Functionals LDA

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

This is the LDA for Ex. If one adds this term to the Thomas-Fermi 

This approximation for xc[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 LDA is systematic error cancellation typically, the LDA underestimates Ac but overestimates Ax, resulting in unexpectedly good values of Axe. This error cancellation is not accidental, but systematic, and is caused by the fact that for any density the LDA exchange-correlation hole satisfies the correct sum rule on the exchange-correlation hole Hxc(r, r ), /d r x ( rO = -1 which is only possible if integrated errors in cancel with those of 

Unlike for the exchange energy, the correlation energy e ° [n] of the uniform electron liquid is not known exactly the determination of the correlation energy of a homogeneous interacting electron system is already a difficult many-body problem on its own Early approximate expressions for were based on apply- 

For many decades the LDA has been applied in, e.g., calculations of band structures and total energies in solid-state physics. In quantum chemistry it is much less popular, because it fails to provide results that are accurate enough to permit a quantitative discussion of the chemical bond in molecules (so-called chemical accuracy requires calculations with an error of not more than about 1 kcal/mol = 0.0434 eV/particle). For the same reason, electrochemical calculations rarely use the LDA, but employ more refined functionals, as described below. 

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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. 

Accurately define the mathematical problem that was solved by specifying the exchange-correlation functional that was used. Many different functionals exist so it is not sufficient to state the kind of functional (LDA, GGA, etc.)—the functional must be identified precisely. 

Joanteguy et al.101 performed a dedicated study of the performance of LDA-, GGA-, and hybrid functionals in determining the ionization potentials of unsaturated molecules. The authors concluded that the accuracy better than 0.1 eV cannot be reached using the considered functionals LDA, BP86, B3P86, B3LYP, and B3PW91. 

The above form includes also the Thomas-Fermi functional (LDA) for which F(s) = const = 1. Fig. 5 shows the considered enhancement factors GEA2 -... 

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... 

Parametru/non-parametric techniques This first distinction can be made between techniques that take account of the information on the population distribution. Non parametric techniques such as KNN, ANN, CAIMAN and SVM make no assumption on the population distribution while parametric methods (LDA, SIMCA, UNEQ, PLS-DA) are based on the information of the distribution functions. LDA and UNEQ are based on the assumption that the population distributions are multivariate normally distributed. SIMCA is a parametric method that constructs a PCA model for each class separately and it assumes that the residuals are normally distributed. PLS-DA is also a parametric technique because the prediction of class memberships is performed by means of model that can be formulated as a regression equation of Y matrix (class membership codes) against X matrix (Gonzalez-Arjona et al., 1999). 

Figure 30 Polarizability (a) and hyperpolarizability (y) for finite polyacetylene oligomers Cniin+2 tts a function of n both from Hartree-Fock (HF) and density-functional (LDA) calculations relative to the Hartree-Fock values for n = 20 (Reproduced with permission from Phys. Rev. Lett., 83, 694 1999 American Physical Society)...

Table 2.11. Correlation energy and electron affinity of H Results obtained by combination of the exact exchange with different correlation functionals (LDA [29], PW91-GGA [30], CS [23], C2 [18] and ISI [21]) in comparison with MP2 [88]...

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. 

Bandgaps (in eV) for the (10,0) SWNT obtained with different functionals (LDA, PBE, and HSE, shown row-wise) and basis sets (3-21G, 3-21G, and 6-31G, shown column-wise). 

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. 

AIMD = ab initio molecular dynamics B-LYP = Becke-Lee-Yang-Parr CVC = core-valence correlation DF = density functional LDA = local density approximation MCLR = multi-configurational linear response MP2 = Mpller-Plesset second order (MRD)CI = multi-reference double-excitation configuration interaction RPA = random phase approximation TD-MCSCF = time-dependent multi-configurational self-consistent field TD-SCF = time-depen-dent self-consistent field. 

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