Landau Zener Transitions Through Small Electronic Eigenvalue Gaps in the Born-Oppenheimer Approximation

Transcription

1 Landau Zener Transitions Through Small Electronic Eigenvalue Gaps in the Born-Oppenheimer Approximation George A. Hagedorn Alain Joye Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia , U.S.A. August 26, 997 Abstract We study the propagation of molecular wave packets through the simplest two types of avoided crossings of electronic energy levels in a limit where the gap between the eigenvalues shrinks as the nuclear masses are increased. For these types of avoided crossings, the electron energy levels essentially depend on only one of the nuclear configuration parameters, as is the case for all diatomic molecules. We find that the transition probabilities are of order and are determined by the Landau Zener formula. CPT-95/P.325 Partially Supported by National Science Foundation Grant DMS 9434 Supported by Fonds National Suisse de la Recherche Scientifique, Grant Permanent address: Centre de Physique Théorique, CNRS Marseille, Luminy Case 97, F-3288 Marseille Cedex 9, France and PHYMAT, Université de Toulon et du Var, B.P.32, F La Garde Cedex, France

2 Introduction Because it is not practical to solve the time dependent Schrödinger equation directly, the time dependent Born Oppenheimer approximation is a principal tool for studying molecular dynamics. This approximation makes use of the smallness of the parameter ɛ, where ɛ 4 is the ratio of the mass of an electron to the average of the masses of the nuclei. In the standard time dependent Born Oppenheimer approximation, the electrons and nuclei are treated separately, but their motions are coupled. The electrons move much faster than the nuclei, and they quickly adjust their motion in response to the relatively slow nuclear motion. The electrons remain approximately in a quantum mechanical bound state as though the nuclei were at fixed classical positions. This is the adiabatic approximation for the electrons. The motion of the nuclei is accurately described by the semiclassical approximation because the nuclei have large masses. The electronic and nuclear motions are coupled because the energy level of the electronic bound state depends on the positions of the nuclei, and the electronic energy level plays the role of an effective potential for the semiclassical dynamics of the nuclei. This physical intuition is the basis for rigorous asymptotic expansions of solutions to the molecular time dependent Schrödinger equation [6, 8,, 5, 6]. However, the validity of the approximation is dependent upon the assumption that the electron energy level of interest is well isolated from the rest of the spectrum of the electronic Hamiltonian. Readers interested in the mathematical literature concerning the validity of Born Oppenheimer approximations should consult [2, 3, 4, 6, 8, 9,,, 2, 3, 5, 6, 7, 2, 32, 34, 35, 36, 38, 39, 4, 4, 42]. The assumption that the electron energy level of interest is isolated from the rest of the spectrum can break down in various ways. For example, two electron energy levels may cross one another for some nuclear configurations. The effects of such crossings on molecular propagation have been studied recently in generic minimal multiplicity situations [3, 7, 6]. Another situation where the standard approximation breaks down is an avoided crossing, where two electron energy levels approach close to one another, but do not actually cross. Generic avoided crossings of energy levels that have the minimal multiplicity allowed by the symmetry group have been classified, and normal forms for the electron Hamiltonian near these avoided crossings have been determined [8]. In [8] it is shown that there are six distinct types of these avoided crossings. In this paper we study molecular propagation through the simplest two types of avoided crossings described in [8]. In these types of crossings, the electron energy levels essentially depend on only one parameter in the nuclear configuration space. In practice, this occurs for diatomic molecules, where the electron energy levels depend only on the distance between the nuclei because of rotational symmetry. There are two types of such avoided crossings because of the possible presence of time reversing operators in the symmetry group of the electron Hamiltonian. The details of the situation dictate whether minimal multiplicity energy levels are of multiplicity Type Avoided Crossings or multiplicity 2 Type 2 Avoided Crossings. Our main result is the determination of what happens when a standard time dependent Born Oppenheimer molecular wave packet propagates through one of these avoided crossings if the gap size is on the order of ɛ. Using matched asymptotic expansions we explicitly 2

3 compute approximate solutions to the molecular Schrödinger equation. We observe that to leading order in ɛ, the Landau Zener formula correctly describes the probabilities for the system to remain in the original electronic level or to make a transition to the other electronic level involved in the avoided crossing. To apply the Landau Zener formula in this case, one treats the nuclei as classical point particles to obtain a time dependent Hamiltonian for the electrons. This leads to the study of the adiabatic limit of an effective time dependent system with two levels isolated in its spectrum. The transition probabilty between the levels of such systems in the adiabatic limit is known for a variety of situations [24, 26, 25, 28, 29, 2, 3, 3]. In particular, when the levels display an avoided crossing, the Landau-Zener formula is valid [4, 27, 22, 37, 23]. In our case, we can apply the Landau Zener formula to the resulting time dependent Schrödinger equation for the electrons alone. More precisely, suppose there is a generic Type or Type 2 avoided crossing at nuclear configuration x =. In an appropriate coordinate system, the gap between the electron energy levels is 2 b x + b 2 ɛ 2 + c 2 ɛ 2 + Ox 2 + ɛ 2,. with b and c 2 non-zero. Suppose that a semiclassical nuclear wave packet passes through the avoided crossing with velocity µ, whose first component is µ. Then the probability of remaining in the same electronic state is e πc2 2 /b µ + Oɛ p,.2 for some p >, and the probability of making a transition to the other electronic level involved in the avoided crossing is e πc2 2 /b µ + Oɛ p..3 This result is completely different from what one obtains for the four other types of avoided crossings, where the corresponding probabilities depend on the particular nuclear wave packet involved [9]. The reason is that in Type and Type 2 avoided crossings, every part of the nuclear wave packet passes through the same size minimum gap between the eigenvalues. In Types 3, 4, 5, and 6 avoided crossings, different parts of the nuclear wave packet feel different size gaps as they pass through the avoided crossing. Our results are also completely different from those obtained in the case of true level crossings [6]. For codimension crossings, transition amplitudes produced by crossings are of order ɛ, not order as in the present paper. Furthermore, the dependence of transition amplitudes on the nuclear velocity µ is exponential as opposed to algebraic in the crossing case. Also, although the problem we study in this paper is somewhat less singular than the case of true crossings, the technical details are more difficult. This stems largely from the complicated ɛ dependence of the classical mechanics in our problem. In [6], the classical mechanics had no ɛ dependence. The Hamiltonian for a molecular system with K nuclei and N K electrons has the form Hɛ = K ɛ4 xj j= 2M j N j=k+ 2m j xj + i<j V ij x i x j..4 3

4 Here x j IR l denotes the position of the j th particle, the mass of the j th nucleus is M j for j K, the mass of the j th electron is m j for K + j N, and V ij is the potential between particles i and j. For convenience we assume M j = for j K. We set n = Kl and let x = x, x 2,..., x Kl IR n denote the nuclear configuration vector. We decompose Hɛ as Hɛ = ɛ4 2 x + hx..5 This defines the electronic Hamiltonian hx that depends parametrically on x. The time dependent Schrödinger equation that we study is i ɛ 2 ψ t = Hɛ ψ,.6 for t in a fixed interval. The factor of ɛ 2 on the left hand side of this equation indicates a particular choice of time scaling. Other choices could be made, but this choice is the distinguished limit [] that produces the most interesting leading order solutions. With this scaling, all terms in the equation play significant roles at leading order, and the nuclear motion has a non-trivial classical limit. This is also the scaling for which the mean initial nuclear kinetic energy is held constant as ɛ tends to zero. In this paper we are interested in the simplest types of electronic transitions that are not associated with level crossings. If the Hamiltonian has the form.5, then to arbitrarily high order in powers of ɛ, the solutions to.6 have no electronic transitions [8, ]. There are no known rigorous results about infinite order processes in the time dependent Born Oppenheimer approximation. However, in real molecular systems, only a single value of ɛ is usually of interest, and hx may have two eigenvalues that approach one another with a minimum gap size that is of the same order of magnitude as the relevant value of ɛ. Under these circumstances, there can be significant transitions between electon energy levels. To generate useful rigorous information about these transitions without the difficulties of going beyond infinte order, we assume the electron Hamiltonian of interest can be embedded in an ɛ dependent family that has a crossing when ɛ =. This is in the same spirit as dealing with quantum mechanical resonances as perturbations of eigenvalues embedded in the continuous spectrum. Thus, we study solutions to.6 where the Hamiltonian has the form Hɛ = ɛ4 2 x + hx, ɛ,.7 with the assumption that hx, ɛ has an Avoided Crossing according to the following definition: Definition Suppose hx, ɛ is a family of self adjoint operators with a fixed domain D in a Hilbert Space H, for x Ω and ɛ [, α, where Ω is an open subset of IR n. Suppose that the resolvent of hx, ɛ is a C 4 function of x and ɛ as an operator from H to D. Suppose hx, ɛ has two eigenvalues E A x, ɛ and E B x, ɛ that depend continuously on x and ɛ and are isolated from the rest of the spectrum of hx, ɛ. Assume 4

5 Γ = { x : E A x, = E B x, } is a single point or non-empty connected proper submanifold of Ω, but that for all x Ω, E A x, ɛ E B x, ɛ when ɛ >. Then we say hx, ɛ has an Avoided Crossing on Γ. Remark: Realistic molecules have Coulomb potentials which give rise to electron Hamiltonians that do not satisfy the smoothness assumptions of this definition. However, one should be able to accommodate Coulomb potentials by using the regularization techniques of [,, 35]. Avoided Crossings of minimal multiplicity energy levels are classified in [8], and normal forms for the electron Hamiltonian near Γ are derived. When Γ has codimension, there are two types. To describe these, we need some notation. Assume without loss of generality that is a generic point of Γ, and decompose with and hx, ɛ = h x, ɛ + h x, ɛ.8 h x, ɛ = hx, ɛp x, ɛ.9 h x, ɛ = hx, ɛi I P x, ɛ. where P x, ɛ is a spectral projector of hx, ɛ associated with E A x, ɛ and E B x, ɛ. In a Type Avoided Crossing, Γ has codimension and the two eigenvalues each have multiplicity. There exists [8] an orthonormal basis {ψ x, ɛ, ψ 2 x, ɛ} of P x, ɛh, which is regular in x, ɛ around,. In this basis, h x, ɛ has the form h x, ɛ = h x, ɛ + Ṽ x, ɛ βx, ɛ γx, ɛ + iδx, ɛ = + Ṽ x, ɛ. γx, ɛ iδx, ɛ βx, ɛ where Ṽ x, ɛ = tracehx, ɛp x, ɛ is a regular function of x, ɛ around the origin and βx, ɛ = b x + b 2 ɛ + O2.2 γx, ɛ = c 2 ɛ + O2 δx, ɛ = O2 Ṽ x, ɛ = O where b >, c 2 >, b 2 IR and m/2 n Om = O, x = ɛ..3 x 2 j j= Type 2 Avoided Crossings are similar, except that the minimal multiplicity of eigenvalues allowed by the symmetry group is 2. Near one of these avoided crossings, one can choose 5

6 an orthonormal basis {ψ x, ɛ, ψ 2 x, ɛ, ψ 3 x, ɛ, ψ 4 x, ɛ, } of P x, ɛh, which is regular in x, ɛ around,. In this basis, h x, ɛ has the form h x, ɛ = h x, ɛ + Ṽ x, ɛ βx, ɛ γx, ɛ + iδx, ɛ γx, ɛ iδx, ɛ βx, ɛ = βx, ɛ γx, ɛ + iδx, ɛ γx, ɛ iδx, ɛ βx, ɛ + Ṽ x, ɛ,.4 where β, γ, δ, and Ṽ are as above [8]. Throughout the paper we assume that the nuclei move transversally through Γ at the point. This means the classical momentum associated with their semiclassical wave packets has a non-trivial component in the x direction when their classical position is passing through Γ. Precise statements of our results require a considerable amount of notation and are presented in Theorems 3. and 4. for Type and Type 2 Avoided Crossings, respectively. We have stated these theorems with the incoming state associated with the lower of the two relevant levels. The analogous results with the incoming state associated with the upper level are also true and proved in the same way, with the obvious changes. Immediate corollaries of the two theorems are that the Landau Zener formula described above gives the correct transition probabilities in each case. The main technique we use is matched asymptotic expansions. We use the standard time dependent Born Oppenheimer approximate solutions to the Schrödinger equation when the nuclei are far enough away from Γ. We match these to inner solutions when the system is near Γ and the standard approximation breaks down. The paper is organized as follows: In Section 2 we discuss the ordinary differential equations whose solutions will be used to describe the semiclassical motion of the nuclei. In Section 3 we discuss semiclassical nuclear wave packets and adiabatic motion of the electrons. We then prove our main result for Type Avoided Crossings, Theorem 3. by using matched asymptotic expansions. In Section 4 we state our main result, Theorem 4., for Type 2 Avoided Crossings and describe the modifications of Section 3 that are required to prove this result. Acknowledgements. George Hagedorn wishes to thank the Centre de Physique Théorique, C.N.R.S., Marseille for its hospitality during July 994. Alain Joye wishes to thank Virginia Polytechnic Institute and State University for its hospitality during August October 994 and July September 995, and the Fonds National Suisse de la Recherche Scientifique for financial support. 6

7 . A Convenient Change of Variables We consider the Schrödinger equation iɛ 2 t ψx, t = ɛ4 ψx, t + hx, ɛψx, t.5 2 In order to get rid of ɛ-dependence in the leading order of βx, ɛ in., we introduce the new variables b 2 ɛ x = b x +., ɛ = c 2 ɛ, t = b 2 /c 2 2t..6 In terms of these new variables, the Schrödinger equation.5 for becomes φx, t = ψxx, ɛ, tt.7 iɛ 2 t φx, t = ɛ 4 2 φx, t + c4 2 hxx, ɛ, ɛɛ φx, t.8 b 2 in the limit ɛ, with x h xx, ɛ, ɛɛ = ɛ ɛ x + O2 + Ṽ xx, ɛ, ɛɛ.9 where Ṽ xx, ɛ, ɛɛ is regular in x, ɛ around, and O2 refers to x and ɛ. We introduce the fixed parameter r = c 4 2/b 2 > and henceforth drop the primes on the new variables. We assume that h x, ɛ has the form. with the following local behavior around x = and ɛ = : with r >. βx, ɛ = rx + O2.2 γx, ɛ = rɛ + O2 δx, ɛ = O2 Ṽ x, ɛ = O 2 Ordinary Differential Equations of Semiclassical Mechanics In Section 3. we introduce semiclassical wave packets for the nuclei. The leading order semiclassical motion for these wave packets is determined by the solutions to certain systems of ordinary differential equations. These involve classical mechanics, the classical action associated with a classical trajectory, and the dynamics of certain matrices that describe the position and momentum uncertainties of the wave packets. The goal of this section is 7

8 to study the small ɛ behavior of these classical quantities that we need for the asymptotic matching procedure that we use in Section 3 to prove our main results. We define V A B x, ɛ = Ṽ x, ɛ ± β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ 2. where x IR n, ɛ >. Let a C t and η C t be the solutions of the classical equations of motion d dt ac t = η C t 2.2 d dt ηc t = V C a C t, ɛ, C = A, B. with initial conditions a C = 2.3 η C = η ɛ, with η ɛ = η + Oɛ, and η >, where the Oɛ term depends on whether C is A or B. Noticing that it follows from.2 that βx, ɛ, γx, ɛ and δx, ɛ are O, and using estimates of the type β/ β 2 + γ 2 + δ 2, we see that V C x, ɛ = O. 2.4 This last condition implies the existence and uniqueness of the solutions of 2.2. The small ɛ perturbation theory for solutions to 2.2 and 2.3 is not simple because of the presence of two different time scales. To overcome the difficulties, we use matched asymptotic expansions. When t is large compared to ɛ, we use a formal perturbation expansion, where the unperturbed problem is the classical mechanics associated with the potential V C x,. This gives rise to the incoming outer solution for t < and the outgoing outer solution for t >. These outer solutions are not valid approximations unless t is large compared to ɛ. When t is small compared to ɛ 2/3, we use a different expansion to construct the inner solution that depends on the rescaled independent variable s = t/ɛ. The technical difficulties are to prove the validity of these approximate solutions and to prove that the inner and outer approximations agree with one another to the appropriate order in ɛ in the matching regimes where ɛ t ɛ 2/3. 2. Small t Asymptotics In order to get started, we need preliminary information about the behavior of the solution of 2.2, 2.3 when both t and ɛ are small. Lemma 2. Let a C t and η C t be the solutions of 2.2 and 2.3. If ɛ and t are small enough, we have { a C t = η ɛt + Ot 2 as t, uniformly in ɛ. η C t = η ɛ + Ot 8

9 Proof: We mimic the proof of [6], p.82. Let us drop the index C in the notation. We want to show that there exists T >, such that at = η ɛt + ut, ɛ 2.5 bt = η ɛ + vt, ɛ where ut, ɛ/t 2 and vt, ɛ/t are uniformly bounded if ɛ 2 + t 2 < T 2, i.e., if t, ɛ B T. We let Y T be the Banach space of pairs of bounded, continuous in t, vector valued functions for t, ɛ B T, with the norm u v u For Y v T, we define Y T = sup t,ɛ B T ut, ɛ/t 2 + sup vt, ɛ/t. 2.6 t,ɛ B T u t vs, ɛds F = v t V. 2.7 η ɛs + us, ɛ, ɛds Due to the estimate 2.4, F maps Y T into Y T, and furthermore, any fixed point of F gives rise to a solution of 2.2, 2.3 by way of 2.5. If we show that F is a strict contraction, it follows from the contraction mapping principle that the solution of 2.2, 2.3 exists, is unique and satisfies the assertion of the lemma. We show that this is the case for T small enough. Let u v, u2 v 2 Y T and let us estimate the norm of u3 v 3 = F u v F u 3 t, ɛ/t 2 = t v t 2 s, ɛ v 2 s, ɛds 2.8 u2 t s sup v t 2 t, ɛ v 2 t, ɛ ds t,ɛ B T u u v v 2 Y T For the other component we use the mean value theorem V η ɛs + u s, ɛ, ɛ V η ɛs + u 2 s, ɛ, ɛ 2. = V 2 ζs, ɛ, ɛu s, ɛ u 2 s, ɛ where V 2 stands for the Hessian of V and ζs, ɛ = η ɛs + u s, ɛ + θs, ɛu 2 s, ɛ u s, ɛ 2. with θs, ɛ ], [. There will appear several constants, independent of ɛ in the sequel which we shall denote generically by c. By hypothesis, u j s, ɛ cs 2, j =, 2, uniformly in ɛ and η ɛ = η + Oɛ with η >. Consequently, there exists a constant c, independent of ɛ, such that v 2. ζ s, ɛ c s

10 By explicit computation we get, omitting the arguments x, ɛ V 2 = Ṽ 2 ± β β + γ γ + δ δ β 2 + γ 2 + δ 2 /2 +ββ 2 + γγ 2 + δδ 2 β 2 + γ 2 + δ 2 /2 β β + γ γ + δ δ β β + γ γ + δ δ β 2 + γ 2 + δ 2 3/ Using the behaviors.2 again and estimates of the type βγ/β 2 + γ 2 + δ 2, we get V 2 x, ɛ c β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ where c is independent of x, ɛ. Then, 2.4 β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ = r 2 x 2 + ɛ 2 + O3, 2.5 with ζs, ɛ = Os and 2.2 we get β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ x=ζs,ɛ = r 2 ζ 2 s, ɛ + ɛ 2 + Os 3 + ɛ 3 if s is small enough. Hence c ζ 2 s, ɛ + ɛ 2 cs V 2 ζs, ɛ, ɛu 2 s, ɛ u s, ɛ c u 2 s, ɛ u s, ɛ / s c s u u2, 2.7 v v 2 Y T for some c independent of ɛ. Thus v 3 t, ɛ / t c t s t c t u 2 v u v u2 v 2 u2 Choosing T small enough so that c t /2 /2, we get the result. v 2 Y T Y T. 2.8 If we replace the potential V C x, ɛ by Ṽ x, ɛ which is regular as x and ɛ, we can go further in the asymptotics, as is easily checked. Lemma 2.2 Let at and ηt be the solutions of 2.2 and 2.3 with V C x, ɛ If ɛ and t are small enough, we have Ṽ x, ɛ. { at = η ɛt Ṽ, ɛ t2 2 + Ot3 ηt = η ɛ Ṽ, ɛt + Ot2 as t, uniformly in ɛ.

11 2.2 Asymptotics for Inner Solution To construct the inner solution, we define rescaled quantities s and z C s by t = ɛs a C t = ɛz C s In terms of these new variables, equations 2.2 and 2.3 become d 2 ds 2 zc s = ɛ V C ɛz C s, ɛ 2.2 d ds zc = η ɛ z C =. We construct the inner solution by making the ansatz that z C s = z C s + ɛz C s +... and computing the functions z C s, ɛz C s,.... For our purposes, it is sufficient to truncate the expansion after two terms and to use a C t = ɛz C t/ɛ + ɛ 2 z C t/ɛ as our inner solution. The following proposition gives a precise statement that this is a valid approximation to the exact solution in the appropriate time regime defined by the conditions: s, ɛ and ɛs 3. Proposition 2. Let z C s be the solution of 2.2. behaviors for small values of ɛ and ɛs 3 : We have the following asymptotic where z C s = z C s + ɛz C s + Oɛ 2 s 3 d ds zc s = d ds zc s + ɛ d ds zc s + Oɛ 2 s 2, z A B s = η ɛs, z A B s = Ṽ, ɛs2 2 ± [ r η ɛ s r 2ηɛ s ηɛs r ] 2ηɛ ln ηɛs + ηɛs Proof: The idea is to expand the potential V C ɛz C s, ɛ around the origin in IR n+ and to insert an a priori expansion of the type z C s = z C s + ɛz C s + ɛ 2 z C 2 s

12 in 2.2. Then we solve the resulting equations order by order in ɛ, using η ɛ = Oɛ and lemma 2.. We let w C s denote the first component of z C s and write V A B ɛz C s, ɛ = Ṽ, ɛ 2.23 ɛw A B s { r. + O2 ɛ 2 w A 2 /2 B s + ɛ 2 + O3} + O, where On is to be understood with ɛz in place of x. Note that ɛs 3 implies ɛs 2 and ɛs. Since by lemma 2., z C s = η ɛs + Oɛs 2, 2.24 where ɛs 2 is small, we have On = Oɛ n s n, and the equation reads d 2 ds z A 2 B s = ɛ Ṽ, ɛ + Oɛ2 s 2.25 w A B s { } rɛ. + Oɛ2 s 2 w A 2 /2 B s + + Oɛs 3 as ɛs 3 and ɛ. Thus the leading order of the equation for the lowest term in ɛ of the expansion 2.22 becomes d 2 ds 2 zc s = 2.26 so that, with the initial conditions z C = and d ds zc = η ɛ, see 2.2, we get, as expected, z C s = η ɛs Then the leading order of the equation for the next term in the expansion 2.22 yields d 2 ds z A B 2 s = Ṽ, ɛ r ηɛs ηɛs The initial conditions are now z A B = and d z A B ds =, which gives z A B s = Ṽ, ɛs2 2 ± [ r ηɛ s r 2ηɛ s ηɛs r ] + 2ηɛ ln ηɛs + ηɛs

13 Let us estimate the difference between the actual solution of 2.2 and the formal approximation just constructed. We compute d 2 z AB A s z B ds 2 s ɛz A B s 2.3 w = ɛr. A B s η ɛs w A 2 B η s + + Oɛs3 ɛs 2 + Oɛ 2 s 2 +. w A B 2 s + + Oɛs3 From lemma 2. and the property ɛs, we have w A 2 /2 B s + + Oɛs 3 = ηɛs Oɛs /2 and finally d 2 ds 2 z AB s z A B s ɛz A B s = η ɛs 2 + /2 + Oɛ, 2.3 = Oɛ 2 s By integration, and taking the initial conditions into account, we get d z AB A s z B s ɛz A B ds s = Oɛ 2 s 2, 2.33 z A A B s z B s ɛz A B s = Oɛ 2 s Corollary 2. Further expanding we get in the same regime in the limit s z A B s = η ɛs Ṽ, ɛɛs2 2 ± r ηɛ ɛs. [ rsigns ɛ s2 ln s + ɛ 2 2η 2 ɛ + ɛ 4η 2 ɛ + ɛ ] ln2η ɛ 2η 2 ɛ +Oɛ/s 2 + Oɛ 2 s The asymptotics for d ds zc s in the same regime are obtained by termwise differentiation of the above formulae up to errors Oɛ/s 3 + Oɛ 2 s 2. Remark: The remainders in these formulae tend to zero in the inner regime, since we assume ɛs 3. So, we only make use of the inner solution when s ɛ ξ, with < ξ < /

14 2.3 Asymptotics for the Outer Solutions We now construct the outer solutions to 2.2 and 2.3 that are valid approximations to the exact solutions when t is large compared to ɛ. To do so, we assume an expansion of the form a C t = a C t + g ɛa C t + g 2 ɛa C 2t and substitute it into the equation. We expand V C a C t, ɛ accordingly, using the condition ɛ/t and lemma 2. again, and solve the resulting equations for a C j t, order by order, the dependence in ɛ of the g j ɛ being determined in the course of the computation by matching the inner solution. We have to deal with the two cases t > separately. The constants of < integration are determined by matching with the asymptotics obtained for the inner solution. For our purposes we define the outer solutions by truncation of the expansions after the terms of order ɛ 2. Proposition 2.2 Let a C t be the solution of 2.2 and 2.3. In the regime ɛ, t, t /ɛ and t 3 /ɛ 2 we have the asymptotics a A B t = Ṽ, ɛt2 2 + η ɛt ± r ηɛ ɛt. signt [ r t Ot 3 + Oɛ 4 /t 2 ɛ2 ln t 2ηɛ 2 + ɛ 2 ] 4ηɛ ln2η ɛ ɛ2 ln ɛ 2ηɛ 2. The asymptotics for η C t in the same regime is obtained by termwise differentiation of the above formulae up to errors Ot 2 + Oɛ 4 /t 3. Proof: For convenience, we drop the ɛ-dependence of η in the notation. We have V A B a C t, ɛ = ± Ṽ, ɛ + O 2.38 a C t r. + O2 a C t + ɛ/a C t 2 + O3/a C t 2 where On with a C t = Ot in place of x and ɛ/t gives On = Ot n With the property a C t c t 2.4 4

15 for a constant c uniform in ɛ, we get for the leading order r d 2 dt a A B 2 t = Ṽ, ɛ signt By integration r a A B t = Ṽ, ɛt2 2 signt t C ɛt + C ɛ 2.42 where C and C are time independent vectors. Comparing with the expansion of the inner solution in corollary 2. we deduce C ɛ = η ɛ ± ɛr ηɛ C ɛ = Note that the ɛ-dependence of the a A B j t will cause no problem, see below. Let us determine the next order. Using 2.39 and 2.4, we further expand + ɛ/a C t 2 + O3/a C t 2 / to get the equation = ɛ2 2a C2 t + Ot + Oɛ4 /t 4 d 2 dt a A B 2 t + g ɛa A B t = Ṽ, ɛ signtr. ɛ ±signt. 2 r + O t + Oɛ 4 /t 4. 2 a A B t + g ɛa A B t + 2 Thus, it follows from 2.4 and 2.42 that the next order must satisfy g ɛ d2 dt a A B ɛ 2 t = ±signt. 2 r 2ηt

16 Comparison with the asymptotics of the inner solution shows that we must choose g ɛ = ɛ 2 and integration yields r a A B t = signt. r 2 ln t + D ɛt + D ɛ η where D ɛ and D ɛ are constant vectors in IR n. By matching with corollary 2., we get D ɛ = 2.49 [ r D ɛ = signt. 2η 2 /2 + ln2η r ln ɛ ] 2η 2. We now have to show that equations 2.42 and 2.48 correctly give the asymptotic behavior of a C t in the chosen regime. We compute d 2 dt a A A 2 B t a B t ɛ 2 a A B t 2.5 = Ṽ a A B t, ɛ + Ṽ, ɛ r. + ɛ/a C t 2 + Ot + ɛ2 2ηt 2 +Ot/ + Oɛ 2 /t 2 + Ot. Using lemma 2. again we have /2 + ɛ2 a C2 t + Ot where ɛ 2 /t = Ot. Finally d 2 = ɛ2 2a C2 t + Ot + Oɛ4 /t = ɛ2 2η t 2 + Oɛ2 /t + Ot + Oɛ 4 /t 4 dt a A A 2 B t a B t ɛ 2 a A B t = Ot + Oɛ 4 /t from which the result follows. 2.4 Classical Action Integrals In Section 4 we construct quantum mechanical wave functions by using matched asymptotic expansions. To do so, we need the small t asymptotics of classical action integrals associated 6

17 with the outer solutions. Let t S C t = ηc2 t V C a C t, ɛ dt = t η C2 t dt η 2 ɛ t 2 V C a C, ɛt, 2.54 and let St be the same quantity for V C x, ɛ Ṽ x, ɛ. From lemma 2.2 we easily deduce Lemma 2.3 As t, St = η 2 ɛ t 2 Ṽ, ɛt η ɛ Ṽ, ɛt2 + Ot 3, 2.55 uniformly in ɛ. From proposition 2.2, and the formula V A B, ɛ = Ṽ, ɛ ± ɛr + Oɛ Lemma 2.4 In the regime ɛ, t, t /ɛ and t 3 /ɛ 2 we have the asymptotics S A B t = S A B ɛ, signt Ṽ, ɛt + η2 ɛ t 2 η ɛ Ṽ, ɛt2 ± rɛt signt rηɛt 2 + r ηɛ ɛ2 ln t + Ot 3 + Oɛ 4 /t 2 + Oɛ 3 ln t. 2.5 Different Initial Momenta For later purposes, we assume from now on that the solution at of 2.2 with V C x, ɛ Ṽ x, ɛ is subject to the initial conditions a = 2.57 η = η whereas the solutions a C t satisfy a C = 2.58 η C = η ɛ = η + Oɛ. The Oɛ term must be included in our calculations because when the electrons makes a transition from one energy level surface to another, the nuclei must compensate by making a change in their kinetic energy in order to conserve the total energy of the whole system. We easily get the estimates Corollary 2.2 When ɛ, t, t 3 /ɛ 2 and t /ɛ we have η A B tat a A B t = rɛt η ɛ η η ɛt [ rη ±signt ɛ t 2 + r ɛ 2 ln t + ɛ 2 ln2η 2 2η ɛ + /2 ɛ 2 ln ɛ ] ɛ +Ot 3 + Oɛ 4 /t 2 + Otɛ 2 ln ɛ 7

18 and Corollary 2.3 When ɛ, t, t 3 /ɛ 2 and t /ɛ we have S A A B t = S B ɛ, signt + St + η 2 ɛ η 2 t 2 ± rɛt signt rηɛt 2 + r ηɛ ɛ2 ln t +Ot 3 + Oɛ 4 /t 2 + Oɛ 3 ln t. 2.6 Matrices A C t and B C t The construction of the semiclassical wave packets that describe the nuclei requires the computation of matrices which are defined by means of classical quantities. Let A C t and B C t be the matrix solutions of the linear system d dt AC t = ib C t 2.59 d dt BC t = iv 2 a C t, ɛa C t where a C t is the solution of 2.2 and 2.3, with initial conditions A C = A B C = B. 2.6 To do asymptotic matching, we need the small t asymptotics of A C t and B C t. We first determine the leading order behavior of V 2 a C t, ɛ for small t and ɛ. From 2.3 and.2 it is easily seen that the Hessian matrix β 2 + γ 2 + δ 2 2 = δ 2 + γ 2 β β + O3 β 2 + γ 2 + δ 2 3/2, 2.6 where we have used the same notation as earlier. More explicitly, 2 rɛ 2 P + O3 β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ = 2.62 x 2 + ɛ 2 + O3 3/2 where P = When x is replaced by a C t = η ɛt + Ot 2, the error terms O3 become O t + ɛ 3 and using a C t c t, we obtain a C t 2 + ɛ 2 + O3 3/2 = η ɛ t 2 + ɛ 2 3/2 + O t + ɛ

19 This last estimate allows us to find the leading order behavior of V 2 a C t, ɛ as ɛ and t : V 2 a A B t, ɛ = ± rɛ 2 η ɛt 2 + ɛ 2 3/2 P + Oɛ + t Consequently, for any positive ɛ, the equation defining A C and B C is regular as t. Moreover, Proposition 2.3 Let A C t and B C t be the solutions of 2.59 and 2.6. For t and ɛ small enough, we have A A B t = A + Ot B A B t = B ± irp A t η ɛt 2 + ɛ 2 + Ot, uniformly in ɛ. Proof: Equations 2.59 and 2.6 are equivalent to the linear system d dt A C t B C = i t IO V 2 a C t, ɛ I IO A C t B C, t A C A B C =, 2.66 B where IO andi are the n n zero and unity matrices. Moreover, the generator is continuous and uniformly bounded for all t small enough, so that the solution exists, is unique, and satisfies the integral equation A C t A B C = + i t B Introducing the norm D = t B C t V 2 a C t, ɛa C t dt sup Dt, ɛ 2.68 t,ɛ B T on the set of matrices depending on t and ɛ, we get t B C t dt B C t, 2.69 t t V 2 a C t, ɛa C t dt A C V 2 a C t, ɛ dt, 2.7 where, by virtue of 2.65, t V 2 a C t, ɛ dt c t ɛ 2 3/2 + dt t 2 + ɛ 2 t = c + ct c + T C 2.7 t2 + ɛ2 9

20 uniformly in ɛ. Consequently, we get the estimates A C A t B C B + t B 2.72 B C B C A C A + C A 2.73 which imply A C A Ct A C A + t C A + B Hence, if t is so small that Ct /2, and A C A 2t C A + B 2.75 A C t = A + Ot 2.76 uniformly in ɛ. The use of this estimate and 2.65 in 2.67 yields t B A B t = B ± iɛ 2 dt rp A + Ot 2.77 ηt 2 + ɛ 2 3/2 t ɛ 2 t dt +O ηt 2 + ɛ 2 3/2 t = B ± irp A ηt 2 + ɛ + Ot + O ɛt 2 ηt 2 + ɛ 2 t = B ± irp A ηt 2 + ɛ + Ot. 2 Corollary 2.4 In the regime ɛ, t and t /ɛ we have B A B t = B signt ir η ɛ P A + Ot + Oɛ 2 /t 2 B A B signt + Ot + Oɛ 2 /t 2. 3 Type Avoided Crossings We now have all the ingredients required to construct an asymptotic solution to the equation iɛ 2 t ψx, t = ɛ4 ψx, t + hx, ɛψx, t 3. 2 as ɛ. Let us describe the building blocks and give their main properties. 2

21 3. Semiclassical Nuclear Wave Packets The semiclassical motion of the nuclei is described by means of wave packets to be thought of as centered in phase space on the classical trajectory, and of width Oɛ. These are the same wave packets that are used in [7, 6]. Let n denote the dimension of the space of nuclear configurations. A multi-index l = l, l 2,..., l n is an ordered n-tuple of non negative integers. The order of l is defined to be l = n j= l j, and the factorial of l is defined to be l! = l!l 2! l n!. The symbol D l denotes the differential operator D l = l, and the symbol x l x 2 l 2 x n ln xl denotes the monomial x l = x l x l 2 2 x ln n. We denote the gradient of a function f by f and the matrix of second partial derivatives by f 2. We view IR n as a subset of IC n, and let e i denote the i th standard basis vector in IR n or IC n. The inner product on IR n or IC n is v, w = n j= v j w j. The semiclassical wave packets we use are products of complex Gaussians and generalizations of Hermite polynomials. The generalizations of the zeroth and first order Hermite polynomials are H x = 3.2 and H x = 2 v, x, 3.3 where v is an arbitrary non zero vector in IC n. The generalizations of the higher order Hermite polynomials are defined recursively as follows: Let v, v 2,..., v m be m arbitrary non zero vectors in IC n. Then H m v, v 2,..., v m ; x = 2 v m, x H m v, v 2,..., v m ; x 2 m i= v m, v i H m 2 v,..., v i, v i+,..., v m ; x. One can prove [7] that these functions do not depend on the ordering of the vectors v, v 2,..., v m. Furthermore, if the space dimension is n = and the vectors v, v 2,..., v m are all equal to IC, then H m v, v 2,..., v m ; x is equal to the usual Hermite polynomial H m x. Now suppose A is a complex invertible n n matrix. We define A = [AA ] /2, where A denotes the adjoint of A. By the polar decomposition theorem, there exists a unique unitary matrix U A, such that A = A U A. Given a multi-index l, we define the polynomial H l A; x = H l U A e,..., U A e }{{, U } A e 2,..., U A e 2,..., U }{{} A e n,..., U A e n ; x 3.4 }{{} l entries l 2 entries l n entries We can now define the semiclassical wave packets ϕ l A, B, h, a, η, x. In the Born Oppenheimer approximation, the role of h is played by ɛ 2. Definition. Let A and B be complex n n matrices with the following properties: A and B are invertible; 3.5 BA is symmetric [real symmetric] + i[real symmetric]; 3.6 Re BA = 2 [BA + BA ] is strictly positive definite; 3.7 Re BA = AA

22 Let a IR n, η IR n, and h >. Then for each multi-index l we define ϕ l A, B, h, a, η, x = 2 l /2 l! /2 π n/4 h n/4 [det A] /2 H l A; h /2 A x a exp { x a, BA x a /2 h + i η, x a / h }. The choice of the branch of the square root of [det A] in this definition depends on the context, and is determined by initial conditions and continuity in time. Remarks.. We never use the functions ϕ l A, B, h, a, η, x unless conditions are satisfied. 2. Condition 3.8 is equivalent to the more symmetrical condition A B + B A = 2I, I For fixed A, B, h, a, and η, the functions ϕ l A, B, h, a, η, x form an orthonormal basis of L 2 IR n. The proof can be found in [7]. 4. Generically U A and U B are complex unitary matrices, and A and B are Hermitian. In the special cases when they all happen to be real, the functions ϕ l A, B, h, a, η, x are simply rotated and dilated eigenfunctions of the n dimensional harmonic oscillator. 5. The utility of the functions ϕ l A, B, h, a, η, x stems from their remarkably beautiful behavior under Fourier transforms. If we define the scaled Fourier transform to be then [ F h Ψ ]ξ = 2π h n/2 IR n Ψx e i ξ, x / h dx, 3. [ F h ϕ l A, B, h, a, η, ]ξ = i l e i η, a / h ϕ l B, A, h, η, a, ξ. 3. The only proof we know of this formula involves a messy induction on l. The details may be found in [7]. 6. The functions ϕ l A, B, h, a, η, x separate the position and momentum uncertainties from one another. For any given l, the position uncertainty depends only on A and the momentum uncertainty depends only on B. For example, when the space dimension is n =, the position and momentum uncertainites of ϕ l A, B, h, a, η, x are given by [l + 2 h]/2 A and [l + 2 h]/2 B, respectively. 7. For technical reasons related to Remark 6, it is crucial that the matrix B only appear in the exponent in the definition of ϕ l A, B, h, a, η, x, and not in the polynomial. By Remark 5, the matrix A only occurs in the exponent of the Fourier transform. This turns out to be technically crucial, also. The fulfillment of these technical requirements makes the Fourier transform formula in Remark 5 all the more amazing. The formulas for the functions ϕ l A, B, h, a, η, x are rather complicated, but the leading order semiclassical propagation of these wave packets is very simple. Under mild hypotheses e.g., V C 3 and bounded below, the Schrödinger equation i h Ψ t = h2 2 2 Ψ x 2 + V x Ψ

23 has an approximate solution of the form e ist/ h ϕ l At, Bt, h, at, ηt, x + O h / Here O h /2 means that the exact solution and the approximate solution agree up to an error whose norm is bounded by an l dependent constant times h /2 for t in a fixed bounded interval [ T, T ]. The vectors at and ηt satisfy the classical equations of motion a t t = ηt, 3.4 η t t = V at. 3.5 The function St is the classical action integral associated with the classical path, t ηs 2 St = V as ds. 3.6 T 2 The matrices At and Bt satisfy A t t = ibt, 3.7 B t t = iv 2 at At. 3.8 If A T and B T satisfy conditions , then so do At and Bt for each t. The proofs of the claims we have made about the ϕ l A, B, h, a, η, x and other properties of the quantities introduced can be found in [7]. Consider ϕ l defined in 3.9. Since the vectors and matrices a, η, A and B will be replaced in these functions by a C t, η C t, A C t and B C t solutions of 2.2 and 2.59, we need to control the changes of ϕ l induced by changes in a, η, A and B. Lemma 3. We have, in the L 2 IR n sense, ϕ l A, B, ɛ 2, a,, x = ϕ l A, B, ɛ 2, a,, x Proof: Let us introduce the temporary notation so that Consider +O A A + B B + a a /ɛ. ψa, B, y = ϕ l A, B,,,, y 3.9 ϕ l A, B, ɛ 2, a,, x = ɛ n/2 ψa, B, x a/ɛ. 3.2 ψa + A, B + B, x a a /ɛ ψa, B, x a/ɛ = ψa + A, B + B, x a a /ɛ ψa + A, B, x a a /ɛ +ψa + A, B, x a a /ɛ ψa, B, x a a /ɛ +ψa, B, x a a /ɛ ψa, B, x a/ɛ

24 We separately estimate each of the three differences on the right hand side of this equation. We compute the Fréchet differential, D B ψa, B, yb = 2 ψa, B, y y B A y, 3.22 so that by the mean value theorem, ψa + A, B + B, x a a /ɛ ψa + A, B, x a a /ɛ 3.23 sup ψa + A, B + tb, x a a /ɛ x a a 2 A + A B. t [,] ɛ 2 But, since by assumption Re BA is strictly positive definite, and ha, A y = 2 l /2 l! /2 π n/4 det A /2 H la, A y 3.24 is a polynomial of degree l in its second variable, we have, for A, B small enough, and for any positive p, ɛ n exp x a a B + B A + A x a a /2ɛ 2 2 = IR n IR n n p A, B, x a a /ɛ p dx exp y B + B A + A y /2 2 y p dy uniformly in A, B, a and a. Thus, for some constant depending only on A and B, ɛ n/2 ψa + A, B + B, a a /ɛ 3.25 ψa + A, B, a a /ɛ L 2 IR n CA, B B This estimate together with the Plancherel identity and 3. yield ɛ n/2 ψa + A, B, a a /ɛ ψa, B, a a /ɛ L 2 IR n ɛ n/2 [ F h Ψ ]A + A, B, a a /ɛ [ F h Ψ ]A, B, a a /ɛ L 2 IR n = ɛ n/2 ψb, A + A, + a + a /ɛ ψb, A, + a + a /ɛ L 2 IR n CB, A A Finally, using the notation given in 3.24, where ψa, B, y ψa, B, y + y sup D y ψa, B, y + ty y 3.28 t [,] D y ψa, B, yy = 2 ψa, B, y y BA y + y BA y + exp y BA y /2 D 2 ha, A y A y

25 Thus we can write ɛ n/2 ψa, B, x a a /ɛ ψa, B, x a/ɛ ɛ n/2 sup exp x a ta /ɛ BA x a ta /ɛ /2 t [,] P A, A x a ta /ɛ a /ɛ = ɛ n/2 exp x a t a /ɛ BA x a t a /ɛ /2 P A, A x a t a /ɛ a /ɛ 3.3 where P A, y is a polynomial of order l in y and [, ] t = t A, B, a, a, ɛ is the point where the supremum is reached. Hence, using 3.25 again, we deduce ɛ n/2 ψa, B, x a a /ɛ ψa, B, x a/ɛ L 2 IR n C A, B a /ɛ 3.3 for some constant C A, B uniform in a, a and t. 3.2 The Choice of Eigenvectors In this section we present a procedure for choosing the electronic eigenvectors and their phases. Although the electronic hamiltonian is independent of time, it is convenient, since we deal with the time dependent Schrödinger equation, to choose specific time dependent electronic eigenvectors. Indeed, the electrons follow the motion of the nuclei in an adiabatic way, so the suitable instantaneous electronic eigenvectors must satisfy some parallel transport condition to take into account the geometric phases arising in this situation. These eigenvectors thus depend on the classical trajectories. Since they may become singular when the corresponding eigenvalues are degenerate, or almost degenerate, we shall define them for t in the outer regime. We shall have two sets of eigenvectors, denoted by Φ ± C x, t, ɛ, where the label ±, refers to positive and negative times. Let η C t be the momentum solution of the classical equations of motion 2.2 and 2.3. The normalized eigenvectors Φ ± C x, t, ɛ are the solutions of Φ ± C x, t, ɛ / t + η C t Φ ± C x, t, ɛ 3.32 for C = A, B and t >. Since the eigenvalues E < Ax, ɛ and E B x, ɛ are non-degenerate for any time t, t small enough, such vectors exist, are unique up to an overall time independent phase factors and are eigenvectors of h x, ɛ associated with E C x, ɛ for any time, see [6]. Let us make the construction of these eigenvectors more specific and give their small t and small ɛ asymptotics. We define the angles ϕx, ɛ and θx, ɛ by βx, ɛ = β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ cosθx, ɛ 3.33 γx, ɛ = β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ sinθx, ɛ cosϕx, ɛ 3.34 δx, ɛ = β 2 x, ɛ + γ 2 x, ɛ + δ 2 x, ɛ sinθx, ɛ sinϕx, ɛ and construct static eigenvectors. Let Φ Ax, ɛ = e iϕx,ɛ cosθx, ɛ/2ψ x, ɛ + sinθx, ɛ/2ψ 2 x, ɛ 3.36 Φ B x, ɛ = e iϕx,ɛ cosθx, ɛ/2ψ 2 x, ɛ sinθx, ɛ/2ψ x, ɛ

26 be the eigenvectors of h x, ɛ associated with E C x, ɛ, C = A, B for π/2 < θx, ɛ π and Φ + Ax, ɛ = cosθx, ɛ/2ψ x, ɛ + e iϕx,ɛ sinθx, ɛ/2ψ 2 x, ɛ 3.38 Φ + B x, ɛ = cosθx, ɛ/2ψ 2 x, ɛ e iϕx,ɛ sinθx, ɛ/2ψ x, ɛ 3.39 be the eigenvectors of h x, ɛ for θx, ɛ < π/2. The solutions of 3.32 are of the form Φ ± C x, t, ɛ = Φ ± C x, ɛe iλ± C x,t,ɛ, { t > t <, 3.4 where λ ± C x, t, ɛ is a real valued function satisfying the equation i t λ± C x, t, ɛ + iη C t λ ± C x, t, ɛ + Φ C x, ɛ η C t Φ C x, ɛ =. 3.4 Lemma 3.2 Let η C t be the momentum solution of the classical equations of motion 2.2 and 2.3. There exist eigenvectors Φ ± C x, t, ɛ, C = A, B, such that Φ ± C x, t, ɛ / t + η C t Φ ± C x, t, ɛ. If x = Oɛ κ >, x, with 2/3 < κ <, < Φ ± C x, t, ɛ Φ ± C x, ɛ = Ot/ɛ κ. Proof: We give a proof for Φ + Ax, t, ɛ only; the other cases are similar. In order to simplify the notation, we drop the indices A and +. Thus we need determine λx, t, ɛ such that where Φx, t, ɛ = e iλx,t,ɛ Φx, ɛ, 3.42 Φx, ɛ = cosθx, ɛ/2ψ x, ɛ + e iϕx,ɛ sinθx, ɛ/2ψ 2 x, ɛ Also, let at a A t and ηt η A t for t >. We introduce the new variable ω x at 3.44 and the notation λω, t, ɛ λω + at, t, ɛ, 3.45 Φω, t, ɛ Φω + at, ɛ In terms of these new variables, equation 3.4 for λ reads i λω, t, ɛ = Φω, t, ɛ Φω, t, ɛ 3.47 t t with Φω, t, ɛ = ηt Φω + at, ɛ t 26

27 We compute, dropping the arguments, Φ = sinθ/2ψ + e iϕ cosθ/2ψ 2 θ/2 + cosθ/2 ψ + e iϕ sinθ/2 ψ 2 i ϕe iϕ sinθ/2ψ Since the ψ j, j =, 2, are orthonormal, we get Φ t Φ = iη ϕ sin 2 θ/2 + cos 2 θ/2 ψ η ψ + sin 2 θ/2 ψ 2 η ψ 2 + sinθ/2 cosθ/2 e iϕ ψ η ψ 2 + e iϕ ψ 2 η ψ. 3.5 As η, ψ j, ψ j are all O, we can write t λω, t, ɛ = ηt ϕ sin 2 θω + at, ɛ/2 + O dt + λ ω. 3.5 It remains to estimate ϕ. ϕx, ɛ = atan δx, ɛ/γx, ɛ 3.52 provided γx, ɛ is different from zero. Hence, using.2 we get ϕx, ɛ = = γx, ɛ δx, ɛ δx, ɛ γx, ɛ ɛo + O2O = γ 2 x, ɛ + δ 2 x, ɛ r 2 ɛ 2 + O3 O/ɛ + O3/ɛ O3/ɛ 2 The above computations are legitimate for x = Oɛ κ, 2/3 < κ <, 3.54 which implies γx, ɛ = rɛ + O2 = rɛ + Oɛ + x + x 2 /ɛ = rɛ + Oɛ 2κ cɛ, 3.55 for some uniform constant c, and O/ɛ = O + x /ɛ = O/ɛ κ, 3.56 O3/ɛ 2 = Oɛ + x + x 2 /ɛ + x 3 /ɛ as ɛ. Hence, if ω + at = Oɛ κ, λω, t, ɛ = λ ω + Ot/ɛ κ,

28 uniformly in t and ω. Taking the integration constant λ ω, we obtain λx, t, ɛ = Ot/ɛ κ. We get the desired result by means of 3.42 and the fact that Φx, ɛ = O The nuclear wave function is localized around the classical trajectory in the semiclassical regime. Thus, in the outer temporal region, because of the genericity condition η >, the major part of the nuclear wave function will be supported away from the neighborhood where the levels almost cross. Hence we can introduce a cutoff function which does not significantly alter the solution and forces the support of the wave function to be away from this neighborhood. We choose the support of the cutoff so that the above lemma applies. Let F be a C cutoff function F : IR + IR, such that { F r = r F r = r The wave functions we construct below in the outer regime will be multiplied by the regulating factor F x a C t /ɛ δ 3.62 where δ < ξ, for C = A, B see below. Remark: If ξ < 2/3 < κ < ξ, the correction terms in the above lemma tend to zero and the set x = Oɛ κ includes the set x a C t = Oɛ δ. On the support of F, x = η ɛt + Oɛ δ + t and since η ɛ = η + Oɛ, where η >, we deduce x > c t 3.64 uniformly in ɛ. Using this property we have Lemma 3.3 For x in the support of F x a C t /ɛ δ, C = A, B, we have for t <, Φ Ax, ɛ = ψ 2 x, ɛ + Oɛ/t /2, Φ B x, ɛ = ψ x, ɛ + Oɛ/t /2, and for t >, Φ + Ax, ɛ = ψ x, ɛ + Oɛ/t /2, Φ + B x, ɛ = ψ 2 x, ɛ + Oɛ/t /2. 28

29 Proof: It follows from.2, 3.63 and 3.64 that for t > <, Similarly, hence cosθx, ɛ = tanϕx, ɛ = x + O2 x 2 + ɛ 2 + O3 = ± + Ot + Oɛ 2 /t 2 + Ot /2 = ± + Oɛ 2 /t O2 ɛ + O2 = Ot2 /ɛ + Ot 2 /ɛ = Ot2 /ɛ, 3.66 ϕx, ɛ = Ot 2 /ɛ The result then follows from the identity sin 2 θx, ɛ/2 = and ψ j =, j =, 2. cosθx, ɛ 2 = ± 2 + Ot 2 /ɛ 3.68 We can describe the solution of the Schrödinger equation in the different regimes introduced. 3.3 The Incoming Outer Solution We assume that at the initial time T, the wave function is given by a semiclassical nuclear wave packet times an electronic function associated with the B level. We consider the associated classical quantities determined by the following initial conditions at t = : a B = η B = η A B = A B B = B S B =, 3.69 where A and B satisfy the hypotheses in the definition of the functions ϕ l. Away from the avoided crossing of the electronic levels, the solution of the Schrödinger equation is well approximated by standard time dependent Born Oppenheimer wave packets. Close to the avoided crossing these standard wave packets fail to approximate the solution. The following lemma tells us how close to the avoided crossing time these standard wave packets can be used as approximations. 29

30 Lemma 3.4 In the incoming outer time regime, T t ɛ ξ, there is an approximation ψ IO x, t of a solution of the Schrödinger equation ψx, t of the form ψ IO x, t = F x a B t /ɛ δ exp i SB t + i ηb tx a B t ɛ 2 ɛ 2 such that where ϕ l A B t, B B t, ɛ 2, a B t,, xφ B x, t, ɛ 3.7 ψx, t = ψ IO x, t + Rx, t, ɛ 3.7 Rx, t, ɛ = Oɛ ξ 3.72 in the L 2 IR n sense, as ɛ. Proof: The proof of this Lemma is very similar to that of Lemma 6.4 of [6]. For T t ɛ ξ and x in the support of the cut off function F x a B t /ɛ δ, E A x E B is bounded by x a multiple of ɛ +ξ. Furthermore, for such x and t, a rather tedious calculation shows that + t ηb x Φ B x, t, ɛ is bounded by a multiple of ɛ +ξ. Thus, up to errors on the order of ɛ 2ξ, ψ IO x, t is equal to Ψx, t = F x a B t /ɛ δ exp i SB t ɛ 2 where and + i ηb tx a B t ɛ 2 Ψ x, t + ɛ 2 Ψ 2 x, t 3.73 Ψ x, t = ϕ l A B t, B B t, ɛ 2, a B t,, xφ B x, t, ɛ 3.74 Ψ 2 x, t = iϕ l A B t, B B t, ɛ 2, a B t,, x Φ Ax, t, ɛ E A x E B x Φ Ax, t, ɛ, t + ηb x Φ B x, t, ɛ + r BA x, ɛp AB x, ɛ t + ηb x Φ B x, t, ɛ, 3.75 where r BA x, ɛ is the restriction of hx, ɛ E B x, ɛ to the range of P AB x. The expression 3.73 is a standard leading order Born-Oppenheimer wave packet, and it suffices to show that it agrees with a solution of the Schrödinger equation up to errors on the order of ɛ ξ. To prove this, we explicitly compute ζx, t, ɛ = iɛ 2 Ψ t HɛΨ

31 For T t ɛ ξ, terms in ζx, t, ɛ that contain derivatives of F are easily seen to have norms that grow at worst like powers of / t times factors that are exponentially small in ɛ. Most of the remaining terms in ζx, t, ɛ are formally given on pages 5 8 of [6], with t + ηb x in place of ηb x in many expressions. A few other terms arise from the iɛ 2 t acting on the electronic eigenfunctions in Our error term ζx, t, ɛ differs formally from that of Section 6 of [6] because our eigenfunctions have time dependence and the ones in Section 6 of [6] do not. After noting that numerous terms in ζx, t, ɛ cancel with one another, we estimate the remaining terms individually. This process is very similar to that described on pages 8 of [6], except that there are a few extra terms that contain time derivatives of the electronic eigenfunctions. To estimate these terms we use arguments similar to those used in the proof of lemma 3.2. Our avoided crossing problem is slightly less singular than the crossing problem treated in [6]. So, it is not surprising that our error term satisfies the same estimate ζ, t, ɛ C ɛ 3 t as the corresponding error term in Section 6 of [6]. We obtain the desired estimate by using the estimate 3.77 and lemma 3.3 of [6]. 3.4 The Inner Solution In the inner time regime, we look for an approximation constructed by means of the classical quantities associated with the potential Ṽ x, ɛ. Let at and St be the corresponding classical quantities satisfying the initial conditions a = η = η S =. In the rescaled variables { y = x at/ɛ,, s = t/ɛ the Schrödinger equation reads iɛ s ψ iɛηɛs yψ = ɛ2 2 yψ + haɛs + ɛy, ɛψ. 3.8 We seek an approximate solution of the form F y ɛ δ exp i Sɛs + i ηɛsy χy, s, ɛ, 3.8 ɛ 2 ɛ with χy, s, ɛ = {fy, s, ɛψ aɛs + ɛy, ɛ + gy, s, ɛψ 2 aɛs + ɛy, ɛ ψ aɛs + ɛy, ɛ}, 3

32 where ψ x, ɛ I I P x, ɛh and f, g are scalar functions. Anticipating exponential fall off of the solution, we insert 3.8 in 3.8 and neglect the derivatives of F. Making use of the decomposition. and eliminating the overall classical phase and cut-off F, we get dropping the arguments iɛψ s f + iɛψ 2 s g + iɛ s ψ + Ṽ a + ɛy xṽ afψ + gψ 2 + ψ +iɛ 2 fη x ψ + iɛ 2 gη x ψ 2 + iɛ 2 η x ψ = ɛ 4 x ψ /2 ɛ 4 f x ψ /2 ɛ 4 g x ψ 2 /2 ɛ 3 y f x ψ ɛ 3 y g x ψ 2 ɛ 2 ψ y f ɛ 2 ψ 2 y g + h ψ + h fψ + gψ 2 +Ṽ a + ɛyfψ + gψ 2 + ψ We assume the solutions have expansions of the form fy, s, ɛ = gy, s, ɛ = ψ x, t, ɛ = ν j ɛf j y, s, j= ν j ɛg j y, s, j= ν j ɛψ j x, t, 3.84 j= with asymptotic scales ν j ɛ to be determined by matching. We insert these expansions in Using the behaviors.2 and lemma 2.2 we verify that the lowest order terms yield iɛν ɛ s f y, sψ aɛs + ɛy, ɛ + iɛν ɛ s g y, sψ 2 aɛs + ɛy, ɛ = h aɛs + ɛy, ɛψ aɛs + ɛy, ɛs, ɛ +ɛν ɛh y, sf y, sψ aɛs + ɛy, ɛ + g y, sψ 2 aɛs + ɛy, ɛ 3.85 on the support of F, where h y, s is the operator on the span of { ψ aɛs + ɛy, ɛ, ψ 2 aɛs + ɛy, ɛ } whose matrix in the basis { ψ aɛs + ɛy, ɛ, ψ 2 aɛs + ɛy, ɛ } is given by η r s + y ηs y By matching the incoming outer solution, we obtain ν j ɛh aɛs + ɛy, ɛψ j aɛs + ɛy, ɛs = 3.87 if ν j ɛ << ɛν ɛ, j =,,, m. The spectrum of h x, ɛ is bounded away from in a neighborhood of,. This implies ψ j x, t =, j =,,, m We split the remaining equation for the order ɛν ɛ = ν m ɛ without arguments iɛν ɛ s f ψ + iɛν ɛ s g ψ 2 = ν m ɛh ψ + ɛν ɛh f ψ + g ψ