LDA calculations

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

The exchange-correlation energy density can be split into two parts exchange component Ex n) and correlation component e Cn). The explicit expression for the exchange component is known from Hartree-Fock theory but the correlation component is known only numerically. Several parametrisations exist for the exchange-correlation energy and potential of a homogeneous gas system which can be used for the LDA calculations within DFT. 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

Figure 1. The valence-electron CPs of Li along the three principal symmetry directions. The solid curves represent the FLAPW-LDA calculations. The dots represent the experimental results measured by Sakurai etal. [33],... 

Figure 10. The valence-electron CPs of Li along the three principal symmetry directions. The solid and dotted curves represent the FLAPW-GWA and FLAPW-LDA calculations, respectively. The dashed curves represent the FLAPW-LDA calculations including correlation effects according to Lundqvist and Lyden [27], The EXPI and EXPII represent the experimental results measured by Sakurai etal. [33] and Schillke etal. [32], respectively (after Kubo [13]).

Aformamide-water complex109 was studied by Sim et al. by means of the DFT(SVWN), DFT(PW86/P86), and DFT(B88/P86) calculations. The LDA led to qualitatively wrong results for conformational energies of this hydrogen-bonded complex. Furthermore, the poor performance of the LDA calculations was observed in studies of conformational equilibria in malonaldehyde, a molecule with the internal hydrogen bond. 

Stanton and Merz studied the reaction of carbon dioxide addition to zinc hydroxide, as a model for zinc metallo-enzyme human carbonic anhydrase IIJ 36. It was shown that the LDA calculations (DFT(SVWN)) were not reliable for locating transition state structures whereas the post-LDA ones (DFT(B88/P86)) led to the transition state structures and ener-... 

In subsection 3.1, we will present GGA and LDA calculations for Au clusters with 6first principles method outlined in section 2, which employs the same scalar-relativistic pseudo-potential for LDA and GGA (see Fig 1). These calculations show the crucial relevance of the level of density functional theory (DFT), namely the quality of the exchange-correlation functional, to predict the correct structures of Au clusters. Another, even more critical, example is presented in subsection 3.2, where we show that both approaches, LDA and GGA, predict the cage-like tetrahedral structure of Au2o as having lower energy than amorphous-like isomers, whereas for other Au clusters, namely Auig, Au ... 

The total energy versus volume for fcc-Pu and different values of the interaction U is presented in Fig. 4. The curve U = 0 corresponds to a LDA calculation. As previously found in several works the minimum of this curve is very low ( 7.70 ua ) compared to the experimental value of the 5 phase (8.60 ua) and closer to the a phase value (8.0 ua). In fact there is no sign of the correlated (5 phase in the U = 0 calculation. As we turn on the correlations, a new feature appears in the curves, almost instantly. We observe a new energy minimum close to the experimental volume of the 5... 

Figure 4- Total energy of fcc-Pu versus volume for different values of the interaction U. U = 0 corresponds to a LDA calculation.

Figure 7.12 compares the theoretical predictions with the experimental values across the 4d series, assuming one valence s electron per atom and taking x = 12 corresponding to close-packed lattices. The experimental values of the bandwidth are taken from the first principles LDA calculations in Table 7.1. The ratio b2 a is obtained by fitting a bandwidth of 10 eV for Mo with Nd = 5, so that from eqn (7.42) b2/a = eV. The skewed parabolic behaviour of the observed equilibrium nearest-neighbour distance is found to be fitted by values of the inverse decay length that vary linearly across the series as... 

Table 9.7 Spin-spin coupling constants (Hz) from LDA calculations and experiment...

---