LDA approximation

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

Nonlocal density gradient corrections (GC)-local spin density (LDA) approximation. 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

A complete treatment, including the solute self-polarization contribution, may be developped in the context of the KS theory. It was shown that within the LDA approximation, simple expressions for the effective KS potential may be obtained. 

Fig. 2. Exchange-correlation potential and its LDA approximation along the bond axis of LiH as functions of the distance z from the bond midpoint. The Li nucleus is at z = — 1.508 a.u. and H is at z = 1.508 a.u.

Most modern band-calculations for non-magnetic solids are performed in the LDA approximation, which is extended also to the relativistic formulation (see Chap. F). Care is taken in the choice of the set of o )i s. A particular problem exists in connecting the atomic wave functions ipi s, calculated in a central potential, in the inter-core region (see Fig. 12) of the solid. It is beyond the scope of this Chapter to go deeper into these details, which will be discussed further in Chap. F. 

The success and the shortcomings of band calculations performed in the LDA approximation have been excellently reviewed by Koelling Essentially, the discussion follows points which have been already qualitatively described, concerning the competition between U and W. It is worthwhile to resume shortly these arguments here. 

Figure 2 (a) VMC (solid lines) and LDA (dashed lines) snxc(r,s) plotted for an electron moving along . Arrows on the electron density (plotted on top), mark the position of the electron, (b) Exact sh frjs) plotted in the same direction and at the same points as in (a) (solid lines) and the corresponding LDA approximation (dashed lines). 

We now turn to our results for nxc and exc. The spherically averaged exchange-correlation hole, nxc(r,s), obtained from our adiabatic calculations is shown in figure 2(a) together with the LDA approximation (28) to this quantity... 

Two factors may be responsible for specific routes of adsorption and transformation of a hydrocarbon molecule on the catalyst surface a geometrical factor and an electronic factor. Certainly no ultimate conclusions with respect to adsorption geometry may be drawn on the basis of very small models of the site. We would rather show interrelations between the electronic structure of the host cluster or molecule and the preferable interaction geometry with the guest hydrocarbon molecule and point out the consequences regarding further reaction routes. The results discussed in this paragraph will be based on DFT geometry optimization within the LDA approximation... 

Comparison of selected atom charges q, bond orders p, and distances do-Mo, (A) of two surface clusters representing MoO2(011) and MoOsfOlO) surfaces, respectively. The results refer to DFT calculations using the LDA approximation. 

One finds that the LDA overestimates the calculated binding energy. At the BHF minimum the difference is about 85 MeV for the total nucleus, which is close to the surface correction contribution in the Bethe-Weizsacker mass formula for (as 118MeV). This suggests that the LDA approximation misses the major contribution to the surface tension and also the major contribution of the surface effects to the compressibility of a finite nucleus. This is supported by the observation that the LDA result for as a function of density is only slightly above the energy versus density curve of nuclear matter. 

Figure 4 also displays a LDA result (short dashes) which is obtained according to Eq. (2) but ignoring the density dependence of the Brueckner G-matrix in 8nm(p), i e- using the G-matrix as calculated for the density p = 0.5 fm Comparing the two curves for the LDA approximation one can see that the density dependence of the G-matrix is a very important ingredient to obtain the saturation point of finite nuclei as it is for nuclear matter. In addition, however, an additional... 

Figure 24 a) Dependence of the BSSE on the number of neighboring atoms included in the CP correction for crystalline urea, with the LDA approximation (SVWN) and a 6-31G(d,p) basis set. (b) Urea molecule surrounded by a star of 63 neighboring ghost atoms. 

Figure 31 Isothermal phase diagram of CaO (volume, A, versus pressure, GPa), as obtained with HF and LDA approximations. Vertical lines represent the transition pressure.

For instance, in LDA approximation, the temperature at a point is assumed as a function of the density in that point, )3(r) = j8(p(r)) this may be easily reached out by employing the scaling transformation to be (Ou-Yang Levy, 1990)... 

The local density approximation (LDA) is valid only in the region of slowly varying electron density. The LDA approximation is obviously an oversimplification of the actual density distribution and is well-known to lead to calculated bond and binding energies that are over-predicted. 

For the actual quasi-particle excitations E one has to solve the Dyson equation (2.1) or p.2) with a non-local energy-dependent self-energy 2. sham and Kohn have suggested LDA approximations for which is also a functional of density. To this they split 2 into a local and non-local component... 

It describes the gradient, or the force on the atom j Fj = )>. n electronic contribution (the first term) and the nuclear repulsion (the second term). The Hellmann-Feynman theorem is rigorously satisfied in the DFT (see [51, 52]), as well as for the KS-LDA approximation [53]. Thus, the electronic part of the force Ff) may be written as a sum of orbital contributions ... 

The fuUy relativistic LCAO method for soUds, based on the DKS scheme in the LDA approximation was represented in [541]. The basis set consists of the numerical-type orbitals constructed by solving the DKS equations for atoms. This choice of basis set allows the spurious mixing of negative-energy states known as variational collapse to be overcome. Furthermore, the basis functions transform smoothly to those in the nonrelativistic limit if one increases the speed of light gradually in a hypothetical way. 

The exchange part in a uniform electron gas is nothing else that the exchange functional that we met in the Thomas-Fermi-Dirac method (2.27). Thus, the exchange energy Ex in this LDA approximation is also given by ... 

The interpolation formula that gives the electron-positron correlation energy functional was found by Borohski and Nieminen [98]. The positron annihilation rate A is proportional to the overlap between the positron and electron densities. In the LDA approximation [72],... 

Figure 432 Dependence of the lifetime of a positron trapped at a vacancy cluster on the number of vacancies N in the cluster. The tnv values were calculated with the LDA approximation. The solid lines are only included to guide the eye. The deviations from the smooth curve reflect the discrete structures of the clusters [72]...

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