X. The Born-Oppenhemer Approxmaton The Born- Oppenhemer (BO) approxmaton s probably the most fundamental approxmaton n chemstry. From a practcal pont of vew t wll allow us to treat the ectronc structure of molecules very accuraty wthout worryng too much about the nucle. owever n a more fundamental way t underpns the way that most chemsts thnk about molecules. Any tme you see a chemst draw a pcture lke the one at rght you are mplctly makng use of the framework suggested by the Bon-Oppenhemer approxmaton. So we are gong to spend some tme talkng about ths approxmaton and when we do and do not expect t to be vald. a. The Adabatc Approxmaton For any molecule we can wrte down the amltonan n atomc unts ( ħ = m e = e = ) as (defnng rαβ rα rβ etc.) : = nucle ectrons nucle ectrons j rj Z Z Z r uclear Knetc nergy lectronc Knetc nergy uclear- uclear epulson lectron- lectron epulson lectron- uclear Attracton
The physcal motvaton behnd the Born-Oppenhemer Approxmaton s that the nucle are much heaver than the ectrons (e.g. a proton s 800 tmes as heavy as an ectron). At any gven nstant the ectrons wll fe a amltonan that depends on the poston of the nucle at that nstant: ectrons nucle ectrons Z Z Z ( ) = r r. Where denotes the dependence of Ĥ on all of the nuclear postons { } at once. n the lmt that the nucle are nfnty massve they wll never move and the postons n the above expresson wll be fxed;.e. the molecule wll be frozen n some partcular confguraton. n ths case the s can be consdered as parameters (rather than operators) that defne the effectve amltonan for the ectrons. For any fxed confguraton of the molecule then one s nterested n solvng a Schrödnger equaton that nvolves only the ectronc degrees of freedom: ( ) Ψ ( ) = ( ) Ψ ( ) where we have noted explctly that the amltonan ts egenstates and egenvalues depend on the partcular nuclear confguraton. Ths s the key ement of the BO approxmaton; t allows one to compute the ectronc structure of a molecule wthout sayng anythng about the quantum mechancs of the nucle. Once we have solved the ectronc Schrödnger equaton we can wrte down the effectve amltonan for the nucle by smply addng back n the terms that were left out of Ĥ : nucle = ence the nucle move on an effectve potental surface that s defned by the ectronc energy and we can defne wavefunctons for the nucle alone that are egenfunctons of ths amltonan: Ψ = nucle j ( ) j ( ) Ψ = Ψ
Thus another way to thnk about the BO approxmaton s that t s vald whenever the ectronc and nuclear wavefunctons approxmaty decouple. otce that the states we are usng do not treat the nucle and ectrons as ndependent partcles; the parametrc dependence of the ectronc egenstates ntroduces a non-trval couplng between the two and so the decouplng need not be complete for the BO approxmaton to be vald. Fnally we note that the ectronc Schrödnger equaton can also be derved by assumng that the hgh masses of the nucle mean that they can be treated classcally. Then the nucle are complety descrbed by a trajectory (t). Ĥ can then be though of as dependng ether on or on tme. f we take the latter approach and assume the nucle move nfnty slowly we have a amltonan t that s changng very slowly wth tme and hence f the ( ) ectrons start out n an egenstate of ( 0) they wll adabatcally follow ths egenstate along the trajectory and end up n an egenstate of ( t). Thus f the nucle are slow-movng classcal partcles the ectronc Schrödnger equaton falls out naturally. For ths reason the BO approxmaton s sometmes called the adabatc approxmaton. ote however that the BO approxmaton does not treat the nucle classcally. t descrbes nucle that move quantum mechancally on an effectve potental defned by the ectrons. b. The Coupled Chann epresentaton By tsf the BO approxmaton s exceedngly accurate whch accounts for ts wdespread use throughout chemstry. ndeed n most cases where t fals one can usually explan the result by assumng that the system s adabatc almost all the tme wth only a few solated regons where correctons need to be accounted for. ence t s extremy useful to consder the exact Schrödnger equaton expressed n the bass defned by the BO approxmaton. Frst we note that whle t s usually convenent to consder the ectronc wavefuncton n lbert space (.e. Ψ ( ) ) t s usually Ψ ). Ths convenent to specfy the nuclear part n real space ( ( )
notaton s a bt unusual but t s the most convenent for the problem at hand. There are two ponts that should be made. Frst note that the dependence on n Ψ ( ) s fundamentally dfferent than that n Ψ ( ). Ψ ( ) s properly thought of as the ampltude for fndng the nucle at n gven confguraton: Ψ = Ψ so that ( ) ( ) Ψ has the nterpretaton of a probablty. On the other hand there s no state and ( ) Ψ that gves Ψ ( ) = Ψ? Ψ s not the probablty of the outcome of any physcal measurement. The dependence of ( ) Ψ on mery reflects the fact that the adabatc states depend on where the nucle are at that nstant. The second pont s that we wll now begn talkng about unusual objects lke: Ψ ( ) Ψ ( ) by whch we mean that the nucle are descrbed by the (real space) wavefuncton Ψ ( ) whle the ectrons are n the (lbert space) wavefuncton Ψ ( ). The rules of quantum mechancs proceed as before n ths mxed representaton. ow n order to represent the full amltonan n the BO bass we note that the ectronc egenstates for any fxed choce of the s forms a complete bass. That s the wavefunctons ( ) that satsfy ( ) ( ) = ( ) ( ) form a complete bass for the ectrons. Lkewse once we have sected a partcular ectronc state the vbratonal egenstates on ths potental surface form a complete bass for the nucle; thus the that satsfy wavefunctons ( ) nucle ( ) ( ) = ( ) for any fxed form a complete bass for expandng any nuclear wavefuncton. Therefore applyng our experence wth many partcles we conclude mmedaty that the set of products
( ) ( ) form a complete bass for any wavefuncton that descrbes the ectrons and the nucle at once. ence we can wrte any wavefuncton for the molecule as: ( ) ( ) ( ) = Ψ C. We can use ths bass to examne where the errors n the BO approxmaton come from. We fnd ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = δ ow we note that we can use the matrx ements as an effectve amltonan for the nucle alone: ( ) ( ) ( ) ( ) ( ) ( ) eff = f we make the defntons ( ) eff = = δ P ( ) ( ) ( ) ( ) ( ) eff = P. Ths s termed the coupled chann Schrödnger equaton because t descrbes the dynamcs of the molecule n terms of several possble adabatc states or channs. The dagonal ( = ) term s just the
BO amltonan; the product bass functons we have chosen are egenfunctons of the BO amltonan by constructon. The offdagonal ( ) terms on the second lne are the correctons to the BO approxmaton; they arse because the ectronc wavefuncton depends (parametrcally) on the nuclear coordnates and the magntude of the correctons wll depend on the rate of change (gradent) of the ectronc wavefuncton as we change our nuclear confguraton. f the ectronc state changes rapdly over a small dstance we expect these terms to be large. Before we move on to dscuss when ths happens we note that ( ) ( ) s usually qute small and we wll not be concerned wth t n what follows. ence we wll use the approxmaton eff ( ) ( ) P The remanng term couples the nucle and ectrons and can be qute large. t s usually called the non-adabatc couplng. c. on-adabatc ffects When does the BO approxmaton fal? Ths s a trcky queston. One mght be tempted to conclude that t wll fal whenever the nucle are lght but ths turns out not to be the major problem. Let us consder the coupled chann equaton and take the frst term (the BO result) as the zeroth order amltonan and treat the second term as a perturbaton. The zeroth order amltonan for the nucle can easly be wrtten n operator form: ( ) P ( ) 0 = δ where we stress that the nuclear momentum operator (by conventon) does not act on the parametrc dependence of the ectronc wavefuncton on. Our perturbaton s gven by: d. ( ) P V = Where we have defned the non-adabatc couplng matrx by: d ( ) ( ) ( )
Ths s a rather unusual matrx n that each of ts ements s a vector. What does ths vector tl us? Wl frst we note that t comes from the gradent of the ectronc wavefuncton wth respect to the th nuclear coordnate. The drecton of ths gradent tls us the drecton n whch the ectronc wavefuncton s changng the fastest whle ts magntude tls us how large ths change s n an absolute sense. One then takes the overlap of ths gradent wth the ectronc functon. Ths tls us as we vary how much the change n looks lke a change from the current ectronc state ( ) to another ( ). ence there s a wealth of nformaton here; d tls us how lky non-adabatc events are (through ts magntude) what physcal motons t can be assocated wth (through ts drecton) and whch ectronc states are nvolved (because of the overlap of the gradent of wth ). Ths s a rather unusual perturbaton as t depends on the momentum of the nucle n addton to the dependence on d. Ths means that the probablty of a non-adabatc event wll depend on how fast the nucle are gong. f they are movng rapdly the perturbaton s larger and non-adabatc effects are expected to be larger. What does ths tl us about when the BO approxmaton s expected to break down? Wl our BO states are egenstates of Ĥ 0 and ths wll cease to be a good approxmaton to the egenstates of Ĥ when V s large. We have already encountered the most common cause for ths: f two egenstates of Ĥ 0 are degenerate Degenerate BO states then V s always
large. What does ths mean? Wl f we plot the adabatc ectronc energes as curves that are functons of then a degeneracy can easly occur f the curves cross as shown n the fgure above. n ths case our physcal pcture of the nuclear moton occurrng on only one potental surface wll fal and we need a lnear combnaton of BO states on both surfaces to get a reasonable startng pont Ψ( r ) c ;( ) ( ) c ; ( ) ( ) t can be shown that these ntersectons never (or at least almost never) occur f there s only one nuclear dmenson; one needs at least two degrees of freedom and even then the ntersecton only occurs at a pont. Ths pont s called a concal ntersecton because ths s the characterstc shape of two surfaces that touch at a pont. A more common Adabatc Bass occurrence s for two surfaces to almost nonpolar polar ntersect but not qute. Ths s shown at rght and s usually referred to as an avoded crossng. ow f one examnes the character of the nonpolar ectronc wavefuncton near the polar avoded crossng one often fnds that the lower state on the left hand sde s more smlar to the excted state on the rght than to the ground state. For example the molecule mght be polar on the left hand sde of the lower state and non-polar on the rght and the excted state mght have the reverse: polar on the rght and non-polar on the left. Ths means that the adabatc ectronc wavefunctons are changng rapdly n the vcnty of the avoded crossng whch means the non-adabatc couplng s large. Lookng at ths from a dynamcal pont of vew unless the nucle move through the avoded crossng regon very slowly the
ectrons wll not have tme to rearrange (e.g. from polar to non-polar) and we wll get sgnfcant probablty transfer from one surface to the other. Ths s often called a non-adabatc transton between the two adabatc surfaces. t s dsconcertng that the adabatc ectronc wavefunctons change so rapdly n the vcnty of an avoded crossng; t makes the chemstry look unnecessarly complcated. For ths reason one often nvokes the concept of a dabatc bass. n ths bass t s the ectrons that are hd fxed and the nucle are allowed to move frey; thus n the above case one would have two dabatc surfaces. One would reman polar through the crossng regon whle the other would reman non-polar. The adabatc ectronc states wll be lnear combnatons of the dabats. The nonadabatc transton would be easly polar Dabatc Bass nonpolar descrbed n the dabatc bass because the movement from the lower adabatc surface to the upper one wll correspond to stayng on the same dabatc surface. Thus f the nucle move quckly through the avoded crossng regon the ectronc wavefuncton wll not have tme to react and a dabatc pcture s approprate. By defnton the dabatc ectronc states do not depend on and so d ( ) = 0. ence the terms nvolvng the nuclear knetc energy are exceedngly smple n the dabatc representaton. The ectronc amltonan s more complcated however because the dabatc states do not dagonalze Ĥ. Thus the full amltonan n the dabatc bass s gven by:
P da da = Vj ( ) j j where the matrx ements of the ectronc amltonan are gven by da V =. j ( ) ( ) da n the adabatc bass ths matrx would be dagonal but n the dabatc bass t s the source of transtons between the surfaces. n practce the dabatc bass s most useful very near a concal ntersecton or avoded crossng. n practce t s usually not possble to fnd strctly dabatc ectronc states for whch d ( ) vanshes everywhere and ndeed such states are not terrbly useful snce a large number of strctly dabatc states would be requred to descrbe the ectronc structure. n practce one nstead wshes to fnd the lnear combnaton of a small set of adabatc states that s maxmally dabatc: ~ da ada = c k ( ) ( ) ( ) = where s the number of adabatc states (often ) that we are nterested n. f there s one nuclear degree of freedom we can do ths by choosng our maxmal dabatc states so that d ( ) s dagonal; that s so that ~ da ~ da j ( ) k ( ) δ jk Thus n ths case the dabatc bass becomes the set of ectronc states that dagonalze the nuclear knetc energy operator whereas the adabatc bass dagonalzes the ectronc amltonan. k j