PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.

Transcription

1 POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and nuclea wavefunctions and then use those equations to establish that the Bon-Oppenheime enegy is a lowe bound to the molecula gound state enegy. A. o deive the Bon-Oppenheime and adiabatic appoximations to the gound state enegy of a molecula system the total wavefunction is fist witten as the tenso poduct of an tonic wavefunction which depends explicitly on the ton coodinates {} and paametically/implicitly on the nuclea coodinates {} and a nuclea wavefunction which depends explicitly on the nuclea coodinates {} ; often efeed to as the Bon-Oppenheime poduct. he tonic wavefunction in is one element of the set of adiabatic tonic wavefunctions { ;... } which coespond to the eigenfunctions of the tonic Hamiltonian H : V ; H ; ;. o find the coesponding nuclea wavefunctions substitute into the time-independent Schödinge equation IS poect on the left with and integate ove tonic coodinates which will yield the conditions that detemine the nuclea wavefunctions as: [ W ]. 3 In 3 W ; ; and is the total adiabatic enegy fo state. In the Bon-Oppenheime appoximation W is set to zeo and the esulting conditions that detemine the nuclea wavefunctions can now be witten as a slight modification of 3:

2 ] [ 4 whee is the total Bon-Oppenheime enegy fo state. heefoe the conditions that detemine the tonic and nuclea wavefunctions ae given by and 4 espectively. Poposition : he Bon-Oppenheime gound state enegy is a lowe bound to the molecula gound state enegy. Poof. o pove that the Bon-Oppenheime enegy is a lowe bound to the molecula gound state enegy one may poceed as follows. Fom the IS we have: O O H 5 whee the subscipts and on the scala poduct indicates integation ove the tonic and nuclea coodinates espectively. Fo a fixed we can use the exact gound state wavefunction as a tial wavefunction in which yields V 6 by application of the vaiation theoem. Fom the ealie patitioning of the Hamiltonian 6 implies c.f. 5: V H O V H. 7 heefoe O. 8

3 Now fo a fixed we can use as a tial wavefunction in 4 which yields 9 by a second application of the vaiation theoem. Using 9 in 8 yields the desied esult O that the Bon-Oppenheime enegy is a lowe bound to the molecula gound state enegy. QD emas. In the above poof thee is essentially one point that equies caeful consideation. In deiving 6 the vaiation theoem was applied to the equation that detemines the gound state tonic wavefunction fo a fixed. So in deiving 7 and 8 which is nothing moe than a compact estatement of 7 I used the fact that if fo a fixed V then integating ove the nuclea coodinates i.e. the iemann summation ove will yield. V Although not ased of you in Q. many of you focused you effots in the above poof by fist showing that the Bon-Oppenheime gound state enegy is a lowe bound to the adiabatic gound state enegy then showing that the adiabatic gound state enegy is a lowe bound to the molecula gound state enegy. While the Bon-Oppenheime gound state enegy is indeed a lowe bound to the adiabatic gound state enegy simply aguing that the diagonal second-ode coupling tem is positive definite does not constitute a coect poof of this poposition. Moe impotantly the adiabatic gound state enegy is NO a lowe bound to the molecula gound state enegy instead the adiabatic gound

4 state enegy is actually an uppe bound to the molecula gound state enegy. So I hope that the following digession will be of inteest to most students in the class. Poposition : he Bon-Oppenheime gound state enegy is a lowe bound to the adiabatic gound state enegy. Poof. o pove that the Bon-Oppenheime enegy is a lowe bound to the adiabatic gound state enegy one may poceed as follows. he conditions that detemine the nuclea wavefunctions within the Bon-Oppenheime appoximation ae given by 4 as ] [ whee the gound state Bon-Oppenheime nuclea wavefunction has been labeled with a supescipt to denote that this function is an eigenfunction of the opeato [ ] with the Bon-Oppenheime gound state enegy as its coesponding eigenvalue. Analogously the conditions that detemine the nuclea wavefunctions within the adiabatic appoximation ae given by 3 as W ] [ whee the gound state adiabatic nuclea wavefunction has also been labeled to denote the fact that this function is an eigenfunction of a slightly diffeent opeato than that given by namely W ]. he coesponding eigenvalue fo the gound state adiabatic [ nuclea wavefunction is the adiabatic gound state enegy. Using as a tial wavefunction in yields [ ] 3

5 by the vaiation theoem. Poecting with on the left of and using 3 yields: [ [ W ] ] W 4 W WLOG assuming that the gound state adiabatic nuclea wavefunction is nomalized futhe simplifies 4 to the following fom: W 5 wheein utilizing the fact that W completes the poof that the Bon- Oppenheime gound state enegy is a lowe bound to the adiabatic gound state enegy. QD emas. In the above poof thee is essentially one point that equies caeful consideation. It is not sufficient to ust state that W and then conclude that the Bon- Oppenheime gound state enegy is a lowe bound to the adiabatic gound state enegy. he poof equies use of the vaiation theoem egading the Bon-Oppenheime and adiabatic nuclea wavefunctions which ae in geneal distinct eigenfunctions with distinct eigenvalues. Poposition 3: he adiabatic gound state enegy is an uppe bound to the molecula gound state enegy. Summay of Poof. he poof that the adiabatic gound state enegy is an uppe bound to the molecula gound state enegy equies use of the vaiation theoem. Since the exact

6 wavefunction can be witten as a linea combination of adiabatic tonic wavefunctions the eigenfunctions of the tonic Hamiltonian fom a complete othonomal set namely ; 6 one quicly ealizes that using a single Bon-Oppenheime poduct to epesent the exact wavefunction educes the degee of vaiational flexibility in the wavefunction. Application of the vaiation theoem to the single Bon-Oppenheime poduct appoximation to the exact wavefunction yields the desied esult namely. Using the esults fom Poposition gives the coect elative enegetic odeing:. Q. Odes of magnitude in the Bon-Oppenheime sepaation. You will use H as a case that lets you poduce some specific numbes and also poceed based on moe geneal consideations in tems of fundamental constants chaacteistic intenuclea spacing a chaacteistic nuclea mass M etc. A. houghout this poblem m e will epesent an ode of magnitude appoximation to the mass of the ton i.e. ~ -3 g M will epesent an ode of magnitude appoximation to the nuclea masses i.e. ~ -7 g and will epesent an ode of magnitude appoximation to the aveage intenuclea sepaation i.e. ~ - m Å. a stimate the ode of magnitude of the ton binding enegy and thus the spacing between tonic enegy levels specifically poducing a numbe fo H & also a geneal expession. If the linea dimension of a molecula system is given as then the Heisenbeg uncetainty elationship ΔpΔ h allows us to appoximate the momentum of the tons as p h /. Since e p / we can mae an ode of magnitude estimate fo the binding enegy m e and the spacing between tonic enegy levels as ε h /. Fo H - ev - cm -. m e ε J

7 b stimate the ode of magnitude of the ational enegy spacings specifically poducing a numbe fo H & also a geneal expession. o fist ode the ational enegy spacings can be consideed as the diffeence between states of the hamonic oscillato i.e. ε hω. At a displacement of the potential enegy of the molecula system will incease by ~ Mω which coesponds to an enegetic penalty appoximating ε. Fom a Mω ε h / m theefoe e ε h / m e M. Fo H ε J ev - cm -. c stimate the ode of magnitude of the otational enegy spacings by estimating the moment of inetia specifically poducing a numbe fo H & also a geneal expession. Using the igid oto model ε ot h / I whee I M is the moment of inetia. heefoe ε h / M. Fo H ot ε ot J ev - cm -. d Fom the elative magnitudes of the level spacings which coespond to classical fequencies give the atio of the fequencies fo tonic ational and otational motion. Fom pats a c we concluded that ε > ε > ε. Since ε hν the ot coesponding fequencies have the following elative ode: ν < ν < ν. Fom the estimates of ε ε and ot ε fo H 4 above we have ν ν : :. : :ν ot Q3. ecall fom lectue that in the Bon-Oppenheime appoximation we neglect a coupling tem that I asseted was small in the equations fo nuclea motion on a single potential enegy suface. A3. o complete the deivations below we need the following identities which I will now deive fom the othonomality conditions on the adiabatic tonic wavefunctions assumed to be eal functions: ot δ. 7

8 Applying the gad opeato on 7 yields the fist two identities: 8 and. 9 aing the divegence of 8 yields the thid and final identity: ] [ ] [. a Constuct an ode of magnitude agument to show that this neglected tem is oughly on the ode of a otational enegy spacing fo a typical stable molecule. he neglected tem efeed to in this question is nothing moe than the diagonal element of the second-ode coupling tem given as: M W. Using the identity povided by this tem can be ewitten as follows: M M W. o fist ode the integal can be eplaced by 3 whee is the distance necessay to tansfom into a function that is othogonal to it. Using 3 we can now ewite as M W h 4

9 which is a positive quantity on the ode of magnitude of a otational enegy spacing c.f. Q3c. N.B.: In 4 W is no longe in atomic units hence the pesence of the h in the numeato. b Show that the non-bon-oppenheime coupling between potential enegy sufaces can divege unde cetain conditions using petubation theoy aguments. he two non-bon-oppenheime coupling tems efeed to in this question ae the fistode and the second-ode coupling tems. he fist-ode o vecto coupling tem is given as: d M. 5 he diagonal vecto coupling tems in which vanish since in which the identity povided by 8 was used. heefoe we only need to concen ouselves with the offdiagonal vecto coupling tems in which. o poceed we simply apply the gad opeato to an off-diagonal matix element of the tonic Hamiltonian i.e. H H H H H H 6 Since H δ it follows that. Using this fact in H 6 povides us with the following elationship afte some algebaic manipulation H 7 which can be inseted into the off-diagonal vecto coupling tem given by 5 d M H. 8

10 his altenative fomulation of the off-diagonal vecto coupling tem explicitly demonstates that this non-bon-oppenheime coupling tem will divege when two potential enegy sufaces appoach each othe i.e. when. he second-ode o matix coupling tem is given as: W. 9 M Since the fist-ode coupling tem is usually lage in magnitude than the second-ode coupling tem the above deivation fo conditions upon which the fist-ode coupling tem diveges would suffice fo this poblem. Fo those of you who ae inteested howeve one could also show that the second-ode coupling tem also diveges at points whee two potential enegy sufaces appoach each othe enegetically. he deivation of this fact is analogous to that above fo the divegence of the fist-ode coupling tem and poceeds by taing the divegence of the gadient of the off-diagonal matix element of the tonic Hamiltonian. he emainde of the deivation will be left as an execise fo the eade. Q4. Poblem 4.3 fom Schatz and atne. A4. he answe to Poblem 4.3 is given in Appendix C. of Schatz and atne pp Q5. Seams of intesection between sufaces. Detemine the conditions unde which the potential enegy sufaces of two states can intesect and thus establish the dimensionality of the suface of intesection. You may assume that all states except these two ae nown. A5. Fo a two state system the enegy eigenvalues can be obtained by solving the quadatic equation esulting fom the secula deteminant. hese enegy eigenvalues ae given as: ± { H H ± H H 4H }. 3

11 If the potential enegy sufaces intesect then which can only happen if the disciminant is zeo. In ode fo the disciminant to vanish the following two conditions must be met simultaneously: and H H 3 H. 3 heefoe the dimensionality of the suface of the intesection is 3N 6 3N 8 fo nonlinea polyatomics and 3N 5 3N 7 fo linea polyatomics.