Infinite Order Discrete Variable Representation for Quantum Scattering

A new approach to multi-dimensional quantum scattering by the infinite order discrete variable representation is presented. Determining the expansion coefficients of the wave function at the asymptotic regions by the solution of the differential Schrödinger equation, we reduce an infinite set of linear equations to a finite one. Application to the benchmark collinear H + H2 → H2 + H reaction is shown to yield precise reaction probabilities. PACS numbers: 34.10.+x, 34.50.-s, 34.50.Pi, 03.65.Nk, 03.80.+r Typeset using REVTEX 1 One of the most common approach to the solution of quantum scattering problem is the application of square-integrable (L) basis functions [1–5]. Of the L basis methods the discrete variable representation(DVR) method [6] is proven to be highly successful [7–9]. The DVR is a grid-point representation in which the potential energy is diagonal and the kinetic energy is a sum of one-dimensional matrices. Hence the Hamiltonian matrix is extremely sparse, which means that iterative linear algebra methods [10] can deal efficiently with extremely large systems. Recently, Eisenberg et. al. [11] has developed an infinite order DVR method for onedimensional quantum reactive scattering problems. They expanded the wave function in terms of an infinite set of L basis functions satisfying the conditions of DVR [6–8,12]. The matrix related to the resulting set of algebraic equations, though infinite, has been shown to have the structure of a Toeplitz matrix [13]. Using the analytical properties of the Toeplitz matrix, they reduced the infinite set of algebraic equations to a finite one and obtained very accurate results for one-dimensional potential scattering. However, their method is not likely to be extended to multi-dimensional systems due to the failure of the Toeplitz properties. In this Letter, we show that the infinite order DVR can be properly converted into a finite DVR even for multi-dimensional reactive systems. Instead of investigating the analytical properties of the Toeplitz matrix, we use the solutions of the differential Schrödinger equations at the asymptotic regions. The full scattering wave function Ψn is decomposed of the incoming distorted wave θ − n and the outgoing wave χ+n , Ψn = −θ n + χ+n , (1) where n is a channel index, i.e., a superindex over the arrangement and rovibrational indices. The distorted wave θ n is any regular scattering solution corresponding to a simple nonreactive (i.e., arrangement conserving) Hamiltonian, H0n, in the asymptotic region in the arrangement channel n: (H0n − E)θ n = 0, (2) 2 while Ψn is the solution of (H − E)Ψn = 0, (3) where H is the full Hamiltonian. For the sake of later convenience, we impose the ”totally incoming” boundary condition on θ n as θ n (r, R) ∼ en √ vn un(r) (4) for large R. Here r denotes all possible internal coordinates of the system, thus excluding the channel radius (relative translation) R; {un(r)}, the channel eigenfunctions; vn the relative velocity for channel n, and kn the corresponding wave vector. The equation for χ + n is then, from (1)-(3), (E −H)χn = (H −E)θ n . (5) χ+n must obey ”totally outgoing” boundary condition, i.e., the only incoming wave part in the full wave function is due to θ n . Using the infinite order uniform DVR [8], (5) is transformed into an infinite set of coupled linear algebraic equations. To do this, we introduce the following convenient sets of DVR basis functions Q,R0,P0, R and P as shown in Fig. 1: Q be the set of N DVR basis functions represented by the N grid points in the reactive (strong) interaction region correspond; R0 = {R0(1), ...,R0(Nr)}, the set of Nr DVR basis functions in the reactant asymptotic (relatively weak interaction) region nearest to the region of reactive interaction; P0 = {P0(1), ...,P0(Np)}, the set of Np DVR basis functions in the product asymptotic regions nearest to the region of reactive interaction; R, the infinite set of DVR basis functions in the reactant asymptotic region except the functions in R0, and P the infinite set of DVR basis functions in the product asymptotic regions except the functions in P0. Here Nr(Np) is the number of open channels in the reactant (product) arrangement. Using these sets, we can rewrite (5) as a set of coupled algebraic equations, AR0R〈R|χ+nr〉+ AR0R0〈R0|χ+nr〉+ AR0Q〈Q|χ+nr〉+ AR0P0〈P0|χ+nr〉+ AR0P〈P|χ+nr〉 = 〈R0|H − E|θ nr〉, (6) 3 AQR〈R|χ+nr〉+ AQR0〈R0|χ+nr〉+ AQQ〈Q|χ+nr〉+ AQP0〈P0|χ+nr〉+ AQP〈P|χ+nr〉 = 〈Q|H −E|θ nr〉, (7) AP0R〈R|χ+nr〉+ AP0R0〈R0|χ+nr〉+ AP0Q〈Q|χ+nr〉+ AP0P0〈P0|χ+nr〉+ AP0P〈P|χ+nr〉 = 〈P0|H − E|θ nr〉, (8) where Aij = (E − Vj)δij − Tij, (9) Tij is the kinetic energy matrix element [8], which is analytically obtained, connecting the DVR grid points i ∈ R0 + Q + P0 and j ∈ R + R0 + Q + P0 + P, Vj is the potential energy at the DVR grid point j, and we omit the summation over the index j such that, e.g., AR0R〈R|χnr〉 means ∑ j∈R+R0+Q+P0+P Aij〈j|χnr〉 for i ∈ R0. In the above we did not write explicitly the similar equations corresponding to the R and P component since we do not use them in order to eliminate 〈R|χnr〉 and 〈P|χnr〉 by searching for an analytical property as in the case of Toeplitz. Instead, for the asymptotic regions, we introduce, 〈P|χ+nr〉 = − Np