The Gradient Expansion

Until now we only considered the formal framework of density functional theory. However, the theory would be of little use if we would not be able to construct good approximate functionals for the exchange-correlation energy and exchange-correlation functional. Historically the first approximation for the exchange-correlation functional to be used was the local density approximation. In this approximation the exchange-correlation functional is taken to be 

The first term Exc[n0] in equation (258) is the exchange-correlation energy of a homogeneous system with constant density n0 and is therefore a function of n0 rather than a functional. We will therefore write this term as Exc(n0). This function is by now well-known from extensive investigations of the homogeneous electron gas [31]. Since the electron gas has translational, rotational, and inversion symmetry the functions functions satisfy 

The function L(1) (often denoted by Kxc in the literature) has been the subject of many investigations [32,33], Let us now go back to the more general case. If we introduce the Fourier transform of a function / as follows  

Note that by taking 8 (r) = 8n0 we violate the condition that 8n(r) integrates to zero. Nevertheless equation (273) can alternatively be derived directly from the properties of the response functions [30] and is related to a generalized form of the well-known compressibility sumrule of the electron gas. An important special case 

We see that the q = 0 value of Z/2) can be directly calculated from the knowledge of vxc( o)- For in 2 relation (273) directly implies the following relation between 

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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. 

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... 

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... 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

In addition, there is interest in further extending the discussion to a variety of situations, that have recently gained much attention in the nonrelativistic case, as time-dependent systems [49], excited states [45] or finite temperature ensembles [110]. As an example of work along these lines we mention the gradient expansion of the noninteracting, relativistic free energy [110], leading to a temperature-dependent relativistic extended Thomas-Fermi model. 

We attribute the failure of the gradient expansion approximation in approximating Tgad[pA, Pb] in the case where pa and pB do not overlap significantly as it is the case of two weakly interacting molecules to the artificial attraction arising from the second-order GEA contribution to T ad[pA, Pb - Interestingly, in the recent ar-... 

The main issue involved in using DFT and the KS scheme pertains to construction of expressions for the XC functional, Exc[n], containing the many-body aspects of the problems (1.38). The main approaches to this issue are (a) local functionals the Thomas Fermi (TF) and LDA, (b)semilocal or gradient-dependent functionals the gradient-expansion approximation (GEA) and generalized gradient approximation (GGA), and (c) nonlocal functionals hybrids, orbital functionals, and SIC. For detailed discussions the reader is referred to the reviews [257,260-272]. 

Langreth and Mehl (LM) ° carefully studied the convergence properties of the gradient expansion and suggested a practical scheme which, for spherical atoms, yielded improved densities and correlation energies. The LM correction is... 

The importance of this formula will become clear if we work out, as an explicit example, the lowest order in the gradient expansion. From the symmetry properties of K(x" one finds that its Fourier transform has the following expansion in powers of q, ... 

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