Molecular Propagation through Small Avoided Crossings of Electron Energy Levels

Transcription

1 Molecular Propagation through Small Avoided Crossings of Electron Energy Levels 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. July 5, 1997 Abstract This is the second of two papers on the propagation of molecular wave packets through avoided crossings of electronic energy levels in a limit where the gap between the levels shrinks as the nuclear masses are increased. An earlier paper deals with the simplest two types of generic, minimal multiplicity avoided crossings, in which the levels essentially depend on only one of the nuclear configuration parameters. The present paper deals with propagation through the remaining four types of generic, minimal multiplicity avoided crossings, in which the levels depend on more than one nuclear configuration parameter. CPT-96/P.337 Partially Supported by National Science Foundation Grant DMS Supported by Fonds National Suisse de la Recherche Scientifique, Grant 8-37 Permanent address: Centre de Physique Théorique, CNRS Marseille, Luminy Case 97, F-1388 Marseille Cedex 9, France and PHYMAT, Université de Toulon et du Var, B.P.13, F La Garde Cedex, France 1

2 1 Introduction This is the second of a pair of papers on the propagation of molecular wave packets through avoided crossings of electronic energy levels in a limit where the gap between the levels shrinks as the nuclear masses are increased. Generic, minimal multiplicity avoided crossings can be classified into six types [19]. Our first paper [] deals with Types 1 and ; the present paper deals with Types 3, 4, 5, and 6. In Type 1 and Type avoided crossings, the electron energy levels essentially depend on only one nuclear configuration parameter. Because of rotational symmetry, this is the case for all diatomic molecules. The results of [] show that in Type 1 and Type avoided crossings, transitions between the levels are correctly described by the Landau Zener formula. In Type 3, 4, 5, and 6 avoided crossings, the electron energy levels essentially depend respectively on,, 3, and 4 nuclear configuration parameters [19]. In practice, these arise in polyatomic molecules, or in diatomic or polyatomic systems in external fields. Molecular propagation through these more complicated avoided crossings is not governed directly by the Landau Zener formula, and electronic transition probabilites depend on the shape of the nuclear wave packet. Intuitively, this is because different pieces of the wave packet feel different size minimum gaps between the electronic levels. As we explain in more detail below, the correct transition probability can be determined by decomposing the nuclear wave packet into infinitesimal pieces, using a different Landau Zener formula for each infinitesimal piece, and then doing an integration. Electron energy levels are functions of the nuclear configuration parameters, determined by discrete quantum mechanical bound state energies of the electrons for each fixed classical nuclear configuration. Their relevance to molecular propagation is tied to the time dependent Born Oppenheimer approximation, which exploits the smallness of the dimensionless parameter ǫ, where ǫ 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 quantum state in response to the relatively slow nuclear motion. They 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 position of the nuclei and the electronic energy level plays the role of an effective potential for the semiclassical dynamics of the nuclei. This intuition is the basis for rigorous asymptotic expansions of solutions to the molecular time dependent Schrödinger equation [7, 9, 1, 16, 17]. However, the validity of the approximation depends on the assumption that the electron energy level of interest is well isolated from the rest of the spectrum of the electronic Hamiltonian. This assumption is violated near an Avoided Crossing. Readers interested in the mathematical literature concerning the validity of Born Oppenheimer approximations should consult [3, 4, 5, 7, 9, 1, 11, 1, 13, 14, 16, 17, 18,, 1,

3 33, 35, 36, 37, 39, 4, 41, 4]. The Hamiltonian for a molecular system with K nuclei and N K electrons has the form Hǫ = K ǫ4 xj j=1 M j N j=k+1 1 m j xj + i<j V ij x i x j. 1.1 Here x j IR l denotes the position of the j th particle, the mass of the j th nucleus is ǫ 4 M j for 1 j K, the mass of the j th electron is m j for K + 1 j N, and V ij is the potential between particles i and j. In realistic systems, different nuclei may have different masses, but for convenience, we assume M j = 1 for 1 j K. We set n = Kl and let x = x 1, x,..., x Kl IR n denote the nuclear configuration vector. We decompose Hǫ as Hǫ = ǫ4 x + hx. 1. This defines the electronic Hamiltonian hx that depends parametrically on x. The time dependent Schrödinger equation that we study is i ǫ ψ t = Hǫ ψ, 1.3 for t in a fixed interval. The factor of ǫ 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 ǫ tends to zero. Nowhere in this paper do we require the Hamiltonian to have the particular form 1.1. We only require the Hamiltonian to be of the form 1.. In some realistic systems, 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 ǫ. For this reason, we generalize the form 1. to allow the electron Hamiltonian to depend on ǫ as well as x. We study solutions to equation 1.3 with Hamiltonians of the form Hǫ = ǫ4 x + hx, ǫ, 1.4 with the assumption that hx, ǫ has an Avoided Crossing according to the following definition: Definition Suppose hx, ǫ is a family of self adjoint operators with a fixed domain D in a Hilbert Space H, for x Ω and ǫ [, α, where Ω is an open subset of IR n. Suppose that the resolvent of hx, ǫ is a C 4 function of x and ǫ as an operator from H to D. Suppose hx, ǫ has two eigenvalues E A x, ǫ and E B x, ǫ that depend continuously on x and ǫ and are isolated from the rest of the spectrum of hx, ǫ. Assume 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, ǫ E B x, ǫ when ǫ >. Then we say hx, ǫ has an Avoided Crossing on Γ. 3

4 Remarks: 1. 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 [11, 1, 36].. The set Ω in the definition plays no interesting role, so we henceforth assume Ω = IR n and drop any further reference to it. The wave packets we construct are supported on sets in which the nuclear coordinates are restricted to a neighborhood of a compact classical nuclear orbit. Our techniques apply to any Ω and any classical path, provided the time interval is restricted to keep the nuclei inside Ω. In realistic systems, Ω may be a proper subset of IR n, since electron energy levels may cross one another or be absorbed into the continuous spectrum as the nuclei are moved. Precise statements of our results require a considerable amount of notation and are presented in Theorems 3.1, 4.1, 4., and 4.3 for Type 3, 4, 5, and 6 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. 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 avoided crossings couple the two levels to leading order in ǫ, and the interesting transitions between the levels occur in the time interval in which the inner solution is valid. To leading order, the Schrödinger equation for the inner solution is hyperbolic. As a result, it makes sense to describe the motion of each infinitesimal piece of the wave function along its characteristic. Along the various characteristics, the wave function feels various different minimum gaps between the eigenvalues. The Landau Zener formula describes the correct transition probability along each individual characteristic. The transition probability for the entire wave function is correctly obtained by applying the Landau Zener formula for each infinitesimal piece of the wave function and then integrating over the nuclear configuration space. The transition probability depends on the shape of the wave packet as it encounters the avoided crossing. To describe the various Types of Avoided Crossings, we recall several results from [19]. For convenience, we use the notation m/ n Om = O x j + ǫ. 1.5 j=1 Assume without loss of generality that is a generic point of Γ, and that we are given a classical nuclear momentum vector η that is transversal to Γ. We decompose with hx, ǫ = h x, ǫ + h x, ǫ 1.6 h x, ǫ = hx, ǫp x, ǫ 1.7 4

5 and h x, ǫ = hx, ǫi I P x, ǫ 1.8 where P x, ǫ is a spectral projector for hx, ǫ associated with E A x, ǫ and E B x, ǫ. Type 1 and Avoided Crossings have the codimension of Γ equal to 1. We will not discuss them here since they have already been studied in []. Type 3 and 4 Avoided Crossings have the codimension of Γ equal to. For these two types of Avoided Crossings, we choose an orthogonal coordinate system for the nuclear configurations in which the x 1 and x coordinate directions are perpendicular to Γ at x = ; the x 3, x 4,... x n coordinate directions are parallel to Γ at x = ; and so that the vector η has the form η = η 1 η 3 η 4. η n. 1.9 The eigenvalues involved in a Type 3 Avoided Crossing are non-degenerate. One can choose [19] an orthonormal basis {ψ 1 x, ǫ, ψ x, ǫ} of P x, ǫh, which is regular in x, ǫ around,, such that h x, ǫ = h 1 x, ǫ + Ṽ x, ǫ, 1.1 where h 1 x, ǫ is represented in this basis by the matrix βx, ǫ γx, ǫ + iδx, ǫ γx, ǫ iδx, ǫ βx, ǫ 1.11 and Ṽ x, ǫ is represented by 1 tracehx, ǫp x, ǫ times the identity matrix. The function Ṽ x, ǫ is regular in x, ǫ near the origin, and βx, ǫ = b 1 x 1 + b x + b 3 ǫ + O 1.1 γx, ǫ = c x + c 3 ǫ + O δx, ǫ = d 3 ǫ + O Ṽ x, ǫ = O with b 1 >, c >, b IR, c 3 IR, and d 3 IR. Generically, d 3 is non-zero, which we henceforth assume. The two energy levels involved in the Avoided Crossing are thus E A B = Ṽ x, ǫ ± βx, ǫ + γx, ǫ + δx, ǫ = Ṽ x, ǫ ± b 1 x 1 + b x + b 3 ǫ + c x + c 3 ǫ + d 3 ǫ + O Figure 1 shows graphs of two electron energy levels near a typical Type 3 Avoided Crossing. Our results for molecular propagation through Type 3 Avoided Crossings are stated precisely in theorem 3.1. To illustrate these results, we present a simple example with two 5

6 Figure 1: Graphs of electron energy levels with a Type 3 Avoided Crossing. nuclear degrees of freedom represented by the coordinates x 1 and x. The electrons belong to a two level system with electron Hamiltonian x1 x + iǫ x iǫ x 1. We choose an initial wave packet associated with the lower electronic level. The nuclei approach the origin along the negative x 1 -axis in a state that is a Gaussian to leading order in ǫ. The nuclear position and momentum uncertainties are proportional to ǫ, and the center of the Gaussian approximately follows a classical path at. Figure gives a graphical representation of this situation immediately before the nuclei encounter the avoided crossing. The left half of the figure corresponds to the lower electronic level, and the right half corresponds to the upper electronic level. For each of the two levels, the figure shows contour plots of the nuclear probability density as functions of the variable y = x at/ǫ. This is a natural variable that moves along the classical path with the nuclei. Immediately after the nuclei have moved through the avoided crossing, the system has non-trivial components in each of the electronic levels, as depicted in figure 3. To leading order, these final nuclear probability densities on the upper and lower surfaces are the initial nuclear probability density depicted on the left side of figure multiplied by e fy and 1 e fy, respectively. Here fy = π 4η 1 y + 1, where the nuclear momentum is concentrated near η η = 1 6

7 Figure : Probability densities before encountering a Type 3 Avoided Crossing. Figure 3: Probability densities after encountering a Type 3 Avoided Crossing. 7

8 at the time when at =. The physical intuition associated with this result is the following: The nuclei pass through the avoided crossing during a time interval whose length is on the order of ǫ. During that short time, to leading order, the nuclei simply translate through the avoided crossing. For each fixed value of y, the electron state propagates independently, and at time t, the electron Hamiltonian has an energy gap of size η1t + y 1 + y + 1. The traditional Landau Zener formula for a time dependent Hamiltonian with this size gap predicts an electronic transition probability of e fy as the system moves through the avoided crossing. Theorem 3.1 confirms this intuition. Type 4 Avoided Crossings are similar, except that the minimal multiplicity of eigenvalues allowed by the symmetry group is. Near one of these avoided crossings, one can choose an orthonormal basis {ψ 1 x, ǫ, ψ x, ǫ, ψ 3 x, ǫ, ψ 4 x, ǫ} of P x, ǫh, which is regular in x, ǫ around,. The operator h satisfies h x, ǫ = h 1 x, ǫ + Ṽ x, ǫ, 1.14 where h 1 x, ǫ is represented in this basis by the matrix βx, ǫ γx, ǫ + iδx, ǫ γx, ǫ iδx, ǫ βx, ǫ βx, ǫ γx, ǫ iδx, ǫ, 1.15 γx, ǫ + iδx, ǫ βx, ǫ and Ṽ x, ǫ is represented by 1 tracehx, ǫp x, ǫ times the 4 4 identity matrix. The 4 function Ṽ x, ǫ is regular in x, ǫ near the origin, and βx, ǫ = b 1 x 1 + b x + b 3 ǫ + O 1.16 γx, ǫ = c x + c 3 ǫ + O δx, ǫ = d 3 ǫ + O Ṽ x, ǫ = O. The two energy levels involved in the Avoided Crossing are thus E A B = Ṽ x, ǫ ± βx, ǫ + γx, ǫ + δx, ǫ + O = Ṽ x, ǫ ± b 1 x 1 + b x + b 3 ǫ + c x + c 3 ǫ + d 3 ǫ + O Type 5 Avoided Crossings have the codimension of Γ equal to 3 and the multiplicity of the eigenvalues equal to. We choose an orthogonal coordinate system for the nuclear configurations in which the x 1, x, and x 3 coordinate directions are perpendicular to Γ at x = ; the x 4, x 5,... x n coordinate directions are parallel to Γ at x = ; and so that the vector η has the form η = η 1 η4 η5. ηn

9 We can choose an orthonormal basis {ψ 1 x, ǫ, ψ x, ǫ, ψ 3 x, ǫ, ψ 4 x, ǫ} of P x, ǫh, which is regular in x, ǫ around,. The operator h x, ǫ satisfies 1.14, where h 1 x, ǫ is represented in this basis by the matrix βx, ǫ γx, ǫ + iδx, ǫ ζx, ǫ + iξx, ǫ βx, ǫ ζx, ǫ + iξx, ǫ γx, ǫ iδx, ǫ γx, ǫ iδx, ǫ ζx, ǫ iξx, ǫ βx, ǫ, ζx, ǫ iξx, ǫ γx, ǫ + iδx, ǫ βx, ǫ 1.19 and Ṽ x, ǫ is represented by 1 tracehx, ǫp x, ǫ times the 4 4 identity matrix. The 4 function Ṽ x, ǫ is regular in x, ǫ near the origin, and βx, ǫ = b 1 x 1 + b x + b 3 x 3 + b 4 ǫ + O γx, ǫ = c x + c 4 ǫ + O δx, ǫ = d 3 x 3 + d 4 ǫ + O ζx, ǫ = e 4 ǫ + O ξx, ǫ = O Ṽ x, ǫ = O. We assume the generically satisfied condition e 4. The two energy levels involved in the Avoided Crossing are thus E A B = βx, Ṽ x, ǫ ± ǫ + γx, ǫ + δx, ǫ + ζx, ǫ + ξx, ǫ 3 = Ṽ x, ǫ ± b j x j + b 4 ǫ + c x + c 4 ǫ + d 3 x 3 + d 4 ǫ + e 4 ǫ + O3. j=1 1. Type 6 Avoided Crossings have the codimension of Γ equal to 4 and the multiplicity of the eigenvalues equal to. We choose an orthogonal coordinate system for the nuclear configurations in which the x 1, x, x 3, and x 4 coordinate directions are perpendicular to Γ at x = ; the x 5, x 6,... x n coordinate directions are parallel to Γ at x = ; and so that the vector η has the form η = η 1 η5 η 6. η n

10 We can choose an orthonormal basis {ψ 1 x, ǫ, ψ x, ǫ, ψ 3 x, ǫ, ψ 4 x, ǫ} of P x, ǫh, which is regular in x, ǫ around,. The operator h x, ǫ satisfies 1.14, where h 1 x, ǫ is represented in this basis by the matrix βx, ǫ γx, ǫ + iδx, ǫ ζx, ǫ + iξx, ǫ βx, ǫ ζx, ǫ + iξx, ǫ γx, ǫ iδx, ǫ γx, ǫ iδx, ǫ ζx, ǫ iξx, ǫ βx, ǫ, ζx, ǫ iξx, ǫ γx, ǫ + iδx, ǫ βx, ǫ 1. and Ṽ x, ǫ is represented by 1 tracehx, ǫp x, ǫ times the 4 4 identity matrix. The 4 function Ṽ x, ǫ is regular in x, ǫ near the origin, and βx, ǫ = b 1 x 1 + b x + b 3 x 3 + b 4 x 4 + b 5 ǫ + O γx, ǫ = c x + c 5 ǫ + O δx, ǫ = d 3 x 3 + d 5 ǫ + O ζx, ǫ = e 4 x 4 + e 5 ǫ + O ξx, ǫ = f 5 ǫ + O Ṽ x, ǫ = O. We assume the generically satisfied condition f 5. The two energy levels involved in the Avoided Crossing are thus E A B = Ṽ x, ǫ ± βx, ǫ + γx, ǫ + δx, ǫ + ζx, ǫ + ξx, ǫ = Ṽ x, ǫ 4 ± b j x j + b 5 ǫ + c x + c 5 ǫ + d 3 x 3 + d 5 ǫ + e 4 x 4 + e 5 ǫ + f 5 ǫ + O3. j=1 1.3 The paper is organized as follows: In Section 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 state and prove our main result for Type 3 Avoided Crossings, Theorem 3.1. In Section 4 we state our main results, Theorems 4.1, 4., and 4.3, for Type 4, 5, and 6 Avoided Crossings, respectively. Acknowledgements. George Hagedorn wishes to thank the Centre de Physique Théorique, C.N.R.S., Marseille for its hospitality during July 1994 and May Alain Joye wishes to thank Virginia Polytechnic Institute and State University for its hospitality during August October 1994 and July September 1995, and the Fonds National Suisse de la Recherche Scientifique for financial support. 1

11 1.1 Convenient Changes of Variables We begin with Type 3 Avoided Crossings. It is convenient to remove the ǫ-dependence in the leading order of βx, ǫ and γx, ǫ in 1.11, so we introduce new variables that are implicitly defined by the relations b 1 x 1 + b x = b 1 x 1 + b x + b 3 ǫ and 1.4 c x = c x + c 3 ǫ. 1.5 Explicitly, this change of variables is given by x 1 = x 1 + ǫ b 3 b c 3, b 1 c 1.6 x = x + c 3ǫ, c 1.7 x j = x j j >. 1.8 When we change variables, the Schrödinger equation iǫ t ψx, t = ǫ4 ψx, t + hx, ǫψx, t 1.9 for ψx, t becomes for iǫ t φx, t = ǫ4 φx, t + hxx, ǫ, ǫφx, t, 1.3 φx, t = ψxx, ǫ, t, 1.31 with h xx b1 x, ǫ, ǫ = 1 + b x c x + id 3ǫ c x id 3 ǫ b 1 x 1 b x + Ṽ xx, ǫ, ǫ I + O, 1.3 where Ṽ xx, ǫ, ǫ is regular in x, ǫ around, and O refers to x and ǫ. We henceforth drop the primes on the new variables, and assume that h 1 x, ǫ has the form 1.11 with the following local behavior around x = and ǫ = : βx, ǫ = b 1 x 1 + b x + O 1.33 γx, ǫ = c x + O δx, ǫ = d 3 ǫ + O Ṽ x, ǫ = O. In the new variables, the two relevant energy levels are E A B = Ṽ x, ǫ ± βx, ǫ + γx, ǫ + δx, ǫ = Ṽ x, ǫ ± b 1 x 1 + b x + c x + d 3 ǫ + O

12 For Type 4 Avoided Crossings, we make the same change of variables, which leads to the matrix 1.15 for h 1 with leading behavior for β, γ, δ, and Ṽ given by In the new variables, the two relevant energy levels are E A B = Ṽ x, ǫ ± βx, ǫ + γx, ǫ + δx, ǫ = Ṽ x, ǫ ± b 1 x 1 + b x + c x + d 3 ǫ + O For Type 5 Avoided Crossings, we make the change of variables defined implicitly by the following: b 1 x 1 + b x + b 3x 3 = b 1 x 1 + b x + b 3 x 3 + b 4 ǫ c x = c x + c 4 ǫ d 3 x 3 = d 3 x 3 + d 4 ǫ This change of variables leads to the Schrödinger equation 1.3 with h 1 represented by 1.19 with the following local behavior around x = and ǫ = : βx, ǫ = b 1 x 1 + b x + b 3 x 3 + O 1.37 γx, ǫ = c x + O δx, ǫ = d 3 x 3 + O ζx, ǫ = e 4 ǫ + O ξx, ǫ = O Ṽ x, ǫ = O. In the new variables, the two relevant energy levels are E A B = Ṽ x, ǫ ± βx, ǫ + γx, ǫ + δx, ǫ + ζx, ǫ + ξx, ǫ = Ṽ x, ǫ ± b 1 x 1 + b x + b 3 x 3 + c x + d 3 x 3 + e 4 ǫ + O For Type 6 Avoided Crossings, we make the change of variables defined implicitly by the following: b 1 x 1 + b x + b 3 x 3 + b 4 x 4 = b 1 x 1 + b x + b 3 x 3 + b 4 x 4 + b 5 ǫ c x = c x + c 5 ǫ d 3 x 3 = d 3 x 3 + d 5 ǫ e 4 x 4 = e 4 x 4 + e 5 ǫ This change of variables leads to the Schrödinger equation 1.3 with h 1 represented by 1. with the following local behavior around x = and ǫ = : βx, ǫ = b 1 x 1 + b x + b 3 x 3 + b 4 x 4 + O 1.4 γx, ǫ = c x + O 1

13 δx, ǫ = d 3 x 3 + O ζx, ǫ = e 4 x 4 + O ξx, ǫ = f 5 ǫ + O Ṽ x, ǫ = O. In the new variables, the two relevant energy levels are E A B = βx, Ṽ x, ǫ ± ǫ + γx, ǫ + δx, ǫ + ζx, ǫ + ξx, ǫ 4 = Ṽ x, ǫ ± b j x j + c x + d 3 x 3 + e 4 x 4 + f 5 ǫ + O j=1 Ordinary Differential Equations of Semiclassical Mechanics In Section 3.1 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 to study the small t and ǫ behavior that we need for the asymptotic matching procedure that we use in the later sections to prove our main results. The small t and ǫ behavior of these quatities is not quite simple, due to the presence of two natural time scales. We present the detailed analysis for Type 3 Avoided Crossings. At the end of this section we describe what modifications need to be made to handle the other types of avoided crossings. We define V A B x, ǫ = Ṽ x, ǫ ± β x, ǫ + γ x, ǫ + δ x, ǫ.1 where x IR n, ǫ >, and β, γ, and δ satisfy Let a C t and η C t be the solutions of the classical equations of motion d dt ac t = η C t. d dt ηc t = V C a C t, ǫ, C = A, B. with initial conditions a C =.3 η C = η ǫ,.4 where η ǫ = η + Oǫ, η has the form described by 1.9, and the Oǫ term depends on whether C is A or B. 13

14 It follows from 1.33 that βx, ǫ, γx, ǫ and δx, ǫ are O, so using estimates of the type β/ β + γ + δ 1, we see that V C x, ǫ = O. This implies the existence and uniqueness of the solutions to. for small times..1 Small t and ǫ Asymptotics.5 We wish to compute the asymptotics of the solution of.,.3 when both t and ǫ tend to zero. We first need the following Lemma.1 Let a C t and η C t be the solutions of. and.3. If ǫ and t are sufficiently small, we have { a C t = η ǫt + Ot as t, uniformly in ǫ. η C t = η ǫ + Ot Proof: We use the contraction mapping principle argument used to prove Lemma.1 of []. The argument goes through without change except that equation.15 of [] must be replaced by β x, ǫ + γ x, ǫ + δ x, ǫ x=ζs,ǫ.6 = b 1 ζ 1 s, ǫ + b ζ s, ǫ + c ζ s, ǫ + d 3 ǫ + O3. Because η has the form 1.9, ζ 1 s, ǫ > c s, and ζ s, ǫ = Os. It follows that β x, ǫ + γ x, ǫ + δ x, ǫ x=ζs,ǫ cs,.7 for sufficiently small s. This estimate replaces estimate.16 of [], and the rest of the proof follows exactly as in []. We also have the following result, which is standard because Ṽ x, ǫ is a regular function: Lemma. Let at and ηt be the solutions of. and.3 with V C x, ǫ Ṽ x, ǫ. If ǫ and t are small enough, we have { at = η ǫt Ṽ, ǫ t + Ot3 ηt = η ǫ Ṽ, ǫt + Ot as t, uniformly in ǫ. We can now get further in the asymptotics of the classical motion. To simplify notation, we define b 1 b c b =, c =, and ρǫ =.. b 1 η1ǫ >..8 d 3 14

15 Proposition.1 Let a C t and η C t be the solutions of. and.3, subject to the initial conditions a C = and η C = η ǫ. For t and ǫ small enough, we have the asymptotics a A B t = Ṽ, ǫt + η ǫt + O t 3 + ǫt.9 [ 1 b t ρǫt ρǫ + ǫ + ǫ ] ρǫ lnρǫt + ρǫt + ǫ ǫ lnǫ ǫt ρǫ The asymptotics for η C t in the same regime is obtained by termwise differentiation of the above formulae up to errors Ot + ǫ t. In the sequel, we will actually need such asymptotic behaviors for matching in the time regime defined by t such that ǫ, t, t /ǫ and t 3 /ǫ. We will refer to this regime as the matching regime. Corollary.1 Further expanding, we get in the matching regime ǫ, t, t /ǫ and t 3 /ǫ a A B t = Ṽ, ǫt + η ǫt ± ǫt ρǫ b [ t signt + ǫ ln t ρǫ + ǫ ] 4 ρǫ 1 + lnρǫ ǫ ln ǫ ρǫ + Ot 3 + Oǫ 4 /t b The asymptotics for η C t in the same regime are obtained by termwise differentiation of this formulae up to errors Ot + Oǫ 4 /t 3. Proof: The proposition is proved along the same line as Proposition.1 of []. explicitly, omitting the arguments x, ǫ V A B = Ṽ We compute β β + γ γ + δ δ β + γ + δ..1 Then we replace x by a C t above and expand the result making use of the local behaviors 1.33, Lemma.1 so that On = O t n + ǫ n, and the explicit form 1.9 for η ǫ η 1ǫ η η3 ǫ = ǫ η..11 4ǫ. ηn ǫ 15

16 Thus we get d dt ac t = Ṽ, ǫ + Ot b 1 η1 ǫtb + Ot + ǫ b 1 η 1ǫt + d 3 ǫ + O t 3 + ǫ 3 = Ṽ, ǫ ρǫtb + Ot + ǫ ρǫt + ǫ 1 + O t + ǫ + Ot.1 = Ṽ, ǫ ρǫtb + O t + ǫ. ρǫt + ǫ We get the result by explicit integration, taking into account the initial conditions.3.. Classical Action Integrals We now determine the asymptotics of classical action integrals which determine phases when we construct quantum mechanical wave functions for the nuclei. Let t η S C C t t = V C a C t, ǫ dt.13 = η ǫ t V C a C, ǫt + t η C t dt,.14 and let t ηt St = Ṽ at, ǫ dt.15 t = ηt dt η ǫ t Ṽ a, ǫt,.16 From lemma. we easily deduce Lemma.3 As t, St = η ǫ t Ṽ, ǫt η ǫ Ṽ, ǫt + Ot 3,.17 uniformly in ǫ. From corollary.1 and the formula V A B, ǫ = Ṽ, ǫ ± d 3ǫ + Oǫ,.18 we obtain Lemma.4 In the regime ǫ, t, t /ǫ and t 3 /ǫ we have the asymptotics S A B t = S A B ǫ, signt Ṽ, ǫt + η ǫ t η ǫ Ṽ, ǫt ± ǫ d 3 t signt η1 ǫb 1t + d 3 ǫ ln t η1ǫb 1 + Ot 3 + Oǫ 4 /t + Oǫ 3 ln t. 16

17 .3 Different Initial Momenta We assume from now on that the solution at of. with V C x, ǫ Ṽ x, ǫ is subject to the initial conditions a =.19 η = η whereas the solutions a C t satisfy a C =. η C = η ǫ = η + Oǫ. The Oǫ 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. When ǫ, t, t 3 /ǫ and t /ǫ we have and η A B tat a A B t = d 3 ǫt η ǫ η η ǫt [ b1 η ±signt 1ǫ t + ǫ d 3 b1 η ln 1ǫ t + 1 ] b 1 η1ǫ d 3 ǫ +Ot 3 + Oǫ 4 /t + Otǫ ln ǫ Corollary.3 When ǫ, t, t 3 /ǫ and t /ǫ we have S A A B t = S B ǫ, signt + St + η ǫ η t ± ǫ d 3 t signt η 1ǫb 1 t + d 3 ǫ η 1ǫb 1 ln t +Ot 3 + Oǫ 4 /t + Oǫ 3 ln t..4 Matrices A C t and B C t The semiclassical wave packets for the nuclei depend on 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.1 d dt BC t = iv C a C t, ǫa C t, 17

18 with the initial conditions A C = A, B C = B,. where a C t is the solution of. and.3. Let us determine the leading order behavior of V C a C t, ǫ. By explicit computation we get, omitting the arguments x, ǫ V C = Ṽ ± β β + γ γ + δ δ β + γ + δ 1/ +ββ + γγ + δδ β + γ + δ 1/ β β + γ γ + δ δ β β + γ γ + δ δ β + γ + δ 3/..3 Thus, from 1.33 and some simple estimates, we see that V C = Ṽ.4 ± γ + δ β β + β + δ γ γ βγ β γ + γ β + O3. β + γ + δ 3/ Explicitly, this has the form ± V C = Ṽ.5 b 1 b 1 b... b 1 β + γ + δ 3/ O3 + γ + δ 1 b b b 1 c... c + β + δ... b 1 c b 1 c β γ Using this expression, we obtain the following result for the solutions to.1 and.: Proposition. Let A C t and B C t be the solutions of.1 and.. In the regime ǫ, t, ǫ/ t, and t ln ǫ, we have the following asymptotics: where A A B t = A + O t ln ǫ,.6 B A B t = B ± i signt [ ] ρǫ t MA + ln NA + O t ln ǫ + ǫ, ρǫ ǫ t.7 M = 1 d 3 b b = b 1/ d 3 b 1 b / d 3... b 1 b / d 3 b / d

19 and N = 1 d 3 c c =... c / d Proof: To study solutions to.1 and. we first consider various quantities that occur in.5 with x replaced by a C t = η ǫt + Ot. By tedious, but straightforward calculations, we obtain the following estimates β + δ β + γ + δ 3/ = γ + δ β + γ + δ 3/ = β γ β + γ + δ 3/ = O t + ǫ. 1 d 3 ρǫ t + ǫ + O t + ǫ ǫ d 3 ρǫ t + ǫ 3/ + O t + ǫ From these estimates, it follows that for small t and ǫ, V A B a A ǫ B t, ǫ = ± ρǫ t + ǫ M + 1 3/ ρǫ t + ǫ N + O t + ǫ..3 1/ We can solve.1 and. if and only if we can solve A C t, ǫ = A + itb t s V C a C r, ǫ A C r, ǫ dr..31 For each fixed positive ǫ, this equation can be solved by iteration for small t. For T >, we define a norm on bounded operator valued functions of t and ǫ by D, = sup Dt, ǫ. {ǫ T, t ln t T, t ln ǫ T } From.31, we see that for positive ǫ, A C t s t, ǫ A = itb ds V C a C r, ǫa C r, ǫ dr t s t B + ds V C a C r, ǫ A C, A dr.3 t s + ds V C a C r, ǫ A dr t B t s + ds V C a C r, ǫ dr A C, A + A. 19

20 From.3 and explicit calculation, we see that for positive ǫ, s V C a C r, ǫ dr C 1 s + C s ǫ dr ρǫ r + ǫ 3/ + C 3 = C 1 s + C s ρǫ s + ǫ + C 3 1 ρǫ ln s ρǫ s ǫ dr ρǫ r + ǫ.33 ρǫ s ǫ By explicit integration, we see from this estimate that t s ds V C a C r, ǫ dr ρǫ t + ǫ ǫ C 4 t + C 5.34 ρǫ 1 + C 6 ρǫ ρǫ ǫ t + ǫ + ρǫ t ln ρǫ t + ρǫ t + ǫ ln ǫ. For sufficiently small T, t ln t T, t ln ǫ T, and ǫ T, this quantity is bounded by t s ds V C a C r, ǫ dr C 7 t + C 8 t ln t + C 9 t ln ǫ.35 O T. It now follows from.3 that for sufficiently small T, A C, A is bounded. This,.3, and.35 imply conclusion.6 of the proposition. To prove.7, we use B C t = B + i = B + i t t V C a C s, ǫ A C s ds.36 V C a C s, ǫ A ds + i t V C a C s, ǫ A C s A ds From.3 and explicit integration, t V C a C s, ǫ A ds.37 t = signt ρǫ t + ǫ MA + 1 ρǫ ln ρǫt ρǫ t NA ǫ ǫ + O t. Furthermore, from.3,.33, and.35, t V C a C s, ǫ A C s A ds.38 t V C a C s, ǫ A C s A ds.39 [ C 1 t + C + Oln t + ln ǫ ] [ C 7 t + O t ln t + t ln ǫ ]..4 For t and ǫ in the regime covered by the proposition, this last quantity is O t ln ǫ. Conclusion.7 is now an easy consequence of.36,.37, and.38.

21 .5 Alterations for Other Types of Avoided Crossings The analysis presented above is identical for Type 4 Avoided Crossings. For Type 5 Avoided Crossings, one must make the following substitutions in the above results, but the techniques of proof remain the same. b = b 1 b b 3., ρǫ = b 1η1 ǫ, M = 1 1 b b, and N = e 4 e 4 e 4 c c + d d. Similarly, for Type 6 Avoided Crossings, one must make the following substitutions, but the techniques of proof remain the same. b 1 b b 3 b = b 4, ρǫ = b 1η1 ǫ, M = 1 1 b b, and N = f 5 f 5 f 5. c c + d d + e e. 3 Propagation through Type 3 Avoided Crossings We now have all the ingredients to construct an asymptotic solution to the equation iǫ t ψx, t = ǫ4 ψx, t + hx, ǫψx, t 3.1 as ǫ. We first present the building blocks and give their main properties. 3.1 Semiclassical Nuclear Wave Packets The semiclassical motion of the nuclei is described by wave packets which correspond to the classical phase space trajectory, and of width Oǫ. These are the same wave packets that are used in [8, 17, ]. Let n denote the dimension of the nuclear configuration space. A multi-index l = l 1, l,..., l n is an ordered n-tuple of non negative integers. The order of l is defined to be l = n j=1 l j, and the factorial of l is defined to be l! = l 1!l! l n!. The symbol D l denotes the differential operator D l = l, and the symbol x 1 l 1 x l x n ln xl denotes the monomial x l = x l 1 1 x l x ln n. We denote the gradient of a function f by f 1 and the matrix of second partial derivatives by f. 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=1 v j w j. 1

22 The semiclassical wave packets are products of complex Gaussians and generalizations of Hermite polynomials. The generalizations of the zeroth and first order Hermite polynomials are defined by and H x = 1 3. H 1 x = 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 1, v,..., v m be m arbitrary non zero vectors in IC n. Then H m v 1, v,..., v m ; x = v m, x H m 1 v 1, v,..., v m 1 ; x m 1 i=1 v m, v i H m v 1,..., v i 1, v i+1,..., v m 1 ; x. One can prove [8] that these functions do not depend on the ordering of the vectors v 1, v,..., v m. Let A be a complex invertible n n matrix. We define A = [AA ] 1/, 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 1,..., U A e }{{ 1, U } A e,..., U A e,..., U }{{} A e n,..., U A e n ; x 3.4 }{{} l 1 entries l 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 ǫ. Definition. Let A and B be complex n n matrices with the following properties: A and B are invertible; 3.5 BA 1 is symmetric [real symmetric] + i[real symmetric]; 3.6 Re BA 1 = 1 [BA 1 + BA 1 ] is strictly positive definite; 3.7 Re BA 1 1 = AA. 3.8 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 = l / l! 1/ π n/4 h n/4 [det A] 1/ H l A; h 1/ A 1 x a exp { x a, BA 1 x a / h + i η, x a / h }. The choice of the branch of the square root of [det A] 1 in this definition depends on the context, and is determined by initial conditions and continuity in time. We encourage the reader to consult Section 3.1 of [] for several remarks concerning this definition. The formulas for the functions ϕ l A, B, h, a, η, x are rather complicated, but the

23 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 = h has an approximate solution of the form Ψ + V x Ψ 3.9 e ist/ h ϕ l At, Bt, h, at, ηt, x + O h 1/. 3.1 Here O h 1/ 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 1/ for t in a fixed bounded interval [ T, T ]. The vectors at and ηt satisfy the classical equations of motion a t t = ηt, 3.11 η t t = V 1 at. 3.1 The function St is the classical action integral associated with the classical path, t ηs St = V as ds T The matrices At and Bt satisfy A t t = ibt, 3.14 B t t = iv at At 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 [8]. In order to control certain errors, we introduce a cutoff function that is supported near the classical path. Let F be a C cutoff function F : IR + IR, such that { F r = 1 r 1 F r = r The wave functions we construct below contain the following cutoff function as a factor: F x a C t /ǫ 1 δ 3.18 where < δ is chosen below, and C = A, B. Multiplication of our semiclassical wave packets by this function leads to exponentially small errors in ǫ. 3

24 3. Choice of Eigenvectors In this section we construct the electronic eigenvectors and their phases on the support of the cutoff function F x a C t /ǫ 1 δ. Although the electronic hamiltonian is independent of time, it is convenient to choose specific time dependent electronic eigenvectors. The electrons follow the motion of the nuclei in an adiabatic way, so the suitable instantaneous electronic eigenvectors must satisfy a parallel transport condition to take into account geometric phases that arise. The 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, t > ǫ 1 ξ for suitable ξ, with < δ < ξ < 1. We have two sets of dynamic eigenvectors, denoted by Φ ± C x, t, ǫ, where the label ±, refers to positive and negative times. Let a C t and η C t be the solution of the classical equations of motion. and.3. Our goal is to construct normalized eigenvectors Φ ± C x, t, ǫ that solve Φ ± C x, t, ǫ, / t + η C t Φ ± C x, t, ǫ 3.19 for C = A, B and t > <. Since the eigenvalues E Ax, ǫ and E B x, ǫ are non-degenerate for any non-zero time t, these vectors exist, are unique up to constant overall phase factors, and are eigenvectors of h 1 x, ǫ associated with E C x, ǫ for any time, see [17]. We need asymptotic information about these eigenvectors, so we construct them explicitly. We define polar coordinates r, θ, φ by the relations βx, ǫ = r cos θ 3. γx, ǫ = r sin θ cos φ 3.1 δx, ǫ = r sin θ sin φ. 3. Then, in the basis { ψ 1 x, ǫ, ψ x, ǫ}, we construct the following static eigenvectors for h 1 x, ǫ: e Φ A = iφ cos θ/ 3.3 sin θ/ sin θ/ Φ B = e iφ 3.4 cos θ/ for π/ < θ π, and cos θ/ Φ + A = e iφ sin θ/ e Φ + B = iφ sin θ/ cos θ/ for θ π/ Lemma 3.1 Suppose < δ < ξ < 1, and t > ǫ 1 ξ. Define µǫ, t = max { t, ǫ 1 δ / t }, and let a C t and η C t solve equations. and.3. For x in the support of F x a C t /ǫ 1 δ, C = A, B, we have for t <, Φ A x, ǫ = ψ x, ǫ + Oµǫ, t, Φ B x, ǫ = ψ 1x, ǫ + Oµǫ, t, 4

25 and for t >, Φ + A x, ǫ = ψ 1x, ǫ + Oµǫ, t, Φ + B x, ǫ = ψ x, ǫ + Oµǫ, t. Proof: Let x belong to the support of F x a C t /ǫ 1 δ. Then x = a C t + ω, where ω ǫ 1 δ t, so Thus, Therefore, x 1 = η 1 t + Ot + Oǫ 1 δ, x = Ot + Oǫ 1 δ, x j = O t + Oǫ 1 δ, for j 3. x 1 = η1 t + O t µǫ, t, x = O t µǫ, t, x j = O t, for j 3. βx, ǫ = b 1 η 1t + O t µǫ, t, 3.7 γx, ǫ = O t µǫ, t, 3.8 δx, ǫ = d 3 ǫ + O t µǫ, t = O t µǫ, t. 3.9 From these estimates, it follows that βx, ǫ r γx, ǫ r δx, ǫ r = signt 1 + γ +δ β = Oµǫ, t, = Oµǫ, t. = signt 1 + O t µǫ, 1/ t = signt + O µǫ, t, Ot Since sin θ/ = 1 1 cos θ, and cos θ/ = 1 sin θ/, we see that sin θ/ = cos θ/ = The lemma follows. { } if t > + O µǫ, t and if t < { } 1 if t > + O µǫ, t. if t < We now construct the dynamic eigenvectors. 5

26 Lemma 3. Suppose < δ < ξ < 1 3, t > ǫ1 ξ, and t < ǫ κ for some κ > 1. Let ac t and η C t solve the classical equations of motion. and.3. There exist eigenvectors Φ ± C x, t, ǫ, C = A, B, that solve 3.19 and satisfy Φ ± C x, t, ǫ Φ± C x, ǫ = O t for all x in the support of F x a C t /ǫ 1 δ. Proof: We give a proof for Φ + Ax, t, ǫ only; the other cases are similar. In order to simplify the notation, we drop the indices A and +, and let at a A t and ηt η A t for t >. We again let µǫ, t = max { t, ǫ 1 δ / t }. Solutions of 3.19 have the form Φx, t, ǫ = Φx, ǫe iλx,t,ǫ, 3.31 where λx, t, ǫ is a real valued function that satisfies the equation i λx, t, ǫ + iηt λx, t, ǫ + Φx, ǫ, ηt Φx, ǫ =. 3.3 t We introduce and define ω x at 3.33 λω, t, ǫ λω + at, t, ǫ, 3.34 Φω, t, ǫ Φω + at, ǫ In the new variables, equation 3.3 for λ is where i t λω, t, ǫ = Φω, t, ǫ, Φω, t, ǫ 3.36 t Φω, t, ǫ = ηt Φω + at, ǫ t Dropping the arguments, we have Φ = sinθ/ψ1 + e iϕ cosθ/ψ θ/ + cosθ/ ψ 1 + e iϕ sinθ/ ψ i ϕe iϕ sinθ/ψ Since the ψ j, j = 1,, are orthonormal, we get Φ, Φ = iη ϕ sin θ/ + cos θ/ ψ 1 η ψ 1 + sin θ/ ψ η ψ t + sinθ/ cosθ/ e iϕ ψ 1 η ψ + e iϕ ψ η ψ

27 As η, ψ j, ψ j are all O, we have λω, t, ǫ = t ηt ϕ sin θω + at, ǫ/ + O dt, 3.4 where we have now fixed the constant of integration. We claim that η ϕ sin θ/ is bounded on the support of F x a C t /ǫ 1 δ. We note that ϕx, ǫ = cot 1 γx, ǫ/δx, ǫ, 3.41 provided δx, ǫ is different from zero. However, on the support of F x a C t /ǫ 1 δ with t < ǫ κ, δx, ǫ = d 3 ǫ + Ox + ǫ = d 3 ǫ + Ot + ǫ Cǫ, for some positive C. By the mean value theorem, ζ y + z = y + z for some ζ [, z ] y + ζ Thus, y + z. sin θ/ = 1 cos θ = r β = r satisfies sin θ/ γ + δ β + γ + δ. β + γ + δ β r From 3.41 we therefore see that η ϕ sin θ/ γ η δ + δ η γ γ + δ β + γ + δ η γ + η δ β + γ + δ However, on the support of F x a C t /ǫ 1 δ, β + γ + δ C t, for some C, η δ = O t, and η γ = η c + O t 3.4 = O t, since η c =. Thus, η ϕ sin θ/ is bounded, and by virtue of 3.4, λω, t, ǫ = λ ω. + O t. This implies the lemma since we can arbitrarily take λ ω. 7

28 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, S B =, A B = A, and B B t B + i d [ 3 b1 η ] MA b 1 η1 + ln 1 t NA, d 3 ǫ where A and B satisfy the hypotheses in the definition of the functions ϕ l, and the asymptotic condition for B B t is to hold for small values of ǫ, t, ǫ/ t, and t ln ǫ. 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 next lemma tells us how close to the avoided crossing time these standard wave packets can be used as approximations. Lemma 3.3 Suppose < δ < ξ < 1. For any T t ǫ 1 ξ, there is an approximation ψ OI x, t to the exact solution ψx, t to the Schrödinger equation, such that x a B t ψ OI x, t = F e isb t/ǫ ϕ l A B t, B B t, ǫ, a B t, η B t, x Φ B x, t, ǫ and ǫ 1 δ 3.43 ψx, t = ψ OI x, t + Oǫ ξ, 3.44 in the L IR n sense, as ǫ. Proof: The proof of this Lemma is very similar to that of Lemma 6.4 of [17]. For T t ǫ 1 ξ and x in the support of the cut off function F x a B t /ǫ 1 δ 1, E A x E B is bounded by x a multiple of ǫ 1+ξ. Furthermore, for such x and t, a rather tedious calculation shows that + t ηb x Φ B x, t, ǫ is bounded by a multiple of ǫ 1+ξ. Thus, up to errors on the order of ǫ ξ, ψ OI x, t is equal to where Ψx, t = F x a B t /ǫ 1 δ e isb t/ǫ Ψ x, t + ǫ Ψ x, t 3.45 Ψ x, t = ϕ l A B t, B B t, ǫ, a B t, η B t, x Φ B x, t, ǫ

29 and Ψ x, t = iϕ la B t, B B t, ǫ, a B t, η B t, x Φ 1 Ax, t, ǫ E A x E B x Φ Ax, t, ǫ, t + ηb x Φ B x, t, ǫ Φ B x, t, ǫ + r BA x, ǫp AB x, ǫ t + ηb x, 3.47 where r BA x, ǫ is the restriction of hx, ǫ E B x, ǫ 1 to the range of P ABx. The expression 3.45 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 ǫ ξ. To prove this, we explicitly compute ζx, t, ǫ = iǫ Ψ t HǫΨ For T t ǫ 1 ξ, terms in ζx, t, ǫ that contain derivatives of F are easily seen to have norms that grow at worst like powers of 1/ t times factors that are exponentially small in ǫ. Most of the remaining terms in ζx, t, ǫ are formally given on pages of [17], with t + ηb x in place of ηb x in many expressions. A few other terms arise from the iǫ t acting on the electronic eigenfunctions in Our error term ζx, t, ǫ differs formally from that of Section 6 of [17] because our eigenfunctions have time dependence and the ones in Section 6 of [17] do not. After noting that numerous terms in ζx, t, ǫ cancel with one another, we estimate the remaining terms individually. This process is very similar to that described on pages of [17], 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.. Our avoided crossing problem is slightly less singular than the crossing problem treated in [17]. So, it is not surprising that our error term satisfies the same estimate ζ, t, ǫ C ǫ 3 t 3.49 as the corresponding error term in Section 6 of [17]. We obtain the desired estimate by using the estimate 3.49 and lemma 3.3 of [17]. Lemma 3.4 Suppose < δ < ξ < 1, t < 3 ǫ1 ξ, and t < ǫ κ for some κ > 1, and let y = x at. Then the approximate solution of lemma 3.3 has the following asymptotics: ǫ [ ψ OI x, t = F ǫ δ y exp i St + i SB ǫ, i b 1η 1 1 t + i d 3 ln ] d 3 ǫt b 1 η1 ǫ ǫ ǫ b 1 η1 exp i d [ 3 b1 η1 y, M + ln t ] N y b 1 η1 d 3 ǫ exp i d 3 b y i t b y b 1 η1 ǫ 9

30 exp i η y ǫ n/ ϕ l A, B, 1,,, y ψ 1 x, ǫ ǫ + Oǫ α, 3.5 for some α >. Proof: We make a sequence of replacements in formula With each replacement, we make an error that is acceptably small because of earlier results. We introduce the notation y C = x a C t/ǫ, and using corollary.1, we replace F ǫ δ y B by F ǫ δ y. Next we use corollary.1 and corollary. to replace the factor exp i ηb t y B by ǫ exp i η y i d 3 b y i t b y + i d 3 t + i b 1η1 t d 3 + i ǫ b 1 η1 ǫ ǫ ǫ b 1 η1 ln b 1η 1 t d 3 ǫ + 1. We use lemma 3.1 of [] and corollary.1 to replace ϕ l A B t, B B t, ǫ, a B t,, x by ϕ l A B t, B B t, ǫ, at,, x. We then use lemma 3.1 of [] and proposition. to replace ϕ l A B t, B B t, ǫ, at,, x by exp i d [ 3 b1 η1 y, M + ln t ] N b 1 η1 d 3 ǫ y ǫ n/ ϕ l A, B, 1,,, y. By corollary.3 we can replace exp i SB t by exp i St + i SB ǫ, i d 3 t i b 1η1t i d 3 ln t. ǫ ǫ ǫ ǫ ǫ η1b 1 Finally, using lemmas 3.1 and 3., we replace Φ B x, t, ǫ by ψ 1 x, ǫ The lemma follows. 3

31 3.4 The Inner Solution For small times we use a different approximation that is constructed by means of the classical quantities associated with the potential Ṽ x, ǫ. Let at, ηt, and St be the classical quantities that satisfy the initial conditions a = η = η S =. In the rescaled variables { y = x at/ǫ, s = t/ǫ, the Schrödinger equation is iǫ s ψ iǫηǫs yψ = ǫ yψ + haǫs + ǫy, ǫψ We look for an approximate solution of the form ψy, s, ǫ = F y ǫ δ exp i Sǫs + i ηǫsy χy, s, ǫ, 3.54 ǫ ǫ with χy, s, ǫ = {fy, s, ǫψ 1 aǫs + ǫy, ǫ + gy, s, ǫψ aǫs + ǫy, ǫ ψ aǫs + ǫy, ǫ}, where ψ x, ǫ I I P x, ǫh and f, g are scalar functions. Anticipating exponential spatial fall off of the solution, we insert 3.54 in 3.53 and neglect the derivatives of F. Making use of the decomposition 1.1 and eliminating the overall classical phase and cut-off F, we have iǫψ 1 s f + iǫψ s g + iǫ s ψ + Ṽ a + ǫy xṽ afψ 1 + gψ + ψ +iǫ fη x ψ 1 + iǫ gη x ψ + iǫ η x ψ = ǫ 4 x ψ / ǫ 4 f x ψ 1 / ǫ 4 g x ψ / ǫ 3 y f x ψ 1 ǫ 3 y g x ψ ǫ ψ 1 y f ǫ ψ y g + h ψ + h 1 fψ 1 + gψ +Ṽ a + ǫyfψ 1 + gψ + ψ We assume the solutions have expansions of the form fy, s, ǫ = gy, s, ǫ = ψ x, t, ǫ = ν j ǫf j y, s, j= ν j ǫg j y, s, j= ν j ǫψ j x, t, 3.57 j= 31

32 with asymptotic scales ν j ǫ to be determined by matching. We insert these expansions in Using the behaviors 1.33 and lemma. we see that the lowest order terms yield iǫν ǫ s f y, sψ 1 aǫs + ǫy, ǫ + iǫν ǫ s g y, sψ aǫs + ǫy, ǫ = h aǫs + ǫy, ǫψ aǫs + ǫy, ǫs, ǫ +ǫν ǫh 11 y, s f y, sψ 1 aǫs + ǫy, ǫ + g y, sψ aǫs + ǫy, ǫ 3.58 on the support of F, where h 11 y, s is the operator on the span of { ψ 1 aǫs + ǫy, ǫ, ψ aǫs + ǫy, ǫ } whose matrix in the basis { ψ 1 aǫs + ǫy, ǫ, ψ aǫs + ǫy, ǫ } is given by b1 η1s + b 1 y 1 + b y c y + id 3 c y id 3 b 1 η1 s b. 1y 1 b y By matching the incoming outer solution, we obtain ν j ǫh aǫs + ǫy, ǫψ j aǫs + ǫy, ǫs = 3.59 if ν j ǫ << ǫν ǫ, j =, 1,, m 1. The spectrum of h x, ǫ is bounded away from in a neighborhood of,. This implies ψ j x, t =, j =, 1,, m By projecting with P x, ǫ and I I P x, ǫ, we split the remaining equation for order ǫν ǫ = ν m ǫ, into iǫν ǫ s f ψ 1 + iǫν ǫ s g ψ = ν m ǫh ψ + ǫν ǫh 11 f ψ 1 + g ψ, 3.61 iǫν ǫ s f ψ 1 + iǫν ǫ s g ψ = ǫν ǫh 11 f ψ 1 + g ψ and ν m ǫh ψ m =. 3.6 The second equation gives ψ m =. The first one is equivalent to i f y, s b1 η1 = s + b 1y 1 + b y c y + id 3 s g y, s c y id 3 b 1 η1s b 1 y 1 b y f y, s g y, s The general solution of this equation can be found exactly in terms of parabolic cylinder functions [6]. 1 ic f y, s y +id 3 1+i D pu 1 b 1 η b = C g y, s 1 y 1 η1 b1 η1 1 s + b 1y 1 + b y 1+i D pu b b1 η1 1 η1s b 1 y 1 + b y + C y 1 i D pl b1 η1 1 ic y id 3 D b 1 η1 pl 1 3 b 1 η1s + b 1 y 1 + b y 1 i b1 η 1 b 1 η 1 s + b 1y 1 + b y,

33 where p l = p u = i c y + d 3. b 1 η1 We define our inner approximation by ψ I y, s = F y ǫ δ exp i Sǫs + i ηǫsy ǫ ǫ [ ν ǫf y, sψ 1 aǫs + ǫy, ǫ + ν ǫg y, sψ aǫs + ǫy, ǫ ], 3.65 with f and g as above. To match 3.5 on the B level, we need g y, s, as s. This forces us to take C 1 y, 3.66 because the second component of the first term on the right hand side of 3.64 does not tend to zero as s. We see this by using the asymptotic formula 1 of 9.46 of [6], D q z = e z /4 z q q 1 O, for arg z < 3π z 4, 3.67 and noting that y /s = Oǫ ξ δ due to the cutoff. The second component of the second term on the right hand side of 3.64 does tend to zero as s. Formula 3.67 and some simple estimates show that on the support of F y ǫ δ, D pl 1 1 i b 1 η 1 b 1 η1s + b 1 y 1 + b y = O s 1. Thus, the first component of the second term on the right hand side of 3.64 determines C y by matching 3.5. As s, that component has the following asymptotics by 3.67: 1 i D pl b 1 η1s + b 1 y 1 + b y b 1 η1 = exp i b 1η1 s + b y + πc y + d 3 b 1 η1 8b 1 η1 exp i c y + d [ ] 3 ln + ln b b 1 η1 b 1 η1 1 η1 s b y 1 + O s. 33

34 In the matching regime, y / s, so by a fairly lengthy calculation, this quantity matches 3.5 if we choose ν ǫ = 1 and C y to be πc C y = ǫ n/ ϕ l A, B, 1,,, y exp y + d 3 8b 1 η1 exp i SB ǫ, i d 3 d 3 d 3 ln ǫ + i i ln d ǫ b 1 η1 4b 1 η1 b 1 η1 3 + i 3d 3 ln b 4b 1 η1 1 η1 exp i c y ln b 4b 1 η1 1 η1 + i c y ln d b 1 η1 3 i d 3 b y b 1 η1 The following result deals with the validity of the inner approximate solution. Lemma 3.5 Suppose < δ < ξ < 1/3, κ > 1, and 1 ξ > κ. The function Ψ Ix, t = ψ I x at/ǫ, t/ǫ is a valid approximation of a solution ψx, t to the Schrödinger equation for ǫ 1 ξ t ǫ 1 ξ, in the sense that as ǫ, ψ, t Ψ I, t = Oǫ 1 3ξ. Furthermore, in the matching regime, ǫ κ < t < ǫ 1 ξ, the inner and outer solutions agree in the sense that Ψ IO, t Ψ I, t = Oǫ α for some α >. Proof: The first result is proved by mimicking the proofs of proposition 3.1 of [] and lemma 3.5 of []. The second result is proved by combining lemma 3.4 and simple estimates based on the calculations presented above. We need the large positive s asymptotics of the inner solution to match it to an outgoing outer solution. To obtain these asymptotics, we use formula 3 of 9.46 of [6], D q z = e z /4 z q q 1 O π z Γ q e qπi e z /4 z q 1 q 1 O, z for 5π < arg z < π. 4 4 From this formula, the first component of the second term in 3.64 has large s asymptotics omitting the factor C y 1 i D pl b 1 η b 1 η1 1 s + b 1y 1 + b y = exp i b 1η1 s + b y 3πc y + d 3 b 1 η1 8b 1 η1 exp i c y + d 3 ln b 1 η1 b 1 η1 1 + O s + O s b 1 η 1 s + b y 3.69