Pseudospectral approximation

If one evaluates this product on a grid instead of using atom-centered Gaussian functions for the auxiliary functions Pi r), one then obtains the very similar pseudospectral approximation.Minimizing the self-interaction error, the four-index electron repulsion integrals,... 

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

In a recent survey, it was concluded by Tanner ([39]) that smooth viscoelastic flows are now computable. In such a geometry, the use of highly accurate numerical techniques (based in [30] on a pseudospectral and finite difference approximation) allows the computation of the... 

Orszag SA (1972) Comparison of pseudospectral and spectral approximations. Stud Appl Math 51 253-259... 

In this chapter, we discuss some new developments in TDDFT beyond the linear response regime for accurate and efficient nonperturbative treatment of multiphoton dynamics and very-high-order nonlinear optical processes of atomic and molecular systems in intense and superintense laser fields. In Section 2, we briefly describe the time-dependent optimized effective potential (OEP) method and its simplified version, i.e., the time-dependent Krieger-Li-Iafrate (KLI) approximation, along with self-interaction correction (SIC). In Section 3, we present the TDDFT approaches and the time-dependent generalized pseudospectral (TDGPS) methods for the accurate treatment of multiphoton processes in diatomic and triatomic molecules. In Section 4, we describe the Floquet formulation of TDDFT. This is followed by a conclusion in Section 5. Atomic units will be used throughout this chapter. 

Sophisticated numerical techniques have been devised to study this standard model within mean field approximation. They exploit that the mean field problem of a Gaussian chain in an external field can be described by a modified diffusion equation in an external field [60]. The latter leads to a partial differential equation that can be solved by efficient computational techniques. Advanced real-space, spectral, and pseudospectral algorithms have been devised to this end [28, 78-80]. ... 

In numerical methods a function is replaced in one or more dimensions by an approximation determined by its numerical values on a grid or mesh of points. Numerical approximations on a grid can be employed in the context of BSE methods. For example, in BSE-DFT methods integrals involving the exchange-correlation potential usually must be evaluated this way. Introduction of a grid also is an essential element of pseudospectral methods (see Pseudospectral Methods in Ab Initio Quantum Chemistry). However, this article is restricted to methods which attempt to determine the orbitals numerically on a grid by approximate solution of the PDE (equation 3). 

Adiabatic approximation, 53, 56 Adiabatic connection formula, 409 Adiabatic Connection Model (ACM), 187 Aliasing, in pseudospectral methods, 174 Allowed reaction, Woodward-Hoffmann rules, 356... 

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