber of boud levels Aother effect is that trasitio momets ca be ozero eve whe Δ ±. This leads to the observatio of overtoes. A ofte used form for the potetial for a diatomic is the Morse potetial Chapter 5 Vibratioal Motio 74

11 Fall 4 ( x ) V( x) = D e β V x x = V D eβ x e = = k x = equilibrium geometry β = k D D ca be used from experimet ( e D measurable) e This is still a approximate potetial Today: Potetial eergy curves ca be calculated usig electroic structure programs. Chapter 5 Vibratioal Motio 75