The Born-Oppenheimer approximation

Transcription

1 The Bor-Oppeheimer approximatio 1 Re-writig the Schrödiger equatio We will begi from the full time-idepedet Schrödiger equatio for the eigestates of a molecular system: [ P 2 + ( Pm 2 + e m 2m m m 4πɛ 0 2 r m,m m r m )] Z r m, m R + 1 Z Z m Ψ = E Ψ. 2 R, R (1) The m idex rus over electros, ad over uclei. Here I have igored relativistic correctios to the Hamiltoia H, ad assumed that the uclei are poitlike. The Hamiltoia cotais o spi-depedet terms, ad so the eigestates Ψ will be factorizable ito spi ad coordiate parts from ow o, I will igore the spi state ad deal with wavefuctios Ψ(R, r), where R ad r are the set of uclear ad electroic positios respectively. This is obviously a isoluble problem without makig approximatios. To get started, cosider the solutio of the problem if the uclei were fixed: [ ( Pm 2 + e m m m 4πɛ 0 2 r m,m m r m )] Z r m, m r + 1 Z Z m φ(r) = Eφ(r). (2) 2 r, r This Hamiltoia will produce a complete orthogoal set of eigestates φ R s (r) with eergies E s (R), where s labels the eigestate ad R remids us that the eigestates will be depedet o where we have fixed the uclear positios. It is importat to see that if R is chaged, the eigestaes φ R s (r) will chage cotiuously, so that eigestates of the same s at differet R are liked, ad form a family of states see Figure 1. We oly ru ito problems if the families get very close to each other at ay poit - this is oe of the criteria by which the Bor-Oppeheimer approximatio ca break dow. Note that for ow, we do t actually care about this cotiuity of families issue, it oly arises whe we make the adiabatic approximatio later. So, as φ Ra s (r) is a complete basis set for electros if the uclei are fixed at R a, ay state with such a fixed ucleus ca be writte: Ψ a (R, r) = s c s δ(r R a )φ Ra s (r) (3) 1

2 E(R)% Family%3:%E 3 (R)% Family%2:%E 2 (R)% Family%1:%E 1 (R)% R% Figure 1: Electro eergy eigevalues as a fuctio of uclear separatio. Note that that they exist i cotiuous families. Do t paic about the δ-fuctio: this is just sayig that the uclei are fixed. Now, for our real system i which uclei ca move, a geeral state will be a sum over all possible Ψ a (R, r) with appropriate coefficiets (i.e., we take a superpositio of all possible locatios where the uclei could be). As R a is a cotiuous variable, our sum becomes a itegral: Ψ(R, r) = dr a c s (R a )δ(r R a )φ Ra s (r) (4) s Note that the factor c s (R a ) varies with positio of the uclei this allows the wavefuctio to vary with R a, both i terms of its total magitude ad the relative cotributios of differet s states. The δ-fuctio does it s usual job withi the itegral, ad we are left with: Ψ(R, r) = s c s (R)φ R s (r) = s ψ s (R)φ R s (r) = s φ R s (r)ψ s (R). (5) So far, we have made o approximatios, ad I have reamed c s (R) as ψ s (R) as we ow have somethig that looks like a wavefuctio for the uclei. The ext step is to substitute Equatio 5 ito Equatio 1: + [ P 2 m s m 2m m + e2 H ( s φr s (r)ψ s (R) ) = s ( 1 4πɛ 0 2 m,m 1 r m r m m, P ( 2 2m φ R s (r)ψ s (R) ) Z + 1 r m R 2, )] Z Z m R R φ R s (r)ψ s (R) = E s φr s (r)ψ s (R) (6) 2

3 I have simply split the Hamiltoia ito two ad swapped the order of summatios. Helpfully, we kow that φ R s (r) are eigestates of the secod part of the Hamiltoia: P 2 ( φ R 2m s (r)ψ s (R) ) + s s E s (R)φ R s (r)ψ s (R) = E s φ R s (r)ψ s (R). (7) But where do we go from here? It would be ice to get rid of the φ R s we ca try to do this by multiplyig by φ R s ad itegratig through with respect to r, as we kow that the differet φr s are orthogoal at give R. Mathematically, this ca be expressed as: drφ s R (r)φ R s (r) = δ s,s. (8) Whe we do this, the summatio ad φ R s are ideed elimiated from the secod two terms of Equatio 7. Ufortuately the first remais difficult, particularaly as P is a differetial operator: [ ( ) ] drφ P 2 s R (r) φ R s (r) + δ s,s E s (R) ψ s (R) = Eψ s (R). (9) 2m s 2 The adiabatic approximatio Now comes our first approximatio. We would like the sum i Equatio 9 to disappear if you remember, this sum comes from the sum i Equatio 5. Could we justify gettig rid of the sum here? I geeral, the uclei are far more massive tha the electros, ad hece will move comparatively slowly. We make the approximatio that the uclei move so slowly that the electros respod adiabatically. Remember the particle i a box from elemetary QM. Now imagie chagig the locatio of the walls if we do this slowly eough, it seems reasoable eough to assume that a particle i the groudstate of the system will cotiuosly evolve ito the ew groudstate. Of course it s eergy will ot be costat i fact, it is doig work o whatever is holdig the walls i place. But the poit is that it moves betwee cotiuously coected groudstates. The Wikipedia page for Adiabatic theorem is quite good, if you wat more details. The argumet also holds for ay state, provided that eergy levels do ot get too close ad that the chage i Hamiltoia is slow eough. How is this relevat to us? Let us step back, ad view the uclei as a exteral potetial actig o the electros. We already have a set of electroic eigestates φ R s (r) for ay cofiguratio of uclei. By makig the adiabatic approximatio, we are sayig that the uclei move slowly eough that the electros will ot chage which family of states they are i whe this movemet occurs (i.e., the s idex does ot chage). This is show graphically i Figure 2. Or, mathematically, if the uclei move from R R, electros i a eigestate will chage as: φ R s (r) φ R s (r) (10) 3

4 (a)$ (b)$ Figure 2: Adiabatic approximatio for (a) a particle i a well ad (b) electros i a diatomic molecule. If the walls/uclei move slowly eough, the particle / electros will remai i a give cotiuously coected family of states. I other words, at the ed of the process they will be i a state of the ew system which correspods to the state of the old system i which they started. Now, what does this mea whe we wat to cosider the uclei as part of the system, rather tha a exteral potetial? Let us imagie we start i a state which cotais cotributios from oly oe s-state family, which we shall call s = 0: Ψ(R, r) = ψ(r)φ R 0 (r) (11) What happes as time progresses? This is ot ecessarily a eigestate of the whole system, ad so the state will ted to chage. However, havig made the adiabatic approximatio, we kow that allowig the uclei to move will ot cause the electros to chage which s-family they are i! So we kow that the Hamiltoia does ot coect differet s-families, ad therefore that eigestates will be cofied to oe s-family. I other words, we ca write that a geeral eigestate of the system is ecessarily give by: Ψ E s,t(r, r) = ψ s,t (R)φ R s (r). (12) Here, s ad t label the particular eigestate we have the t th eigestate of the s-family of electro states. So we have got rid of our sum, which is what we wated. Now Equatio 9 becomes: [ ( ) ] drφ sr P 2 (r) φ R s (r) + E s (R) ψ s,t (R) = E s,t ψ s,t (R). (13) 2m 4

5 3 Fully simplifyig the equatio for diatomic molecules The first term is still a bit aoyig it cotais a differetial operator o uclear positios, which is actig o both φ R s (r) ad ψs,t(r). R It would be ice to be able to assume that R φ R s (r) 0 so that we could take the differetial operator out of the itegral (which would the simply be uity), leavig: [ ] P 2 + E s (R) ψ s,t (R) = E s,t ψ s,t (R), (14) 2m which looks like a Schrödiger equatio for uclei, with a potetial depedet o electroic cofiguratios. For a diatomic molecule, we ca describe the two uclei usig oly a relative coordiate i the reduced mass formalism (here we are assumig that the cetre of mass frame is the cetre of mass frame of the uclei - reasoable, give how light the electros are). The cetre of mass evolutio is the separable ad trivial, ad I shall igore it. I this case, Equatio 13 reduces to: [ drφ sr (r) ( 2 2 R 2µ ) φ R s (r) + E s (R) ] ψ s,t (R) = E s,t ψ s,t (R), (15) with R ow represetig the relative coordiate of the uclei, ad µ the reduced mass. Note that the electroic eergy ca oly deped o the spearatio of ucleii i the diatomic case. We ca split the differetial operator ito radial ad agular parts: 2 2 R 2µ = 2 2 R 2µ + L2 R 2µR 2, (16) Where L R is the orbital agular mometum of the uclei. Physically, we kow that the distace betwee uclei i a molecule varies very little, ad so the uclear part of the wavefuctio will be very strogly peaked aroud some R. By cotrast, the electroic state φ R s (r) will be relatively isesitive to R i this regio: the electro orbitals will care very little about these small chages i uclear separatio. So it seems reasoable that the radial compoet of the differetial operator would essetially oly act o ψ s,t (R). What about the agular part? We ca t say that the uclear wavefuctio is strogly peaked aroud oe directio of R. However, if you follow the argumet i pages of Brasde ad Joachai, you will see that through some maipulatio, the agular term ca be show to be: [ ( ) ] [ ] L drφ sr 2 (r) R φ R Kt (K t + 1) 2 2µR 2 s (r) ψ s,t (R) = + 2 g(s, R) ψ 2µR 2 2µR2 s,t (R) (17) where g(s, R) is a fuctio with modulus 1 that depeds o the electroic cofiguratio ad the uclear separatio, ad K t is the quatum umber of total orbital agular mometum K 2 = (L R + L r ) 2. Here L r is the orbital agular mometa of the electros. The fial term, i 5

6 effect, modifies E s (R) by a small amout (small, as E s (R) 2 2m er ), ad so it ca be swallowed 2 ito this term without much worry. So fially, we have: [ 2 2 R + K ] t(k t + 1) 2 + E 2µ 2µR s(r) ψ 2 s,t (R) = E s,t ψ s,t (R). (18) Note that this is differet from the form preseted i lectures, where the secod term is purely due to the orbital agular mometum of the uclei. We should be suspicious of the result i the lecture otes, because it implies that the orbital agular mometum of the uclei is a good quatum umber, which ca oly be true if the uclei system experieces o exteral torque. But this ca t be, because we kow that the electroic orbital agular mometum is ot a good quatum umber (except the compoet alog the iteruclear axis, Λ) due to its couplig with the uclei! I other words, if the uclei exert a torque o the electros, the electros must exert a torque o the uclei. Oly the total orbital agular mometum of the system K is a good quatum umber, as it must be i the absece of spi-orbit couplig. The lecturer s approach differs by assumig that L2 R 2µR 2 φ R s (r) is egligible, which is ot strictly justified. 4 Coclusio Thus we have arrived at the much simpler (ad physically meaigful) equatios of the Bor- Oppeheimer approximatio We first solve for the electroic eigestates φ R s (r) at fixed uclear positios (Equatio 2), ad the use the eigevalues to fid uclear eigestates ψ s,t (R) (Equatio 18). Havig foud the states φ R s (r) ad ψ s,t (R), our overall eigestates are simply: Ψ E s,t(r, r) = ψ s,t (R)φ R s (r). (19) At this stage, our approximatios are the adiabatic assumptio, ad that the uclear wavefuctio depeds much more strogly o R tha the electroic cofiguratio. To go further, we assume that the potetial well E s(r) is approximately quadratic about R = R s, givig us our eergy spectrum E s,t = E s (R s ) + K t(k t + 1) 2 + ω 2µRs 2 s (ν t + 1/2). (20) The s ad t labels idicate quatities that deped o which electroic cofiguratio ad uclear wavefuctio you are i, respectively. They ca be left out if this distictio is ot required. Previously I made a bit of a fuss about our result beig differet from the lecturer s. This aswer for the eergy spectrum, however, looks very similar. Is there ay actual distictio? I fact, there is. For the lecturer, the J quatum umber (orbital agular mometum of uclei) ca be ay positive iteger. For us, however, K Λ for a give electroic state (as the uclei do ot cotribute to this compoet of agular mometum ad hece caot cacel it out). Therefore certai levels that would be expected from the lecturer s discussio actually do ot appear for Λ 0. Of course, you will oly be expected to repeat the fudged derivatio i your exams, but it might be worth otig that you kow that K rather tha J should strictly be used. 6