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 24061-0123, U.S.A.

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.

∗Partially Supported by National Science Foundation Grants DMS–9403401 and DMS–9703751 †Supported by Fonds National Suisse de la Recherche Scientifique, Grant 8220-037200 ‡Permanent address: Institut Fourier, Université Grenoble 1, BP 74, 38402 Saint-Martin d’Hères Cedex, France

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 [20] deals with Types 1 and 2; the present paper deals with Types 3, 4, 5, and 6.

In Type 1 and Type 2 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 [20] show that in Type 1 and Type 2 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 2, 2, 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 ϵ, where ϵ^{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, 12, 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, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 33, 35, 36, 37, 39, 40, 41, 42].

The Hamiltonian for a molecular system with K nuclei and N −K electrons has the form

H(ϵ) = ∑_{j=1}^{K} − \frac{ϵ^{4}}{2M_{j}} Δ_{x_{j}} − ∑_{j=K+1}^{N} \frac{1}{2m_{j}} Δ_{x_{j}} + ∑_{i<j} V_{ij}(x_{i} − x_{j}). (1.1)

Here x_{j} ∈ IR^{l} denotes the position of the jth particle, the mass of the jth nucleus is ϵ^{−4}M_{j} (for 1 ≤ j ≤ K), the mass of the jth 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_{2}, …, x_{Kl}) ∈ IR^{n} denote the nuclear configuration vector. We decompose H(ϵ) as

H(ϵ) = − \frac{ϵ^{4}}{2} Δ_{x} + h(x). (1.2)

This defines the electronic Hamiltonian h(x) that depends parametrically on x. The time–dependent Schrödinger equation that we study is

i ϵ^{2} \frac{∂ψ}{∂t} = H(ϵ) ψ, (1.3)

for t in a fixed interval. The factor of ϵ^{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” [2] 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 ϵ 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 (1.2). In some realistic systems, h(x) 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 ϵ. For this reason, we generalize the form (1.2) to allow the electron Hamiltonian to depend on ϵ as well as x. We study solutions to equation (1.3) with Hamiltonians of the form

H(ϵ) = − \frac{ϵ^{4}}{2} Δ_{x} + h(x, ϵ), (1.4)

with the assumption that h(x, ϵ) has an Avoided Crossing according to the following definition:

Definition Suppose h(x, ϵ) is a family of self–adjoint operators with a fixed domain D in a Hilbert Space H, for x ∈ Ω and ϵ ∈ [0, α), where Ω is an open subset of IR^{n}. Suppose that the resolvent of h(x, ϵ) is a C^{4} function of x and ϵ as an operator from H to D. Suppose h(x, ϵ) has two eigenvalues E_{A}(x, ϵ) and E_{B}(x, ϵ) that depend continuously on x and ϵ and are isolated from the rest of the spectrum of h(x, ϵ). Assume Γ = { x : E_{A}(x, 0) = E_{B}(x, 0) } is a single point or non-empty connected proper submanifold of Ω, but that for all x ∈ Ω, E_{A}(x, ϵ) ≠ E_{B}(x, ϵ) when ϵ > 0. Then we say h(x, ϵ) has an Avoided Crossing on Γ.

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, 12, 36].

2. 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, 4.2, 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 ϵ, 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

O(m) = O ({(∑_{j=1}^{n} x_{j}^{2}) + ϵ^{2}}^{m/2}) . (1.5)

Assume without loss of generality that 0 is a generic point of Γ, and that we are given a classical nuclear momentum vector η^{0} that is transversal to Γ. We decompose

h(x, ϵ) = h_{∥}(x, ϵ) + h_{⊥}(x, ϵ) (1.6)

with

h_{∥}(x, ϵ) = h(x, ϵ)P (x, ϵ) (1.7) and

h_{⊥}(x, ϵ) = h(x, ϵ)(I I − P (x, ϵ)) (1.8)

where P (x, ϵ) is a spectral projector for h(x, ϵ) associated with E_{A}(x, ϵ) and E_{B}(x, ϵ).

Type 1 and 2 Avoided Crossings have the codimension of Γ equal to 1. We will not discuss them here since they have already been studied in [20].

Type 3 and 4 Avoided Crossings have the codimension of Γ equal to 2. For these two types of Avoided Crossings, we choose an orthogonal coordinate system for the nuclear configurations in which the x_{1} and x_{2} coordinate directions are perpendicular to Γ at x = 0; the x_{3}, x_{4}, … x_{n} coordinate directions are parallel to Γ at x = 0; and so that the vector η^{0} has the form

η^{0} = ( η_{1}^{0} ; 0 ; η_{3}^{0} ; η_{4}^{0} ; ⋮ ; η_{n}^{0} ) . (1.9)

The eigenvalues involved in a Type 3 Avoided Crossing are non-degenerate. One can choose [19] an orthonormal basis {ψ_{1}(x, ϵ), ψ_{2}(x, ϵ)} of P (x, ϵ)H, which is regular in (x, ϵ) around (0, 0), such that

h_{∥}(x, ϵ) = h_{1}(x, ϵ) + Ṽ (x, ϵ), (1.10)

where h_{1}(x, ϵ) is represented in this basis by the matrix

( \begin{matrix}β(x, ϵ) & γ(x, ϵ) + iδ(x, ϵ) \\ γ(x, ϵ) − iδ(x, ϵ) & −β(x, ϵ) \end{matrix} )  (1.11)

and Ṽ (x, ϵ) is represented by \frac{1}{2} trace(h(x, ϵ)P (x, ϵ)) times the 2 × 2 identity matrix. The function Ṽ (x, ϵ) is regular in (x, ϵ) near the origin, and

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + b_{3}ϵ + O(2)  (1.12)
γ(x, ϵ) = c_{2}x_{2} + c_{3}ϵ + O(2)
δ(x, ϵ) = d_{3}ϵ + O(2)
Ṽ (x, ϵ) = O(0)

with b_{1} > 0, c_{2} > 0, b_{2} ∈ 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 \atop B}} = Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2}}
= Ṽ (x, ϵ) ± \sqrt{(b_{1}x_{1} + b_{2}x_{2} + b_{3}ϵ)^{2} + (c_{2}x_{2} + c_{3}ϵ)^{2} + (d_{3}ϵ)^{2} + O(3)}.  (1.13)

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

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_{2}. The electrons belong to a two level system with electron Hamiltonian

( \begin{matrix}x_{1} & x_{2} + iϵ \\ x_{2} − iϵ & −x_{1} \end{matrix} ) .

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 ϵ. The nuclear position and momentum uncertainties are proportional to ϵ, and the center of the Gaussian approximately follows a classical path a(t). Figure 2 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 − a(t))/ϵ. 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 2 multiplied by e^{−f(y_{2})} and 1 − e^{f(y_{2})}, respectively. Here

f(y_{2}) = \frac{π}{4η_{1}^{0}} (y_{2}^{2} + 1),

where the nuclear momentum is concentrated near

η(0) = ( \begin{matrix}η_{1}^{0} \\ 0 \end{matrix} )

Figure 2: Probability densities before encountering a Type 3 Avoided Crossing. Figure 3: Probability densities after encountering a Type 3 Avoided Crossing. at the time when a(t) = 0. 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 ϵ. 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 2\sqrt{(η_{1}^{0}t + y_{1})^{2} + y_{2}^{2} + 1}. The traditional Landau–Zener formula for a time dependent Hamiltonian with this size gap predicts an electronic transition probability of e^{−f(y_{2})} 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 2. Near one of these avoided crossings, one can choose an orthonormal basis {ψ_{1}(x, ϵ), ψ_{2}(x, ϵ), ψ_{3}(x, ϵ), ψ_{4}(x, ϵ)} of P (x, ϵ)H, which is regular in (x, ϵ) around (0, 0). The operator h_{∥} satisfies

h_{∥}(x, ϵ) = h_{1}(x, ϵ) + Ṽ (x, ϵ), (1.14)

where h_{1}(x, ϵ) is represented in this basis by the matrix

( \begin{matrix}β(x, ϵ) & γ(x, ϵ) + iδ(x, ϵ) & 0 & 0 \\ γ(x, ϵ) − iδ(x, ϵ) & −β(x, ϵ) & 0 & 0 \\ 0 & 0 & β(x, ϵ) & γ(x, ϵ) − iδ(x, ϵ) \\ 0 & 0 & γ(x, ϵ) + iδ(x, ϵ) & −β(x, ϵ) \end{matrix} ) , (1.15)

and Ṽ (x, ϵ) is represented by \frac{1}{4} trace(h(x, ϵ)P (x, ϵ)) times the 4 × 4 identity matrix. The function Ṽ (x, ϵ) is regular in (x, ϵ) near the origin, and

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + b_{3}ϵ + O(2)  (1.16)
γ(x, ϵ) = c_{2}x_{2} + c_{3}ϵ + O(2)
δ(x, ϵ) = d_{3}ϵ + O(2)
Ṽ (x, ϵ) = O(0).

The two energy levels involved in the Avoided Crossing are thus

E_{{A \atop B}} = Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + O(2)}
= Ṽ (x, ϵ) ± \sqrt{(b_{1}x_{1} + b_{2}x_{2} + b_{3}ϵ)^{2} + (c_{2}x_{2} + c_{3}ϵ)^{2} + (d_{3}ϵ)^{2} + O(3)}.  (1.17)

Type 5 Avoided Crossings have the codimension of Γ equal to 3 and the multiplicity of the eigenvalues equal to 2. We choose an orthogonal coordinate system for the nuclear configurations in which the x_{1}, x_{2}, and x_{3} coordinate directions are perpendicular to Γ at x = 0; the x_{4}, x_{5}, … x_{n} coordinate directions are parallel to Γ at x = 0; and so that the vector η^{0} has the form

η^{0} = ( η_{1}^{0} ; 0 ; 0 ; η_{4}^{0} ; η_{5}^{0} ; ⋮ ; η_{n}^{0} ) . (1.18)

We can choose an orthonormal basis {ψ_{1}(x, ϵ), ψ_{2}(x, ϵ), ψ_{3}(x, ϵ), ψ_{4}(x, ϵ)} of P (x, ϵ)H, which is regular in (x, ϵ) around (0, 0). The operator h_{∥}(x, ϵ) satisfies (1.14), where h_{1}(x, ϵ) is represented in this basis by the matrix

( \begin{matrix}β(x, ϵ) & 0 & γ(x, ϵ) + iδ(x, ϵ) & ζ(x, ϵ) + iξ(x, ϵ) \\ 0 & β(x, ϵ) & −ζ(x, ϵ) + iξ(x, ϵ) & γ(x, ϵ) − iδ(x, ϵ) \\ γ(x, ϵ) − iδ(x, ϵ) & −ζ(x, ϵ) − iξ(x, ϵ) & −β(x, ϵ) & 0 \\ ζ(x, ϵ) − iξ(x, ϵ) & γ(x, ϵ) + iδ(x, ϵ) & 0 & −β(x, ϵ) \end{matrix} ) ,

(1.19)

and Ṽ (x, ϵ) is represented by \frac{1}{4} trace(h(x, ϵ)P (x, ϵ)) times the 4 × 4 identity matrix. The function Ṽ (x, ϵ) is regular in (x, ϵ) near the origin, and

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}ϵ + O(2)
γ(x, ϵ) = c_{2}x_{2} + c_{4}ϵ + O(2)
δ(x, ϵ) = d_{3}x_{3} + d_{4}ϵ + O(2)
ζ(x, ϵ) = e_{4}ϵ + O(2)
ξ(x, ϵ) = O(2)
Ṽ (x, ϵ) = O(0).

We assume the generically satisfied condition e_{4} ≠ 0. The two energy levels involved in the Avoided Crossing are thus

E_{{A \atop B}}
= Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + ζ(x, ϵ)^{2} + ξ(x, ϵ)^{2}}
= Ṽ (x, ϵ) ± \sqrt{(∑_{j=1}^{3} b_{j}x_{j} + b_{4}ϵ)^{2} + (c_{2}x_{2} + c_{4}ϵ)^{2} + (d_{3}x_{3} + d_{4}ϵ)^{2} + (e_{4}ϵ)^{2} + O(3)}.

(1.20)

Type 6 Avoided Crossings have the codimension of Γ equal to 4 and the multiplicity of the eigenvalues equal to 2. We choose an orthogonal coordinate system for the nuclear configurations in which the x_{1}, x_{2}, x_{3}, and x_{4} coordinate directions are perpendicular to Γ at x = 0; the x_{5}, x_{6}, … x_{n} coordinate directions are parallel to Γ at x = 0; and so that the vector η^{0} has the form

η^{0} = ( η_{1}^{0} ; 0 ; 0 ; 0 ; η_{5}^{0} ; η_{6}^{0} ; ⋮ ; η_{n}^{0} ) . (1.21)

( \begin{matrix}β(x, ϵ) & 0 & γ(x, ϵ) + iδ(x, ϵ) & ζ(x, ϵ) + iξ(x, ϵ) \\ 0 & β(x, ϵ) & −ζ(x, ϵ) + iξ(x, ϵ) & γ(x, ϵ) − iδ(x, ϵ) \\ γ(x, ϵ) − iδ(x, ϵ) & −ζ(x, ϵ) − iξ(x, ϵ) & −β(x, ϵ) & 0 \\ ζ(x, ϵ) − iξ(x, ϵ) & γ(x, ϵ) + iδ(x, ϵ) & 0 & −β(x, ϵ) \end{matrix} ) ,

(1.22)

and Ṽ (x, ϵ) is represented by \frac{1}{4} trace(h(x, ϵ)P (x, ϵ)) times the 4 × 4 identity matrix. The function Ṽ (x, ϵ) is regular in (x, ϵ) near the origin, and

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}x_{4} + b_{5}ϵ + O(2)
γ(x, ϵ) = c_{2}x_{2} + c_{5}ϵ + O(2)
δ(x, ϵ) = d_{3}x_{3} + d_{5}ϵ + O(2)
ζ(x, ϵ) = e_{4}x_{4} + e_{5}ϵ + O(2)
ξ(x, ϵ) = f_{5}ϵ + O(2)
Ṽ (x, ϵ) = O(0).

We assume the generically satisfied condition f_{5} ≠ 0. The two energy levels involved in the Avoided Crossing are thus

E_{{A \atop B}}

= Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + ζ(x, ϵ)^{2} + ξ(x, ϵ)^{2}}
= Ṽ (x, ϵ)

± \sqrt{(∑_{j=1}^{4} b_{j}x_{j} + b_{5}ϵ)^{2} + (c_{2}x_{2} + c_{5}ϵ)^{2} + (d_{3}x_{3} + d_{5}ϵ)^{2} + (e_{4}x_{4} + e_{5}ϵ)^{2} + (f_{5}ϵ)^{2} + O(3)}.

(1.23)

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 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, 4.2, 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 1996. 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.

1.1 Convenient Changes of Variables

We begin with Type 3 Avoided Crossings. It is convenient to remove the ϵ-dependence in the leading order of β(x, ϵ) and γ(x, ϵ) in (1.11), so we introduce new variables that are implicitly defined by the relations

b_{1}x′_{1} + b_{2}x′_{2} = b_{1}x_{1} + b_{2}x_{2} + b_{3}ϵ  and  (1.24)
c_{2}x′_{2} = c_{2}x_{2} + c_{3}ϵ.  (1.25)

Explicitly, this change of variables is given by

x′_{1} = x_{1} + \frac{ϵ}{b_{1}} (b_{3} − \frac{b_{2}c_{3}}{c_{2}} ) ,  (1.26)
x′_{2} = x_{2} + \frac{c_{3}ϵ}{c_{2}},  (1.27)
x′_{j} = x_{j}  (j > 2).  (1.28)

When we change variables, the Schrödinger equation

iϵ^{2} \frac{∂}{∂t}ψ(x, t) = −\frac{ϵ^{4}}{2} Δψ(x, t) + h(x, ϵ)ψ(x, t) (1.29)

for ψ(x, t) becomes

iϵ^{2} \frac{∂}{∂t}ϕ(x′, t) = −\frac{ϵ^{4}}{2} Δϕ(x′, t) + h(x(x′, ϵ), ϵ)ϕ(x′, t), (1.30)

for

ϕ(x′, t) = ψ(x(x′, ϵ), t), (1.31)

with

h_{∥}(x(x′, ϵ), ϵ) = ( \begin{matrix}b_{1}x′_{1} + b_{2}x′_{2} & c_{2}x′_{2} + id_{3}ϵ \\ c_{2}x′_{2} − id_{3}ϵ & −b_{1}x′_{1} − b_{2}x′_{2} \end{matrix} ) + Ṽ (x(x′, ϵ), ϵ) I I + O(2), (1.32)

where Ṽ (x(x′, ϵ), ϵ) is regular in (x′, ϵ) around (0, 0) and O(2) refers to x′ and ϵ. We henceforth drop the primes on the new variables, and assume that h_{1}(x, ϵ) has the form (1.11) with the following local behavior around x = 0 and ϵ = 0:

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + O(2)  (1.33)
γ(x, ϵ) = c_{2}x_{2} + O(2)
δ(x, ϵ) = d_{3}ϵ + O(2)
Ṽ (x, ϵ) = O(0).

In the new variables, the two relevant energy levels are

E_{{A \atop B}} = Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2}}
= Ṽ (x, ϵ) ± \sqrt{(b_{1}x_{1} + b_{2}x_{2})^{2} + (c_{2}x_{2})^{2} + (d_{3}ϵ)^{2} + O(3)}.  (1.34)

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 (1.33). In the new variables, the two relevant energy levels are

E_{{A \atop B}} = Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2}}
= Ṽ (x, ϵ) ± \sqrt{(b_{1}x_{1} + b_{2}x_{2})^{2} + (c_{2}x_{2})^{2} + (d_{3}ϵ)^{2} + O(3)}.  (1.35)

For Type 5 Avoided Crossings, we make the change of variables defined implicitly by the following:

b_{1}x′_{1} + b_{2}x′_{2} + b_{3}x′_{3} = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}ϵ
c_{2}x′_{2} = c_{2}x_{2} + c_{4}ϵ
d_{3}x′_{3} = d_{3}x_{3} + d_{4}ϵ.  (1.36)

This change of variables leads to the Schrödinger equation (1.30) with h_{1} represented by (1.19) with the following local behavior around x = 0 and ϵ = 0:

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + O(2)  (1.37)
γ(x, ϵ) = c_{2}x_{2} + O(2)
δ(x, ϵ) = d_{3}x_{3} + O(2)
ζ(x, ϵ) = e_{4}ϵ + O(2)
ξ(x, ϵ) = O(2)
Ṽ (x, ϵ) = O(0).

In the new variables, the two relevant energy levels are

E_{{A \atop B}}
= Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + ζ(x, ϵ)^{2} + ξ(x, ϵ)^{2}}
= Ṽ (x, ϵ) ± \sqrt{(b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3})^{2} + (c_{2}x_{2})^{2} + (d_{3}x_{3})^{2} + (e_{4}ϵ)^{2} + O(3)}.

(1.38)

For Type 6 Avoided Crossings, we make the change of variables defined implicitly by the following:

b_{1}x′_{1} + b_{2}x′_{2} + b_{3}x′_{3} + b_{4}x′_{4} = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}x_{4} + b_{5}ϵ
c_{2}x′_{2} = c_{2}x_{2} + c_{5}ϵ
d_{3}x′_{3} = d_{3}x_{3} + d_{5}ϵ
e_{4}x′_{4} = e_{4}x_{4} + e_{5}ϵ.  (1.39)

This change of variables leads to the Schrödinger equation (1.30) with h_{1} represented by (1.22) with the following local behavior around x = 0 and ϵ = 0:

β(x, ϵ) = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}x_{4} + O(2)  (1.40)
γ(x, ϵ) = c_{2}x_{2} + O(2)

δ(x, ϵ) = d_{3}x_{3} + O(2)
ζ(x, ϵ) = e_{4}x_{4} + O(2)
ξ(x, ϵ) = f_{5}ϵ + O(2)
Ṽ (x, ϵ) = O(0).

In the new variables, the two relevant energy levels are

E_{{A \atop B}}
= Ṽ (x, ϵ) ± \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + ζ(x, ϵ)^{2} + ξ(x, ϵ)^{2}}
= Ṽ (x, ϵ) ± \sqrt{(∑_{j=1}^{4} b_{j}x_{j})^{2} + (c_{2}x_{2})^{2} + (d_{3}x_{3})^{2} + (e_{4}x_{4})^{2} + (f_{5}ϵ)^{2} + O(3)}. (1.41)

2 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 ϵ 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 ϵ 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 \atop B}} (x, ϵ) = Ṽ (x, ϵ) ± \sqrt{β^{2}(x, ϵ) + γ^{2}(x, ϵ) + δ^{2}(x, ϵ)} (2.1)

where x ∈ IR^{n}, ϵ > 0, and β, γ, and δ satisfy (1.33). Let a^{C}(t) and η^{C}(t) be the solutions of the classical equations of motion

\frac{d}{dt}a^{C}(t) = η^{C}(t)  (2.2)
\frac{d}{dt}η^{C}(t) = −∇V^{C}(a^{C}(t), ϵ),  C = A, B.

with initial conditions

a^{C}(0) = 0  (2.3)
η^{C}(0) = η^{0}(ϵ),  (2.4)

where η^{0}(ϵ) = η^{0} + O(ϵ), η^{0} has the form described by (1.9), and the O(ϵ) term depends on whether C is A or B.

It follows from (1.33) that |β(x, ϵ)|, |γ(x, ϵ)| and |δ(x, ϵ)| are O(0), so using estimates of the type β/\sqrt{β^{2} + γ^{2} + δ^{2}} ≤ 1, we see that

∥∇V^{C}(x, ϵ)∥ = O(0). (2.5)

This implies the existence and uniqueness of the solutions to (2.2) for small times.

2.1 Small t and ϵ Asymptotics

We wish to compute the asymptotics of the solution of (2.2), (2.3) when both t and ϵ tend to zero. We first need the following

Lemma 2.1 Let a^{C}(t) and η^{C}(t) be the solutions of (2.2) and (2.3). If ϵ and t are sufficiently small, we have

\{ \begin{matrix}a^{C}(t) = η^{0}(ϵ)t + O(t^{2}) \\ η^{C}(t) = η^{0}(ϵ) + O(t) \end{matrix}

as t → 0, uniformly in ϵ.

Proof: We use the contraction mapping principle argument used to prove Lemma 2.1 of [20]. The argument goes through without change except that equation (2.15) of [20] must be replaced by

β^{2}(x, ϵ) + γ^{2}(x, ϵ) + δ^{2}(x, ϵ)|_{x=ζ(s,ϵ)}  (2.6)
= (b_{1}ζ_{1}(s, ϵ) + b_{2}ζ_{2}(s, ϵ))^{2} + (c_{2}ζ_{2}(s, ϵ))^{2} + (d_{3}ϵ)^{2} + O(3).

Because η^{0} has the form (1.9), |ζ_{1}(s, ϵ)| > c|s|, and ζ_{2}(s, ϵ) = O(s^{2}). It follows that

β^{2}(x, ϵ) + γ^{2}(x, ϵ) + δ^{2}(x, ϵ)|_{x=ζ(s,ϵ)} ≥ cs^{2}, (2.7)

for sufficiently small s. This estimate replaces estimate (2.16) of [20], and the rest of the proof follows exactly as in [20].

We also have the following result, which is standard because Ṽ (x, ϵ) is a regular function:

Lemma 2.2 Let a(t) and η(t) be the solutions of (2.2) and (2.3) with V^{C}(x, ϵ) ≡ Ṽ (x, ϵ). If ϵ and t are small enough, we have

\{ \begin{matrix}a(t) = η^{0}(ϵ)t − ∇Ṽ (0, ϵ)\frac{t^{2}}{2} + O(t^{3}) \\ η(t) = η^{0}(ϵ) − ∇Ṽ (0, ϵ)t + O(t^{2}) \end{matrix}

as t → 0, uniformly in ϵ.

We can now get further in the asymptotics of the classical motion. To simplify notation, we define

b = ( b_{1} ; b_{2} ; 0 ; ⋮ ; 0 ) ,  c = ( 0 ; c_{2} ; 0 ; ⋮ ; 0 ) ,  and  ρ(ϵ) = \frac{b_{1} η_{1}^{0}(ϵ)}{|d_{3}|} > 0.  (2.8)

Proposition 2.1 Let a^{C}(t) and η^{C}(t) be the solutions of (2.2) and (2.3), subject to the initial conditions a^{C}(0) = 0 and η^{C}(0) = η^{0}(ϵ). For t and ϵ small enough, we have the asymptotics

a^{{A \atop B}} (t) = −∇Ṽ (0, ϵ)\frac{t^{2}}{2} + η^{0}(ϵ)t + O(|t|^{3} + ϵt^{2})  (2.9)

∓b\frac{1}{2ρ(ϵ)} [t\sqrt{(ρ(ϵ)t)^{2} + ϵ^{2}} + \frac{ϵ^{2}}{ρ(ϵ)} ln(ρ(ϵ)t + \sqrt{(ρ(ϵ)t)^{2} + ϵ^{2}}) − \frac{ϵ^{2} ln(ϵ)}{ρ(ϵ)} − 2ϵt]

The asymptotics for η^{C}(t) in the same regime is obtained by termwise differentiation of the above formulae up to errors O(t^{2} + ϵ|t|).

In the sequel, we will actually need such asymptotic behaviors for matching in the time regime defined by t such that ϵ → 0, t → 0, |t|/ϵ → ∞ and t^{3}/ϵ^{2} → 0. We will refer to this regime as the matching regime.

Corollary 2.1 Further expanding, we get in the matching regime ϵ → 0, t → 0, |t|/ϵ → ∞ and t^{3}/ϵ^{2} → 0

a^{{A \atop B}} (t) = − ∇Ṽ (0, ϵ)\frac{t^{2}}{2} + η^{0}(ϵ)t ± \frac{ϵt}{ρ(ϵ)} b
∓ sign(t) [\frac{t^{2}}{2} + \frac{ϵ^{2} ln |t|}{2 (ρ(ϵ))^{2}} + \frac{ϵ^{2}}{4 (ρ(ϵ))^{2}} (1 + 2 ln(2ρ(ϵ))) − \frac{ϵ^{2} ln ϵ}{2 (ρ(ϵ))^{2}} ] b
+ O(t^{3}) + O(ϵ^{4}/t^{2})

The asymptotics for η^{C}(t) in the same regime are obtained by termwise differentiation of this formulae up to errors O(t^{2}) + O(ϵ^{4}/t^{3}).

Proof: The proposition is proved along the same line as Proposition 2.1 of [20]. We compute explicitly, (omitting the arguments (x, ϵ))

−∇V ^{{A \atop B}} = −∇Ṽ ∓ \frac{β∇β + γ∇γ + δ∇δ}{\sqrt{β^{2} + γ^{2} + δ^{2}}} . (2.10)

Then we replace x by a^{C}(t) above and expand the result making use of the local behaviors (1.33), Lemma 2.1 (so that O(n) = O(|t|^{n} + ϵ^{n})), and the explicit form (1.9) for η_{0}(ϵ)

η^{0}(ϵ) = ( η_{1}^{0}(ϵ) ; 0 ; η_{3}^{0}(ϵ) ; η_{4}^{0}(ϵ) ; ⋮ ; η_{n}^{0}(ϵ) ) . (2.11)

Thus we get

\frac{d^{2}}{dt^{2}} a^{C}(t) = −∇Ṽ (0, ϵ) + O(t) ∓ \frac{b_{1}η_{1}^{0}(ϵ)tb + O(t^{2} + ϵ^{2})}{\sqrt{(b_{1}η_{1}^{0}(ϵ)t)^{2} + (d_{3}ϵ)^{2} + O(|t|^{3} + ϵ^{3})}}  (2.12)
= −∇Ṽ (0, ϵ) ∓ \frac{ρ(ϵ)tb + O(t^{2} + ϵ^{2})}{\sqrt{((ρ(ϵ)t)^{2} + ϵ^{2})(1 + O(|t| + ϵ))}} + O(t)
= −∇Ṽ (0, ϵ) ∓ \frac{ρ(ϵ)tb}{\sqrt{(ρ(ϵ)t)^{2} + ϵ^{2}}} + O(|t| + ϵ).

We get the result by explicit integration, taking into account the initial conditions (2.3).

2.2 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

S^{C}(t) = ∫_{0}^{t} (\frac{η^{C}(t′)^{2}}{2} − V^{C}(a^{C}(t′), ϵ)) dt′  (2.13)
= η^{0}^{2}(ϵ)\frac{t}{2} − V^{C}(a^{C}(0), ϵ)t + ∫_{0}^{t} η^{C}(t′)^{2} dt′,  (2.14)

and let

S(t) = ∫_{0}^{t} (\frac{η(t′)^{2}}{2} − Ṽ (a(t′), ϵ)) dt′  (2.15)
= ∫_{0}^{t} η(t′)^{2}dt′ − η^{0}^{2}(ϵ)\frac{t}{2} − Ṽ (a(0), ϵ)t,  (2.16)

From lemma 2.2 we easily deduce Lemma 2.3 As t → 0,

S(t) = η^{0}^{2}(ϵ)\frac{t}{2} − Ṽ (0, ϵ)t − η^{0}(ϵ)∇Ṽ (0, ϵ)t^{2} + O(t^{3}), (2.17)

uniformly in ϵ.

From corollary 2.1 and the formula

V ^{{A \atop B}} (0, ϵ) = Ṽ (0, ϵ) ± d_{3}ϵ + O(ϵ^{2}), (2.18) we obtain Lemma 2.4 In the regime ϵ → 0, t → 0, |t|/ϵ → ∞ and t^{3}/ϵ^{2} → 0 we have the asymptotics

S^{{A \atop B}} (t) = S0^{{A \atop B}} (ϵ, sign(t)) − Ṽ (0, ϵ)t + η^{0}^{2}(ϵ)\frac{t}{2} − η^{0}(ϵ)∇Ṽ (0, ϵ)t^{2} ± ϵ|d_{3}|t

∓sign(t) (η_{1}^{0}(ϵ)b_{1}t^{2} + \frac{d_{3}^{2}ϵ^{2}}{η_{1}^{0}(ϵ)b_{1}} ln |t|) + O(t^{3}) + O(ϵ^{4}/t^{2}) + O(ϵ^{3} ln |t|).

2.3 Different Initial Momenta

We assume from now on that the solution a(t) of (2.2) with V^{C}(x, ϵ) ≡ Ṽ (x, ϵ) is subject to the initial conditions

a(0) = 0  (2.19)
η(0) = η^{0}

whereas the solutions a^{C}(t) satisfy

a^{C}(0) = 0  (2.20)
η^{C}(0) = η^{0}(ϵ) = η^{0} + O(ϵ).

The O(ϵ) 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 ϵ → 0, t → 0, t^{3}/ϵ^{2} → 0 and |t|/ϵ → ∞ we have

η^{{A \atop B}} (t)(a(t) − a^{{A \atop B}} (t))

= ∓|d_{3}|ϵt − (η^{0}(ϵ) − η^{0})η^{0}(ϵ)t

±sign(t) [\frac{b_{1}η_{1}^{0}(ϵ)}{2}t^{2} + \frac{ϵ^{2}d_{3}^{2}}{2b_{1}η_{1}^{0}(ϵ)} (ln (\frac{2b_{1}η_{1}^{0}(ϵ)|t|}{|d_{3}|ϵ}) + \frac{1}{2})]

+O(t^{3}) + O(ϵ^{4}/t^{2}) + O(tϵ^{2} ln ϵ)

and

Corollary 2.3 When ϵ → 0, t → 0, t^{3}/ϵ^{2} → 0 and |t|/ϵ → ∞ we have

S^{{A \atop B}} (t) = S0^{{A \atop B}} (ϵ, sign(t)) + S(t) + (η^{0}^{2}(ϵ) − η^{0}^{2})\frac{t}{2} ± ϵ|d_{3}|t

∓sign(t) (η_{1}^{0}(ϵ)b_{1}t^{2} + \frac{d_{3}^{2}ϵ^{2}}{η_{1}^{0}(ϵ)b_{1}} ln |t|)

+O(t^{3}) + O(ϵ^{4}/t^{2}) + O(ϵ^{3} ln |t|).

2.4 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

\frac{d}{dt}A^{C}(t) = iB^{C}(t)  (2.21)
\frac{d}{dt}B^{C}(t) = i(V^{C})^{(2)}(a^{C}(t), ϵ)A^{C}(t),

with the initial conditions

A^{C}(0) = A_{0},  (2.22)
B^{C}(0) = B_{0},

where a^{C}(t) is the solution of (2.2) and (2.3). Let us determine the leading order behavior of (V^{C})^{(2)}(a^{C}(t), ϵ). By explicit computation we get, (omitting the arguments (x, ϵ))

(V^{C})^{(2)} = Ṽ^{(2)} ± (|∇β⟩⟨∇β| + |∇γ⟩⟨∇γ| + |∇δ⟩⟨∇δ|) (β^{2} + γ^{2} + δ^{2})^{−1/2}
+(ββ^{(2)} + γγ^{(2)} + δδ^{(2)})(β^{2} + γ^{2} + δ^{2})^{−1/2}
−|β∇β + γ∇γ + δ∇δ⟩⟨β∇β + γ∇γ + δ∇δ|(β^{2} + γ^{2} + δ^{2})^{−3/2}.  (2.23)

Thus, from (1.33) and some simple estimates, we see that

(V^{C})^{(2)} = Ṽ^{(2)}  (2.24)

± \frac{(γ^{2} + δ^{2})|∇β⟩⟨∇β| + (β^{2} + δ^{2})|∇γ⟩⟨∇γ| − βγ (|∇β⟩⟨∇γ| + |∇γ⟩⟨∇β| + O(3))}{(β^{2} + γ^{2} + δ^{2})^{3/2}}.

Explicitly, this has the form

(V^{C})^{(2)} = Ṽ^{(2)}  (2.25)

± \frac{1}{(β^{2} + γ^{2} + δ^{2})^{3/2}} [ O(3) + (γ^{2} + δ^{2}) ( \begin{matrix}b_{1}^{2} & b_{1}b_{2} & 0 & … & 0 \\ b_{1}b_{2} & b_{2}^{2} & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} )

+ (β^{2} + δ^{2}) ( \begin{matrix}0 & 0 & 0 & … & 0 \\ 0 & c_{2}^{2} & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} ) − β γ ( \begin{matrix}0 & b_{1}c_{2} & 0 & … & 0 \\ b_{1}c_{2} & 2b_{1}c_{2} & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} )] .

Using this expression, we obtain the following result for the solutions to (2.21) and (2.22):

Proposition 2.2 Let A^{C}(t) and B^{C}(t) be the solutions of (2.21) and (2.22). In the regime ϵ → 0, |t| → 0, ϵ/|t| → 0, and |t ln^{2} ϵ | → 0, we have the following asymptotics:

A^{{A \atop B}} (t) = A_{0} + O (|t| | ln ϵ|) ,  (2.26)
B^{{A \atop B}} (t) = B_{0} ± \frac{i sign(t)}{ρ(ϵ)} [MA_{0} + ln (\frac{2ρ(ϵ)|t|}{ϵ}) NA_{0}] + O (|t| ln^{2} ϵ + \frac{ϵ^{2}}{t^{2}} ) ,

(2.27)

where

M = \frac{1}{|d_{3}|}|b⟩⟨b| = ( \begin{matrix}b_{1}^{2}/|d_{3}| & b_{1}b_{2}/|d_{3}| & 0 & … & 0 \\ b_{1}b_{2}/|d_{3}| & b_{2}^{2}/|d_{3}| & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} ) (2.28)

and

N = \frac{1}{|d_{3}|}|c⟩⟨c| = ( \begin{matrix}0 & 0 & 0 & … & 0 \\ 0 & c_{2}^{2}/|d_{3}| & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} ) . (2.29)

Proof: To study solutions to (2.21) and (2.22) we first consider various quantities that occur in (2.25) with x replaced by a^{C}(t) = η^{0}(ϵ)t + O(t^{2}). By tedious, but straightforward calculations, we obtain the following estimates

\frac{β^{2} + δ^{2}}{(β^{2} + γ^{2} + δ^{2})^{3/2}} = \frac{1}{|d_{3}| \sqrt{ρ(ϵ)^{2}t^{2} + ϵ^{2}}} + O ((t^{2} + ϵ^{2})^{0})

\frac{γ^{2} + δ^{2}}{(β^{2} + γ^{2} + δ^{2})^{3/2}} = \frac{ϵ^{2}}{|d_{3}| (ρ(ϵ)^{2}t^{2} + ϵ^{2})^{3/2}} + O ((t^{2} + ϵ^{2})^{0})
\frac{β γ}{(β^{2} + γ^{2} + δ^{2})^{3/2}} = O ((t^{2} + ϵ^{2})^{0}) .

From these estimates, it follows that for small t and ϵ,

(V ^{{A \atop B}})^{(2)}(a^{{A \atop B}} (t), ϵ) = ± (\frac{ϵ^{2}}{(ρ(ϵ)^{2}t^{2} + ϵ^{2})^{3/2}} M + \frac{1}{(ρ(ϵ)^{2}t^{2} + ϵ^{2})^{1/2}} N) + O ((t^{2} + ϵ^{2})^{0}) .(2.30)

We can solve (2.21) and (2.22) if and only if we can solve

A^{C}(t, ϵ) = A_{0} + itB_{0} − ∫_{0}^{t} ∫_{0}^{s} (V^{C})^{(2)}(a^{C}(r, ϵ)) A^{C}(r, ϵ) dr. (2.31)

For each fixed positive ϵ, this equation can be solved by iteration for small |t|. For T > 0, we define a norm on bounded operator valued functions of t and ϵ by

||| D(·, ·) ||| = sup_{{ϵ≤T, |t ln t|≤T, |t ln ϵ|≤T }} ∥ D(t, ϵ) ∥.

From (2.31), we see that for positive ϵ,

∥A^{C}(t, ϵ) − A_{0}∥ = ∥ itB_{0} − ∫_{0}^{t} ds ∫_{0}^{s}(V^{C})^{(2)}(a^{C}(r, ϵ))A^{C}(r, ϵ) dr ∥
≤ |t| ∥B_{0}∥ + | ∫_{0}^{t} ds ∫_{0}^{s} ∥(V^{C})^{(2)}(a^{C}(r, ϵ))∥ |||A^{C}(·, ·) − A_{0}||| dr |(2.32)
+ | ∫_{0}^{t} ds ∫_{0}^{s} ∥(V^{C})^{(2)}(a^{C}(r, ϵ))∥ ∥A_{0}∥ dr |
≤ |t| ∥B_{0}∥

+ | ∫_{0}^{t} ds ∫_{0}^{s} ∥(V^{C})^{(2)}(a^{C}(r, ϵ))∥ dr | ( |||A^{C}(·, ·) − A_{0}||| + ∥A_{0}∥ ) .

From (2.30) and explicit calculation, we see that for positive ϵ,

| ∫_{0}^{s} ∥(V^{C})^{(2)}(a^{C}(r, ϵ))∥ dr |
≤ C_{1}|s| + C_{2} ∫_{0}^{|s|} \frac{ϵ^{2} dr}{(ρ(ϵ)^{2}r^{2} + ϵ^{2})^{3/2}} + C_{3} ∫_{0}^{|s|} \frac{dr}{\sqrt{ρ(ϵ)^{2}r^{2} + ϵ^{2}}}  (2.33)

= C_{1}|s| + C_{2} \frac{|s|}{\sqrt{ρ(ϵ)^{2}s^{2} + ϵ^{2}}} + C_{3} \frac{1}{ρ(ϵ)} ln (\frac{ρ(ϵ)|s|}{ϵ} + \sqrt{\frac{ρ(ϵ)^{2}s^{2}}{ϵ^{2}} + 1}) .

By explicit integration, we see from this estimate that

| ∫_{0}^{t} ds ∫_{0}^{s} ∥(V^{C})^{(2)}(a^{C}(r, ϵ))∥ dr |

≤ C_{4} t^{2} + C_{5} \frac{\sqrt{ρ(ϵ)^{2}t^{2} + ϵ^{2}} − ϵ}{ρ(ϵ)^{2}}  (2.34)

+ C_{6} \frac{1}{ρ(ϵ)^{2}} | ϵ − \sqrt{ρ(ϵ)^{2}t^{2} + ϵ^{2}} + ρ(ϵ)|t| ( ln (ρ(ϵ)|t| + \sqrt{ρ(ϵ)^{2}t^{2} + ϵ^{2}}) − ln ϵ )| .

For sufficiently small T, | t ln t | ≤ T, | t ln ϵ | ≤ T, and ϵ ≤ T, this quantity is bounded by

| ∫_{0}^{t} ds ∫_{0}^{s} ∥(V^{C})^{(2)}(a^{C}(r, ϵ))∥ dr |
≤ C_{7} |t| + C_{8} |t ln |t| | + C_{9} |t ln ϵ |  (2.35)
≤ O ( T ) .

It now follows from (2.32) that for sufficiently small T, |||A^{C}(·, ·) − A_{0}||| is bounded. This, (2.32), and (2.35) imply conclusion (2.26) of the proposition.

To prove (2.27), we use

B^{C}(t) = B_{0} + i ∫_{0}^{t} (V^{C})^{(2)}(a^{C}(s, ϵ)) A^{C}(s) ds  (2.36)
= B_{0} + i ∫_{0}^{t} (V^{C})^{(2)}(a^{C}(s, ϵ)) A_{0} ds + i ∫_{0}^{t} (V^{C})^{(2)}(a^{C}(s, ϵ)) (A^{C}(s) − A_{0}) ds

From (2.30) and explicit integration,

∫_{0}^{t} (V^{C})^{(2)}(a^{C}(s, ϵ)) A_{0} ds  (2.37)
= sign(t) (\frac{|t|}{\sqrt{ρ(ϵ)^{2}t^{2} + ϵ^{2}}} MA_{0} + \frac{1}{ρ(ϵ)} ln (\frac{ρ(ϵ)t}{ϵ} + \sqrt{\frac{ρ(ϵ)^{2}t^{2}}{ϵ^{2}} + 1}) NA_{0} ) + O(|t|).

Furthermore, from (2.32), (2.33), and(2.35),

∥ ∫_{0}^{t} (V^{C})^{(2)}(a^{C}(s, ϵ)) (A^{C}(s) − A_{0}) ds ∥  (2.38)
≤ ∫_{0}^{t} ∥(V^{C})^{(2)}(a^{C}(s, ϵ))∥ ∥A^{C}(s) − A_{0}∥ ds  (2.39)
≤ [ C_{1}|t| + C_{2} + O(ln |t| + ln ϵ) ] [ C_{7}|t| + O(|t| ln |t| + |t| ln ϵ) ] .  (2.40)

For t and ϵ in the regime covered by the proposition, this last quantity is O( t ln^{2} ϵ). Conclusion (2.27) is now an easy consequence of (2.36), (2.37), and (2.38).

2.5 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_{2} ; b_{3} ; 0 ; ⋮ ; 0 ) , ρ(ϵ) = \frac{b_{1}η_{1}^{0}(ϵ)}{|e_{4}|}, M = \frac{1}{|e_{4}|} |b⟩⟨b|, and N = \frac{1}{|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 = ( b_{1} ; b_{2} ; b_{3} ; b_{4} ; 0 ; ⋮ ; 0 ) , ρ(ϵ) = \frac{b_{1}η_{1}^{0}(ϵ)}{|f_{5}|}, M = \frac{1}{|f_{5}|} |b⟩⟨b|, and N = \frac{1}{|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

iϵ^{2} \frac{∂}{∂t}ψ(x, t) = −\frac{ϵ^{4}}{2} Δψ(x, t) + h(x, ϵ)ψ(x, t) (3.1)

as ϵ → 0. 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 O(ϵ). These are the same wave packets that are used in [8, 17, 20].

Let n denote the dimension of the nuclear configuration space. A multi-index l = (l_{1}, l_{2}, …, l_{n}) is an ordered n-tuple of non–negative integers. The order of l is defined to be |l| = ∑_{j=1}^{n} l_{j}, and the factorial of l is defined to be l! = (l_{1}!)(l_{2}!) ⋯ (l_{n}!). The symbol D^{l} denotes the differential operator D^{l} = \frac{∂^{|l|}}{(∂x_{1})^{l_{1}}(∂x_{2})^{l_{2}}···(∂x_{n})^{ln}}, and the symbol x^{l} denotes the monomial x^{l} = x_{1}^{l_{1}}x_{2}^{l_{2}} ⋯ x_{n}^{l_{n}}. We denote the gradient of a function f by f^{(1)} and the matrix of second partial derivatives by f^{(2)}. We view IR^{n} as a subset of I C^{n}, and let e_{i} denote the ith standard basis vector in IR^{n} or I C^{n}. The inner product on IR^{n} or I C^{n} is ⟨ v, w ⟩ = ∑_{j=1}^{n} \overline{v_{j}}w_{j}.

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

H̃_{0}(x) = 1 (3.2)

and

H̃_{1}(x) = 2 ⟨ v, x ⟩, (3.3)

where v is an arbitrary non–zero vector in I C^{n}. The generalizations of the higher order Hermite polynomials are defined recursively as follows: Let v_{1}, v_{2}, …, v_{m} be m arbitrary non–zero vectors in I C^{n}. Then

H̃_{m}(v_{1}, v_{2}, …, v_{m}; x) = 2 ⟨ v_{m}, x ⟩ H̃_{m−1}(v_{1}, v_{2}, …, v_{m−1}; x)
− 2 ∑_{i=1}^{m−1} ⟨ v_{m}, \overline{v_{i}} ⟩ H̃_{m−2}(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_{2}, …, v_{m}.

Let A be a complex invertible n × n matrix. We define |A| = [AA^{∗}]^{1/2}, 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) = H̃_{|l|}(\underbrace{U_{A}e_{1}, …, U_{A}e_{1}}_{l_{1} entries}, \underbrace{U_{A}e_{2}, …, U_{A}e_{2}}_{l_{2} entries}, …, \underbrace{U_{A}e_{n}, …, U_{A}e_{n}}_{l_{n} entries}; x) (3.4)

We can now define the semiclassical wave packets φ_{l}(A, B, h̄, a, η, x). In the Born–Oppenheimer approximation, the role of h̄ is played by ϵ^{2}.

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} = \frac{1}{2}[(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 h̄ > 0. Then for each multi-index l we define

φ_{l}(A, B, h̄, a, η, x) = 2^{−|l|/2}(l!)^{−1/2}π^{−n/4}h̄^{−n/4} [det A]^{−1/2}
· H_{l}(A; h̄^{−1/2}|A|^{−1}(x − a))

· exp {−⟨ (x − a), BA^{−1} (x − a) ⟩/2h̄ + i ⟨ η, (x − a) ⟩/h̄} .

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 [20] for several remarks concerning this definition. The formulas for the functions φ_{l}(A, B, h̄, 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 h̄ \frac{∂Ψ}{∂t} = − \frac{h̄^{2}}{2} Δ Ψ + V (x) Ψ (3.9)

has an approximate solution of the form

e^{iS(t)/h̄} φ_{l}(A(t), B(t), h̄, a(t), η(t), x) + O(h̄^{1/2}). (3.10)

Here O(h̄^{1/2}) means that the exact solution and the approximate solution agree up to an error whose norm is bounded by an l–dependent constant times h̄^{1/2} for t in a fixed bounded interval [−T, T]. The vectors a(t) and η(t) satisfy the classical equations of motion

\frac{∂a}{∂t}(t) = η(t),  (3.11)
\frac{∂η}{∂t} (t) = −V^{(1)}(a(t)).  (3.12)

The function S(t) is the classical action integral associated with the classical path,

S(t) = ∫_{−T}^{t} (\frac{(η(s))^{2}}{2} − V (a(s))) ds.  (3.13)

The matrices A(t) and B(t) satisfy

\frac{∂A}{∂t}(t) = iB(t),  (3.14)
\frac{∂B}{∂t} (t) = iV^{(2)}(a(t)) A(t).  (3.15)

If A(−T) and B(−T) satisfy conditions (3.5)–(3.8), then so do A(t) and B(t) for each t. The proofs of the claims we have made about the φ_{l}(A, B, h̄, 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, (3.16)

such that

\{ \begin{matrix}F (r) = 1 & 0 ≤ r ≤ 1 \\ F (r) = 0 & r ≥ 2 \end{matrix}  (3.17)

The wave functions we construct below contain the following cutoff function as a factor:

F (∥x − a^{C}(t)∥/ϵ^{1−δ′}) (3.18)

where 0 < δ′ is chosen below, and C = A, B. Multiplication of our semiclassical wave packets by this function leads to exponentially small errors in ϵ.

3.2 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)∥/ϵ^{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| > ϵ^{1−ξ} for suitable ξ, with 0 < δ′ < ξ < 1. We have two sets of dynamic eigenvectors, denoted by Φ_{C}^{±}(x, t, ϵ), 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 (2.2) and (2.3). Our goal is to construct normalized eigenvectors Φ_{C}^{±}(x, t, ϵ) that solve

⟨Φ_{C}^{±}(x, t, ϵ), (∂/∂t + η^{C}(t) · ∇) Φ_{C}^{±}(x, t, ϵ)⟩ ≡ 0 (3.19)

for C = A, B and t{> \atop <}0. Since the eigenvalues E_{A}(x, ϵ) and E_{B}(x, ϵ) 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, ϵ) associated with E_{C}(x, ϵ) 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

β(x, ϵ) = r cos θ  (3.20)
γ(x, ϵ) = r sin θ cos ϕ  (3.21)
δ(x, ϵ) = r sin θ sin ϕ.  (3.22)

Then, in the basis { ψ_{1}(x, ϵ), ψ_{2}(x, ϵ)}, we construct the following static eigenvectors for h_{1}(x, ϵ):

Φ_{A}^{−} = ( \begin{matrix}e^{iϕ} cos θ/2 \\ sin θ/2 \end{matrix} )  (3.23)
Φ_{B}^{−} = ( \begin{matrix}− sin θ/2 \\ e^{−iϕ} cos θ/2 \end{matrix} )  (3.24)

for π/2 < θ ≤ π, and

Φ_{A}^{+} = ( \begin{matrix}cos θ/2 \\ e^{−iϕ} sin θ/2 \end{matrix} )  (3.25)
Φ_{B}^{+} = ( \begin{matrix}−e^{iϕ} sin θ/2 \\ cos θ/2 \end{matrix} )  (3.26)

for 0 ≤ θ ≤ π/2.

Lemma 3.1 Suppose 0 < δ′ < ξ < 1, and |t| > ϵ^{1−ξ}. Define μ(ϵ, t) = max { |t|, ϵ^{1−δ′}/|t| }, and let a^{C}(t) and η^{C}(t) solve equations (2.2) and (2.3). For x in the support of F (∥x − a^{C}(t)∥/ϵ^{1−δ′}), C = A, B, we have for t < 0,

Φ_{A}^{−}(x, ϵ) = ψ_{2}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B}^{−}(x, ϵ) = −ψ_{1}(x, ϵ) + O(μ(ϵ, t)),

and for t > 0,

Φ_{A}^{+}(x, ϵ) = ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B}^{+}(x, ϵ) = ψ_{2}(x, ϵ) + O(μ(ϵ, t)).

Proof: Let x belong to the support of F (∥x − a^{C}(t)∥/ϵ^{1−δ′}). Then x = a^{C}(t) + ω, where ∥ω∥ ≤ 2ϵ^{1−δ′} ≪ |t|, so

x_{1} = η_{1}^{0}t + O(t^{2}) + O(ϵ^{1−δ′}),
x_{2} = O(t^{2}) + O(ϵ^{1−δ′}),
x_{j} = O(|t|) + O(ϵ^{1−δ′}),  for j ≥ 3.

Thus,

x_{1} = η_{1}^{0}t + O(|t|μ(ϵ, t)),
x_{2} = O(|t|μ(ϵ, t)),
x_{j} = O(|t|),  for j ≥ 3.

Therefore,

β(x, ϵ) = b_{1}η_{1}^{0}t + O(|t|μ(ϵ, t)),  (3.27)
γ(x, ϵ) = O(|t|μ(ϵ, t)),  (3.28)
δ(x, ϵ) = d_{3}ϵ + O(|t|μ(ϵ, t)) = O(|t|μ(ϵ, t)).  (3.29)

From these estimates, it follows that

\frac{β(x, ϵ)}{r} = \frac{sign(t)}{\sqrt{1 + \frac{γ^{2}+δ^{2}}{β^{2}}}} = sign(t) ( 1 + \frac{O ((|t|μ(ϵ, t))^{2})}{O(t^{2})})^{−1/2} = sign(t) + O (μ(ϵ, t)^{2}) ,

\frac{γ(x, ϵ)}{r} = O(μ(ϵ, t)),
\frac{δ(x, ϵ)}{r} = O(μ(ϵ, t)).

Since sin^{2}(θ/2) = \frac{1}{2}(1 − cos (θ)), and cos^{2}(θ/2) = 1 − sin^{2}(θ/2), we see that

sin^{2}(θ/2) = \{ \begin{matrix}0 & if t > 0 \\ 1 & if t < 0 \end{matrix} \} + O (μ(ϵ, t)^{2})  and  (3.30)
cos^{2}(θ/2) = \{ \begin{matrix}1 & if t > 0 \\ 0 & if t < 0 \end{matrix} \} + O (μ(ϵ, t)^{2}) .

The lemma follows.

We now construct the dynamic eigenvectors.

Lemma 3.2 Suppose 0 < δ′ < ξ < \frac{1}{3}, |t| > ϵ^{1−ξ}, and |t| < ϵ^{κ} for some κ > \frac{1}{2}. Let a^{C}(t) and η^{C}(t) solve the classical equations of motion (2.2) and (2.3). There exist eigenvectors Φ_{C}^{±}(x, t, ϵ), C = A, B, that solve (3.19) and satisfy

|Φ_{C}^{±}(x, t, ϵ) − Φ_{C}^{±}(x, ϵ)| = O(|t|)

for all x in the support of F (∥x − a^{C}(t)∥/ϵ^{1−δ′}).

Proof: We give a proof for Φ_{A}^{+}(x, t, ϵ) only; the other cases are similar. In order to simplify the notation, we drop the indices A and +, and let a(t) ≡ a^{A}(t) and η(t) ≡ η^{A}(t) for t > 0. We again let μ(ϵ, t) = max { |t|, ϵ^{1−δ′}/|t| }.

Solutions of (3.19) have the form

Φ(x, t, ϵ) = Φ(x, ϵ)e^{iλ(x,t,ϵ)}, (3.31)

where λ(x, t, ϵ) is a real valued function that satisfies the equation

i\frac{∂}{∂t}λ(x, t, ϵ) + iη(t) · ∇λ(x, t, ϵ) + ⟨Φ(x, ϵ), η(t) · ∇Φ(x, ϵ)⟩ = 0. (3.32)

We introduce

ω ≡ x − a(t) (3.33)

and define

λ̃(ω, t, ϵ) ≡ λ(ω + a(t), t, ϵ),  (3.34)
Φ̃(ω, t, ϵ) ≡ Φ(ω + a(t), ϵ).  (3.35)

In the new variables, equation (3.32) for λ̃ is

i\frac{∂}{∂t}λ̃(ω, t, ϵ) = −⟨Φ̃(ω, t, ϵ), \frac{∂}{∂t}Φ̃(ω, t, ϵ)⟩ (3.36)

where

\frac{∂}{∂t}Φ̃(ω, t, ϵ) = η(t) · ∇Φ(ω + a(t), ϵ). (3.37)

Dropping the arguments, we have

∇Φ =  (− sin(θ/2)ψ_{1} + e^{−iφ} cos(θ/2)ψ_{2}) ∇θ/2
+ cos(θ/2)∇ψ_{1} + e^{−iφ} sin(θ/2)∇ψ_{2}
−i∇φe^{−iφ} sin(θ/2)ψ_{2}.  (3.38)

Since the ψ_{j}, j = 1, 2, are orthonormal, we get

⟨Φ̃, \frac{∂}{∂t}Φ̃⟩ = − iη∇φ sin^{2}(θ/2) + cos^{2}(θ/2)⟨ψ_{1}|η∇ψ_{1}⟩ + sin^{2}(θ/2)⟨ψ_{2}|η∇ψ_{2}⟩
+ sin(θ/2) cos(θ/2) (e^{−iφ}⟨ψ_{1}|η∇ψ_{2}⟩ + e^{iφ}⟨ψ_{2}|η∇ψ_{1}⟩) .  (3.39)

As η, ψ_{j}, ∇ψ_{j} are all O(0), we have

λ̃(ω, t, ϵ) = ∫_{0}^{t} (η(t′)∇φ sin^{2}(θ(ω + a(t′), ϵ)/2) + O(0)) dt′, (3.40)

where we have now fixed the constant of integration.

We claim that |η · ∇φ sin^{2} θ/2| is bounded on the support of F (∥x − a^{C}(t)∥/ϵ^{1−δ′}). We note that

φ(x, ϵ) = cot^{−1} (γ(x, ϵ)/δ(x, ϵ)), (3.41)

provided δ(x, ϵ) is different from zero. However, on the support of F (∥x − a^{C}(t)∥/ϵ^{1−δ′}) with |t| < ϵ^{κ},

δ(x, ϵ) = d_{3}ϵ + O(x^{2} + ϵ^{2})
= d_{3}ϵ + O(t^{2} + ϵ^{2})
≥ Cϵ,  for some positive C.

By the mean value theorem,

\sqrt{y^{2} + z^{2}} = |y| + |z| \frac{ζ}{\sqrt{y^{2} + ζ^{2}}}  for some ζ ∈ [ 0, |z|]
≤ |y| + |z|.

Thus,

2 sin^{2} θ/2 = 1 − cos θ = \frac{r − β}{r} = \frac{\sqrt{β^{2} + γ^{2} + δ^{2}} − β}{r}

satisfies

| 2 sin^{2} θ/2 | ≤ \begin{matrix}√ & \overline{γ^{2} + δ^{2}} \\ √ & \overline{β^{2} + γ^{2} + δ^{2}} \end{matrix} .

From (3.41) we therefore see that

| η · ∇φ sin^{2} θ/2 | ≤ \frac{| γ η · ∇δ | + | δ η · ∇γ |}{\sqrt{γ^{2} + δ^{2}} \sqrt{β^{2} + γ^{2} + δ^{2}}}
≤ \frac{| η · ∇γ | + | η · ∇δ |}{\sqrt{β^{2} + γ^{2} + δ^{2}}}

However, on the support of F (∥x − a^{C}(t)∥/ϵ^{1−δ′}),

\sqrt{β^{2} + γ^{2} + δ^{2}} ≥ C|t|,  for some C,
| η · ∇δ | = O(|t|),  and
| η · ∇γ | = η_{0} · c + O(|t|)  (3.42)
= O(|t|),  since η_{0} · c = 0.

Thus, η · ∇φ sin^{2} θ/2 is bounded, and by virtue of (3.40),

λ̃(ω, t, ϵ) = λ̃_{0}(ω). + O(|t|).

This implies the lemma since we can arbitrarily take λ̃_{0}(ω) ≡ 0.

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 = 0:

a^{B}(0) = 0,
η^{B}(0) = η^{0},
S^{B}(0) = 0,
A^{B}(0) = A_{0},  and
B^{B}(t) ∼ B_{0} + \frac{i|d_{3}|}{b_{1}η_{1}^{0}} [MA_{0} + ln (\frac{2b_{1}η_{1}^{0}|t|}{|d_{3}|ϵ}) NA_{0}] ,

where A_{0} and B_{0} 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 ϵ, |t|, ϵ/|t|, and |t ln^{2} ϵ |. 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 0 < δ′ < ξ < 1. For any −T ≤ t ≤ −ϵ^{1−ξ}, there is an approximation ψ_{OI}(x, t) to the exact solution ψ(x, t) to the Schrödinger equation, such that

ψ_{OI}(x, t) = F (\frac{∥x − a^{B}(t)∥}{ϵ^{1−δ′}}) e^{iS^{B}(t)/ϵ^{2}}φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B}^{−}(x, t, ϵ)

(3.43)

and

ψ(x, t) = ψ_{OI}(x, t) + O(ϵ^{ξ}), (3.44)

in the L^{2}(IR^{n}) sense, as ϵ → 0.

Proof: The proof of this Lemma is very similar to that of Lemma 6.4 of [17]. For −T ≤ t ≤ −ϵ^{1−ξ} and x in the support of the cut off function F (∥x − a^{B}(t)∥/ϵ^{1−δ′}), \frac{1}{E_{A}(x)−E_{B}(x)} is bounded by a multiple of ϵ^{−1+ξ}. Furthermore, for such x and t, a rather tedious calculation shows that (\frac{∂}{∂t} + η^{B} · ∇_{x}) Φ_{B}^{−}(x, t, ϵ) is bounded by a multiple of ϵ^{−1+ξ}. Thus, up to errors on the order of ϵ^{2ξ}, ψ_{OI}(x, t) is equal to

Ψ(x, t) = F (∥x − a^{B}(t)∥/ϵ^{1−δ′}) e^{iS^{B}(t)/ϵ^{2}} (Ψ_{0}(x, t) + ϵ^{2}Ψ_{2}^{⊥}(x, t))  (3.45)

where

Ψ_{0}(x, t) = φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B}^{−}(x, t, ϵ) (3.46) and

Ψ_{2}^{⊥}(x, t) = iφ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x)

× ( Φ_{A}^{−}(x, t, ϵ) \frac{1}{E_{A}(x) − E_{B}(x)}⟨ Φ_{A}^{−}(x, t, ϵ), (\frac{∂}{∂t} + η^{B} · ∇_{x}) Φ_{B}^{−}(x, t, ϵ) ⟩

+ r_{BA}(x, ϵ)P_{⊥}^{AB}(x, ϵ) (\frac{∂}{∂t} + η^{B} · ∇_{x} ) Φ_{B}^{−}(x, t, ϵ) ),  (3.47)

where r_{BA}(x, ϵ) is the restriction of (h(x, ϵ) − E_{B}(x, ϵ))^{−1} to the range of P_{⊥}^{AB}(x).

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 ϵ^{ξ}. To prove this, we explicitly compute

ζ(x, t, ϵ) = iϵ^{2} \frac{∂Ψ}{∂t} − H(ϵ)Ψ. (3.48)

For −T ≤ t ≤ −ϵ^{1−ξ}, terms in ζ(x, t, ϵ) 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 ϵ. Most of the remaining terms in ζ(x, t, ϵ) are formally given on pages 105–108 of [17], with (\frac{∂}{∂t} + η^{B} · ∇_{x}) in place of η^{B} ·∇_{x} in many expressions. A few other terms arise from the iϵ^{2} \frac{∂}{∂t} acting on the electronic eigenfunctions in (3.45). Our error term ζ(x, t, ϵ) 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 ζ(x, t, ϵ) cancel with one another, we estimate the remaining terms individually. This process is very similar to that described on pages 108– 110 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 3.2. 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, ϵ)∥ ≤ C ϵ^{3}|t|^{−2} (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 0 < δ′ < ξ < \frac{1}{3}, t < −ϵ^{1−ξ}, and |t| < ϵ^{κ} for some κ > \frac{1}{2}, and let y = \frac{x−a(t)}{ϵ}. Then the approximate solution of lemma 3.3 has the following asymptotics:

ψ_{OI}(x, t) = − F (ϵ^{δ′}∥y∥) exp (i\frac{S(t)}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{b_{1}η_{1}^{0}t^{2}}{2ϵ^{2}} + i\frac{d_{3}^{2} [\frac{1}{2} − ln (\frac{|d_{3}ϵt|}{2b_{1}η_{1}^{0}} )]}{2b_{1}η_{1}^{0}} )

× exp (−i\frac{|d_{3}|}{2b_{1}η_{1}^{0}} ⟨ y, [ M + ln (\frac{2b_{1}η_{1}^{0}|t|}{|d_{3}|ϵ}) N ] y⟩ )

× exp (−i\frac{|d_{3}|b · y}{b_{1}η_{1}^{0}} − i\frac{t b · y}{ϵ} )

× exp (i\frac{η_{0} · y}{ϵ} ) ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) ψ_{1}(x, ϵ)

+ O(ϵ^{α}),  (3.50)

for some α > 0.

Proof: We make a sequence of replacements in formula (3.43). 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))/ϵ, and using corollary 2.1, we replace

F (ϵ^{δ′}∥y^{B}∥) by F (ϵ^{δ′}∥y∥).

Next we use corollary 2.1 and corollary 2.2 to replace the factor

exp (i\frac{η^{B}(t) · y^{B}}{ϵ} )  by

exp (i\frac{η_{0} · y}{ϵ} − i\frac{|d_{3}|b · y}{b_{1}η_{1}^{0}} − i\frac{t b · y}{ϵ} + i\frac{|d_{3}|t}{ϵ} + i\frac{b_{1}η_{1}^{0}t^{2}}{2ϵ^{2}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} (ln \frac{2b_{1}η_{1}^{0}|t|}{|d_{3}|ϵ} + \frac{1}{2})) .

We use lemma 3.1 of [20] and corollary 2.1 to replace

φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), 0, x) by φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a(t), 0, x).

We then use lemma 3.1 of [20] and proposition 2.2 to replace

φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a(t), 0, x) by

exp (−i\frac{|d_{3}|}{2b_{1}η_{1}^{0}} ⟨ y, [ M + ln (\frac{2b_{1}η_{1}^{0}|t|}{|d_{3}|ϵ}) N ] y⟩ ) ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y).

By corollary 2.3 we can replace

exp (i\frac{S^{B}(t)}{ϵ^{2}})  by  exp (i\frac{S(t)}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{|d_{3}|t}{ϵ} − i\frac{b_{1}η_{1}^{0}t^{2}}{ϵ^{2}} − i\frac{d_{3}^{2} ln |t|}{η_{1}^{0}b_{1}}) .

Finally, using lemmas 3.1 and 3.2, we replace

Φ_{B}^{−}(x, t, ϵ) by − ψ_{1}(x, ϵ)

The lemma follows.

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, ϵ). Let a(t), η(t), and S(t) be the classical quantities that satisfy the initial conditions

\begin{matrix}a(0) = 0 \\ η(0) = η^{0} \\ S(0) = 0. \end{matrix}  (3.51)

In the rescaled variables

\{ \begin{matrix}y = (x − a(t))/ϵ, \\ s = t/ϵ, \end{matrix}  (3.52)

the Schrödinger equation is

iϵ\frac{∂}{∂s}ψ − iϵη(ϵs)∇_{y}ψ = −\frac{ϵ^{2}}{2} △_{y}ψ + h(a(ϵs) + ϵy, ϵ)ψ. (3.53)

We look for an approximate solution of the form

ψ(y, s, ϵ) = F (∥y∥ϵ^{δ′}) exp (i\frac{S(ϵs)}{ϵ^{2}} + i\frac{η(ϵs)y}{ϵ} ) χ(y, s, ϵ), (3.54)

with

χ(y, s, ϵ) = {f(y, s, ϵ)ψ_{1}(a(ϵs) + ϵy, ϵ) + g(y, s, ϵ)ψ_{2}(a(ϵs) + ϵy, ϵ)  (3.55)
+ ψ_{⊥}(a(ϵs) + ϵy, ϵ)} ,

where ψ_{⊥}(x, ϵ) ∈ (I I−P (x, ϵ))H 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 (1.10) and eliminating the overall “classical phase” and cut-off F, we have

iϵψ_{1} \frac{∂}{∂s}f + iϵψ_{2} \frac{∂}{∂s}g + iϵ\frac{∂}{∂s}ψ_{⊥} + (Ṽ (a) + ϵy∇_{x}Ṽ (a))(fψ_{1} + gψ_{2} + ψ_{⊥})

+iϵ^{2}fη∇_{x}ψ_{1} + iϵ^{2}gη∇_{x}ψ_{2} + iϵ^{2}η∇_{x}ψ_{⊥}

= −ϵ^{4}Δ_{x}ψ_{⊥}/2 − ϵ^{4}fΔ_{x}ψ_{1}/2 − ϵ^{4}gΔ_{x}ψ_{2}/2 − ϵ^{3}∇_{y}f∇_{x}ψ_{1}

−ϵ^{3}∇_{y}g∇_{x}ψ_{2} − ϵ^{2}ψ_{1}Δ_{y}f − ϵ^{2}ψ_{2}Δ_{y}g + h^{⊥}ψ_{⊥} + h_{1}(fψ_{1} + gψ_{2})

+Ṽ (a + ϵy)(fψ_{1} + gψ_{2} + ψ_{⊥}).  (3.56)

We assume the solutions have expansions of the form

f(y, s, ϵ) = ∑_{j=0}^{∞} ν_{j}(ϵ)f_{j}(y, s),
g(y, s, ϵ) = ∑_{j=0}^{∞} ν_{j}(ϵ)g_{j}(y, s),
ψ_{⊥}(x, t, ϵ) = ∑_{j=0}^{∞} ν_{j}(ϵ)ψ_{⊥j}(x, t),  (3.57)

with asymptotic scales ν_{j}(ϵ) to be determined by matching. We insert these expansions in (3.56). Using the behaviors (1.33) and lemma 2.2 we see that the lowest order terms yield

iϵν_{0}(ϵ)\frac{∂}{∂s}f_{0}(y, s)ψ_{1}(a(ϵs) + ϵy, ϵ) + iϵν_{0}(ϵ)\frac{∂}{∂s}g_{0}(y, s)ψ_{2}(a(ϵs) + ϵy, ϵ)

= h_{⊥}(a(ϵs) + ϵy, ϵ)ψ_{⊥}(a(ϵs) + ϵy, ϵs, ϵ)
+ϵν_{0}(ϵ)h_{11}(y, s) (f_{0}(y, s)ψ_{1}(a(ϵs) + ϵy, ϵ) + g_{0}(y, s)ψ_{2}(a(ϵs) + ϵy, ϵ))  (3.58)

on the support of F, where h_{11}(y, s) is the operator on the span of { ψ_{1}(a(ϵs) + ϵy, ϵ), ψ_{2}(a(ϵs) + ϵy, ϵ) } whose matrix in the basis { ψ_{1}(a(ϵs) + ϵy, ϵ), ψ_{2}(a(ϵs) + ϵy, ϵ) } is given by

( \begin{matrix}b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2} & c_{2}y_{2} + id_{3} \\ c_{2}y_{2} − id_{3} & −b_{1}η_{1}^{0}s − b_{1}y_{1} − b_{2}y_{2} \end{matrix} ) .

By matching the incoming outer solution, we obtain

ν_{j}(ϵ)h_{⊥}(a(ϵs) + ϵy, ϵ)ψ_{⊥j}(a(ϵs) + ϵy, ϵs) = 0 (3.59)

if ν_{j}(ϵ) << ϵν_{0}(ϵ), j = 0, 1, ⋯, m − 1. The spectrum of h_{⊥}(x, ϵ) is bounded away from 0 in a neighborhood of (0, 0). This implies

ψ_{⊥j}(x, t) = 0, j = 0, 1, ⋯, m − 1. (3.60)

By projecting with P (x, ϵ) and (I I − P (x, ϵ)), we split the remaining equation for order ϵν_{0}(ϵ) = ν_{m}(ϵ),

iϵν_{0}(ϵ)\frac{∂}{∂s}f_{0}ψ_{1} + iϵν_{0}(ϵ)\frac{∂}{∂s}g_{0}ψ_{2}

= ν_{m}(ϵ)h_{⊥}ψ_{⊥} + ϵν_{0}(ϵ)h_{11}(f_{0}ψ_{1} + g_{0}ψ_{2}),  (3.61)

into

iϵν_{0}(ϵ)\frac{∂}{∂s}f_{0}ψ_{1} + iϵν_{0}(ϵ)\frac{∂}{∂s}g_{0}ψ_{2} = ϵν_{0}(ϵ)h_{11}(f_{0}ψ_{1} + g_{0}ψ_{2}) and
ν_{m}(ϵ)h_{⊥}ψ_{⊥m} = 0.  (3.62)

The second equation gives ψ_{⊥m} = 0. The first one is equivalent to

i\frac{∂}{∂s} ( \begin{matrix}f_{0}(y, s) \\ g_{0}(y, s) \end{matrix} ) = ( \begin{matrix}b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2} & c_{2}y_{2} + id_{3} \\ c_{2}y_{2} − id_{3} & −b_{1}η_{1}^{0}s − b_{1}y_{1} − b_{2}y_{2} \end{matrix} ) ( \begin{matrix}f_{0}(y, s) \\ g_{0}(y, s) \end{matrix} ) . (3.63)

The general solution of this equation can be found exactly in terms of parabolic cylinder functions [6].

( \begin{matrix}f_{0}(y, s) \\ g_{0}(y, s) \end{matrix} ) = C_{1}(y) ( \begin{matrix}_{\frac{(1−i)(c_{2}y_{2}+id_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{u}−1} (_{\frac{(−1+i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \\ D_{p_{u}} (_{\frac{(−1+i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \end{matrix} ) (3.64)

+ C_{2}(y) ( \begin{matrix}D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \\ _{\frac{(−1−i)(c_{2}y_{2}−id_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \end{matrix} ) ,

where

p_{l} = −p_{u} = − i \frac{c_{2}^{2}y_{2}^{2} + d_{3}^{2}}{2b_{1}η_{1}^{0}} .

We define our inner approximation by

ψ_{I}(y, s) = F (∥y∥ϵ^{δ′}) exp (i\frac{S(ϵs)}{ϵ^{2}} + i\frac{η(ϵs)y}{ϵ} )  (3.65)
× [ ν_{0}(ϵ)f_{0}(y, s)ψ_{1}(a(ϵs) + ϵy, ϵ) + ν_{0}(ϵ)g_{0}(y, s)ψ_{2}(a(ϵs) + ϵy, ϵ) ] ,

with f_{0} and g_{0} as above. To match (3.50) on the B level, we need

g_{0}(y, s) → 0, as s → −∞.

This forces us to take

C_{1}(y) ≡ 0,  (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 9.246 of [6],

D_{q}(z) = e^{−z^{2}/4} z^{q} ( 1 − O ( |\frac{q}{z} |^{2} ) ) ,  for  arg z < \frac{3π}{4} ,  (3.67)

and noting that ∥y∥/s = O(ϵ^{ξ−δ′}) → 0 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∥ϵ^{δ′}),

D_{p_{l}−1} (\frac{(−1 − i)}{\sqrt{b_{1}η_{1}^{0}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) = O(|s|^{−1}).

Thus, the first component of the second term on the right hand side of (3.64) determines C_{2}(y) by matching (3.50). As s → −∞, that component has the following asymptotics by (3.67):

D_{p_{l}} (\frac{(−1 − i)}{\sqrt{b_{1}η_{1}^{0}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2}))

= exp ( −i\frac{(b_{1}η_{1}^{0}s + b · y)^{2}}{2b_{1}η_{1}^{0}} + \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp ( −i\frac{(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} [ ln (\sqrt{\frac{2}{b_{1}η_{1}^{0}}} ) + ln (b_{1}η_{1}^{0}|s| − b · y) ] )

× ( 1 + O(|s|^{−2}) ) .

In the matching regime, ∥y∥/|s| → 0, so by a fairly lengthy calculation, this quantity matches (3.50) if we choose ν_{0}(ϵ) = 1 and C_{2}(y) to be

C_{2}(y) = − ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp (\frac{−π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp (i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{d_{3}^{2}}{b_{1}η_{1}^{0}} ln ϵ + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} − i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln |d_{3}| + i\frac{3d_{3}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}))

× exp ( −i\frac{c_{2}^{2}y_{2}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}) + i\frac{c_{2}^{2}y_{2}^{2}}{2b_{1}η_{1}^{0}} ln |d_{3}| − i\frac{|d_{3}|b · y}{b_{1}η_{1}^{0}} ) .  (3.68)

The following result deals with the validity of the inner approximate solution.

Lemma 3.5 Suppose 0 < δ′ < ξ < 1/3, κ > \frac{1}{2}, and 1 − ξ > κ. The function Ψ_{I}(x, t) = ψ_{I}((x−a(t))/ϵ, t/ϵ) is a valid approximation of a solution ψ(x, t) to the Schrödinger equation for −ϵ^{1−ξ} ≤ t ≤ ϵ^{1−ξ}, in the sense that as ϵ → 0,

∥ ψ(·, t) − Ψ_{I}(·, t) ∥ = O(ϵ^{1−3ξ}) → 0.

Furthermore, in the matching regime, −ϵ^{κ} < t < −ϵ^{1−ξ}, the inner and outer solutions agree in the sense that

∥ Ψ_{IO}(·, t) − Ψ_{I}(·, t) ∥ = O(ϵ^{α})

for some α > 0.

Proof: The first result is proved by mimicking the proofs of proposition 3.1 of [20] and lemma 3.5 of [20]. 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 9.246 of [6],

D_{q}(z) = e^{−z^{2}/4} z^{q} ( 1 − O ( |\frac{q}{z} |^{2} ) ) − \frac{\sqrt{2π}}{Γ(−q)} e^{−qπi} e^{z^{2}/4} z^{−q−1} ( 1 − O ( |\frac{q}{z} |^{2} ) ) ,

for −\frac{5π}{4} < arg z < \frac{π}{4}.

From this formula, the first component of the second term in (3.64) has large s asymptotics (omitting the factor C_{2}(y))

D_{p_{l}} (\frac{(−1 − i)}{\sqrt{b_{1}η_{1}^{0}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2}))

= exp ( −i\frac{(b_{1}η_{1}^{0}s + b · y)^{2}}{2b_{1}η_{1}^{0}} − \frac{3π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp ( −i\frac{(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} ln (\sqrt{\frac{2}{b_{1}η_{1}^{0}}} (b_{1}η_{1}^{0}s + b · y) ) )

× ( 1 + O(|s|^{−2}) )

+ O ( |s|^{−1} ) .  (3.69)

The second component of the second term in (3.64) has large s asymptotics (again omitting the factor C_{2}(y))

\frac{(−1 − i)(c_{2}y_{2} − id_{3})}{2\sqrt{b_{1}η_{1}^{0}}} D_{p_{l}−1} (\frac{(−1 − i)}{\sqrt{b_{1}η_{1}^{0}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2}))

= \frac{(1 + i)(c_{2}y_{2} − id_{3})}{2\sqrt{b_{1}η_{1}^{0}}} \frac{\sqrt{2π} e^{iπ} e^{− \frac{π(c^{2}_{2}y^{2}_{2}+d^{2}_{3})}{2b_{1}η^{0}_{1}}}}{Γ (1 + i\frac{(c_{2}^{2}y_{2}^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× exp ( i\frac{(b_{1}η_{1}^{0}s + b · y)^{2}}{2b_{1}η_{1}^{0}} + \frac{3π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp ( i\frac{(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} ln (\sqrt{\frac{2}{b_{1}η_{1}^{0}}} (b_{1}η_{1}^{0}s + b · y) ) )

× ( 1 + O(|s|^{−2}) )

+ O ( |s|^{−1} )

= \frac{(−1 − i)(c_{2}y_{2} − id_{3})}{\sqrt{2b_{1}η_{1}^{0}}} \frac{\sqrt{π}}{Γ (1 + i\frac{(c_{2}^{2}y_{2}^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× exp ( i\frac{(b_{1}η_{1}^{0}s + b · y)^{2}}{2b_{1}η_{1}^{0}} − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp ( i\frac{(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} ln (\sqrt{\frac{2}{b_{1}η_{1}^{0}}} (b_{1}η_{1}^{0}s + b · y) ) )

× ( 1 + O(|s|^{−2}) )

+ O ( |s|^{−1} ) .  (3.70)

3.5 Outgoing Outer Solution

To construct the outer solution for positive times, we use the classical quantities associated with the A and B levels that satisfy the following initial conditions:

a^{A}(0) = 0,  (3.71)
η^{A}(0) = η^{0} − \frac{2|d_{3}|ϵ}{b_{1}η_{1}^{0}} b,
S^{A}(0) = 0,
A^{A}(0) = ( (A_{0}A_{0}^{∗})^{−1} + \frac{π}{b_{1}η_{1}^{0}} |c⟩⟨c| )^{−1/2}
B^{A}(t) ∼ ( B_{0}A_{0}^{−1} + \frac{π}{b_{1}η_{1}^{0}} |c⟩⟨c| ) A^{A}(0) + \frac{i|d_{3}|}{b_{1}η_{1}^{0}} [MA^{A}(0) + ln (\frac{2b_{1}η_{1}^{0}t}{|d_{3}|ϵ}) NA^{A}(0)] ,

a^{B}(0) = 0,  (3.72)

η^{B}(0) = η^{0},
S^{B}(0) = 0,
A^{B}(0) = A_{0},
B^{B}(t) ∼ B_{0} − \frac{i|d_{3}|}{b_{1}η_{1}^{0}} ln (\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ) NA_{0} − \frac{i|d_{3}|}{b_{1}η_{1}^{0}} [MA_{0} + ln (\frac{2b_{1}η_{1}^{0}t}{|d_{3}|ϵ}) NA_{0}] .

Here the asymptotic conditions for B^{C}(t) are to be satisfied in the regime where t, ϵ, ϵ/t, and t ln^{2}(ϵ) are all small. It is convenient to define the quantity

B_{1}^{A}(t) = B_{0} + \frac{i|d_{3}|}{b_{1}η_{1}^{0}} [MA_{0} + ln (\frac{2b_{1}η_{1}^{0}t}{|d_{3}|ϵ}) NA_{0}] .

One should note the difference in initial momentum with the classical quantities associated with the B level, (3.43). As explained earlier, the corresponding loss in kinetic energy equals the gain in potential energy, to leading order.

One should also note that the complicated expressions for the A^{C}’s and B^{C}’s are determined in order to satisfy conditions (3.5)–(3.8).

The following lemma describes the outgoing outer solution.

Lemma 3.6 Suppose 0 < δ′ < ξ < 1/3, and 1/2 < κ < 1 − ξ. For ϵ^{1−ξ} ≤ t ≤ T, the outgoing outer solution ψ_{OO}(x, t) agrees with an exact solution of the Schrödinger equation up to O(ϵ^{ξ}) errors and matches the inner solution for ϵ^{κ} < t < ϵ^{1−ξ}. It has the form

ψ_{OO}(x, t)  (3.73)
= F (\frac{∥x − a^{A}(t)∥}{ϵ^{1−δ′}}) e^{iS^{A}(t)/ϵ^{2}} ∑_{|m|≤|l|} d_{m}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x) Φ_{A}^{+}(x, t, ϵ)

+ F (\frac{∥x − a^{B}(t)∥}{ϵ^{1−δ′}}) e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B}^{+}(x, t, ϵ),

where the first summation is finite, and the d_{m}^{B} in the second summation decrease more rapidly than any inverse power of |m|.

Proof: To prove this lemma, we must do two things: We must establish the validity of the approximation and check that formula (3.73) matches the inner solution. We do the first of these tasks by mimicking the proof of lemma 3.3. This is straightforward except for the subtlety of estimating the errors generated by the infinite sum in (3.73). These error terms can be estimated because the growth in m of the error terms generated by the φ_{m} grow polynomially in |m|, uniformly in ϵ, and we prove below that the numerical factors d_{m}^{B} decay faster than |m|^{−N} for any N.

The second task of checking the matching is straigtforward, but very tedious. We assume ϵ^{κ} < t < ϵ^{1−ξ} and separately show that the first and second components of (3.65) match the first and second terms on the right hand side of (3.73), respectively. We show these two results by sequences of approximate equalities. At each step we make an error that is O(ϵ^{α}) for some α > 0.

The first component of (3.65) is

F (ϵ^{δ′}∥y∥) exp (i\frac{S}{ϵ^{2}} + i\frac{η · y}{ϵ} ) C_{2}(y) D_{p_{l}} (\frac{(−1 − i)}{\sqrt{b_{1}η_{1}^{0}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) .

By the explicit formula for C_{2}(y) and formula (3.69), this approximately equals

− F (ϵ^{δ′}∥y∥) exp (i\frac{S}{ϵ^{2}} + i\frac{η · y}{ϵ} ) ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp ( − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp (i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{d_{3}^{2}}{b_{1}η_{1}^{0}} ln ϵ + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} − i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln |d_{3}| + i\frac{3d_{3}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}))

× exp ( −i\frac{c_{2}^{2}y_{2}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}) + i\frac{c_{2}^{2}y_{2}^{2}}{2b_{1}η_{1}^{0}} ln |d_{3}| − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} )

× exp ( −i\frac{b_{1}η_{1}^{0}s^{2}}{2} − is b · y − i\frac{(b · y)^{2}}{2b_{1}η_{1}^{0}} − \frac{3π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp ( −i\frac{(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} ln ( \sqrt{\frac{2}{b_{1}η_{1}^{0}}} (b_{1}η_{1}^{0}s + b · y) ) ) .  (3.74)

By making a small error, we can drop the b · y term in the logarithm in the final term in this expression. Also, by corollary 2.1, we make a small error by replacing F (ϵ^{−δ′}∥y∥) by F (ϵ^{−δ′}∥y^{A}∥). Thus, simple calculations show that (3.74) approximately equals

− F (ϵ^{δ′}∥y^{A}∥) ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp ( − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} )

× exp ( −i\frac{|d_{3}|}{2b_{1}η_{1}^{0}} ⟨ y, M y ⟩ − i\frac{c_{2}^{2}y_{2}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{2b_{1}η_{1}^{0}s}{|d_{3}|} ) )

× exp ( i\frac{η · y}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} − is b · y )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{2b_{1}η_{1}^{0}}{|d_{3}|ϵ^{2}s} ) )

= − F (ϵ^{δ′}∥y^{A}∥) ϵ^{−n/2} φ_{l}(A_{0}, B_{1}^{A}(t), 1, 0, 0, y) exp ( − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} )

× exp ( i\frac{η · y}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} − is b · y )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{2b_{1}η_{1}^{0}}{|d_{3}|ϵ^{2}s} ) ) .

By lemma 3.1 of [20] and proposition 2.2, this approximately equals

− F (ϵ^{δ′}∥y^{A}∥) ϵ^{−n/2} ∑_{|m|≤|l|} d̃_{m}^{A} φ_{m}(A^{A}(t), B^{A}(t), 1, 0, 0, y)

× exp ( i\frac{η · y}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} − is b · y )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{2b_{1}η_{1}^{0}}{|d_{3}|ϵ^{2}s} ) ) ,  (3.75)

where the numbers d̃_{m}^{A} are defined by the relation

exp (−\frac{πd_{3}^{2}}{2b_{1}η_{1}^{0}} ) H_{l}(A_{0}; |A_{0}|^{−1}y) = [\frac{det A^{A}(0)}{det A_{0}}]^{1/2} ∑_{|m|≤|l|} d̃_{m}^{A} H_{m}(A^{A}(0); |A^{A}(0)|^{−1}y).

One may see that the sum is finite and compute the d̃_{m}^{A} by equating coefficients of the monomials involved in this expression.

We now note the following estimates, the last of which follows from corollary 2.1:

( η^{A}(0) − η^{0} ) = −2|d_{3}|ϵ + O(ϵ^{2}),  (3.76)
η^{A}(0)^{2} − η^{0}^{2} = −4|d_{3}|ϵ + O(ϵ^{2}),  (3.77)
η^{A}(t) − η(t) = \frac{ϵ|d_{3}|}{b_{1}η_{1}^{0}} b − t b + O(ϵ^{2}/t) + O(t^{2}) + O(ϵ^{4}/t^{3}).  (3.78)

Using these, lemma 3.1 of [20], and corollary 2.2, we see that (3.75) approximately equals

− F (ϵ^{δ′}∥y^{A}∥) ϵ^{−n/2} ∑_{|m|≤|l|} d̃_{m}^{A} φ_{m}(A^{A}(t), B^{A}(t), 1, 0, 0, y^{A})

× exp ( i\frac{η^{A} · y^{A}}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} − is b · y + i\frac{( η(t) − η^{A}(t)) · y}{ϵ} )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{2b_{1}η_{1}^{0}}{|d_{3}|ϵ^{2}s} ) )

× exp ( i\frac{|d_{3}|t}{ϵ} + i\frac{(η^{A}(0) − η^{0}) · η^{A}(0)t}{ϵ^{2}} − i\frac{b_{1}η_{1}^{A}(0)}{2ϵ^{2}}t^{2} − i\frac{d_{3}^{2}}{2b_{1}η_{1}^{A}(0)} [ \frac{1}{2} + ln (\frac{2b_{1}η_{1}^{A}(0)t}{|d_{3}|ϵ})]) .

Using (3.76), (3.78), and the definition of η^{A}(0), we see that this quantity approximately equals

− F (ϵ^{δ′}∥y^{A}∥) ∑_{|m|≤|l|} d̃_{m}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x)

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − ib_{1}η_{1}^{0}s^{2} + i\frac{|d_{3}|t}{ϵ} − i\frac{d_{3}^{2}}{b_{1}η_{1}^{A}(0)} ln t ) .  (3.79)

We define d_{m}^{A} = − e^{i \frac{S^{B}_{0}(ϵ,−)−S^{A}_{0}(ϵ,+)}{ϵ^{2}}} d̃_{m}^{A}, apply corollary 2.3, and recall that we are studying the first component of the inner solution. By using lemmas 3.1 and 3.2, we see that (3.79) matches the following approximate solution to the full Schrödinger equation.

F (ϵ^{δ′}∥y^{A}∥) e^{iS^{A}(t)/ϵ^{2}} ∑_{|m|≤|l|} d_{m}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x) Φ_{A}^{+}(x, t, ϵ).

This is the first term of (3.73).

We now concentrate on the second component of (3.65), which equals

F (ϵ^{δ′}∥y∥) exp (i\frac{S}{ϵ^{2}} + i\frac{η · y}{ϵ} ) C_{2}(y)

× \frac{(−1 − i)(c_{2}y_{2} − id_{3})}{2\sqrt{b_{1}η_{1}^{0}}} D_{p_{l}−1} (\frac{(−1 − i)}{\sqrt{b_{1}η_{1}^{0}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) .

By the explicit formula for C_{2}(y) and formula (3.70), this approximately equals

− F (ϵ^{δ′}∥y∥) exp (i\frac{S}{ϵ^{2}} + i\frac{η · y}{ϵ} ) ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp ( − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp (i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{d_{3}^{2}}{b_{1}η_{1}^{0}} ln ϵ + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} − i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln |d_{3}| + i\frac{3d_{3}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}))

× exp ( −i\frac{c_{2}^{2}y_{2}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}) + i\frac{c_{2}^{2}y_{2}^{2}}{2b_{1}η_{1}^{0}} ln |d_{3}| − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} )

× \frac{(−1 − i)(c_{2}y_{2} − id_{3})}{\sqrt{2b_{1}η_{1}^{0}}} \frac{\sqrt{π}}{Γ (1 + i\frac{(c_{2}^{2}y_{2}^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× exp ( i\frac{(b_{1}η_{1}^{0}s + b · y)^{2}}{2b_{1}η_{1}^{0}} − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{8b_{1}η_{1}^{0}} )

× exp ( i\frac{(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{2b_{1}η_{1}^{0}} ln (\sqrt{\frac{2}{b_{1}η_{1}^{0}}} (b_{1}η_{1}^{0}s + b · y) ) ) .  (3.80)

As with (3.74), we drop the b·y in the logarithm in the final factor and we replace F (ϵ^{−δ′}∥y∥) by F (ϵ^{−δ′}∥y^{B}∥). Thus, simple calculations show that (3.80) approximately equals

F (ϵ^{δ′}∥y^{B}∥) \sqrt{\frac{π}{2b_{1}η_{1}^{0}}} \frac{(1 + i)(c_{2}y_{2} − id_{3})}{Γ (1 + i\frac{(c_{2}^{2}y_{2}^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp ( − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{4b_{1}η_{1}^{0}} )

× exp ( i\frac{|d_{3}|}{2b_{1}η_{1}^{0}} ⟨ y, M y ⟩ + i\frac{c_{2}^{2}y_{2}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{2b_{1}η_{1}^{0}s}{|d_{3}|} ) + i\frac{c_{2}^{2}y_{2}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ) )

× exp ( i\frac{η · y}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} + is b · y )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} + i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln (\frac{4b_{1}^{2}η_{1}^{0}^{2}s}{|d_{3}|ϵ^{2}}) ) .

By lemma 3.1 of [20] and proposition 2.2, this approximately equals

F (ϵ^{δ′}∥y^{B}∥) \sqrt{\frac{π}{2b_{1}η_{1}^{0}}} \frac{(1 + i)(c_{2}y_{2} − id_{3})}{Γ (1 + i\frac{(c_{2}^{2}y_{2}^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× ϵ^{−n/2} φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y) exp ( − \frac{π(c_{2}^{2}y_{2}^{2} + d_{3}^{2})}{4b_{1}η_{1}^{0}} )

× exp ( i\frac{η · y}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} + is b · y )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} + i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln (\frac{4b_{1}^{2}η_{1}^{0}^{2}s}{|d_{3}|ϵ^{2}}) ) .

Using lemma 3.1 of [20], lemmas 2.1 and 2.2, and corollary 2.2, we see that this approximately equals

F (ϵ^{δ′}∥y^{B}∥) \sqrt{\frac{π}{2b_{1}η_{1}^{0}}} \frac{(1 + i)(c_{2}y_{2}^{B} − id_{3})}{Γ (1 + i\frac{(c_{2}^{2}(y_{2}^{B})^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× ϵ^{−n/2} φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}) exp ( − \frac{π(c_{2}^{2}(y_{2}^{B})^{2} + d_{3}^{2})}{4b_{1}η_{1}^{0}} )

× exp ( i\frac{η^{B} · y^{B}}{ϵ} − i\frac{|d_{3}| b · y}{b_{1}η_{1}^{0}} + is b · y + i\frac{(η(t) − η^{B}(t)) · y}{ϵ} )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} + i\frac{b_{1}η_{1}^{0}s^{2}}{2} + i\frac{d_{3}^{2}}{4b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln (\frac{4b_{1}^{2}η_{1}^{0}^{2}s}{|d_{3}|ϵ^{2}}) )

× exp ( −i\frac{|d_{3}|t}{ϵ} + i\frac{b_{1}η_{1}^{0}}{ϵ^{2}} t^{2} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} [ \frac{1}{2} + ln (\frac{2b_{1}η_{1}^{0}t}{|d_{3}|ϵ}) ] )

= F (ϵ^{δ′}∥y^{B}∥) \sqrt{\frac{π}{2b_{1}η_{1}^{0}}} \frac{(1 + i)(c_{2}y_{2}^{B} − id_{3})}{Γ (1 + i\frac{(c_{2}^{2}(y_{2}^{B})^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})}

× φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) exp ( − \frac{π(c_{2}^{2}(y_{2}^{B})^{2} + d_{3}^{2})}{4b_{1}η_{1}^{0}} )

× exp ( i\frac{S}{ϵ^{2}} + i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} + ib_{1}η_{1}^{0}s^{2} − i\frac{|d_{3}|t}{ϵ} + i\frac{d_{3}^{2}}{b_{1}η_{1}^{0}} ln t )

× exp ( i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{8b_{1}^{3}η_{1}^{0}^{3}}{d_{3}^{2}} ) )

From corollary 2.3 we see that this approximately equals

exp ( i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{S_{0}^{B}(ϵ, +)}{ϵ^{2}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} + i\frac{d_{3}^{2}}{2b_{1}η_{1}^{0}} ln ( \frac{8b_{1}^{3}η_{1}^{0}^{3}}{d_{3}^{2}} ) )

× \sqrt{\frac{π}{2b_{1}η_{1}^{0}}} \frac{(1 + i)(c_{2}y_{2}^{B} − id_{3})}{Γ (1 + i\frac{(c_{2}^{2}(y_{2}^{B})^{2}+d_{3}^{2})}{2b_{1}η_{1}^{0}})} exp ( − \frac{π(c_{2}^{2}(y_{2}^{B})^{2} + d_{3}^{2})}{4b_{1}η_{1}^{0}} )

× F (ϵ^{δ′}∥y^{B}∥) e^{iS^{B}(t)/ϵ^{2}} φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x).  (3.81)

Here, the first line is simply a constant phase, and the product of the first two lines, which we call G(y_{2}^{B}), is an entire function of y_{2}^{B}. From the asymptotics of the Gamma and Polygamma functions [1], G and all its derivatives grow at worst polynomially for real y_{2}^{B}. From this it follows that

G(y_{2}^{B}) φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x)

is a Schwartz function. It consequently [43] has a convergent series expansion of the form

G(y_{2}^{B}) φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x)
= ∑_{m} k_{m}^{B}(t) φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x),  (3.82)

where the coefficients

k_{m}^{B}(t) = ⟨ φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}), G(y_{2}^{B}) φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B})⟩

decay faster than any negative power of |m| for each fixed t. Similarly, we see that the numbers

d_{m}^{B} = ⟨ φ_{m}(A^{B}(0), B^{B}(0), 1, 0, 0, y^{B}), G(y_{2}^{B}) φ_{l}(A^{B}(0), B^{B}(0), 1, 0, 0, y^{B})⟩

decay faster than any negative power of |m|.

The semiclassical wave packets φ_{j}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}) satisfy i\frac{∂ψ}{∂t} = K(t) ψ, where K(t) is the operator −\frac{1}{2} Δ_{y^{B}} + \frac{1}{2}⟨y^{B}, E_{B}^{(2)}(a^{B}(t)) y^{B}⟩. Using this, we compute the derivative of k_{m}^{B}(t),

\frac{∂k_{m}^{B}}{∂t} (t)

= i ⟨ K(t) φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}), G(y_{2}^{B}) φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B})⟩

− i⟨ φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}), G(y_{2}^{B}) K(t) φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B})⟩

= − ⟨ φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}), \frac{i}{2} ( Δ_{y^{B}} G(y_{2}^{B})) φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B})⟩

− ⟨ φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B}), i ( ∇_{y^{B}} G(y_{2}^{B})) · ∇_{y^{B}} φ_{l}(A^{B}(t), B^{B}(t), 1, 0, 0, y^{B})⟩.

(3.83)

The first inner product on the right hand side of (3.83) is bounded by C_{N}(1 + |m|)^{−N}, where N is arbitrary and C_{N} can be chosen independent of t. The second inner product on the right hand side of (3.83) is bounded by C_{N} | ln(t/ϵ)|(1 + |m|)^{−N} where N is arbitrary and C_{N} can be chosen independent of t.

By integrating these estimates with respect to t, we see that for arbitrarily large N,

|k_{m}^{B}(t) − d_{m}^{B}| ≤ ∫_{0}^{t} C_{N} | ln(t/ϵ)| (1 + |m|)^{−N} dt
≤ C′_{N} (1 + |m|)^{−N} |t| | ln(t/ϵ)|.

Thus, d_{m}^{B} = lim_{t→0} k_{m}^{B}(t), and for t in the matching regime, the right hand side of (3.82) equals

∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x),

up to an O(ϵ^{ξ}) error.

Since we are studying the second component of the inner solution, lemmas 3.1 and 3.2 now show that (3.81) matches the following wave packet for t in the matching regime:

F (ϵ^{δ′}∥y^{B}∥) e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B}^{+}(x, t, ϵ).  (3.84)

This is the second term of (3.73).

It remains to show that (3.84) is an approximate solution of the Schrödinger equation up to O(ϵ^{ξ}) errors. It is convenient to rewrite the Schrödinger equation in terms of the dependent variable χ(y, t) and independent variables y = y^{B} and t, where χ is defined by rewriting the expression (3.84) as

e^{iS^{B}(t)/ϵ^{2}} e^{iη^{B}(t)·y/ϵ} χ(y, t).

Explicitly, the Schrödinger equation is

iϵ^{2} \frac{∂χ}{∂t} = −\frac{ϵ^{2}}{2} Δ_{y}χ + h(a^{B}(t) + ϵy, ϵ)χ − [ E_{B}(a^{B}(t), ϵ) + ϵE_{B}^{(1)}(a^{B}(t), ϵ) · y ] χ,

and χ(y, t) is

χ(y, t) = F (ϵ^{δ′}∥y^{B}∥) ∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y) Φ_{B}^{+}(a^{B}(t) + ϵy, t, ϵ).

We explicitly write out the error term

ζ(y, t) = iϵ^{2} \frac{∂χ}{∂t} + \frac{ϵ^{2}}{2} Δ_{y}χ − h(a^{B}(t) + ϵy, ϵ)χ + [E_{B}(a^{B}(t), ϵ) + ϵE_{B}^{(1)}(a^{B}(t), ϵ) · y] χ,

(3.85)

and cancel the low order terms, as in the proof of lemma 3.3.

We first consider those remaining terms that do not involve derivatives of the cut-off function F (ϵ^{δ′}∥y^{B}∥). For these terms, we expand the sum over m and estimate each resulting term separately. Each term in the sum in (3.84) is an approximate solution, and the only issue is whether we can control the growth of the errors with m. Since the d_{m}^{B} decrease faster than any negative power of |m|, it is sufficient to show that we can estimate these individual terms in ζ(y, t) by C_{m}ϵ^{2+ξ} errors, where C_{m} grows at worst polynomially in |m|. We mimick the proof of lemma 3.3, but keep track of the m dependence of the error terms.

These terms contain first or second derivatives of the functions φ_{m} or functions with at worst cubic growth in ∥ϵy∥ times functions φ_{m}. We now give the crucial estimate for one type of factor that occurs in such terms. The arguments for the others are similar, although the quadratic and cubic ones are longer and substantially more messy.

Consider the m dependence of

J_{m}^{2} = ∫_{IR_{n}} |v · ∇_{y}φ_{m}(A, B, 1, 0, 0, y)|^{2} dy,

for some vector v ∈ IR^{n}. We claim that J_{m} grows like m^{1/2}. By Fourier transforming,

J_{m}^{2} = ∫_{IR_{n}} |⟨v, k⟩ φ_{m}(B, A, 1, 0, 0, k)|^{2} dk.

Using the definition (3.4) and the shorthand notation

H_{m}(B; k) = H_{B}^{♯}(m_{1}, m_{2}, …, m_{n}, k),

we have

2 ⟨U_{B}e_{l}, κ⟩ H_{B}^{♯}(m_{1}, m_{2}, …, m_{n}, κ)
= H_{B}^{♯}(m_{1}, m_{2}, …, m_{l} + 1, …, m_{n}, κ)

+ 2 ∑_{j=1}^{n} ⟨U_{B}e_{l}, \overline{U_{B}e_{j}}⟩ m_{j} H_{B}^{♯}(m_{1}, m_{2}, …, m_{j} − 1, …, m_{n}, κ).

Hence, using u = ∑_{j=1}^{n} ⟨U_{B}e_{j}, u⟩ U_{B}e_{j} ≡ ∑_{j=1}^{n} u_{j}(B)U_{B}e_{j}, we obtain

2 ⟨u, κ⟩ H_{B}^{♯}(m_{1}, m_{2}, …, m_{n}, κ)
= ∑_{j=1}^{n} \overline{u_{j}(B)} H_{B}^{♯}(m_{1}, m_{2}, …, m_{j} + 1, …, m_{n}, κ)

+ 2 ∑_{j=1}^{n} ⟨u, \overline{U_{B}e_{j}}⟩ m_{j} H_{B}^{♯}(m_{1}, m_{2}, …, m_{j} − 1, …, m_{n}, κ).

We apply this with u = |B|v, κ = |B|^{−1}k, and multiply by the other factors in the definition of φ_{m}. This yields

2 ⟨v, k⟩ φ_{m}(B, A, 1, 0, 0, k)
= 2 ⟨u, κ⟩ 2^{−|m|/2}(m!)^{−1/2}H_{m}(B; κ)φ_{0}(B, A, 1, 0, 0, k)
= 2^{−|m|/2}(m!)^{−1/2}φ_{0}(B, A, 1, 0, 0, k) [ ∑_{j=1}^{n} \overline{u_{j}(B)}H_{B}^{♯}(m_{1}, m_{2}, …, m_{j} + 1, …, m_{n}, κ)

+ 2 ∑_{j=1}^{n} ⟨u, \overline{U_{B}e_{j}}⟩ m_{j} H_{B}^{♯}(m_{1}, m_{2}, …, m_{j} − 1, …, m_{n}, κ)]

= ∑_{j=1}^{n} \sqrt{2(m_{j} + 1)}\overline{u_{j}(B)}φ_{m′(j)}(B, A, 1, 0, 0, k)

+ ∑_{j=1}^{n} \sqrt{2^{3}m_{j}}⟨u, \overline{U_{B}e_{j}}⟩φ_{m′′(j)}(B, A, 1, 0, 0, k),

where m′(j) has m_{j} + 1 in place of m_{j} and m′′(j) has m_{j} − 1 in place of m_{j}. By using the orthonormality of distinct φ_{l}, it follows from this expression that J_{m} grows like |m|^{1/2} for B and v restricted to compact sets on which (3.5)–(3.8) are satisfied.

The other terms that do not contain derivatives of F (ϵ^{δ′}∥y^{B}∥) are estimated by similar (but sometimes more lengthy) calculations. The terms in the error, ζ, that grow most rapidly in m grow like |m|^{3/2}. Such growth occurs, for example, from the errors incurred by replacing the effective potential by its second order Taylor series.

Two remaining terms contain derivatives of the cut-off function F (ϵ^{δ′}∥y^{B}∥). Both of these terms arise from \frac{ϵ^{2}}{2} Δ_{y} in the expression (3.85) for ζ(y, t). For these terms we do not expand the sum over m.

One term is

\frac{ϵ^{2+2δ′}}{2} [ΔF](ϵ^{δ′}∥y^{B}∥) ∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y) Φ_{B}^{+}(a^{B}(t) + ϵy, t, ϵ),

whose norm is clearly bounded by a constant times ϵ^{2+2δ′}.

The other term is

ϵ^{2+δ′} [∇F](ϵ^{δ′}∥y^{B}∥) · ∇_{y} { ∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), 1, 0, 0, y) Φ_{B}^{+}(a^{B}(t) + ϵy, t, ϵ) } .

(3.86)

We compute the second gradient in this expression with the help of the formulas used to study terms that did not contain derivatives of F. The gradient introduces growth like |m|^{1/2} inside the sum, but because of the rapid fall off of the d_{m}^{B}, we see that the norm of the second gradient is bounded, independent of ϵ. Since the gradient of F is bounded, the entire term (3.86) is bounded by a constant times ϵ^{2+δ′}.

Thus, we see that ζ(y, t) defined by (3.85) has norm bounded by ϵ^{2+ξ} for some ξ > 0. The result then follows from Lemma 3.3 of [17].

Remarks.

1. Suppose one chooses l = 0 so that the incoming outer nuclear wave packet is a Gaussian. Then the part of the outgoing outer solution that is associated with the A surface also has a Gaussian nuclear wave packet. The absolute square of the nuclear wave packet for the part associated with the B surface is the difference between two Gaussians. The infinite series expansion is required because the G(y_{2}) factor gives rise to complicated phases.

2. Our calculations show that to leading order, the probability of an electronic transition from the B surface to the A surface is

P_{B→A} ≈ ∑_{|m|≤|l|} |d̃_{m}^{A}|^{2} = ∑_{|m|≤|l|} |d_{m}^{A}|^{2}

= ∫_{IR_{n}} |φ_{l}(A_{0}, B_{0}, 1, 0, 0, y)|^{2} e^{− \frac{π(c^{2}_{2}y^{2}_{2}+d^{2}_{3})}{b_{1}η^{0}_{1}}} dy.

For the case l = 0, we have the simple expression

d̃_{0}^{A} = [\frac{det A_{0}}{det A^{A}(0)}]^{1/2} exp (−\frac{πd_{3}^{2}}{2b_{1}η_{1}^{0}} ) .

So, to leading order the transition probability is

P_{B→A} ≈ [\frac{det A_{0}}{det A^{A}(0)}] exp (−\frac{πd_{3}^{2}}{b_{1}η_{1}^{0}} ) .

For codimension 2 crossings we obtain the same formula [17], except that d_{3} is zero for codimension 2 crossings. Thus, the leading order transition probability depends algebraically on η_{1}^{0} in the crossing case, but exponentially in the avoided crossing case. Although our error estimates are not uniform in η_{1}^{0} as η_{1}^{0} → 0, this might provide a way to distinguish true crossings from avoided crossings.

We now summarize the main results of this section.

Theorem 3.1 Let h(x, ϵ) be a hamiltonian such that h_{∥}(x, ϵ) is characterized by (1.10), (1.11), and (1.33). Let ψ(x, t) be a solution of the corresponding Schrödinger equation (1.29) such that

ψ(x, −T) = e^{iS^{B}(−T)/ϵ^{2}} φ_{l}(A^{B}(−T), B^{B}(−T), ϵ^{2}, a^{B}(−T), η^{B}(−T), x) Φ_{B}^{−}(x, −T, ϵ)
+ O(ϵ^{q})

for some positive q. Then, for any 0 < ξ < 1/3, there exists a positive p such that in the limit ϵ → 0, we have

ψ(x, t) = e^{iS^{B}(t)/ϵ^{2}} φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B}^{−}(x, t, ϵ) + O(ϵ^{p}),

for −T ≤ t ≤ −ϵ^{1−ξ}. For −ϵ^{1−ξ} ≤ t ≤ ϵ^{1−ξ}, we have

ψ(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} ( f_{0}(y, s) ψ_{1}(x, ϵ) + g_{0}(y, s) ψ_{2}(x, ϵ) ) + O(ϵ^{p}),

where f_{0} and g_{0} are defined by (3.64), (3.66), and (3.68) with y = (x − a(t))/ϵ and s = t/ϵ. Finally, for ϵ^{1−ξ} ≤ t ≤ T,

ψ(x, t) = e^{iS^{A}(t)/ϵ^{2}} ∑_{|m|≤|l|} d_{m}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x) Φ_{A}^{+}(x, t, ϵ)

+ e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m}^{B} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B}^{+}(x, t, ϵ)

+ O(ϵ^{p}),

where the d_{m}^{A} and d_{m}^{B} are specified in the proof of lemma 3.6.

4 Type 4, 5, and 6 Avoided Crossings

In this section we make precise statements of our results for Type 4, 5, and 6 Avoided Crossings. The proofs of these results are obtained by simply mimicking arguments from Section 3, so we do not present the details.

4.1 Type 4 Avoided Crossings

Since the matrix (1.15) is the direct sum of (1.11) and its conjugate, the only substantial difference between Type 3 Avoided Crossings and Type 4 Avoided Crossings is the degeneracy of the electronic states. For Type 4 Avoided Crossings, the following lemma plays the role of lemmas 3.1 and 3.2. The proof of this lemma is an extension of the proofs of lemmas 3.1 and 3.2. Similar arguments may also be found in Section 4.2 of [20].

Lemma 4.1 Consider a system with a Type 4 Avoided Crossing. Suppose 0 < δ′ < ξ < \frac{1}{3}, |t| > ϵ^{1−ξ}. Define μ(ϵ, t) = max { |t|, ϵ^{1−δ′}/|t| }, and let a^{C}(t) and η^{C}(t) solve equations (2.2) and (2.3). For x in the support of F (∥x−a^{C}(t)∥/ϵ^{1−δ′}), there exist orthonormal eigenvectors Φ_{C,j}^{±}(x, t, ϵ) for C = A, B, t{> \atop <}0, and j = 1, 2, that correspond to E_{C}(x, ϵ) and solve

⟨Φ_{C,j}^{±}(x, t, ϵ), (∂/∂t + η^{C}(t) · ∇) Φ_{C,k}^{±}(x, t, ϵ)⟩ ≡ 0  (j, k = 1, 2).  (4.1)

If in addition, |t| < ϵ^{κ} for some κ > \frac{1}{2}, then for t < 0,

Φ_{A,1}^{−}(x, t, ϵ) = ψ_{2}(x, ϵ) + O(μ(ϵ, t)),
Φ_{A,2}^{−}(x, t, ϵ) = ψ_{4}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,1}^{−}(x, t, ϵ) = −ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,2}^{−}(x, t, ϵ) = −ψ_{3}(x, ϵ) + O(μ(ϵ, t)),

Φ_{A,1}^{+}(x, t, ϵ) = ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{A,2}^{+}(x, t, ϵ) = ψ_{3}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,1}^{+}(x, t, ϵ) = ψ_{2}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,2}^{+}(x, t, ϵ) = ψ_{4}(x, ϵ) + O(μ(ϵ, t)).

Our main result for Type 4 Avoided Crossings is the following:

Theorem 4.1 Let h(x, ϵ) be a hamiltonian such that h_{∥}(x, ϵ) is characterized by (1.14), (1.15), and (1.33). Let a^{C}(t), η^{C}, A^{C}, B^{C}, and S^{C} be defined as in Section 2. For j = 1, 2, let ψ_{j}(x, t) be a solution of the Schrödinger equation (1.29) such that

ψ_{j}(x, −T) = e^{iS^{B}(−T)/ϵ^{2}} φ_{l}(A^{B}(−T), B^{B}(−T), ϵ^{2}, a^{B}(−T), η^{B}(−T), x) Φ_{B,j}^{−}(x, −T, ϵ)
+ O(ϵ^{q})

for some positive q. Then, for any 0 < ξ < 1/3, there exists a positive p such that in the limit ϵ → 0, we have

ψ_{j}(x, t) = e^{iS^{B}(t)/ϵ^{2}} φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,j}^{−}(x, t, ϵ) + O(ϵ^{p}),

for −T ≤ t ≤ −ϵ^{1−ξ}. For −ϵ^{1−ξ} ≤ t ≤ ϵ^{1−ξ}, in the basis { ψ_{1}(x, ϵ), ⋯, ψ_{4}(x, ϵ) }, we have

ψ_{1}(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} C_{2}(y)

× ( \begin{matrix}D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \\ _{\frac{(−1−i)(c_{2}y_{2}−id_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \\ 0 \\ 0 \end{matrix} )

+ O(ϵ^{p}) ,

and

ψ_{2}(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} C_{2}(y)

× ( \begin{matrix}0 \\ 0 \\ D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \\ _{\frac{(−1−i)(c_{2}y_{2}+id_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \end{matrix} )

+ O(ϵ^{p}) ,

where C_{2} is defined by (3.68), p_{l} = − i\frac{c_{2}^{2}y_{2}^{2}+d_{3}^{2}}{2b_{1}η_{1}^{0}} , y = (x − a(t))/ϵ, and s = t/ϵ. Finally, for ϵ^{1−ξ} ≤ t ≤ T,

ψ_{j}(x, t) = e^{iS^{A}(t)/ϵ^{2}} ∑_{|m|≤|l|} d_{m,j}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x) Φ_{A,j}^{+}(x, t, ϵ)

+ e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m,j}^{B} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,j}^{+}(x, t, ϵ)

+ O(ϵ^{p}),

where the d_{m,j}^{A} and d_{m,j}^{B} are computed by the process used in the proof of lemma 3.6.

4.2 Type 5 Avoided Crossings

Type 5 Avoided Crossings have more complicated electronic structure, so we present a few more of the details. To specify the static eigenvectors analogous to those in lemma 3.1, we define a coordinate system by the following relations, with 0 ≤ r, 0 ≤ θ < π, 0 ≤ ϕ < \frac{π}{2}, and 0 ≤ μ < 2π:

r = \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + ζ(x, ϵ)^{2}},
β(x, ϵ) = r cos θ,
γ(x, ϵ) = r sin θ sin ϕ cos μ,
δ(x, ϵ) = r sin θ sin ϕ sin μ,
ζ(x, ϵ) = r sin θ cos ϕ,

where β(x, ϵ), γ(x, ϵ), δ(x, ϵ), and ζ(x, ϵ) are given by (1.37). The static eigenvectors are then given by

Φ_{A,1}^{−}(x, ϵ) = ( \begin{matrix}cos \frac{θ}{2} cos ϕ \\ e^{−iμ} cos \frac{θ}{2} sin ϕ \\ 0 \\ sin & \frac{θ}{2} \end{matrix} ) ,

Φ_{A,2}^{−}(x, ϵ) = ( \begin{matrix}−e^{iμ} cos \frac{θ}{2} sin ϕ \\ cos \frac{θ}{2} cos ϕ \\ − sin \frac{θ}{2} \\ 0 \end{matrix} ) ,

Φ_{B,1}^{−}(x, ϵ) = ( \begin{matrix}− sin \frac{θ}{2} \\ 0 \\ e^{−iμ} cos \frac{θ}{2} sin ϕ \\ cos \frac{θ}{2} cos ϕ \end{matrix} ) ,

Φ_{B,2}^{−}(x, ϵ) = ( \begin{matrix}0 \\ − sin \frac{θ}{2} \\ − cos \frac{θ}{2} cos ϕ \\ e^{iμ} cos \frac{θ}{2} sin ϕ \end{matrix} ) ,

for π/2 < θ ≤ π, and

Φ_{A,1}^{+}(x, ϵ) = ( \begin{matrix}cos & \frac{θ}{2} \\ 0 \\ e^{−iμ} sin \frac{θ}{2} sin ϕ \\ sin \frac{θ}{2} cos ϕ \end{matrix} ) ,

Φ_{A,2}^{+}(x, ϵ) = ( \begin{matrix}0 \\ cos & \frac{θ}{2} \\ − sin \frac{θ}{2} cos ϕ \\ e^{iμ} sin \frac{θ}{2} sin ϕ \end{matrix} ) ,

Φ_{B,1}^{+}(x, ϵ) = ( \begin{matrix}−e^{iμ} sin \frac{θ}{2} sin ϕ \\ sin \frac{θ}{2} cos ϕ \\ cos & \frac{θ}{2} \\ 0 \end{matrix} ) ,

Φ_{B,2}^{+}(x, ϵ) = ( \begin{matrix}− sin \frac{θ}{2} cos ϕ \\ −e^{−iμ} sin \frac{θ}{2} sin ϕ \\ 0 \\ cos & \frac{θ}{2} \end{matrix} ) ,

for 0 ≤ θ < π/2. Using these definitions and mimicking the proofs of lemmas 3.1 and 3.2, we have the following analog of lemma 4.1:

Lemma 4.2 Consider a system with a Type 5 Avoided Crossing. Suppose 0 < δ′ < ξ < \frac{1}{3}, |t| > ϵ^{1−ξ}. Define μ(ϵ, t) = max { |t|, ϵ^{1−δ′}/|t| }, and let a^{C}(t) and η^{C}(t) solve equations (2.2) and (2.3). For x in the support of F (∥x−a^{C}(t)∥/ϵ^{1−δ′}), there exist orthonormal eigenvectors Φ_{C,j}^{±}(x, t, ϵ) for C = A, B, t{> \atop <}0, and j = 1, 2, that correspond to E_{C}(x, ϵ) and solve (4.1). If in addition, |t| < ϵ^{κ} for some κ > \frac{1}{2}, then for t < 0,

Φ_{A,1}^{−}(x, t, ϵ) = ψ_{4}(x, ϵ) + O(μ(ϵ, t)),
Φ_{A,2}^{−}(x, t, ϵ) = −ψ_{3}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,1}^{−}(x, t, ϵ) = −ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,2}^{−}(x, t, ϵ) = −ψ_{2}(x, ϵ) + O(μ(ϵ, t)),

and for t > 0,

Φ_{A,1}^{+}(x, t, ϵ) = ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{A,2}^{+}(x, t, ϵ) = ψ_{2}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,1}^{+}(x, t, ϵ) = ψ_{3}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,2}^{+}(x, t, ϵ) = ψ_{4}(x, ϵ) + O(μ(ϵ, t)).

Our main result for Type 5 Avoided Crossings is the following:

Theorem 4.2 Let h(x, ϵ) be a hamiltonian such that h_{∥}(x, ϵ) is characterized by (1.14), (1.19), and (1.37). Let a^{C}(t), η^{C}, A^{C}, B^{C}, and S^{C} be the quantities from Section 2.7 appropriate for Type 5 Avoided Crossings. For j = 1, 2, let ψ_{j}(x, t) be a solution of the Schrödinger equation (1.29) such that

ψ_{j}(x, −T) = e^{iS^{B}(−T)/ϵ^{2}} φ_{l}(A^{B}(−T), B^{B}(−T), ϵ^{2}, a^{B}(−T), η^{B}(−T), x) Φ_{B,j}^{−}(x, −T, ϵ)
+ O(ϵ^{q})

for some positive q. Then, for any 0 < ξ < 1/3, there exists a positive p such that in the limit ϵ → 0, we have

ψ_{j}(x, t) = e^{iS^{B}(t)/ϵ^{2}} φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,j}^{−}(x, t, ϵ) + O(ϵ^{p}),

for −T ≤ t ≤ −ϵ^{1−ξ}. For −ϵ^{1−ξ} ≤ t ≤ ϵ^{1−ξ}, in the basis { ψ_{1}(x, ϵ), ⋯, ψ_{4}(x, ϵ) }, we have

ψ_{1}(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} C_{2}(y)

× ( \begin{matrix}D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ 0 \\ _{\frac{(−1−i)(c_{2}y_{2}−id_{3}y_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ _{\frac{(−1−i)e_{4}}{2\sqrt{b_{1}η_{1}^{0}}}} D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \end{matrix} )

+ O(ϵ^{p}) ,

and

ψ_{2}(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} C_{2}(y)

× ( \begin{matrix}0 \\ D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ _{\frac{(1+i)e_{4}}{2\sqrt{b_{1}η_{1}^{0}}}} D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ _{\frac{(−1−i)(c_{2}y_{2}+id_{3}y_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \end{matrix} )

+ O(ϵ^{p}) ,

where C_{2}(y) is defined by

C_{2}(y) = − ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp (\frac{−π(c_{2}^{2}y_{2}^{2} + d_{3}^{2}y_{3}^{2} + e_{4}^{2})}{8b_{1}η_{1}^{0}} )

× exp (i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{e_{4}^{2}}{b_{1}η_{1}^{0}} ln ϵ + i\frac{e_{4}^{2}}{4b_{1}η_{1}^{0}} − i\frac{e_{4}^{2}}{2b_{1}η_{1}^{0}} ln |e_{4}| + i\frac{3e_{4}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}))

× exp ( −i\frac{c_{2}^{2}y_{2}^{2} + d_{3}^{2}y_{3}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}) + i\frac{c_{2}^{2}y_{2}^{2} + d_{3}^{2}y_{3}^{2}}{2b_{1}η_{1}^{0}} ln |e_{4}| − i\frac{|e_{4}|b · y}{b_{1}η_{1}^{0}} ) ,

p_{l} = − i\frac{c_{2}^{2}y_{2}^{2}+d_{3}^{2}y_{3}^{2}+e_{4}^{2}}{2b_{1}η_{1}^{0}} , y = (x − a(t))/ϵ, and s = t/ϵ. Finally, for ϵ^{1−ξ} ≤ t ≤ T,

ψ_{j}(x, t) = e^{iS^{A}(t)/ϵ^{2}} ∑_{|m|≤|l|} d_{m,j}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x) Φ_{A,j}^{+}(x, t, ϵ)

+ e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m,j}^{B,1} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,1}^{+}(x, t, ϵ)

+ e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m,j}^{B,2} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,2}^{+}(x, t, ϵ)

+ O(ϵ^{p}),

where the d_{m,j}^{A} and d_{m,j}^{B,k} are computed by the process used in the proof of lemma 3.6. The initial conditions for the semiclassical quantities in the expression for ψ_{j}(x, t) for t ≥ ϵ^{1−ξ} are the analogs of formulas (3.71) and (3.72) after one has taken into account the appropriate alterations from section 2.7.

4.3 Type 6 Avoided Crossings

Type 6 Avoided Crossings have even more complicated electronic structure. To specify the static eigenvectors analogous to those in lemma 3.1, we define a coordinate system by the following relations, with 0 ≤ r, 0 ≤ θ < π, 0 ≤ ϕ < \frac{π}{2}, 0 ≤ μ < 2π, and 0 ≤ ω < 2π:

r = \sqrt{β(x, ϵ)^{2} + γ(x, ϵ)^{2} + δ(x, ϵ)^{2} + ζ(x, ϵ)^{2} + ξ(x, ϵ)^{2}},
β(x, ϵ) = r cos θ,
γ(x, ϵ) = r sin θ sin ϕ cos μ,
δ(x, ϵ) = r sin θ sin ϕ sin μ,
ζ(x, ϵ) = r sin θ cos ϕ cos ω,
ξ(x, ϵ) = r sin θ cos ϕ sin ω,

where β(x, ϵ), γ(x, ϵ), δ(x, ϵ), ζ(x, ϵ), and ξ(x, ϵ) are given by (1.40). The static eigenvectors are then given by

Φ_{A,1}^{−}(x, ϵ) = ( \begin{matrix}e^{iω} cos \frac{θ}{2} cos ϕ \\ e^{−iμ} cos \frac{θ}{2} sin ϕ \\ 0 \\ sin & \frac{θ}{2} \end{matrix} ) ,

Φ_{A,2}^{−}(x, ϵ) = ( \begin{matrix}−e^{iμ} cos \frac{θ}{2} sin ϕ \\ e^{−iω} cos \frac{θ}{2} cos ϕ \\ − sin \frac{θ}{2} \\ 0 \end{matrix} ) ,

Φ_{B,1}^{−}(x, ϵ) = ( \begin{matrix}− sin \frac{θ}{2} \\ 0 \\ e^{−iμ} cos \frac{θ}{2} sin ϕ \\ e^{−iω} cos \frac{θ}{2} cos ϕ \end{matrix} ) ,

Φ_{B,2}^{−}(x, ϵ) = ( \begin{matrix}0 \\ − sin \frac{θ}{2} \\ −e^{iω} cos \frac{θ}{2} cos ϕ \\ e^{iμ} cos \frac{θ}{2} sin ϕ \end{matrix} ) ,

for π/2 < θ ≤ π, and

Φ_{A,1}^{+}(x, ϵ) = ( \begin{matrix}cos & \frac{θ}{2} \\ 0 \\ e^{−iμ} sin \frac{θ}{2} sin ϕ \\ e^{−iω} sin \frac{θ}{2} cos ϕ \end{matrix} ) ,

Φ_{A,2}^{+}(x, ϵ) = ( \begin{matrix}0 \\ cos & \frac{θ}{2} \\ −e^{iω} sin \frac{θ}{2} cos ϕ \\ e^{iμ} sin \frac{θ}{2} sin ϕ \end{matrix} ) ,

Φ_{B,1}^{+}(x, ϵ) = ( \begin{matrix}−e^{iμ} sin \frac{θ}{2} sin ϕ \\ e^{−iω} sin \frac{θ}{2} cos ϕ \\ cos & \frac{θ}{2} \\ 0 \end{matrix} ) ,

Φ_{B,2}^{+}(x, ϵ) = ( \begin{matrix}−e^{iω} sin \frac{θ}{2} cos ϕ \\ −e^{−iμ} sin \frac{θ}{2} sin ϕ \\ 0 \\ cos & \frac{θ}{2} \end{matrix} ) ,

for 0 ≤ θ < π/2. Using these definitions and mimicking the proofs of lemmas 3.1 and 3.2, we have the following analog of lemma 4.1:

Lemma 4.3 Consider a system with a Type 6 Avoided Crossing. Suppose 0 < δ′ < ξ < \frac{1}{3}, |t| > ϵ^{1−ξ}. Define μ(ϵ, t) = max { |t|, ϵ^{1−δ′}/|t| }, and let a^{C}(t) and η^{C}(t) solve equations (2.2) and (2.3). For x in the support of F (∥x−a^{C}(t)∥/ϵ^{1−δ′}), there exist orthonormal eigenvectors Φ_{C,j}^{±}(x, t, ϵ) for C = A, B, t{> \atop <}0, and j = 1, 2, that correspond to E_{C}(x, ϵ) and solve (4.1). If in addition, |t| < ϵ^{κ} for some κ > \frac{1}{2}, then for t < 0,

Φ_{A,1}^{−}(x, t, ϵ) = ψ_{4}(x, ϵ) + O(μ(ϵ, t)),
Φ_{A,2}^{−}(x, t, ϵ) = −ψ_{3}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,1}^{−}(x, t, ϵ) = −ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,2}^{−}(x, t, ϵ) = −ψ_{2}(x, ϵ) + O(μ(ϵ, t)),

and for t > 0,

Φ_{A,1}^{+}(x, t, ϵ) = ψ_{1}(x, ϵ) + O(μ(ϵ, t)),
Φ_{A,2}^{+}(x, t, ϵ) = ψ_{2}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,1}^{+}(x, t, ϵ) = ψ_{3}(x, ϵ) + O(μ(ϵ, t)),
Φ_{B,2}^{+}(x, t, ϵ) = ψ_{4}(x, ϵ) + O(μ(ϵ, t)).

Our main result for Type 6 Avoided Crossings is the following:

Theorem 4.3 Let h(x, ϵ) be a hamiltonian such that h_{∥}(x, ϵ) is characterized by (1.14), (1.22), and (1.40). Let a^{C}(t), η^{C}, A^{C}, B^{C}, and S^{C} be the quantities from Section 2.7 appropriate for Type 6 Avoided Crossings. For j = 1, 2, let ψ_{j}(x, t) be a solution of the Schrödinger equation (1.29) such that

ψ_{j}(x, −T) = e^{iS^{B}(−T)/ϵ^{2}} φ_{l}(A^{B}(−T), B^{B}(−T), ϵ^{2}, a^{B}(−T), η^{B}(−T), x) Φ_{B,j}^{−}(x, −T, ϵ)
+ O(ϵ^{q})

for some positive q. Then, for any 0 < ξ < 1/3, there exists a positive p such that in the limit ϵ → 0, we have

ψ_{j}(x, t) = e^{iS^{B}(t)/ϵ^{2}} φ_{l}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,j}^{−}(x, t, ϵ) + O(ϵ^{p}),

for −T ≤ t ≤ −ϵ^{1−ξ}. For −ϵ^{1−ξ} ≤ t ≤ ϵ^{1−ξ}, in the basis { ψ_{1}(x, ϵ), ⋯, ψ_{4}(x, ϵ) }, we have

ψ_{1}(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} C_{2}(y)

× ( \begin{matrix}D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ 0 \\ _{\frac{(−1−i)(c_{2}y_{2}−id_{3}y_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ _{\frac{(−1−i)(e_{4}y_{4}−if_{5})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \end{matrix} )

+ O(ϵ^{p}) ,

and

ψ_{2}(x, t) = e^{iS(t)/ϵ^{2}+iη(t)(x−a(t))/ϵ^{2}} C_{2}(y)

× ( \begin{matrix}0 \\ D_{p_{l}} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ _{\frac{(1+i)(e_{4}y_{4}+if_{5})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b · y)) \\ _{\frac{(−1−i)(c_{2}y_{2}+id_{3}y_{3})}{2\sqrt{b_{1}η_{1}^{0}}}}D_{p_{l}−1} (_{\frac{(−1−i)}{\sqrt{b_{1}η_{1}^{0}}}}(b_{1}η_{1}^{0}s + b_{1}y_{1} + b_{2}y_{2})) \end{matrix} )

+ O(ϵ^{p}) ,

where C_{2} is defined by

C_{2}(y)

= − ϵ^{−n/2} φ_{l}(A_{0}, B_{0}, 1, 0, 0, y) exp (\frac{−π(c_{2}^{2}y_{2}^{2} + d_{3}^{2}y_{3}^{2} + e_{4}^{2}y_{4}^{2} + f_{5}^{2})}{8b_{1}η_{1}^{0}} )

× exp (i\frac{S_{0}^{B}(ϵ, −)}{ϵ^{2}} − i\frac{f_{5}^{2}}{b_{1}η_{1}^{0}} ln ϵ + i\frac{f_{5}^{2}}{4b_{1}η_{1}^{0}} − i\frac{f_{5}^{2}}{2b_{1}η_{1}^{0}} ln |f_{5}| + i\frac{3f_{5}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}))

× exp ( −i\frac{c_{2}^{2}y_{2}^{2} + d_{3}^{2}y_{3}^{2} + e_{4}^{2}y_{4}^{2}}{4b_{1}η_{1}^{0}} ln (2b_{1}η_{1}^{0}) + i\frac{c_{2}^{2}y_{2}^{2} + d_{3}^{2}y_{3}^{2} + e_{4}^{2}y_{4}^{2}}{2b_{1}η_{1}^{0}} ln |f_{5}| − i\frac{|f_{5}|b · y}{b_{1}η_{1}^{0}} ) ,

p_{l} = − i\frac{c_{2}^{2}y_{2}^{2}+d_{3}^{2}y_{3}^{2}+e_{4}^{2}y_{4}^{2}+f_{5}^{2}}{2b_{1}η_{1}^{0}} , y = (x − a(t))/ϵ, and s = t/ϵ. Finally, for ϵ^{1−ξ} ≤ t ≤ T,

ψ_{j}(x, t) = e^{iS^{A}(t)/ϵ^{2}} ∑_{|m|≤|l|} d_{m,j}^{A} φ_{m}(A^{A}(t), B^{A}(t), ϵ^{2}, a^{A}(t), η^{A}(t), x) Φ_{A,j}^{+}(x, t, ϵ)

+ e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m,j}^{B,1} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,1}^{+}(x, t, ϵ)

+ e^{iS^{B}(t)/ϵ^{2}} ∑_{m} d_{m,j}^{B,2} φ_{m}(A^{B}(t), B^{B}(t), ϵ^{2}, a^{B}(t), η^{B}(t), x) Φ_{B,2}^{+}(x, t, ϵ)

+ O(ϵ^{p}),

where the d_{m,j}^{A} and d_{m,j}^{B,k} are computed by the process used in the proof of lemma 3.6. The initial conditions for the semiclassical quantities in the expression for ψ_{j}(x, t) for t ≥ ϵ^{1−ξ} are the analogs of formulas (3.71) and (3.72) after one has taken into account the appropriate alterations from section 2.7.

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