Ronnie Kosloff

Department of Physical Chemistry, Hebrew University, Jerusalem 91904, ISRAEL

A faithful representation of a wavefunction is a necessary step in quantum simulations. Grid based methods have the advantage of a direct local representation in coordinate space. The Fourier method overcomes the difficulty in representing non-local operators by a fast transformation to a grid in momentum space. For a wavepacket which is localized in phase space the convergence of the method is exponential. The drawback of the Fourier method is that it requires a rectangular shape in phase space. Using the restriction of an energy shell one finds that in most molecular encounters the energy shell has a convoluted shape. A general coordinate mapping based on a semiclassical idea of local momentum, optimizes the representation. Reduction of the number of grid points by more than an order of magnitude is shown for cold atom collisions where long range interactions dominate. For multi-dimensional problems the enhancement due to coordinate mapping is much larger. For free molecular encounters the elimination of the center of mass leads to the use of curvilinear coordinates. For this reason the Fourier method has been found inferior to either DVR or basis set representations which are optimized to these coordinates. A reformulation of hyperspherical coordinates allows to use a Cartesian representation. Benchmarks using a Fourier representation for NaFH and H₃⁺ in Cartesian/hyperspherical, Jacobi and bond coordinates are shown, maintaining the simple implementation and exponential convergence.