Gradient correction

Density Gradient Corrections.—Mucci and March50 have discussed the corrections expected to the result (84) of the simplest density description. As they emphasize, three steps are involving in deriving equations (116) and (84) for molecules at equilibrium  

Their argument for the corrections due to (a) and (b) is briefly summarized below. Correction (c) will be considered in Section 15 and Appendix 4. The inhomogeneity correction (b) above, and the non-zero chemical potential, are both incorporated in the generalized Euler equation (49). Multiplying this by the electron density p and integrating through space yields 

Combining this with equation (81) for the eigenvalue sum and noting that T=l tr dr leads immediately to the result 

Using result (86) to lowest order in a density gradient expansion gives 

Finally, using T= - E at equilibrium yields the generalization of equation (84) as 

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Philipsen P H T, te Velde G and Baerends E J 1994 The effect of density-gradient corrections for a molecule-surface potential energy surface. Slab calculations on Cu(100)c(2x2)-C0 Chem. Phys. Lett. 226 583... 

Garcia A, Eisasser C, Zhu J, Louie S G and Cohen M L 1992 Use of gradient-corrected functionais in totai-energy caicuiations for soiids Phys. Rev. B 46 9829... 

Fiiatov M and Thiei W 1997 A new gradient-corrected exchange-correiation density functionai Moi. Phys. 91 847 van Voorhis T and Scuseria G E 1998 A novei form for the exchange-correiation energy functionai J. Chem. Phys. 

Becke-Lee-Yang-Parr gradient-corrected functional for use with... 

I he function/(r) is usually dependent upon other well-defined functions. A simple example 1)1 j functional would be the area under a curve, which takes a function/(r) defining the curve between two points and returns a number (the area, in this case). In the case of ni l the function depends upon the electron density, which would make Q a functional of p(r) in the simplest case/(r) would be equivalent to the density (i.e./(r) = p(r)). If the function /(r) were to depend in some way upon the gradients (or higher derivatives) of p(r) then the functional is referred to as being non-local, or gradient-corrected. By lonlrast, a local functional would only have a simple dependence upon p(r). In DFT the eiK igy functional is written as a sum of two terms ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

BLYP Becke correlation functional with Lee, Yang, Parr exchange Gradient-corrected... 

Density functional theory calculations have shown promise in recent studies. Gradient-corrected or hybrid functionals must be used. Usually, it is necessary to employ a moderately large basis set with polarization and diffuse functions along with these functionals. 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

The HE, GVB, local MP2, and DFT methods are available, as well as local, gradient-corrected, and hybrid density functionals. The GVB-RCI (restricted configuration interaction) method is available to give correlation and correct bond dissociation with a minimum amount of CPU time. There is also a GVB-DFT calculation available, which is a GVB-SCF calculation with a post-SCF DFT calculation. In addition, GVB-MP2 calculations are possible. Geometry optimizations can be performed with constraints. Both quasi-Newton and QST transition structure finding algorithms are available, as well as the SCRF solvation method. 

OPW (orthogonalized plane wave) a band-structure computation method P89 (Perdew 1986) a gradient corrected DFT method parallel computer a computer with more than one CPU Pariser-Parr-Pople (PPP) a simple semiempirical method PCM (polarized continuum method) method for including solvation effects in ah initio calculations... 

PW91 (Perdew, Wang 1991) a gradient corrected DFT method QCI (quadratic conhguration interaction) a correlated ah initio method QMC (quantum Monte Carlo) an explicitly correlated ah initio method QM/MM a technique in which orbital-based calculations and molecular mechanics calculations are combined into one calculation QSAR (quantitative structure-activity relationship) a technique for computing chemical properties, particularly as applied to biological activity QSPR (quantitative structure-property relationship) a technique for computing chemical properties... 

All three terms are again functionals of the electron density, and functionals defining the two components on the right side of Equation 57 are termed exchange functionals and correlation functionals, respectively. Both components can be of two distinct types local functionals depend on only the electron density p, while gradient-corrected functionals depend on both p and its gradient, Vp. ... 

Becke formulated the following gradient-corrected exchange functional based on the LDA exchange functional in 1988, which is now in wide use ... 

Note that this use of the term local does not coincide with the use of the term in mathematics both local and gradient-corrected functionals are local in the mathematical sense. 

Similarly, there are local and gradient-corrected correlation functionals. For example, here is Perdew and Wang s formulation of the local part of their 1991 correlation functional ... 

In an analogous way to the exchange functional we examined earlier, a local correlation functional may also be improved by adding a gradient correction. 

Pure DFT methods are defined by pairing an exchange functional with a correlation functional. For example, the well-known BLYP functional pairs Becke s gradient-corrected exchange functional with the gradient-corrected correlation functional of Lee, Yang and Parr. 

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