Reduced density gradient

The argument of the function F is the reduced density gradient for spin o... 

The second class of GGA exchange functionals use for F a rational function of the reduced density gradient. Prominent representatives are the early functionals by Becke, 1986 (B86) and Perdew, 1986 (P), the functional by Lacks and Gordon, 1993 (LG) or the recent implementation of Perdew, Burke, and Emzerhof, 1996 (PBE). As an example, we explicitly write down F of Perdew s 1986 exchange functional, which, just as for the more recent PBE functional, is free of semiempirical parameters ... 

Wesolowski, T. A., Parisel, O., Ellinger, Y., Weber, J., 1997, Comparative Study of Benzene---X (X = 02, N2, CO) Complexes Using Density Functional Theory The Importance of an Accurate Exchange-Correlation Energy Density at High Reduced Density Gradients , J. Phys. Chem. A, 101, 7818. 

Locally, the magnitude of this contribution might be larger than that of the zeroth order. This might happen far from nuclear centers. So-called reduced density gradient s(f) defined as ... 

Fig. 4. Variation of the reduced density gradient s(r) = — lVpL along the Civ axis in the...

Except for the TK9291 functional0, all of them behave similarly at small reduced density gradients ... 

Bulk solids typically have well-bounded values of the reduced density gradient s, and so might be particularly amenable to GGA description. However, the full verdict is not yet in. We emphasize here the importance of performing full-potential all-electron calculations in tests of GGA s, and avoiding as much as possible further approximations, such as pseudopotential or shape approximations, which may lead to errors which mask the effects of the GGA, relative to LSD. 

It is customary to discuss gradient expansions for exchange in terms of dimensionless reduced density gradients... 

The second-order density gradient expansion for exchange performs well only in the limit of small reduced density gradients x. The assumption that x is small may be justified for an infinite electron gas but not for finite systems, where x diverges in far-out regions. This can be demonstrated by assuming a spherically symmetric exponential density p(r) =... 

The other class contains all the functionals which can be expressed as polynomial function of the reduced density gradient. Among them, we can mention those introduced by Becke in 1986 (B86) [22,23], by Perdew and Wang in 1986 (P) [24], by Lacks and Gordon in 1993 (LG) [27] and the very recent functional of Gill (G) [30], either in its original form or in the modified form proposed by Handy. For instance, the enhancement factor of the P functional is ... 

Fig. 2 Exact and Thomas-Fermi reduced density gradient s of Eq. (9) as a function of position z for the Airy gas model with force F = 0.10 (atomic units)...

Start with a density that is so slowly varying that the second-order gradient expansions are valid for Ex (Eq. (16)) and for Exc (Eq. (20)). This requires that the reduced density gradients on both length scales of Section 5 must be small. Thus p and q of Eqs. (10) and (11) have magnitudes much less than 1, and so do pc = k.F/ksf p and = kp/ksfq- Moreover, the second-order contribution to the correlation energy... 

Fig. 7.5 The enhancement factors Fxc (for rg = 0 bohr) with respect to the LDA exchange energy as a function of s, the reduced density gradient, is shown for the four functionals PBE, WC, AMOS, and PBEsol. Additional plots for other rg values are given in [33]...

Fig. 7.6 A two-dimensional plot of the reduced density gradient s is shown for diamond in the (110) plane. The color coding indicating the value of s is specified in the insert...

FIGURE 7.9 (a) Structure 01-126. (b) Structure 01-127. Noncovalent interaction (NCI) plot of predicted crystal structure. Reduced density gradient isosurface (IS) cutoff set to 0.25. Isosurfaces represent either unfavorable, weak, or strong favorable interactions, (see insert for color representation of the figure.)... 

Behavior of Reduced Density Gradient as a Function of Density..304... 

The objectives of the present work are to analyze the behavior of GGA functionals from the perspective of the constraints related to the low and large reduced density gradient regions, taking as a starting point the approximation of Perdew, Burke, and Emzerhof (PBE) to analyze the behavior of the reduced density gradient as a function of the density and to establish some of the implications it may have in the development of improved GGA functionals. 

Most of the PBE modifications have essentially the same behavior at large values of the reduced density gradient, that is, they are built in such a way that F (s) 1 +... 

FIGURE 14.1 Plot of the enhancement factor as a function of the reduced density gradient for PBE, VMT, VT 8,4, and PBE-LS. (Inset) Zoom in the range of interest for real systems. 

BEHAVIOR OF REDUCED DENSITY GRADIENT AS A FUNCTION OF DENSITY... 

FIGURE 14.3 Plot of the reduced density gradient versus the electronic density for water molecule (top), its dimer (middle), and its tetramer (bottom). 

Let us consider now the density of the reduced density gradient, g(s). It provides, for a given electronic density, the number of electrons with the values of s that lie between s and s + ds. Zupan et al. - have defined this distribution as... 

FIGURE 14.5 Plot of the density of the reduced density gradient as a function of s for the water systems considered in this work. 

The study of the GGA functionals considered in this work, and the analysis of the behavior of the reduced density gradient as a function of the electronic density, together with its density, allows one to observe two important aspects for the possible improvement of exchange energy functionals that only depend on the density and its gradient. 

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