5. THE ADIABATIC APPROXIMATION

Transcription

1 5. THE ADIABATIC APPROXIMATION In quantum mechancs, the adabatc approxmaton refers to those solutons to the Schrödnger equaton that make use of a tme-scale separaton between fast and slow degrees of freedom, and use ths to fnd approxmate solutons as product states n the fast and slow degrees of freedom. Perhaps the most fundamental and commonly used verson s the Born Oppenhemer (BO) approxmaton, whch underles much of how we conceve of molecular electronc structure and s the bass of potental energy surfaces. The BO approxmaton assumes that the moton of electrons s much faster than nucle due to ther large dfference n mass, and therefore electrons adapt very rapdly to any changes n nuclear geometry. That s, the electrons adabatcally follow the nucle. As a result, we can solve for the electronc state of a molecule for fxed nuclear confguratons. Gradually steppng nuclear confguratons and solvng for the energy leads to a potental energy surface, or adabatc state. Much of our descrptons of chemcal reacton dynamcs s presented n terms of propagaton on these potental energy surfaces. The barrers on these surfaces are how we descrbe the rates of chemcal reactons and transton state. The trajectores along these surfaces are used to descrbe mechansm. More generally, the adabatc approxmaton can be appled n other contexts n whch there s a tme-scale separaton between fast and slow degrees of freedom. For nstance, n the study of vbratonal dynamcs when the bond vbratons of molecules occur much faster than the ntermolecular motons of a lqud or sold. It s also generally mplct n a separaton of the Hamltonan nto a system and a bath, a method we wll often use to solve condensed matter problems. As wdely used as the adabatc approxmaton s, there are tmes when t breaks down, and t s mportant to understand when ths approxmaton s vald, and the consequences of when t s not. Ths wll be partcularly mportant for descrbng tme-dependent quantum mechancal processes nvolvng transtons between potental energy sources Born Oppenhemer Approxmaton As a startng pont, t s helpful to revew the Born Oppenhemer Approxmaton (BOA). For a molecular system, the Hamltonan can be wrtten n terms of the knetc energy of the nucle (N) and electrons (e) and the potental energy for the Coulomb nteractons of these partcles. Hˆ Tˆ Tˆ Vˆ Vˆ Vˆ e N ee NN en n N 1 me 1M n N n N 1 e 1 ZIZe 1 Ze 4 r r 4 R R 4 r R, j1 j I, 1 I 1 1 j I (5.1) Andre Tokmakoff, 11/7/14

2 5- Here and n the followng, we wll use lowercase varables to refer to electrons and uppercase to nucle. The varables n,, r, r, and me refer to the number, ndex, poston, Laplacan, and mass of electrons, respectvely, and N,, R, and M refer to the nucle. e s the electron charge, and Z s the atomc number of the nucleus. Note, ths Hamltonan does not nclude relatvstc effects such as spn-orbt couplng. The tme-ndependent Schrödnger equaton s ˆ ˆ E ˆ Hˆ r, Rˆ r, Rˆ r, R ˆ (5.) r, ˆ Rˆ s the total vbronc wavefuncton, where vbronc refers to the combned electronc and nuclear egenstates. Exact solutons usng the molecular Hamltonan are ntractable for most problems of nterest, so we turn to smplfyng approxmatons. The BO approxmaton s motvated by notng that the nucle are far more massve than an electron (me MI). Wth ths crteron, and when the dstances separatng partcles s not unusually small, the knetc energy of the nucle s small relatve to the other terms n the Hamltonan. Physcally, ths means that the electrons move and adapt rapdly adabatcally n response to shftng nuclear postons. Ths offers an avenue to solvng for by fxng the poston of the nucle, solvng for the electronc wavefunctons, and then teratng for varyng R to obtan effectve electronc potentals on whch the nucle move. Snce t s fxed for the electronc calculaton, we proceed by treatng R as a parameter rather than an operator, set T ˆN to, and solve the electronc TISE: Hˆ r,r ˆ r,r ˆ U R rˆ, R (5.3) el U are the electronc energy egenvalues for the fxed nucle, and the electronc Hamltonan n the BO approxmaton s Hˆ Tˆ Vˆ Vˆ (5.4) el e ee en In (5.3), s the electronc wavefuncton for fxed R, wth = referrng to the electronc ground state. Repeatng ths calculaton for varyng R, we obtan U(R), an effectve or mean-feld potental for the electronc states on whch the nucle can move. These effectve potentals are known as Born Oppenhemer or adabatc potental energy surfaces (PES). For the nuclear degrees of freedom, we can defne a Hamltonan for the h electronc PES: H ˆ T ˆ U ( R ˆ ) (5.5) Nuc, N whch satsfes a TISE for the nuclear wave functons ( R) :

3 5-3 Hˆ ( R) E ( R) (5.6) Nuc, Here refers to the th egenstate for nucle evolvng on the h PES. Equaton (5.5) s referred to as the BO Hamltonan. The BOA effectvely separates the nuclear and electronc contrbutons to the wavefuncton, allowng us to express the total wavefuncton as a product of these contrbutons (, rr) ( R) (, rr ) and the egenvalues as sums over the electronc and nuclear contrbuton: E E E. The BOA does not treat the nucle classcally. However, t s the bass for N e semclasscal dynamcs methods n whch the nucle evolve classcally on a potental energy surface, and nteract wth quantum electronc states. If we treat the nuclear dynamcs classcally, then the electronc Hamltonan can be thought of as dependng on R or on tme as related by velocty or momenta. If the nucle move nfntely slowly, the electrons wll adabatcally follow the nucle and systems prepared n an electronc egenstate wll reman n that egenstate for all tmes. 5.. Nonadabatc Effects Even wthout the BO approxmaton, we note that the nuclear-electronc product states form a complete bass n whch to express the total vbronc wavefuncton: (, rr) c ( R) (, rr ) (5.7),,,

4 5-4 We can therefore use ths form to nvestgate the consequences of the BO approxmaton. For a gven vbronc state, the acton of the Hamltonan on the wavefuncton n the TISE s, N el, Hˆ Tˆ R Hˆ R ( R) ( R ) (5.8) Expandng the Laplacan n the nuclear knetc energy va the chan rule as AB ( A) B ( A) B A B, we obtan an expresson wth three terms Hˆ ( R) Tˆ R U ( R) ( R),, N R, ( R) R ( R) M M, ( R) R ( R) Ths expresson s exact for vbronc problems, and s referred to as the coupled channel Hamltonan. Note that f we set the last two terms n (5.9) to zero, we are left wth H ˆ T U, whch s just the Hamltonan we used n the Born-Oppenhemer approxmaton, eq. (5.5). Therefore, the last two terms descrbe devatons from the BO approxmaton, and are referred to as nonadabatc terms. These depend on the spatal gradent of the wavefuncton n the regon of nterest, and act to couple adabatc Born Oppenhemer states. The coupled channel Hamltonan has a form that s remnscent of a perturbaton theory Hamltonan n whch the Born Oppenhemer states play the role of the zero-order Hamltonan beng perturbed ay a nonadabatc couplng V: ˆN (5.9) Hˆ Hˆ Vˆ (5.1) To nvestgate ths relatonshp further, t s helpful to wrte ths Hamltonan n ts matrx form. We obtan the matrx elements by sandwchng the Hamltonan between two projecton operators and evaluatng I,, j, I, j, BO Hˆ drr d (, rr) Hˆ (, rr) (, rr ). (5.11) Makng use of eq. (5.9) we fnd that the Hamltonan can be expressed n three terms where Hˆ dr ( R) Tˆ R U ( R) ( R) I,, j, I, N j j,, j d I, ( ) R j, ( ) j( ) M R R R F R I I d I, ( ) j, ( ) j( ) M R R R G R I I (5.1)

5 5-5 F R r r R r R * j ( ) d (, ) R j (, ) G R r r R r R * j ( ) d (, ) R j (, ) (5.13) The frst term n eqn. (5.1) gves the BO Hamltonan. In the latter two terms, F s referred to as the nonadabatc, frst-order, or dervatve couplng, and G s the second-order nonadabatc couplng or dagonal BO correcton. Although they are evaluated by ntegratng over electronc degrees of freedom, both depend parametrcally on the poston of the nucle. In most crcumstances the last term s much smaller than the other two, so that we can concentrate on the second term n evaluatng couplngs between adabatc states. For our purposes, we can wrte the nonadabatc couplng n equaton (5.1) as ˆ V R dr ( R) ( R) F ( R) I,, j, I, R j, j I M I (5.14) Ths emphaszes that the couplng between surfaces depends parametrcally on the nuclear postons, the gradent of the electronc and nuclear wavefunctons, and the spatal overlap of those wavefunctons Dabatc and Adabatc States Although the Born Oppenhemer surfaces are the most straghtforward and commonly calculated, they may not be the most chemcally meanngful states. As an example consder the potental energy curves for the datomc NaCl. The chemcally dstnct potental energy surfaces one s lkely to dscuss have dstnct atomc or onc character at large separaton between the atoms. These dabatc curves focus on physcal effects, but are not egenstates. In the fgure, the onc state a s nfluenced by the Coulomb attracton between ons that draws them together, leadng to a stable confguraton at Req once these attractve terms are balanced by nuclear repulsve forces. However, the neutral atoms (Na and Cl ) have a potental energy surface b whch s domnated by repulsve nteractons. The adabatc potentals from the BO Hamltonan wll reflect sgnfcant couplng between the dabatc electronc states. BO states of the same symmetry wll exhbt an avoded crossng where the electronc energy between correspondng dabatc states s equal. As expected from our earler dscusson, the splttng at the crossng for ths onedmensonal system would be Vab, twce the couplng between dabatc states.

6 5-6 The adabatc potental energy surfaces are mportant n nterpretng the reacton dynamcs, as can be llustrated wth the reacton between Na and Cl atoms. If the neutral atoms are prepared on the ground state at large separaton and slowly brought together, the atoms are weakly repelled untl the separaton reaches the transton state R. Here we cross nto the regme where the onc confguraton has lower energy. As a result of the nonadabatc couplngs, we expect that an electron wll transfer from Na to Cl, and the ons wll then feel an attractve force leadng to an onc bond wth separaton Req. Dabatc states can be defned n an endless number of ways, but only one adabatc surface exsts. In that respect, the term nonadabatc s also used to refer to all possble dabatc surfaces. However, dabatc states are generally chosen so that the nonadabatc electronc couplngs n eq. (5.13) are zero. Ths can be accomplshed by makng the electronc wavefuncton ndependent of R. As seen above, for coupled states wth the same symmetry the couplngs repel the adabatc states and we get an avoded crossng. However, t s stll possble for two adabatc states to cross. Mathematcally ths requres that the energes of the adabatc states be degenerate (E = E ) and that the couplng at that confguraton be zero (V =V =). Ths sn t possble for a onedmensonal problem, such as the NaCl example above, unless symmetry dctates that the nonadabatc couplng vanshes. To accomplsh ths for a Hermtan couplng operator you need two ndependent nuclear coordnates, whch enable you to ndependently tune the adabatc splttng and couplng. Ths leads to a sngle pont n the two-dmensonal space at whch degeneracy exsts, whch s known as a concal ntersecton (an mportant topc that s not dscussed further here) Adabatc and Nonadabatc Dynamcs OK, so what does the dscusson above mean for dynamcs? The BO approxmaton never explctly addresses electronc or nuclear dynamcs, but neglectng the nuclear knetc energy to obtan potental energy surfaces has mplct dynamcal consequences. As we dscussed for our NaCl example, movng the neutral atoms together slowly allows electrons to completely equlbrate about each forward step, resultng n propagaton on the adabatc ground state. Ths s the essence of the adabatc approxmaton. If you prepare the system n, an egenstate of H at the ntal tme t, and propagate slowly enough, that wll evolve as an egenstate for all tmes: H t t E t t (5.15) Equvalently ths means that the n th egenfuncton of Ht wll also be the n th egenfuncton of H t. In ths lmt, there are no transtons between BO surfaces, and the dynamcs only reflect the phases acqured from the evolvng system. That s the tme propagator can be expressed as

7 5-7 Utt (, ) adabatc exp dte t t (5.16) In the opposte lmt, we also know that f the atoms were ncdent on each other so fast (wth such hgh knetc energy) that the electron dd not have tme to transfer at the crossng, that the system would pass smoothly through the crossng along the dabatc surface. In fact t s expected that the atoms would collde and recol. Ths mples that there s an ntermedate regme n whch the velocty of the system s such that the system wll splt and follow both surfaces to some degree. In a more general sense, we would lke to understand the crtera for adabatcty that enable a tme-scale separaton between the fast and slow degrees of freedom. Speakng qualtatvely about any tme-dependent nteracton between quantum mechancal states, the tme-scale that separates the fast and slow propagaton regmes s determned by the strength of couplng between those states. We know that two coupled states exchange ampltude at a rate dctated by the Rab frequency R, whch n turn depends on the energy splttng and couplng of the states. For systems n whch there s sgnfcant nonperturbatve transfer of populaton between two states a and b, the tme scale over whch ths can occur s approxmately t 1/R /Vab. Ths s not precse, but s provdes a reasonable startng pont for dscussng slow versus fast. Slow n an adabatc sense would mean that a tme-dependent nteracton act on the system over a perod such that t << /Vab In the case of our NaCl example, we would be concerned wth the tme scale over whch the atoms pass through the crossng regon between dabatc states, whch s determned by the ncdent velocty between atoms. Adabatcty crteron Let s nvestgate these ssues by lookng more carefully at the adabatc approxmaton. Snce the adabatc states () t are orthogonal for all tmes, we can evaluate the tme propagator as e U t and the tme-dependent wavefuncton s E (') t dt (5.17) t b () t e E (') t dt (5.18) Although these are adabatc states we recognze that the expanson coeffcents can be tmedependent n the general case. So, we would lke to nvestgate the factors that govern ths tme-

8 5-8 dependence. To make the notaton more compact, let s defne the tme-rate of change of the egenfuncton as () t (5.19) t If we substtute the general soluton eq. (5.18) nto the TDSE, we get E(') t dt E(') t dt b b E b e be e (5.) Note, the thrd term on the left hand sde equals the rght hand term. Actng on both sdes from the left wth leads to E(') t dt E(') t dt be b e (5.1) We can break up the terms n the summaton nto one for the target state and one for the remanng states. b b b exp dt E ( t') (5.) E () t E () t E () t where. The adabatc approxmaton apples when we can neglect the summaton n eq. (5.), or equvalently when for all. In that case, the system propagates on the adabatc state ndependent of the other states: b b. The evoluton of the coeffcents s t b( t) b ()exp ( t) ( t) dt t b()exp E( t) dt (5.3) Here we note that n the adabatc approxmaton E ( t) ( t) H( t) ( t). Equaton (5.3) ndcates that n the adabatc approxmaton the populaton n the states never changes, only ther phase. The second term on the rght n eq. (5.) descrbes the nonadabatc effects, and the overlap ntegral t (5.4) determnes the magntude of ths effect. s known as the nonadabatc couplng (even though t refers to couplngs between adabatc surfaces), or as the geometrcal phase. Note the

9 5-9 parallels here to the expresson for the nonadabatc couplng n evaluatng the valdty of the Born- Oppenhemer approxmaton, however, here the gradent of the wavefuncton s evaluated n tme rather than the nuclear poston. It would appear that we can make some connectons between these two results by lnkng the gradent varables through the momentum or velocty of the partcles nvolved. So, when can we neglect the nonadabatc effects? We can obtan an expresson for the nonadabatc couplng by expandng H E t (5.5) and actng from the left wth, whch for leads to H (5.6) E E For adabatc dynamcs to hold, and so we can say H E E E (5.7) So how accurate s the adabatc approxmaton for a fnte tme-perod over whch the systems propagates? We can evaluate eq. (5.), assumng that the system s prepared n state and that the occupaton of ths state never vares much from one. Then the occupaton of any other state can be obtaned by ntegratng over a perod as Here I used e 1 e sn b exp dte ( t') H b exp E 1 E H E E e sn E. For 1, we expand the sn term and fnd b (5.8) H E / (5.9) t Ths s the crteron for adabatc dynamcs, whch can be seen to break down near adabatc curve crossngs where E, regardless of how fast we propagate through the crossng. Even away from curve crossng, there s always the possblty that nuclear knetc energes are such that ( H / t) wll be greater than or equal to the energy splttng between adabatc states.

10 Landau Zener Transton Probablty Clearly the adabatc approxmaton has sgnfcant lmtatons n the vcnty of curve crossngs. Ths phenomenon s better descrbed through transtons between dabatc surfaces. To begn, how do we lnk the temporal and spatal varables n the curve crossng pcture? We need a tme-rate of change of the energy splttng, E de / dt. The Landau Zener expresson gves the transton ab probabltes as a result of propagatng through the crossng between dabatc surfaces at a constant E. If the energy splttng between states vares lnearly n tme near the crossng pont, then settng the crossng pont to t = we wrte E E Et (5.3) a If the couplng between surfaces V ab s constant, the transton probablty for crossng from surface a to b for a trajectory that passes through the crossng s b P ba V ab 1exp E (5.31) and P 1 P. Note f V then P, but f the splttng sweep rate E s small as aa determned by ba ab ba E (5.3) Vab then we obtan the result expected for the adabatc dynamcs Pba 1. We can provde a classcal nterpretaton to eq. (5.31) by equatng E wth the velocty of partcles nvolved n the crossng. We defne the velocty as v R/ t, and the slope of the dabatc surfaces at the crossng, F E / R. Recognzng we fnd E E / t vf F (5.33) a b a b P ba Vab 1exp vfa F b (5.34) In the context of potental energy surfaces, what ths approxmaton says s that you need to know the slopes of the potentals at ther crossng pont, the couplng and ther relatve velocty n order to extract the rates of chemcal reactons.

11 5-11 Readngs 1. Truhlar, D. D., Potental Energy Surfaces. In The Encyclopeda of Physcal Scence and Technology, 3rd ed.; Meyers, R. A., Ed. Academc Press: New York, 1; Vol. 13, pp Tully,. C., Nonadabatc Dynamcs Theory.. Chem. Phys. 1, 137, A31.