An illustrative example molecule-surface scattering

The process we are interested in is the rotational energy transfer during the collision as well as the diffraction of the diatomic from the surface that exhibits a two-dimensional corrugation. Since the diffraction is a quantum effect, we treat the whole system classically except for the directions X and Y parallel to the surface. The separation of the total Hamiltonian into a classical and quantum part reads as follows  

The MQCB equations for this problem are thus given by the Schrddinger equation in the (X, T) subspace for i//(X, Y, qa(f), f) that depends parametrically on the classical variables q (r) = Z(f),0(f),(p(f), the corresponding 

As detailed above, the initial conditions for the MQCB equations need to be calculated based on an initial full-dimensional wavefimction. For simplicity, we consider an initially non-rotating molecule approaching the surface at normal incidence with mean momentum P. Assuming a Gaussian shape for the wavefimction in Z-direction centred around Zq, the full-dimensional initial wavefimction is given by 

These values form the initial starting vector for the MQCB equations. 

From a numerical point of view, equation (40) is integrated with a simple first-order integration method in parallel with the quantum propagation within the low-dimensional Hilbert space. Due to the periodicity of the surface potential, the quantum propagation was performed using a two-dimensional Fourier basis for the X and Y degrees of freedom. 

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