Pseudospectral representation

The global approach uses an interpolation based on a family of global functions which span all the sampled space with appropriate boundary conditions. This approach which is due to Gauss, is termed collocation (Sec. III.A). In a more elaborate form, based on orthogonal functions it is termed pseudospectral representation (Sec. III.B) (16). Since any local method is global within a small interval we will start by analyzing global approaches. 

In a first discretization step, we apply a suitable spatial discretization to Schrodinger s equation, e.g., based on pseudospectral collocation [15] or finite element schemes. Prom now on, we consider tjj, T, V and H as denoting the corresponding vector and matrix representations, respectively. The total... 

The goal of pseudospectral methods is to reduce the formal dependence of the Coulomb and Exchange operators in the basis set representation (two-electron integrals, eq. (3.51)) to This can be accomplished by switching between a grid... 

Recently we have proposed more efficient relativistic molecular theory by an application of the pseudospectral (PS) approach [135]. In the PS approach [136,137], we use the mixed basis function between a grid representation in the physical space and spectral representation in the function space. 

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

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