Gradient expansions

Gradient expansions [6,69], which offer systematic corrections to LSD for electron densities that vary slowly over space, might appear to be the natural next step beyond LSD. As we shall see, they are not understanding why not will light the path to the generalized gradient approximations of Sect. 1.6.3. 

As a first measure of inhomogeneity, we define the reduced density gradient 

Other measures of density inhomogeneity, such as p = V n/ 2kp) n, are also possible. Note that s and p are small not only for a slow density variation but also for a density variation of small amplitude (as in Sect. 1.5.4). The slowly-varying limit is one in which p/s is also small [6]. 

Because there is no special direction in the uniform electron gas, there can be no term linear in Vn. Moreover, terms linear in can be recast as terms, since 

Correlation introduces a second length scale, the screening length I/kg, and thus another reduced density gradient 

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Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

Later on Cahn and Hilliard presented some thermodynamic estimates for the nucleation of liquid in vapour. Values of AO and the composition profiles c(r) of the embryos have been estimated using the mean-field and gradient expansion approximations for the free energy functional F c(7 ). A number of qualitative features in variation... 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

This provides partial chemical binding. Note that 7VF[p] and 7 w p constitute the first two terms in a gradient expansion of the kinetic energy. 

Geldart, D. J. W. Nonlocal Energy Functionals Gradient Expansions and Beyond. 190, 31-56 (1996). 

Least square fits were obtained by the gradient-expansion algorithm developed by Marquardt (30). The DC offset (i.e. EPR signal at 20 mtorr) was subtracted from each data point before performing the least squares fit. The inverse of the fitting parameter, k, is the pressure at which the EPR signal reaches half maximum and, presumably, is related to the number oxygen. In these studies, constant at 475 25 torr. 

Before the progress with the relativistic gradient expansion of the kinetic energy took place, and due to a growing interest of applying the Kohn-Sham scheme of density functional theory [19] in the relativistic framework, an explicit functional for the exchange energy of a relativistic electron gas was found [20,21] ... 

Also, the fourth order terms of the gradient expanssion of the kinetic energy have been evaluated [24], leading to more involved expressions. This is one of the problems of the methods based on the gradient expansion the systematic improvement of the results by adding higher orders is not possible because of the asymptotic nature, and... 

Nonlocal Energy Functionals Gradient Expansions and Beyond... 

There are many problems of interest where the particle density l omes very small and may even vanish. Can the information obtained from the gradient expansions be of any use in such cases First, note that an estimate of the nonlocal contributions to the energy can be given by Eq. (5), provided the series is truncated at an optimal point (certainly prior to the occurrence of any... 

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