Chapter 6: Rotational and Rovibrational Spectra. A) General discussion of two- body problem with central potential

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1 Fall 4 Chapte 6: Rotational and Rovibational Specta Diffeent Appoximations... 8 Spectum fo Hamonic Oscillato + Rigid Rotato... 8 Polyatomic Molecules Hamonic Oscillato + Rigid Roto Model to Obtain Quantities fom Statistical Mechanics Moe Conventional Discussion of Rotations fo Diatomics Chapte 6: Rotational and Rovibational Specta A) Geneal discussion of two- body poblem with cental potential Examples: a) Diatomic V( R ) b) Hydogen atom with chage Z : V () = Ze 4πε Let us conside paticles with mass m, m Coodinates: ( ) Ĥ = p + p +V m m,, p = m d p = m " dt = m " Define cente of mass coodinate R cm = m + m, M = m + m m + m P cm = M R" cm = m " + m " Also define the elative coodinate = p = µ " = µ( " " ) mm µ = m + m Chapte 6: Rotational and Rovibational Specta 75

2 Fall 4 Then we will show that T = P cm P cm M + p p µ = µ = M (m " + m " ) (m " + m " ) + µ( ) = M (m + m + m m + m m ( + )) = (m + m ) m (m + m ) + m (m + m ) = m + m = P + P = T m m Hence the Hamiltonian can be witten as Now we can apply a sepaation of vaiables: Ĥ = P cm M + p +V () µ " = M cm + µ +V () ψ ( R cm, ) = φ( R cm )ψ ( ) " M cm φ(r cm ) + φ(r cm ) E, tanslational enegy T µ +V () ψ () = E ψ () total E The enegy we ae eally inteested in If we put the system in a (vey lage) box, the cente of mass poblem just yields the paticle in the 3d- box solutions sin kπ x cm L sin lπ z cm L ( ) sin mπ z cm L E T = π k + l + m ML This descibes tanslational kinetic enegy of the cente of mass motion [used in stat mech, Chem 35, Chem356] Chapte 6: Rotational and Rovibational Specta 76

3 Fall 4 Ou inteest is the elative motion: Because V() only depends on spheical coodinates µ ψ ( " ) +V ()ψ ( " ) = Eψ ( " ) = x + y + z, it is vey convenient to use ( ) = x + y + z x = sinθcosϕ y y = sinθsinϕ ϕ = actan x z = cosθ z θ = accos x + y + z < ; θ π; ϕ π To develop this futhe we need to obtain = + + x y z In spheical coodinates, this is a vey tedious execise using the chain ule. The basic step in the deivation is like this: f(, θϕ, ) f f θ f ϕ = + + x x θ x ϕ x θ ϕ + + x x x θ x ϕ ( ) and x = x + y + z = x x + y + z x cosϕsinθ = = = sinθcosϕ expess eveything in spheical coodinates You might appeciate that deiving the opeato + + x y z ( ) x is a lot of wok (use MathCad?) Let me give you the esult fo the kinetic enegy opeato µ = µ + ˆL µ ˆL = sinθ θ sinθ θ + sin θ ϕ Chapte 6: Rotational and Rovibational Specta 77

4 Fall 4 Let me also give you ˆL z = xp y yp x = i x y y x = i ϕ ˆL = ˆL x + ˆL y + ˆL z So ˆL is pecisely the opeato coesponding to the total angula momentum This kinetic enegy opeato can also be witten as Let us note that µ + + ˆL µ ˆL only depends on the angula coodinates θϕ, while L ˆz only depends on ϕ In a late lectue I will deive the eigenfunctions and eigenvalues of the use the commutation elations to do this (compae hamonic oscillato) ˆL and L ˆz opeatos. We will Fo now I will just list the solutions ˆL Y m l θ,ϕ ( ) = l(l +) Y m l ( θ,ϕ ) ˆL z Y l m (θ,ϕ) = my l m (θ,ϕ) m = l... + l These functions ae the angula pats of the familia hydogen obitals l =, m = s- obitals l =, m =,, p- obitals (3) l =, m =,,,, d- obitals (5) l = 3, m = 3,,,,,,3 f- obitals (7) Always (l + ) l - type obitals These functions ae nomalized as π π m sinθ dθ dϕy l *(θ,ϕ)y m l (θ,ϕ) = δ δ ll mm = fo l = l, m= m othewise The exta facto sinθ is elated to the suface element Chapte 6: Rotational and Rovibational Specta 78

5 Fall 4 Moe geneal fo spheical coodinates dxdydz Volume element sinθddθdϕ f ( x, y, z ) dxdydz d sinθ dθ f (,θ,ϕ) dϕ To solve Hψ = Eψ in spheical coodinates we ty the solution µ + m ψ(, θ, ϕ) = f() Y l (, θ ϕ) π f ()Y m (θ,ϕ) +V () f ()Y m l l (θ,ϕ) + µ f () ˆL Y m l (θ,ϕ) This yields a adial diffeential equation: µ + Togethe with nomalization condition π ( ) = Ef ()Y l m (θ,ϕ) f () + l(l +) f () +V () f () = E µ l f () f () d = This adial equation, depending on l, is a completely geneal esult, valid fo any - body poblem, with a ( ) cental potential V m When discussing chapte 7 (Hydogen atom), I will say moe about the spheical solutions Y ( θϕ,, ) and Ze also discuss the adial poblem fo V() = 4πε l At this point, I want to etun to diatomics. The solutions to the S.E. fo a vibating and otating molecule ae µ + ψ (,θ,ϕ) = f ()Y l m (θ,ϕ) l(l +) f () + µ ( ) f d () = f () +V ( ) f ( ) = E l f () Chapte 6: Rotational and Rovibational Specta 79

6 Fall 4 The angula functions ae the exact solutions elated to otations. The adial equation is the hamonic oscillato in disguise. This takes some wok to show this. Let us bing the equation to a moe familia fom: Define f () = g() and Hence Then O using f () = g() : f( ) f () = f () + f f f f = + = + µ f () + (l(l +) +V () µ f () = g l(l +) + +V () µ µ g() = Eg() ( f ()) d = g() ( ) d = ( f ()) E At this stage we have not yet made any appoximations. This stats to look like a hamonic oscillato equation. Now use x= ( R e ) = x+ Re = = x x ψ µ x + l(l +) +V (x) ψ µ(x + R e ) l (x) = E l ψ l (x) nomalization: ( ()) = ( ()) f d g d ψ ( x) dx At this point we have made a appoximation, changing the inteval of integation. This is a easonable appoximation since fo =, the potential tends to be vey lage. If V( x) = kx we have hamonic oscillato except fo the additional tem l(l +) µ(x + R e ) Chapte 6: Rotational and Rovibational Specta 8

7 Fall 4 Diffeent Appoximations a) Hamonic oscillato + igid oto: V (x) = kx statmech. Set l(l +) µ(x + R e ) l(l +) µ(r e ) Replace l J : common quantum numbe in otational spectoscopy E v,j = ω v + + µr e J(J +) m ψvjm,, = ψv( xy ) l ( θ, ϕ) Simplest solution; often used. This is what we used in chem35, b) Solve fo anhamonic wavefunction Then evaluate ψ (x) µ x B v = +V (x) ψ (x) = E ψ (x) v v v ψ v *(x) µ(x + R e ) ψ (x) dx v EvJ, = Ev+ BvJ( J + ) Rotational constant depends on vibational level (smalle B, lage R, as ν inceases. This is called centifugal distotion). We only need to calculate few vibational wavefunctions c) Solve equation exactly fo each vj;, degeneacy is always J +, m = J... J. This is a vey accuate solution. One would usually use the - vaiable, not x. Spectum fo Hamonic Oscillato + Rigid Rotato E v,j = ω v + + I J(J +) k I = µ R, ω = µ Define B = I Selection ules (diatomic, H.O./R.R.) Chapte 6: Rotational and Rovibational Specta 8

8 Fall 4 Δv =,± Δ J = ± ω...4 cm B... cm a) Pue otational tansitions E J + E J = ((J +)(J + ) J(J +)) I = (J +) = (J +) = h I I h hv obs = (J +) 4π I v obs = h (J +) B(J +) 4π I ω obs = v obs c = h 4π Ic (J +) = B(J +) otational constant B = h 8π I B = (Hetz) h 8π Ic (cm- ) (use c in cm s - ) (J +) 4π I Rotational tansitions Δ J = ± (selection ule; discussed late) Set of Equidistant lines Chapte 6: Rotational and Rovibational Specta 8

9 Fall 4 At oom tempeatue both gound and excited otational levels ae occupied, and we get J J + absoption fo all J. This leads to a set of equidistant lines in the spectum. At vey low T (T=. K fo example), we only find one line, coesponding to the J= to J= tansition, at B Let us now conside tansitions to diffeent vibational state (though absoption of light) Δν = +, Δ J = ± v =, J v =, J +tansitions have enegies: ΔE,J,J + = E,J + E,J = 3 ω + B(J +)(J + ) [ ω + BJ(J +)] = ω + B(J +), J =,,,3,... v =, J v =, J tansitions have enegies: ΔE,J,J = E,J E,J = 3 ω + BJ(J ) [ ω + BJ(J +)] = ω B, J =,,3,... Hee we stat the ΔJ = tansitions fom J =,as J = J = do not exist. At vey low tempeatue only the, - >, tansition with enegy ω + B exists. Chapte 6: Rotational and Rovibational Specta 83

10 Fall 4 If we go beyond H.O., then Δ E = E + B J'( J' + ) B J( J + ) Woking it out: v v+, J J+ v J J' ; J' = J + and B B centifugal distotions E,J + E,J V R = ω + B + (3 B B )J + ( B B )J E,J E,J V P = ω ( B + B )J + ( B B )J ( B B ) < The spacings between R lines deceases to the ight, while the spacing fo the P lines inceases (to the left). (The pictue below is not vey clea) Also the lines in pue otational spectum ae not exactly equally spead. ω J J + B(J +) 4 D(J +) 3 E J = BJ(J +) DJ (J +) Why: solve equation exactly. One can fit enegy levels well to fom of above type. It is an appoximation. Polyatomic Molecules (biefly) Assume we know equilibium geomety, a quadatic foce constant matix, then we can define a cente of mass motion. R cm = m j j / m j And a wavefunction (paticle in the box) associated with R cm j j Anothe 3 coodinates ae associated with oveall otation and we can associate classical igid body moment of inetia: I m R R R m R R αβ = δαβ j( jx + jy + jz ) j jα jβ j j αβ=, xyz,, Chapte 6: Rotational and Rovibational Specta 84

11 Fall 4 This symmetic matix can be diagonalized yielding eigenvalues Ia, Ib, I and a coesponding set of c axis. These coespond to displacement of all the nuclei using a igid otation. The emaining (3N 6) o (3N 5) coodinates define the nomal modes Ĥ ˆ T cm + ˆ Tˆ R T R + ω i i Jˆ Jˆ Jˆ = + + I I I a b c a b c ψ R j,k,m (θ, χ,ϕ) = P jkm (θ)eikχ e imϕ E m= j... j jkm,, k = j... j d dq + i q i The otational poblem fo any Ia, Ib, I can be solved using matix diagonalization: numeically c Staightfowad. We will not woy about details hee. ψtotal = ψcm ψr ϕ( q) ϕ( q)... ϕ3n 6( q3n 6) E = Ecm + ER + Evib This sepaation of vaiables is valid in the H.O./R.R. appoximation. These appoximations ae quite good fo spectoscopy; Vey good fo stat mech. Hamonic Oscillato + Rigid Roto Model to Obtain Quantities fom Statistical Mechanics I want to ecall the basic fomulas fom Statistical Mechanics hee, and show the immense usefulness of H.O./R.R. fo themochemisty. a) Exact fomulation of Stat Mech Define system patition function (e.g. gas of molecules): Eα ( N, V) QNVT (,, ) = exp α kt enegy levels B Connection to themodynamics: ( ) ANVT (,, ) = ktln Q NVT,, B Fom the absolute value of A the Helmholtz fee enegy we can get any themodynamics function. How? da = sdt pdv + µ dn Chapte 6: Rotational and Rovibational Specta 85

12 Fall 4 A A A = S = P = µ T V, N V T, N N TV, U = A+ TS, G = A + PV, H = A + TS + PV C v U = T V Now poceed to independent molecules in the gas phase: ( ) N Q = q m N t V R N e = q q q q q tanslation, vibation, otation, nuclea, electonic qm etc t q : fom paticle in the box quantum solutions Vibational: q t 3 = V T N α α = π Mk B q v = 3N 6 i= e ω i /kt e ω i /kt Fom Hamonic oscillato Q.M. sum each level Rotational + Nuclea spin (complication leads to symmety facto σ ): q R π T T T σ T T T A B C T x = I x I x : Moments of inetia fom igid Roto quantum mechanics De/ kt Ei/ kt qe = e e Δ D : sepaated atom limit e Δ : binding enegies, bottom of well E i i One can calculate Q fo each molecula species in the gas phase Chapte 6: Rotational and Rovibational Specta 86

13 Fall 4 Using a slight extension one can also calculate eaction ates (using tansition state theoy appoximation). tansitionstate Q and fom this one can obtain Simple Quantum models - Paticle in the box - Hamonic oscillato - Rigid oto - Electonic enegies ( no good simple models. Need to do accuate calculations.) Accuate themochemisty fo gas phase eactions Moe Conventional Discussion of Rotations fo Diatomics Two bodies at fixed distance R otating aound cente of mass: Moment of Inetia: µ R Angula momentum: µv = (µvr) µr = L I Geneal fo igid body otation of linea molecules: L= L = µ vr (see Boh atom) L T = (moe geneal fomulas fo polyatomic) I Relative motion in quantum mechanics q = x α α = µk Z = µω, k ω = µ Chapte 6: Rotational and Rovibational Specta 87

14 Fall 4 Ĥ = ω d dq + q + ˆL µr m Solutions: ψ ( qy ) ( θ, ϕ ) v E= E + E v l R = v + ω + J(J +) µr Hamonic Oscillato + Rotational igid body motion. ω ~ cm B = I. - cm Selection ules fo H.O/R.R. Δ V = ± Δ J = ± Molecules needs pemanent dipole to obseve otational tansitions. Note: It is had to see whee the appoximations come in o how to impove on it. This is the eason I discussed the full teatment. Chapte 6: Rotational and Rovibational Specta 88