Application of the spectral element method to the solution of the multichannel Schrödinger equation

Abstract

We apply the spectral element method to the determination of scattering and bound states of the multichannel Schrödinger equation. In our approach, the reaction coordinate is discretized on a grid of points whereas the internal coordinates are described by either purely diabatic or locally diabatic (diabatic-by-sector) bases. Bound levels and scattering matrix elements are determined with spectral accuracy using relatively small number of points. The scattering problem is cast as a linear system solved using state-of-the-art sparse matrix non-iterative packages. Boundary conditions can be imposed so as to compute a single column of the matrix solution. A comparison with log-derivative propagators customarily used in molecular physics is performed. The same discretization scheme can also be applied to bound levels that are computed using direct scalable sparse-matrix solvers.

Fitting ultracold resonances without a fit

“…[22]), algorithmic complexity and computational cost both increase with the derivative order. In this work we exploit the pseudospectral representation termed spectral element method, a spatial grid representation presented for instance in [21] and references therein. In the present work we impose asymptotic R-matrix boundary conditions, but one could equally well work with log-derivative or scattering boundary conditions.…”

“…where the constant vector c on the rhs equal to zero everywhere but at last grid point r max , where it equals the unit matrix of dimension the total number of channels [21]. Elements of the Hamiltonian matrix in the spectral element representation can be found in [21]. By definition, the multichannel wavefunction Ψ at r max equals the R-matrix R.…”

“…The K-matrix is ordinarily computed by imposing the asymptotic form of the wavefunction Ψ ∼ f − gK in terms of regular and irregular solutions f and g for the free particle problem. Since in our algorithm K is not needed, we rather determine directly its inverse M by matching to Ψ ∼ f M − g. The linear matching equations for M simply follow from equation ( 25) of [21] and read :…”

Large amplitude motion within acetylene–rare gas complexes hosted in helium droplets

“…C 2 H 2 is described as a pseudo rigid linear rotor with rotational and centrifugal correction constants B and D. This reduces the dimensionality of the problem to two internal coordinates [R,θ ]: R is the distance between the rare gas and the centre-of-mass (CM) of C 2 H 2 ; θ is the alignment angle between the symmetry axis of C 2 H 2 and the axis which links the rare gas to the CM of C 2 H 2 . The calculations are performed using the spectral element based close-coupling program VRBoundScat developed in the group of Jean-Michel Launay [14][15][16] and the C 2 H 2 −Ar, Kr potential energy surfaces V(R,θ ) derived in the group of Jacky Liévin. 9,10 This provides us with bound states and wave functions.…”

Quantitative analysis of losses close to a d -wave open-channel Feshbach resonance in K39

“…The computational details are too long to convey here, we simply mention that the energy derivatives of the S matrix are obtained analytically in the framework of the spectral element representation of the Hamiltonian [37]. The diagonal terms of Q M correspond to the average time delay experienced during a collision starting in a given channel, whereas its eigenvalues q i are associated with the lifetime of metastable states of the molecular system.…”