J. Phys. B: At. Mol. Opt. Phys. 31 (1998) L891 L897. Printed in the UK PII: S953-475(98)9711- LETTER TO THE EDITOR Wavepacket evolution in an unbound system Xiangyang Wang and Dan Dill Department of Chemistry, Boston University, 59 Commonwealth Avenue, Boston, MA 2215, USA Received 26 August 1998 Abstract. We report a method to efficiently describe wavepacket evolution in an unbound system for arbitrary large times. The problem is difficult using current methods due to the spatial expansion of wavepacket with time. Our approach circumvents this difficulty by solving the time development operator in a small interaction region outside of which the wavepacket evolution is described analytically. The formulation is illustrated by comparing numerical results of wavepacket propagation in a Morse potential with results using the pseudo-spectral Chebyshev expansion method. The evolution of wavepacket states has been studied extensively both theoretically and experimentally (e.g. Mallalieu and Stroud 1995, Cao and Wilson 1997, Bardeen et al 1995, Stapelfeldt et al 1995). It not only provides a uniform model for time-dependent processes such as photon atom interaction (e.g. Buchleitner and Delande 1995, Burke and Burke 1997), but also probes fundamental principles of the quantum world through wavepacket interference and entanglement (e.g. Monroe et al 1996, Noel and Stroud 1995). For a system involving unbound states, such as in electron molecule scattering and molecular photoionization, the experiment occurs in a rather large volume relative to the size of the molecules involved and so the particle can propagate and dissipate to virtually infinite distance during a long time period. To predict the time-dependent result of such experiments, some methods use a bound-system approximation by building a box which is large enough for all physics of interest to be confined within the box, during the interaction time (Zhang and Lambropoulos 1996), while other methods introduce an imaginary absorbing potential (e.g. Grozdanov and McCarrol 1996). Here we instead partition the molecular space into two regions: an interior interaction region (i) and a decomposition region (d), as in the time-independent R-matrix theory (e.g. Aymar et al 1996) and multiple-scattering theory (Dill and Dehmer 1974). The two regions are separated by a spherical surface of radius R. The potential is classified into two parts, a short-range potential V i and a long-range potential V. The radius R is chosen to make V i zero in the decomposition region. The long-range potential V is usually zero for short-range scattering problems and is a Coulomb potential for photoionization problems. The time-development operator U(t,) describes the evolution of the wavepacket ψ(t) = U(t,)ψ() (1) and it satisfies the time-dependent Schrödinger equation U(t,) = H U(t, ) (2) t 953-475/98/21891+7$19.5 c 1998 IOP Publishing Ltd L891
L892 where H is the Hamiltonian of the system H = H + V i (3) H = T + V. (4) First we find the solution U (t, ) for a special case V i =, t U (t, ) = H U (t, ). (5) For some analytical potentials V, this solution is available in path-integral form. Next we write U(t,) = U (t, )U I (t) (6) and substitute this expression into equation (2) to obtain t U I(t) = U 1 (t, )V i U(t,). (7) Integration of this equation yields U I (t) = 1 + 1 U 1 (s, )V i U(s,)ds. (8) Using this result in equation (6) we obtain an integral equation for U(t,), U(t,) = U (t, ) + 1 U (t,s)v i U(s,)ds (9) where we have used the relation U (t, )U 1 (s, ) = U (t, s). (1) Our goal is to exploit the fact that V i vanishes outside the interaction region. First we rewrite equation (9) in matrix form by partitioning the basis set spanning the whole space into two subsets x x = x i x i + x d x d =1 (11) x i x d x where x i denotes a point in the interaction region and x d denotes a point in the decomposition region. With this partitioning we get { } U U(t,) = ii (t, ) U id (t, ) U di (t, ) U dd (12) (t, ) { U ii id } (t, s) U (t, s) U (t, s) = U di (13) dd (t, s) U (t, s) { } V ii V i = i. (14) Substituting these expressions in equation (9), we obtain four integral equations. Because we have defined the interaction potential so that it vanishes outside the interaction region, the first of these equations, U ii (t, ) = U ii 1 (t, ) + U ii (t, s)v ii i U ii (s, ) ds (15)
L893 has the same form as the general equation (9), but depends only on interaction-region quantities. This is the key feature of our approach, for it allows us to work in a limited region of space. The second equation, U di (t, ) = U di 1 (t, ) + U di (t,s)vii i U ii (s, ) ds (16) determines the escape from the interaction region into the decomposition region. We can solve this equation once we know the solution to equation (15). In the example described below, the wavepacket is initially localized in the interaction region and so these two equations are sufficient to describe its evolution. The remaining two equations, U id (t, ) = U id 1 (t, ) + U dd (t, ) = U dd 1 (t, ) + U ii (t,s)vii i U id (s, ) ds (17) U di (t,s)vii i U id (s, ) ds (18) describe propagation from the decomposition region into the interaction region. These equations thus pertain to scattering processes, and we will discuss their application elsewhere. In the following discussion we will first discuss the solution of equation (15). We will drop all the superscripts and assume it is understood that all the operators below are actually their respective projected operators in the interaction region. Generally speaking it is difficult to solve integral equations such as equation (15) either directly or using recursive methods. In the case that both the long-range potential V and short-range potential V i are time independent, we have, U (t, s) = U (t s, ) (19) U(t,) = U (t, ) + 1 U (t s, )V i U(s,)ds (2) this equation can be simplified by applying the Laplace transform technique and the Faltung theorem f(s)= exp( st) f (t) dt (21) { } L f(t s)g(s)ds = f g (22) where f denotes the Laplace transform of function f. Applying the Laplace transform to both sides of equation (15), we have Ũ(s) = U (s) + 1 U (s) V i Ũ(s) (23) { Ũ(s) = U 1 (s) V i } 1. (24) The Laplace transform of U(t,) can be found easily from this equation. Consequently, U(t,) can be calculated from Ũ(s) using an inverse Laplace transform U(t,) = L 1{ Ũ(s) }. (25) As an example, we propagate a wavepacket state in a Morse potential (all parameters are listed in table 1). The initial state is chosen to be the ground state (E = 26.252 au) with a group velocity k = 6. au, ψ(x, t = ) = (x) exp(ikx), (26)
L894 Table 1. Parameters for Morse potential, calculation using our theory and Chebyshev expansion method (atomic unit). Morse potential This work Chebyshev expansion method α = 1. R = 15. R = 15. D = 3. basis = 75 points = 248 r e = 2. truncation error = 1 12 truncation error = 1 12 Figure 1. Morse potential and initial wavepacket. The energy of the wavepacket is 8.252 au. The arrow indicates the initial momentum direction. the minus sign indicates that the wavepacket initially moves toward the negative x-direction (figure 1). Such a state has the same density distribution as the ground state and its energy is { p 2 } E = (x) exp( ikx) 2m + V(x) (x) exp(ikx) dx = E + k 2 /2 p k = 8.252 au. (27)
L895 We choose the radius R of the interaction region to be 15 au, the long-range potential to be { x> V (x) = (28) x and V i (x) to be the Morse potential. The probability for a wavepacket with an energy of 8.252 au to penetrate the half-space x < in this Morse potential is negligible. The time-development operator for V can be found in path-integral form. In the coordinate representation it corresponds to a free-particle propagator with a topological constraint (e.g. Kleinert 199), x 2 U (t, ) x 1 =g (x 2,t;x 1,) = 1 sin(kx 1 ) sin(kx 2 ) exp( ik 2 t/2)dk. (29) 2π Using a function basis is more stable numerically than using the coordinate representation and so we choose the basis functions for the interaction region to be φ k (x) = 2/R sin {( k 1 2) πx/r }. (3) The matrix elements in this basis set are U mn (t, ) = Ũ mn (s) = R For the basis defined in equation (3), we find for n m R φ m (x 2 ) dx 2 φ n (x 1 )g (x 2,t;x 1,)dx 1 (31) U mn (t, ) exp( st)dt. (32) k m k n φ m (R )φ n (R ) Ũ mn (s) = (i + 1) s(km 2 + sr e2 (i 1) } (33) 2is)(k2 n 2is){1 and for n = m where km 2 Ũ mm (s) = φ2 m (R ) (i + 1) {1 + sr s(km 2 e2 (i 1) }+ R2 φ4 m (R ) 2is)2 2i(km 2 2is) (34) k m = ( m 1 2) π/r. (35) The inverse Laplace transform is carried out using Durbin s formula and the ɛ-algorithm is used to accelerate the convergence (Piessens and Huysmans 1984). As a comparison, we also solved the same problem using the pseudo-spectral Chebyshev expansion method (Tal-Ezer and Kosloff 1984, Kosloff 1992). We choose the box length R to be 15 au to cover all the space the particle can reach in a time period of 25 au, and 248 discrete points to acquire a reasonable precision. We analyse the results in terms of the probability of the particle either remaining in the ground state or escaping into the decomposition region. We compute the probability of remaining in the ground state as R 2 P (t) = (x) ψ(x, t) dx. (36)
L896 Figure 2. Absolute value of the wavefunction at t = 25 au for the method of this paper (full curve) and the Chebyshev expansion method (dotted curve). Since the ground state is an eigenstate of H, this probability does not change with time, and we use this property as a test of the numerical precision of our calculation. The probability of escape into the decomposition region is P d (t) = 1 R ψ(x, t) 2 dx. (37) The probability increases with time and describes the time evolution of the escape process. The numerical results clearly indicate that the two independent methods match each other with good precision. It should be noted that for the current problem, our method takes only one third of the CPU time of the Chebyshev expansion method. In theory, the computational effort required for our method is roughly proportional to the evolution time while that of the Chebyshev method is proportional to the evolution time squared. In figure 2 we present the graph of the wavepacket at time 25 au. The results from these two methods are almost identical. The small difference around R = 15. auisdue to the boundary condition of our basis set. In summary, we have shown that the time-development operator for an unbound system can be solved in a small interaction region. We express the operator in terms of the shortrange potential and the propagator of the long-range potential. The formulation is valid
L897 Table 2. Probability of particle in decomposition region (P d ) and particle in the ground state (P ). This work Chebyshev Time (au) P P d P P d..82 215..82 215. 2.5.82 216.38 23.82 215.38 89 5..82 217.164 66.82 215.165 2 7.5.82 216.197 67.82 215.198 1 1..82 215.28 63.82 215.28 69 12.5.82 214.213 33.82 215.213 14 15..82 215.215 56.82 215.215 38 17.5.82 214.216 67.82 215.216 65 2..82 215.217 37.82 215.217 43 22.5.82 215.217 74.82 215.217 95 25..82 215.217 97.82 215.218 29 for both bound and unbound systems. When the potential is time independent, our theory provides an efficient numerical method to describe wavepacket evolution in an unbound potential. We are grateful to the Center of Scientific Computing and Visualization at Boston University where the calculations of this work were carried out. References Aymar M, Greene C H and Luc-Koenig E 1996 Rev. Mod. Phys. 68 115 Bardeen C J, Wang Q and Shank C V 1995 Phys. Rev. Lett. 75 341 Buchleitner A and Delande D 1995 Phys. Rev. Lett. 75 1487 Burke P G and Burke V M 1997 J. Phys. B: At. Mol. Opt. Phys. 3 L383 Cao J and Wilson K 1997 J. Chem. Phys. 17 1441 Dill D and Dehmer J L 1974 J. Chem. Phys. 61 692 Grozdanov T P and McCarrol R 1996 J. Phys. B: At. Mol. Opt. Phys. 29 3373 Kleinert H 199 Path Integral in Quantum Mechanics, Statistics and Polymer Physics (Singapore: World Scientific) p 212 Kosloff R 1992 Time-Dependent Quantum Molecular Dynamics (New York: Plenum) p 97 Mallalieu M and Stroud CRJr1995 Phys. Rev. A 51 1827 Monroe C, Meekhof D M, King B E and Wineland D J 1996 Science 272 1131 Noel M W and Stroud CRJr1995 Phys. Rev. Lett. 75 1252 Piessens R and Huysmans R 1984 ACM Trans. Math. Software 1 348 Stapelfeldt H, Constant E and Corkum P B 1995 Phys. Rev. Lett. 74 378 Tal-Ezer H and Kosloff R 1984 J. Chem. Phys. 81 3967 Zhang J and Lambropoulos P 1996 Phys. Rev. Lett. 77 2186