Local functionals LDA

Historically (and in many applications also practically) the most important type of approximation is the local-density approximation (LDA). To understand the concept of an LDA recall first how the noninteracting kinetic energy Ts [n] is treated in the Thomas-Fermi approximation In a homogeneous system one knows that, per volume45 

45The change from a capital T to a lower-case t is commonly used to indicate quantities per volume. 

This approximation for Exc[n] has proved amazingly successful, even when applied to systems that are quite different from the electron liquid that forms the reference system for the LDA. A partial explanation for this success of the 

46If one adds this term to the Thomas-Fermi expression (35) one obtains the so-called Thomas-Fermi-Dirac approximation to E[n]. It one multiplies it with an adjustable parameter a one obtains the so-called Xa approximation to Exc[n]. These approximations are not much used today in DFT. 

47Sometimes one uses the per-particle instead of the per-volume energy of the homogeneous system in writing the LDA. Since the conversion factor between both is the number of particles per volume, i.e., the density, an additional n(r) then appears under the integrals in (84) and also contributes to (85). 

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Four different Fock/Kohn-Sham operators have been applied to obtain the orbitals, which are subsequently localized by the standard Foster-Boys procedure. In addition to the local/semi-local functionals LDA and PBE, the range-separated hybrid RSHLDA [37, 56] with a range-separation parameter of /r = 0.5 a.u. as well as the standard restricted Hartree-Fock (RHF) method were used. The notations LDA[M] and LDA[0] refer to the procedure applied to obtain the matrix elanents either by the matrix algebra [M] or by the operator algebra [O] method. All calculations were done with the aug-cc-pVTZ basis set, using the MOLPRO quantum chemical program package [57]. The matrix elements were obtained by the MATROP facility of MOLPRO [57] the Cg coefficients were calculated by Mathematica. 

The above conclusions support a rough hierarchy of functionals of ascending accuracy as follows local functionals (LDA), nonlocal functionals (BP, BLYP.), hybrid functionals (B3LYP, B3P,. ..). But this order is purely empirical and there is more than one exception to it. The value of calibration studies in this field can therefore hardly be overestimated. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

Three density functional theories (DFT), namely LDA, BLYP, and B3LYP, are included in this section. The simplest is the local spin density functional LDA (in the SVWN implementation), which uses the Slater exchange functional [59] and the Vosko, Wilk and Nusair [60] correlation functional. The BLYP functional uses the Becke 1988 exchange... 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

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]. 

Helgaker et alP presented a fully analytical implementation of spin-spin coupling constants at the DFT level. They used the standard procedure for linear response theory to evaluate second-order properties of PSO, FC and SD mechanisms. Their calculation involves all four contributions of the nonrelativistic Ramsey theory. They tested three different XC functionals -LDA (local density approximation), BLYP (Becke-Lee-Yang-Parr), " and B3LYP (hybrid BLYP). All three levels of theory represent a... 

This local functional, which hereafter will be referred to as S-RC, gives results, which are considerably better than those of the usual LDA [23, 24]. For example, the mean absolute error (MAE) for the G2 dataset (atomization energies of 148 molecules) is reduced by a factor of 3 [23]. 

This chapter is intended to give the reader an idea of what types of functionals exist, and to describe what their main features are, separately for local functionals (TF, LDA and Xa Sec. 5.1), semilocal, or gradient-dependent, functionals (GEA and GGA Sec. 5.2), and nonlocal functionals (hybrids, orbital functionals such as meta-GGAs, EXX and SIC, and integral-dependent functionals such as ADA Sec. 5.3). This chapter does deal only most superficially with the actual construction of these functionals. For more... 

The density-functional theory (DFT) with spin polarization, even in the local approximations (LDA and LSDA with spin polarization) has been applied to describe many physical properties with good precision. The... 

If the LDA is used for the local field corr tion, which is the last term in (38), becomes a local function. 

Several authors [8,9,15-18], have recently attempted the calculation of c. All those calculations have several points in common a) The use of the density functional formalism [14] with approximate functionals for the kinetic energy of the electrons, like that of Eq. (14), or including higher order gradient corrections, b) A local density (LDA) description of exchange and correlation effects. The LDA exchange energy is... 

Finally, optical properties of conjugated polymers are currently poorly described. This is due to the fact that the local functionals currently in use (LDA and GGAs) fail to capture subtle charge displacements at the ends of a long molecular chain. Recently, a solution to this important problem, using the sophisticated ultra nonlocal Vignale-Kohn xc functional, has come much closer. ... 

Table 12 Vertical excitation energies (in eV) for N2- TDLDA and TDGGA represent results from time-dependent density-functional calculations with either a local-density (LDA) or a generalized-gradient (GGA) approximation, whereas TDHF are similar results from time-dependent Hartree-Fock calculations. MRCCSD, SOPPA, MRTDHF are results from sophisticated configuration-interaction calculations, and CIS are less sophisticated configuration-interaction calculations. The final state and the electronic excitation are also shown and the results are compared with experimental values (Exp.). The results are from ref. 89...

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