Molecular Resonance Raman and Rayleigh Scattering Stimulated by a Short Laser Pulse

Transcription

1 Molecular Resonance Raman and Rayleigh Scattering Stimulated by a Short Laser Pulse George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry Virginia Polytechnic Institute and State University Blacksburg, Virginia , U.S.A. Edward F. Valeev Department of Chemistry and Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry Virginia Polytechnic Institute and State University Blacksburg, Virginia , U.S.A. May 9, 013 Abstract We study a simple model for a molecule subjected to a short laser pulse. We derive very simple expressions for resonance Raman and resonance Rayleigh scattering amplitudes to leading order in the strength µ of the laser and the Born Oppenheimer parameter ɛ for the molecule. We also briefly consider the situation where the incident light is close to, but not exactly at the resonance frequency for an electronic transition of the molecule. Partially Supported by National Science Foundation Grant #DMS Partially Supported by National Science Foundation Grant #CHE and a Camille and Henry Dreyfus Teacher Scholar Award. 1

2 1 Introduction The goal of this paper is to describe resonance Rayleigh and Raman scattering amplitudes for a molecular system subjected to a short laser pulse. We begin by describing the simplest situation we consider. Our simplest molecular system has two nuclei restricted to move only along a line. The electrons have only two states for each configuration of the nuclei. After the removal the center of mass motion, this system is described by the Hamiltonian H(ɛ = ɛ4 x + h(x, where x describes the relative positions of the nuclei and h(x describes the two level system for the electrons. The quantity ɛ 4 is the reduced mass for the nuclei in units where the electron mass is 1. The parameter ɛ is the Born Oppenheimer parameter for the molecule. In practice, it is typically on the order of 1/10. We assume h(x is a real symmetric matrix that depends smoothly on x. We assume its eigenvalues V 0 (x < V 1 (x are separated by a positive minimal gap G = inf [V 1 (x V 0 (x] > 0, and that V 0 has a non-degenerate global minimum of zero x at x = 0, i.e., V 0 (0 = 0, V 0(0 = 0, V 0 (x > 0 for x 0, and the second deriviative V 0 (0 = ω0 > 0 is strictly positive. We further assume lim inf x ɛ, the bottom of the spectrum of H(ɛ is discrete. V 0(x > 0, so that for small We assume γ = V 1 (0 > 0 and β = V 1(0. We further assume β 0, which is the generic situation. If V 1 has a minimum, it is not at x = 0, i.e., it is offset from the minimum of V 0. We let this molecular system interact with a laser pulse that we model classically. This leads us to study solutions to the time dependent Schrödinger equation i ψ ( = ɛ4 ψ ω t + h(x ψ + µ f(t cos g(x ψ, t x where ψ(x, t has two components. We assume the coupling between the laser and the molecule is given by a self-adjoint matrix g(x that depends smoothly on x. We also assume f is real and has support in some interval [, T ]. We assume the laser pulse is short, by which we mean that T is strictly less than the period of classical oscillation for

3 a particle in the potential V 1 (x starting at x = 0 with zero momentum. We note that the factor of that multiplies the time derivative in the Schrödinger equation just indicates a choice of time scale. The time scale we have chosen is the one that yields a non-trivial classical limit for the nuclear motion, even in the absence of the laser pulse. For simplicity, we start in the ground state of the molecule. We denote the ground electron state by Φ 0 (x and the excited electron state by Φ 1 (x. For i, j = 0, 1, we let G i,j (x = Φ i (x, g(x Φ j (x, where the inner product is in the Hilbert space for the electrons. The factor of ɛ in the time dependence of the laser pulse indicates that the laser is tuned to frequency ω in the time scale that is appropriate for the electrons. Throughout much of the paper, we assume ω = γ, so that the laser is tuned to a frequency in resonance with the electrons. This will give rise to resonance Rayleigh and Raman scattering. Rayleigh scattering is a non-trivial interaction of the laser with the molecule for which the final state of the molecule has the same energy as the initial state. We compute the quantum mechanical amplitude for this process to leading order in the parameters µ and ɛ, where both parameters are small, and µ is small compared to. For the simple situation described above, we have the following result: Theorem 1.1. Assume the situation described above and the resonance condtion ω = γ. The quantum mechanical amplitude for Rayleigh scattering starting from the ground state for the molecule is T 0,0 (µ, ɛ = µ 4 ɛ 3 π ω0 G 1,0 (0 β ( µ f(t dt + O ( µ 4 + O. ɛ 8 We take µ 1, so the errors are small compared to the term explicitly presented. For our simple model, Raman scattering corresponds to interactions between the laser and molecule that result in changes of the vibrational state of the molecule. Because of conservation of total energy, this manifests itself in practice by the emission of light of different frequency from the incident light. Because we start in the ground state of the molecule, the emitted light s frequency must necessarily be lower than the initial light frequency. This 3

4 situation is called Stokes Raman scattering. We presume that the same techniques would apply to the anti Stokes case in which the molecule goes from a higher level to a lower level, and the emitted light has higher frequency than the incident light. In our analysis, we just consider the dynamics of the molecule and compute the transition amplitude to the new vibrational state. We denote the transition amplitude from the ground vibrational state to the j th vibrational state by T j,0 (µ, ɛ. In our simple model, our results for this quantity are the following: Theorem 1.. Assume the hypotheses of Theorem 1.1. If j 1 is odd, then T j,0 (µ, ɛ = ( 1 j+1 ( µ + O i ω 0 G 1,0 (0 4 β ( µ 4 + O. ɛ 8 µ ɛ 3 j ( j 1! (j! 1/ e ijω 0s f(s ds If j is even, then T j,0 (µ, ɛ = ( 1 j+ ( µ + O ω0 G 1,0 (0 4 β ( µ 4 + O. ɛ 8 µ ɛ 3 (j! 1/ π [( ] 1/ j! e ijω 0s f(s ds We prove these two theorems in Sections 6. In Section 7 we discuss similar results for an analogous model in which there can be more nuclei and they can move in several dimensions. The ground vibrational state frequencies and the gradient of the potential energy surface V 1 at the origin again enter as the crucial quantities, but to obtain a simple statement of the results, we compute the gradient in a special coordinate system. In Section 8, we consider small relaxations of the resonance condition; i.e., we allow ω γ, but keep ω close to γ. Application of the results of this paper to the Rayleigh and Raman scattering amplitudes for specific systems subjected to a short laser pulse will be published elsewhere in the theoretical chemical physics literature. Nevertheless, it is useful to connect our work here with the traditional approaches used in chemical physics. Description of Raman processes follows one of two pathways. The time independent approach with a classical electromagnetic field was pioneered in 195 by Kramers and Heisenberg [11]; two years later their result was rederived by Dirac for the case of a quantized 4

5 electromagnetic field [4, 5]. The Kramers Heisenberg Dirac (KHD formula expresses the Raman amplitude for transition between the initial (0 and final (j vibrational eigenstates on the ground state surface in terms of all excited state vibrational eigenstates φ n : T j,0 µ n ψ j G 0,1 (x φ n φ n G 1,0 (x ψ 0, (1.1 E 0 + ω E n + i Γ where E 0 and E n are the energies of ψ 0 and φ n, and Γ avoids the singularity at the resonance. The drawback of this approach is the need to know all vibrational eigenstates in the excited electronic level; this is particularly troublesome when there are no bound states in the excited electronic level. Nevertheless, the time independent approach can yield useful predictions about the Raman spectra in complex molecules [1]. The time dependent approaches are more suitable in practice because for the finite time dynamics, the knowledge of all states in (1.1 should not be necessary. The time dependent treatment of the Raman process was initiated by Lee and Heller [1] and Tannor and Heller [14]; for more details we refer the reader to the work by Williams and Imre [15]. The central idea of Heller and co-workers was that in the short time (or semiclassical limit only the Franck Condon region needs to be considered; this allows computation of Raman spectra for molecules with many degrees of freedom, needing only the excited state forces at the ground state equilibrium geometry [10]. Among the technical weaknesses of the existing semiclassical approaches is the lack of explicit treatment of the laser pulse and the lack of rigorous error estimates. Our approach corrects these deficiencies. Note that the short time approximation is not sufficient to help the analysis of the resonance Raman excitation profiles, obtained by varying the incident light frequency. The state-of-the-art treatment of resonance Raman profiles [13, ] expands the excited state wave packet in terms of harmonic vibrational eigenstates on the excited state surface and involves numerical integration of the transition dipole integrals between the ground and excited vibrational states. Without access to the vibrational states of the excited state (for example, if the excited state is repulsive explicit time dependent propagation of the vibronic wave function is required [3]. Our approach can be extended to the computation of resonance Raman profiles by explicit evaluation of the higher order terms in Theorem 1.; these results will be presented elsewhere in the near future. 5

6 The paper is organized as follows: We recall some preliminary molecular quantum mechanics results in Section. In Section 3, we do a formal perturbation expansion in powers of µ. In Section 4, we obtain small ɛ approximations for the coefficients of the perturbation expansion of Section 3. Theorem 1.1 is proved in Section 5. Theorem 1. is proved in Section 6. Higher dimensional analogs are discussed in Section 7, and relaxation of the resonance condition is discussed in Section 8. Preliminaries We shall make use of semiclassical wave packets, the time independent Born Oppenheimer approximation, and time dependent Born Oppenheimer approximation..1 Semiclassical Wave Packets In Sections 1 6, we just deal with n = 1 dimension for the nuclear motion. In one dimension the semiclassical wave packets can be defined as follows [8]: Let > 0, a R, η R, A C, and B C, with Re AB = 1. We define ϕ 0 (A, B,, a, η, x = π 1/4 1/4 A 1/ B (x a exp ( + i A η (x a Here, the choice of square root of A is arbitrary. It will ultimately be determined by initial conditions and continuity in time. Then for j = 1,, 3,..., we define ϕ j (A, B,, a, η, x = j/ (j! 1/ ( A A where H j denotes the j th order Hermite polynomial. j/ ( (x a H j 1/ A With these definitions, for any allowed, fixed A, B, a, and η,. ϕ 0 (A, B,, a, η, x, {ϕ j (A, B,, a, η, x : j = 0, 1,, 3,... } is an orthonormal basis of L (R, dx.. The Time Independent Born Oppenheimer Approximation for the Molecule Under our assumptions, the low lying bound states of the molecule and their energies have expansions in powers of ɛ [7]. 6

7 Since we have assumed h(x to be real symmetric, we may choose the ground state eigenvector Φ 0 (x of h(x to be real, normalized for each x, and to depend smoothly on x. For sufficiently small ɛ, the bottom of the spectrum of H 0 (ɛ = ɛ4 x + h(x is discrete. The eigenvectors of H 0 (ɛ with energies near the bottom of the spectrum are given by ϕ j (A 0, B 0,, 0, 0, x Φ 0 (x + ɛ k j =1,3 c k,j ϕ k (A 0, B 0,, 0, 0, x Φ 0 (x + O (, (.1 where the error term is measured by the Hilbert space norm. The matrix B 0 is the positive fourth root of V 0 (0, and A 0 = B0 1. The corresponding bound state energies are ( E j (ɛ = j + 1 ω 0 + O ( ɛ 4, where ω 0 = V 0 (0..3 The Time Dependent Born Oppenheimer Approximation for the Molecule The states of the molecule that shall concern us have asymptotic expansions in powers of ɛ [6]. We shall only need these approximations for the upper electronic state V 1, and we shall only need them for specific initial conditions. The initial conditions we shall encounter have the form ψ(ɛ, x, 0 = ϕ j (A 0, B 0,, 0, 0, x Φ 1 (x + ɛ k j =1,3 + O (. c k,j ϕ k (A 0, B 0,, 0, 0, x Φ 1 (x 7

8 Let (A(t, B(t, a(t, η(t, S(t be the unique solution to the system ȧ(t = η(t, η(t = V 1 (a(t, A(t = i B(t, Ḃ(t = i V ( 1 (a(t A(t, Ṡ(t = η(t V 1 (a(t, subject to the initial conditions a(0 = 0, η(0 = 0, A(0 = A 0, B(0 = B 0, and S(0 = 0. The time dependent Schrödinger equation i ψ t = H 0 (ɛ ψ has a solution of the form ( ψ(ɛ, x, t = e is(t/ɛ ψ0 (ɛ, x, t + ɛ ψ 1 (ɛ, x, t + ψ (ɛ, x, t + ɛ 3 ψ 3 (ɛ, x, t + O (. Here, j = 0, 1,,... is fixed, and and ψ 1 (ɛ, x, t = ψ 0 (ɛ, x, t = ϕ j (A(t, B(t,, a(t, η(t, x Φ 1 (x, k j =1,3 ψ (ɛ, x, t = i η(t Φ 0(a(t, ( x Φ 1 (a(t V 1 (a(t V 0 (a(t c k,j (t ϕ k (A(t, B(t,, a(t, η(t, x Φ 1 (x, ϕ j (A(t, B(t,, a(t, η(t, x Φ 0 (x, ψ 3 (ɛ, x, t = i η(t Φ 0(a(t, ( x Φ 1 (a(t V 1 (a(t V 0 (a(t c k,j (t ϕ k (A(t, B(t,, a(t, η(t, x Φ 0 (x k j =1,3 The terms ψ and ψ 3 are required for the proof of this result, but we shall drop them since they can be included in the error term. Thus, the approximation we shall use is ( ψ(ɛ, x, t = e is(t/ɛ ψ0 (ɛ, x, t + ɛ ψ 1 (ɛ, x, t + O (. 8

9 3 The Formal Perturbation Expansion in Powers of µ Our model contains no spontaneous absorption of emission, so there are no transitions before time or after time T. Thus, we shall confine attention to the time interval [, T ]. We do a formal perturbation expansion in powers of µ. ψ(ɛ, x, t = ψ 0 (ɛ, x, t + µ ψ 1 (ɛ, x, t + µ ψ (ɛ, x, t +. If f and g are bounded, and f has compact support in [, T ], this series is convergent for our simple model. (See below. Although we assume g is bounded, one might wish to allow g(x to grow like a constant times x. In that case, depending on the circumstances, the perturbation series in µ may only be asymptotic, rather than convergent. We substitute this expansion into the Schrödinger equation and equate terms of the same order in µ. Order 0 in µ ψ 0 (ɛ, x, t = e i(t+t H 0(ɛ/ ψ 0 (ɛ, x, This shows that we could take ψ 0 (ɛ, x, t to be any solution to the Schrödinger equation for the molecule alone. We shall ultimately choose ψ 0 (ɛ, x, t to be the ground vibrational state of the ground electronic state for the molecule. Order 1 in µ ψ 1 (ɛ, x, t = i t f(s cos e i(t sh 0(ɛ/ g(x ψ 0 (ɛ, x, s ds. Order in µ ψ (ɛ, x, t = i t f(s cos e i(t sh 0(ɛ/ g(x ψ 1 (ɛ, x, s ds (3.1 = 1 ɛ 4 t ds s dr f(s f(r cos ( ω r cos e i(t sh 0(ɛ/ g(x e i(s rh 0(ɛ/ g(x ψ 0 (ɛ, x, r. 9

10 Order N in µ ψ N (ɛ, x, t = i t f(s cos e i(t sh 0(ɛ/ g(x ψ N 1 (ɛ, x, s ds. By using this expression iteratively, it is easy to see that ψ N (ɛ, x, t C ( T N ɛ N ( ( N N max g(x max N! f(t, x t [, T ] where C = ψ 0 (ɛ, x, t. By the ratio test, this guarantees a positive radius of convergence for the perturbation series in µ for each ɛ > 0. Furthermore, one can show that the perturbation series converges to a solution of the Schrödinger equation. This estimate also allows us to estimate errors if we truncate the series after some number of terms. To see this, let a n be a sequence that satisfies a n C bn n!. Then, N b n a n a n C n! n=0 n=0 n=n+1 C bn+1 (N + 1! n=n+1 b (n N 1 (n N 1! = C e b b N+1 (N + 1!. For the perturbation series, we can take b to be a constant multiple of ( µ N+1 the series after N terms yields an O error. N+ µ, so truncating 4 Approximate Computation of the ψ m (ɛ, x, t Let Φ 0 (x denote the ground state of h(x. We can choose this eigenfunction to be real, normalized in C for each x, and to depend smoothly on x. We assume that prior to time, the molecule is in its ground electron state and ground vibrational state. Making a particular choice of phase, we have ψ 0 (ɛ, x, = ϕ 0 (A 0 e iω0t, B 0 e iω0t,, 0, 0, x Φ 0 (x + ɛ c k,0 ϕ k (A 0 e iω0t, B 0 e iω0t,, 0, 0, x Φ 0 (x k=1,3 + O (. 10

11 Here, B 0 is the strictly positive fourth root of V 0 (0 and A 0 = B 1 0, so A 0 = ω 1/ 0 and B 0 = ω 1/ 0. The molecule s energy in this state is E 0 (ɛ = 1 ɛ ω 0 + O ( ɛ 4. As a result of these considerations, for t [, T ], ψ 0 (ɛ, x, t = ϕ 0 (A 0 e iω 0t, B 0 e iω 0t,, 0, 0, x Φ 0 (x + ɛ k=1,3 + O ( = e iω 0t/ ( c k,0 ϕ k (A 0 e iω 0t, B 0 e iω 0t,, 0, 0, x Φ 0 (x ϕ 0 (A 0, B 0,, 0, 0, x Φ 0 (x + ɛ k=1,3 c k,0 ϕ k (A 0, B 0,, 0, 0, x Φ 0 (x + O (. = i = i We next compute the first order term in µ: ψ 1 (ɛ, x, t i ɛ i ɛ t t t i t f(s cos e i(t sh 0(ɛ/ f(s cos f(s cos t e i(t sh 0(ɛ/ g(x ψ 0 (ɛ, x, s ds e i(t sh 0(ɛ/ G 0,0 (x G 0,0 (x ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x Φ 0 (x ds k=1,3 c k,0 ϕ k (A 0 (s, B 0 (s,, 0, 0, xφ 0 (x ds f(s cos e i(t sh 0(ɛ/ G 1,0 (x ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x Φ 1 (x ds f(s cos e i(t sh 0(ɛ/ G 1,0 (x k=1,3 c k,0 ϕ k (A 0 (s, B 0 (s,, 0, 0, xφ 1 (x ds + O ( ɛ 0. (4.1 Here, A 0 (s = A 0 e iω 0s, and B 0 (s = B 0 e iω 0s. 11

12 In the first of these four integrals, we can expand G 0,0 (x ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x = G 0,0 (0 ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x + (G 0,0(0 x ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x = + O ( l=0,1 c l,0 e iω 0s/ ϕ l (A 0, B 0,, 0, 0, x + O (. We then do an integration by parts to see that the first integral in (4.1 is O (ɛ 0. Similarly, the second integral in (4.1 is O (ɛ 1. Although we shall not need the result, we remark that for s T, one can expand to arbitrarily high order and do several integrations by parts to see that the first two terms are O ( ɛ L for any L if s T. From Taylor series and explicit calculations with the wavepackets, G 1,0 (x ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x = G 1,0 (0 ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x + (G 1,0(0 x ϕ 0 (A 0 (s, B 0 (s,, 0, 0, x + O ( = G 1,0 (0 e iω 0s/ ϕ 0 (A 0, B 0,, 0, 0, x + ɛ d (1 1 e iω 0s/ ϕ 1 (A 0, B 0,, 0, 0, x + O (. We use the final expression here in the third integral of (4.1. We use the analogous calculation with zeroth order Taylor series in the final integral of (4.1. Thus, the final two integrals in (4.1 thus can be written as i i ɛ t t + O ( ɛ 0. f(s cos e i(t sh 0(ɛ/ G 1,0 (0 e iω0s/ ϕ 0 (A 0, B 0,, 0, 0, x Φ 1 (x ds f(s cos e i(t sh 0(ɛ/ e iω 0s/ k=1,3 d k ϕ k (A 0, B 0,, 0, 0, x Φ 1 (x ds 1

13 We decompose ψ 1 (ɛ, x, t = ψ 0 1(ɛ, x, t + ψ 1 1(ɛ, x, t, (4. where ψ l 1(ɛ, x, t is the component in the direction of Φ l (x. The estimates above show ψ 0 1(ɛ,, t = O ( ɛ 0. and ψ 1 1(ɛ, x, t = i i ɛ t t f(s cos e i(t sh 0(ɛ/ G 1,0 (0 e iω0s/ ϕ 0 (A 0, B 0,, 0, 0, x Φ 1 (x ds f(s cos e i(t sh 0(ɛ/ e iω 0s/ k=1,3 d k ϕ k (A 0, B 0,, 0, 0, x Φ 1 (x ds + O ( ɛ 0. (4.3 In the next section we shall use these expressions to compute the inner products that involve ψ (ɛ, x, t. 5 The Rayleigh Scattering Amplitude Because of energy conservation, the probability for Rayleigh Scattering is the same as the probability for the system to make a transition from the ground electron state to the excited electron state and back down again, with the final vibrational state the same as the initial vibrational state. This probability is the absolute square of the transition amplitude. If the system starts and ends in the m th vibrational state, we denote the amplitude by T m,m (µ, ɛ. For small µ, the leading order contribution to this amplitude when m = 0 is T 0,0 (µ, ɛ = µ ψ 0 (ɛ, x, T, ψ (ɛ, x, T + O(µ 3, since no transitions occur after time T. We now compute this quantity to leading order for small µ and small ɛ, subject to the restriction µ. 13

14 We use the expression (3.1 to write this inner product in this expression as ψ 0 (ɛ, x, T, ψ (ɛ, x, T = i = i T T f(s cos ψ 0 (ɛ, x, T, e i(t sh 0(ɛ/ g(x ψ 1 (ɛ, x, s ds f(s cos g(x e i(t sh 0(ɛ/ ψ 0 (ɛ, x, T, ψ 1 (ɛ, x, s ds = i T f(s cos g(x ψ 0 (ɛ, x, s, ψ 1 (ɛ, x, s ds In the final expression here, we substitute (.1 for the ψ 0 and (4. and (4.3 for the ψ 1. We can then put some norms inside the integrals to do very simple estimates to rewrite the resulting expression as G 1,0(0 ɛ 4 G 1,0(0 ɛ 3 T T ds ds k=1,3 s s dr f(s f(r cos ( ω r cos e i(s rω 0/ ϕ 0 (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x dr f(s f(r cos ( ω r cos e i(s rω 0/ c k ϕ k (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x G 1,0(0 ɛ 3 + O ( ɛ T ds s dr f(s f(r cos ( ω r cos ϕ 0 (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ k=1,3 e i(s rω 0/ d k ϕ k (A 0, B 0,, 0, 0, x (5.1 In this expression, H 1 0(ɛ = ɛ4 x + V 1(x, and the inner products are in L (R, dx. 14

15 Proposition 5.1. The first term in the expression (5.1 equals 1 4 ɛ 3 π ω0 G 1,0 (0 β The other terms are all O(ɛ. f(t dt + O ( ɛ. ( ω r Proof. In the first term in (5.1, we can replace cos by 1 ( e iωr/ɛ + e iωr/ɛ. In the resulting expression, we can integrate by parts in the r integral on the term that contains e iωr/ɛ. This integration by parts yields a factor of from the integration of e iωr/ɛ. Consequently, the first term in (5.1 equals G 1,0(0 ɛ 4 + O ( ɛ. T ds s dr f(s f(r cos e iωr/ɛ e i(s rω 0/ ϕ 0 (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x Interchanging the order of integration, we next rewrite this expression as G 1,0(0 T T dr ds f(s f(r cos e iωr/ɛ e i(s rω 0/ ɛ 4 r ϕ 0 (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x + O ( ɛ and replace cos by 1 ( e iωs/ɛ + e iωs/ɛ. We then do an integration by parts on the term that contains e iωs/ɛ. This integration by parts yields a factor of from the integration of e iωs/ɛ. So, we see that the first term in (5.1 equals G 1,0(0 4 ɛ 4 T ds s dr f(s f(r e iω(s r/ɛ e i(s rω 0/ ϕ 0 (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x + O ( ɛ. (5. Physically, the reason we can drop three of the four complex exponentials that make up the product of cosines when computing the leading order result is that the electron state first can only make a transition up and then later can only make a transition down. The terms 15

16 we have dropped correspond to going down then up, down then down, and up then up. In our model, such transitions are not possible. The only term that is physically reasonable is the one that first makes a transition up and then makes a transition down. Letting τ = s r, we note that expression (5. contains e iτh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x. By standard semiclassical results [8], this expression can be written as e is(τ/ɛ ϕ 0 (A(τ, B(τ,, a(τ, η(τ, x + ɛ e is(τ/ɛ α k (τ ϕ k (A(τ, B(τ,, a(τ, η(τ, x k=1,3 + O (. (5.3 Here, ȧ(τ = η(τ, η(τ = V 1(a(τ, Ȧ(τ = i B(τ, Ḃ(τ = i V 1 (a(τ A(τ, Ṡ(τ = η(τ V (a(τ, with initial conditions a(0 = 0, η(0 = 0, A(0 = A 0, B(0 = B 0, and S(0 = 0. The solution to this system of ordinary differential equations satisfies a(τ = β τ / + O ( τ 3, η(τ = β τ + O ( τ, A(τ = A 0 + i B 0 τ + O ( τ, B(τ = B 0 + i V 1 (0 A 0 τ + O ( τ, S(τ = β τ 3 /6 ω τ + β τ 3 /6 + O ( τ 4 = ω τ + β τ 3 /6 + O ( τ 4. 16

17 Also, by our assumptions on T a(τ is only near zero when τ is small. being less than a period of oscillation in the upper level, The first term from (5.3 gives rise in (5. to the inner product ϕ 0 (A 0, B 0,, 0, 0, x, e is(τ/ɛ ϕ 0 (A(τ, B(τ,, a(τ, η(τ, x. From Proposition 4 of [9] and the asymptotics above, this equals ( ( ( i ω τ/ɛ e exp A 0 β τ τ O(τ + O. 4 The factor of e i ω τ/ɛ here cancels with the e i ω (s r/ɛ in (5.. Intuitively, on the whole real line, the other factors ( ( ( ζ(τ, ɛ = exp A 0 β τ τ O(τ + O. 4 are an approximation to a multiple of a Dirac delta. Specifically, for τ on the whole real line in the sense of distributions, as ɛ tends to zero, ζ(τ, ɛ approaches ɛ π A 0 β δ(τ. Note that to leading order, ζ(τ, ɛ is an even function of τ, so for τ [0,, we obtain a factor of one half times this Dirac delta. We also note that 0 τ ζ(τ, ɛ dτ = O (. (5.4 To make a direct, rigorous argument, we split the integral s ɛ 3/4 s s dr in (5. into dr + dr. The first integral is O ( ɛ L for any L. When substituted into (5.1, s ɛ 3/4 the second integral gives rise to = + s s ɛ 3/4 s s ɛ 3/4 s s ɛ 3/4 dr f(r e i(s rω0/ ζ(s r, ɛ dr dr f(s ζ(s r, ɛ dr dr ξ(s, r (s r ζ(s r, ɛ dr, 17

18 where ξ(s, r is uniformly bounded. Because of (5.4, the second term here is O (. The first term gives us a contribution to (5.1 of 1 π ω0 G 1,0 (0 f(t dt 4 ɛ 3 β + O ( ɛ. By similar analyses, the other terms from (5.3 yield higher order corrections. We apply all the same techniques to the second and third integrals in (5.1. The result is that they are both O (ɛ. This proves the theorem. The proposition immediately implies Theorem 1.1 with an error term of O ( µ 3 ( ( µ µ 4 + O + O. However, the first of these error terms is dominated by the second since µ 1. This completes the proof of Theorem 1.1. ɛ 8 6 The Raman Scattering Amplitudes Our analysis of the Raman scattering amplitudes follows exactly the same strategy. The only difference is that the Raman amplitude T j,0 (µ, ɛ is the inner product with the j th vibrational state of the molecule instead of the ground state. Thus, T j,0 (µ, ɛ = µ ( ϕ j (A 0 (T, B 0 (T,, 0, 0, x + O(ɛ Φ 0 (x, ψ (ɛ, x, T + O ( µ 3, where j 1. The Rayleigh scattering amplitude looked the same, except that the ϕ j was a ϕ 0. Since the proof strategy is the same, but just some formulas are different, we shall just describe the leading order formal calculations. The leading order contribution to the analog of (5.1 is G 1,0(0 ɛ 4 T ds s dr f(s f(r cos ( ω r cos e is(j+1/ω 0 e irω 0/ ϕ j (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x Mimicking the Rayleigh case, we replace cos ( ω r and then cos by their expressions in terms of complex exponentials. The integration by parts arguments show that to leading 18

19 order for small ɛ, we can again ignore three of the four terms that arise. We are thus left with the following analog of (5., G 1,0(0 4 ɛ 4 + O ( ɛ. T ds s dr f(s f(r e iω(s r/ɛ e is(j+1/ω 0 e irω 0/ ϕ j (A 0, B 0,, 0, 0, x, e i(s rh1 0 (ɛ/ɛ ϕ 0 (A 0, B 0,, 0, 0, x The analog of (5.3 is unchanged since it just involves the propagation of the ϕ 0. However, the inner product we now must consider is ϕ j (A 0, B 0,, 0, 0, x, e is(τ/ɛ ϕ 0 (A(τ, B(τ,, a(τ, η(τ, x. We employ Propositions 4 and 7 of [9] and take a limit of the conclusion of Proposition 7 of [9] to circumvent zeros in denominators. We conclude that this inner product equals ( j ( ( i β e i ω τ/ɛ (j! 1/ A0 τ exp A 0 β τ 1 + O(τ + O ɛ 4 ( τ 3. The factor of e i ω τ/ɛ again cancels with the e iω(s r/ɛ and leaves an extra factor of e ijω 0s in the double integral above. In the region where s > r, ( this inner product thus leads to a complicated j dependent factor times a Dirac delta in s r ɛ jω 0 /. The explicit β expression for the j dependent factor depends on whether j is odd or even. It is ( 1 j 1 i ( j j 1! (j! 1/ ɛ A 0 β if j is odd, and ( 1 j (j! 1/ π [( ɛ ] 1/ j if j is even.! A 0 β This leads immediately to the conclusions of Theorem 1. as the leading order behavior after we again note that O(µ 3 errors are dominated by O(µ / errors. Remarks 1. We note that the Dirac delta has a j dependent time delay that is small because it contains a factor of ɛ. Physically, this delay is reasonable because when the electrons are in the upper state, the nuclear wave function must have time to move before the 19

20 maximum overlap occurs with the j th nuclear vibrational state in the lower electronic level.. The factor of e ijsω 0 in the final time integration is also completely reasonable. The absorption is resonant, but the emission is not because the system is going to the j th level. The extra phase factor arises because of the energy shift. 3. The formula for the complicated j dependent factor looks quite different if j is odd or even, but it is an increasing function of j. One can obtain a single formula for all j by writing the given expressions in terms of gamma functions with arguments that are not necessarily integers. 7 More Degrees of Freedom for the Nuclei In this section we shall simply state our results. The method of proof is the same as in the earlier sections, but is significantly messier because several quantities are vectors instead of scalars. Also, we shall state our results in terms of some quantities that are computed after changes of variables. The transformation back to the original variables is, in principle, trivial, but again, very messy. If the molecule has N nuclei moving in n dimensions, then after the removal of the center of mass motion, the Hamiltonian for the molecule can always be written as d l=1 ɛ 4 m l z l + h 1 (z, where d = (N 1n, z R d, and the m l are some positive numbers. We define y l = m l z l and rewrite this in the new variables as ɛ4 d l=1 yl + h (y. We assume h ( is a smooth matrix valued function with eigenvalues V 0 (y < V 1 (y. We further assume V 0 has a global minimum at the origin, and that V 0 (0 = 0. We assume the Hessian V ( 0 (0 to be strictly positive definite with eigenvalues ω 1, ω,..., ω d. By 0

21 performing a rotation to diagonalize V ( 0 (0 and then a rescaling the l th resulting variable by 1/ ω l, we define a new variable x R d, so that the Hamiltonian becomes ( d ɛ4 ω l + h x 3 (x. l l=1 With an abuse of notation, we denote the eigenvalues of h 3 (x by V 0 (x < V 1 (x. In the x variable, we have V 0 (0 = 0, ( V 0 (0 = 0, and the Hessian matrix V ( 0 (0 equal to the diagonal matrix whose l th diagonal entry is ω l. The benefit of this choice of variables is that the in these variables, the ground electron state Hamiltonian for the nuclei, H 0 0(ɛ, has quadratic approximation near x = 0 equal to d l=1 ω l ( ɛ4 x l + x l We define γ = V 1 (0 and β = ( x V 1 (0. The tilde is to indicate the gradient with respect to x, which is different from the gradient β in the original variables. We assume γ > 0 and that β is a non-zero vector.. The Schrödinger equation we wish to study is i ψ ( ( ω t G0,0 (x G = H 0 (ɛ ψ + µ f(t cos 0,1 (x t G 1,0 (x G 1,1 (x We once ( again assume that the G i,j (x are bounded and that the matrix G0,0 (x G g(x = 0,1 (x is self-adjoint. G 1,0 (x G 1,1 (x The low-lying eigenvalues of H 0 (ɛ are d (j l + 1/ ω l + O ( ɛ 4, l=1 ψ. where j = (j 1, j,..., j d is a multi-index. The corresponding eigenfunctions are ϕ j (I, I,, 0, 0, x + O(ɛ, where I denotes the d d identity matrix and the 0 s denote d component zero vectors. We let j 0 denote the zero multi-index and compute the Rayleigh and Raman scattering amplitudes starting from the ground state of the molecule. The Rayleigh scattering amplitude is T j0,j 0 (µ, ɛ and the Raman scattering amplitude to level j is T j,j0 (µ, ɛ. 1

22 Although the analysis is significantly messier, we simply mimic the one dimensional case. To state the results, we use the notation ω 0 = ω 1 ω. ω d. Theorem 7.1. The resonance Rayleigh scattering amplitude is T j0,j 0 (µ, ɛ = µ π G1,0 (0 ( µ 4 ɛ 3 f(s ds + O β ( µ 4 + O. ɛ 8 Remark The formula in this theorem looks slightly different from the one in Theorem 1.1 because it contains β instead of β and it does not contain a factor of ω 0. However, if one makes the change of variables in the one dimensional case that we have used in this section, one sees that β = β/ ω 0, and that the two formulas produce identical results. The expressions for the Raman scattering amplitudes depend on whether j = l j l is odd or even. Theorem 7.. When j 1 is odd, the resonance Raman scattering amplitude is j! j 1 j T j,j0 (µ, ɛ = µ G 4 ɛ 3 1,0 (0 ( 1 i β β j +1 ( µ + O ( µ 4 + O. ɛ 8 ( j 1 (j! 1/ When j is even, the resonance Raman scattering amplitude is T j,j0 (µ, ɛ = µ 4 ɛ G 1,0(0 ( 1 j + j π β j! 3 (( β j +1 1 (j! 1/ j! ( µ + O ( µ 4 + O. ɛ 8 f(s e is j ω 0 f(s e is j ω 0 ds ds As usual, in these expressions, β j means β j = d l=1 β l j l,

23 and all the results are for small µ and ɛ that satisfy µ 1. Remark For some values of j, the expressions given in Theorem 7. may be zero because we might have β j = 0. In those cases, the true leading order term is of higher order in ɛ. However, for each value of j, there is at least one value of j for which the theorem gives a non-zero leading order term since we have assumed β 0. 8 Results for a Slightly Off Resonance Incident Pulse In this section we again restrict attention to the 1 dimensional model. In the earlier sections we assumed ω = γ. To obtain a rigorous proof of the results for this section, we assume ω = γ + O(ɛ p, for some p > 1. (8.1 We primarily concentrate on the Rayleigh scattering amplitude, but present some Raman scattering results. Under assumption (8.1, the integration by parts argument of Section 4 guarantees that the corresponding term is still the leading term for small ɛ. This uses the assumption p > 1 so that the terms that are dropped cannot be larger than the one that is kept. Everything else is essentially the same, except that a different time intergral appears near the end of the argument from Section 5. Near the end of the calculation for the Rayleigh amplitude, one encounters the double integral with respect to r and s with integrand f(s f(r exp ( i (ω γ(s r/ɛ ( exp β (s r 4 instead of ( f(s f(r exp β (s r. 4 This gives rise to a different factor times the Dirac delta. The new factor is ɛ ( ( ( π i (ω γ (ω γ 1 + erf exp. β ɛ β β instead of simply ɛ π β. 3

24 This leads to the following expression for the leading order Rayleigh scattering amplitude T 0,0 (µ, ɛ when µ 1 : µ 4 ɛ 3 π ω0 G 1,0 (0 β ( ( ( i(ω γ 1 + erf exp ɛ β (ω γ β f(t dt. This formula gives good agreement with the simulations, as seen in the following plot. The value of γ is 1, and the horizontal axis is the variable ω. (Blue shows the simulation for T 0,0 ; red shows the result from the formula above. The value of ɛ was 0.05, so the range of values of ω may be somewhat larger than the set of values for which the proof is appropriate Figure 1: Comparison plot for T 0,0. The analysis for the T j,0 (ɛ, µ follows the same reasoning. For example, for j = 1, the ( time integral of β (s r exp ( i (ω γ (s r/ exp β (s r gives rise to 4 + i ɛ (ω γ ( ( i (ω γ π 1 + erf exp ( β β β ɛ ( times an approximate Dirac delta in s r in the region where s > r. ɛ β (ω γ β The following is a plot that compares the numerical values for T 1,0 with the corresponding formula we obtain (with colors as in the plot above. Again, ɛ = 0.05, γ = 1, and the horizontal axis shows values of ω. The range of ω s shown is likely greater than the range for which the proof applies. 4

25 Figure : Comparison plot for T 1,0. 5

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