Pseudospectral method

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 

Consider the following Coulomb contribution to the element of the Fock matrix (eq. (3.51)) similar considerations hold for the exchange contribution.------------ 

For a specific point in space for coordinate 1, r, the integration over coordinate 2 may be carried out 

The integral is just a three-centre one-electron integral, which can he evaluated analytically. The integration over coordinate 1 may then be approximated as a sum over a finite set of grid points in the physical space. 

As the number of grid points increases, this approximation becomes better. The reduction in the formal scaling from to comes from the fact that the summations involve GM operations, G being the number of grid points, which typically wiU be linearly dependent on the number of basis functions M, i.e. GM M.  

The counterpoise-corrected complexation energy is then given as - AEc . 

For regular basis sets, this typically stabilizes at the basis set limiting value much earher than uncorrected values, but this is not necessarily the case if diffuse functions are included in the basis set. Note that AEc is an approximate correction it gives an estimate of the BSSE effect but does not provide either an upper or lower limit. 

There are variations of this method. For example may it be argued that the full set of ghost orbitals should not be used, since some of the functions in the complex are used for describing the electrons of the other component, and only the virtual orbitals are available for artificial stabilization. However, it appears that the method of full counterpoise correction (using all basis functions as ghost orbitals) gives the best results. 

It is usually observed that the CP correction for methods including electron correlation is larger and more sensitive to the size of the basis set than at the HF (or DFT) level. This is in line with the fact that the HF wave function converges much faster with respect to the size of the basis set than correlated wave functions. 

There have also been attempts at developing methods where the BSSE is excluded explicitly in the computational expressions. An example of this is the Chemical Hamiltonian Approach (CHA), but such methods are not yet commonly used. 

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Friesner R A 1987 Solution of the Flartree-Fock equations for polyatomic molecules by a pseudospectral method J. Chem. Phys. 86 3522-31... 

For an overview of Professor Carter s group s work using pseudospectral methods, see ... 

Martinez T J and Carter E A 1995 Pseudospectral methods applied to the electron correlation problem Modem Electronic Structure Theory yo 2, ed D R Yarkony (Singapore World Scientific) pp 1132-65... 

Fig. 3. Quantum solution of the test system of 3.3 for e = 1/100. computed numerically using Fourier pseudospectral methods in space and a syraplectic discretization in time. Reduced g -density f t)j dg versus t and qF Initial...

Adiabatic Pseudospectral Methods for Multidimensional Vibrational Potential. 

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]... 

Another class of powerful numerical techniques that should be mentioned here are spectral and pseudospectral methods, which are often used in DNS because of... 

Pseudospectral Methods functions with one higher angular momentum than already present. ... 

Xa F X/j) <—(Xa Jy X/j) y l(x a0 g (5.13) -leant compared to the three- and four-centre integrals, pseudospectral methods may i i nrovide energies at the same accuracy as fully analytical methods. For small to medium ... 

In the historical survey of the spectral methods given by Canute et al [22], it was assumed that Lanczos [101] was the first to reveal that a proper choice of trial functions and distribution of collocation points is crucial to the accuracy of the solution of ordinary differential equations. Villadsen and Stewart [203] developed this method for boundary value problems. The earliest applications of the spectral collocation method to partial differential equations were made for spatially periodic problems by Kreiss and Oliger [94] and Orszag [139]. However, at that time Kreiss and Oliger [94] termed the novel spectral method for the Fourier method while Orszag [139] termed it a pseudospectral method. 

Alipanah A, Razzaghi M, Dehghan M (2007) Nonclassical pseudospectral method for the solution of brachistochrone problem. Chaos, Solitons and Fractals 34 1622-1628... 

Fornberg B (1998) A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge... 

T. A. Driscoll and B. Fornberg, Block pseudospectral methods for Maxwell s equations II. Two-dimensional, discontinuous-coefficient case, SIAMJ. Sci. Comput., vol. 21, pp. 1146-1167, 1999.doi 10.1137/S106482759833320X... 

This tool, which they call pseudospectral methods, promises to reduce the CPU, memory and disk storage requirements for many electronic structure calculations, thus permitting their application to much larger molecular systems. In addition to ongoing developments in the underlying theory and computer... 

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