Local Density Methods

In the Local Density Approximation (LDA) it is assumed that the density locally can be treated as a uniform electron gas, or equivalently that the density is a slowly varying 

In the more general case, where the a and p densities are not equal, LDA (where the sum of the a and p densities is raised to the 4/3 power) has been virtually abandoned and replaced by the Local Spin Density Approximation (LSDA) (which is given as the sum of die individual densities raised to the 4/3 power, eq. (6.17)). 

LSDA may also be written in terms of the total density and the spin polarization. 

For closed-shell systems LSDA is equal to LDA, and since this is the most common case, LDA is often used interchangeably with LSDA, although this is not true in the general case (eqs. (6.16) and (6.17)). The method proposed by Slater in 1951 can be considered as an LDA mediod where die correlation energy is neglected and the exchange term is given as 

With a = 2/3 this is identical to the Dirac expression. The original method used a = 1, but a value of 3/4 has been shown to give better agreement for atomic and molecular systems. The name Slater is often used as a synonym for the L(S)DA exchange energy involving die electron density raised to the 4/3 power (1/3 power for the energy density). 

In the Local Density Approximation (LDA) it is assumed that the density locally can be 

For closed-shell systems LSDA is equal to LDA, and since this is the most common case, LDA is often used interchangeably with LSDA, although this is not true in the 

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A later chapter treats local-density methods in more detail. As currently applied they are frequently incorporated into supercell models of the defect, with extended basis sets that include many plane waves. 

The ordering of the reliability of the methods is similar to the results for the G2/97 test set seen previously. As expected from its known tendency for substantial overbinding, the local density method (LDA) performs poorest with a mean absolute deviation of 134 kcal/mol. The BLYP functional has a mean absolute deviation of 9.3 kcal/mol, while the B3LYP functional performs the best with a mean absolute deviation of 4.8 kcal/mol. In our previous study on the G2/97 test set that included seven functionals, the B3LYP function also had the lowest mean absolute deviation. 

Although the initial crystallographic study of the simple heptiptycene 79 did not afford a structural model, 80 recently the structure of the crystalline 1 1 heptiptycene-chloro-benzene clathrate, in which the solvent molecule was packed in the channels between ribbons of the heptiptycene, was ascertained. 81 The calculated molecular geometry via Hartree-Fock (6-31G(D)) and local density methods compared well with the X-ray data. 

Dunlap et a/.111 have considered the first row diatomics by a local density treatment of exchange and correlation, and this work supports the conclusion of Gunnarsson et a/.,98 and of the independent work of Baerends and Roos,112 that, by comparison with experiment, local density methods give excellent answers for most properties of the first row diatomics. 

The necessary derivations with respect to the small displacements can be performed either numerically, or, more recently, also analytically. These analytical methods have developed very rapidly in the past few years, allowing complete ab initio calculation of the spectra (frequencies and intensities) of medium sized molecules, such as furan, pyrrole, and thiophene (Simandiras et al., 1988) however, with this approach the method has reached its present limit. Similar calculations are obviously possible at the semi-empirical level and can be applied to larger systems. Different comparative studies have shown that the precise calculation of infrared and Raman intensities makes it necessary to consider a large number of excited states (Voisin et al., 1992). The complete quantum chemical calculation of a spectrum will therefore remain an exercise which can only be perfomied for relatively small molecule. For larger systems, the classical electro-optical parameters or polar tensors which are calibrated by quantum chemical methods applied to small molecules, will remain an attractive alternative. For intensity calculations the local density method is also increasing their capabilities and yield accurate results with comparatively reduced computer performance (Dobbs and Dixon, 1994). 

I Extended Hiickel (9), Xa Scattered Wave (15), semiempirical SCF methods 16)) gave way to more sophisticated treatments, based both on ab initio Hartree-Fock SCF (17), Cl (18-20)) and on local density methods (DVM-Xa, 22), LCGTO-Xa (23), LCGTO-LSD-MP (24))- These more elaborate studies furnish structural and energetic information on chemisorption bonds -within the restriction of a local cluster model (9), of course. 

In the following we shall illustrate the present status of the local density method as implemented in the LCGTO - Xa approach by applying it to transition metal clusters in both fields mentioned above. The examples will deal with nickel clusters of up to 17 atoms, but larger clusters seem to be within the reach of today s computational possibilities. 

DENSITY FUNCTIONAL THEORY a.1 LOCAL DENSITY METHODS 183... 

FIG. 1. Representative photoemission EDCs for condensed C60 showing the full valence band and modulation with hv oi the cluster features. Those within 5 eV of the highest occupied level are p, derived, those between 5 and 12 eV are hybrids of s-p, character, and features below 12 eV are primarily s derived. The full bandwidth is the same as graphite and diamond, but only Cm has the richness in structure. The bottom curve is the density of states (DOS) calculated with the pseudopotential local-density method. The numbers and vertical lines associate experimental and theoretical features. 

Within the framework of the local density method, which is strictly an orbital theory, the antiferromagnetic state can be attained by reducing the symmetry constraints imposed on spin-polarized calculations, hence allowing the spin orbitals to localize and local magnetic moments to persist, if it is variationally favorable to do so. While I do not know of any formal justification for this type of symmetry breaking (one cannot just mix determinants within DFT to obtain proper spin and space eigenfunctions), the results discussed below for Cr2, M02, and Mn2 certainly indicate that it is a reasonable approach. A rough rationalization can be obtained if one reasons... 

Ag2 has also been calculated several times with local density methods The relativistic DVM-2fa results of Ziegler et = 2.52 A, = 203 cm " and = 2.03 eV, are in satisfactory agree-... 

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