Spectral projection approach to the quantum scattering calculations

Abstract

A new method of implementing scattering calculations is presented. For the S-matrix computation it produces a complete set of solutions of the wave equation that need be valid only inside the interaction region. For problems with small sizes the method is one of several that are practical in the sense that it involves merely a real symmetric Hamiltonian represented in a minimal ℒ2 basis set. For more challenging larger systems it lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. A modified Chebyshev polynomial expansion of (E−Ĥ)−1 is used. This acts on a set of energy independent wave packets located on the edge of the interaction region. The procedure requires minimal storage and is shown to converge rapidly in a manner that is uniform in energy.

Extraction of state-to-state reactive scattering attributes from wave packet in reactant Jacobi coordinates

“…In the CRWP approach, the wave packet is propagated using the following modified Chebyshev recursion relation: 45,46 …”

Chebyshev real wave packet propagation: H+O2 (J=0) state-to-state reactive scattering calculations

“…⌽(0) is a real initial wave packet, and ␥ is a damping operator. Equation ͑2͒ is just the damped Chebyshev propagation, which was developed independently by Mandelshtam et al [8][9][10] Of course, ⌰ ϭ (1/ )sin Ϫ1 (Ĥ norm ) can be used to propagate only the imaginary part of the wave packet, leading to a slightly different propagation. 45 The evolved wave packet ⌽Ј(t) is clearly different from ⌽(t) , even if we choose the same initial wave packet.…”

“…Kouri and co-workers [2][3][4][5][6][7] derived a new, time-independent ͑TI͒ wave packetLippmann-Schwinger equation and presented Chebyshev expansion expressions for both the Green operator and Dirac delta function. Mandelshtam and Taylor [8][9][10] introduced a real damping scheme into the Chebyshev recursion which made the real wave packet method possible for dissipative systems. The real Chebyshev propagation method can be viewed in an alternative way as a modification of the time-dependent Schrödinger equation.…”

Efficient time-independent wave packet scattering calculations within a Lanczos subspace: H+O2 (J=0) state-to-state reaction probabilities

“…Kouri and co-workers 3-5 derived a timeindependent ͑TI͒ wave packet-Lippmann-Schwinger equation and presented a Chebyshev expansion expression of the causal Green's operator. Mandelshtam and Taylor [6][7][8] introduced a very efficient scheme for the representation of a time ͑energy͒-dependent wave packet under dissipative boundary conditions in terms of a polynomial expansion with a real damped Chebyshev recursion. This has allowed very significant computational advances to be made.…”