Approximations local density

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 function. The exchange energy for a uniform electron gas is given by the Dirac formula (eq. (6.2)). 

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 j power) has been virtually abandoned and replaced by the Local Spin Density Approximation (LSDA) (which is given as the sum of the individual densities raised to the 4-power, eq. (6.33)). 

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

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. The method proposed by Slater in 1951 can be considered as an LDA 

With a = % this is identical to the Dirac expression. The original Xa method used a -1, but a value of 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 the electron density raised to the Vs power. 

The term local density approximation (LDA) was originally used to indicate any density functional theory where die value of fixe at some position r could be computed exclusively from the value of p at diat position, i.e., the local value of p. In principle, then, the only requirement on p is that it be single-valued at every position, and it can otherwise be wildly ill-behaved (recall that there are cusps in the density at the nucleus, so some ill-behavior 

The distinction is probably best indicated by example. Following from Eq. (8.7) and the discussion in Section 8.1.2, the exchange energy for the uniform electron gas can be computed exactly, and is given by Eq. (8.23) with the constant a equal to. However, the Slater approach takes a value for a of 1, and the Xa model most typically uses j. All of these models have the same local dependence on the density, but only the first is typically referred to as LDA, while the other two are referred to by name as Slater (S) and Xa.  

The LDA, Slater, and Xa methods can all be extended to the spin-polarized regime using 

The spin-dependent treatment via (4.35) is compared with the solution of (4.27) (on the basis of the LDA). The error of the spin-dependent IP in general is considerably smaller than that of its unpolarized counterpart, most notably for the light atoms and the lanthanides. On the other hand, the error is still substantial for the 3d and 4d elements, so that the question of gradient corrections has to be raised. 

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In a number of classic papers Hohenberg, Kohn and Sham established a theoretical framework for justifying the replacement of die many-body wavefiinction by one-electron orbitals [15, 20, 21]. In particular, they proposed that die charge density plays a central role in describing the electronic stnicture of matter. A key aspect of their work was the local density approximation (LDA). Within this approximation, one can express the exchange energy as... 

Stampfl C, van de Walle C G, Vogel D, Kruger P and Pollmann J 2000 Native defects and impurities in InN First-principles studies using the local-density approximation and self-interaction and relaxation-corrected pseudopotentials Phys. Rev. B 61 R7846-9... 

Figure Cl. 1.6. Minimum energy stmctures for neutral Si clusters ( = 12-20) calculated using density functional theory witli tire local density approximation. Cohesive energies per atom are indicated. Note tire two nearly degenerate stmctures of Si g. Ho K M, Shvartsburg A A, Pan B, Lu Z Y, Wang C Z, Wacher J G, Fye J L and Jarrold M F 1998 Nature 392 582, figure 2.

The local density approximation (LDA) is the oldest and simplest of the functional types stiU in use. It is based on the idea of a imiform electron gas, a homogeneous... 

Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

The following relatively simple expression is commonly used for the exchange-only energy under the local density approximation [Slater 1974] ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

One approach, using a local density approximation for each part, has E - = Es -1- Evwn, where Eg is a Slater functional and Evwn is a correlation functional from Vosko, Wilk, and Nusair (1980). Both functionals in this treatment assume a homogeneous election density. The result is unsatisfactory, leading to enors of more than 50 kcal mol for simple hydrocarbons. 

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. 

Local Density Approximation in Quantum Chemistry and Solid State Physics J. P. Dahl,... 

Transition structures are more dihicult to describe than equilibrium geometries. As such, lower levels of theory such as semiempirical methods, DFT using a local density approximation (LDA), and ah initio methods with small basis sets do not generally describe transition structures as accurately as they describe equilibrium geometries. There are, of course, exceptions to this, but they must be identihed on a case-by-case basis. As a general rule of thumb, methods that are empirically dehned, such as semiempirical methods or the G1 and G2 methods, describe transition structures more poorly than completely ah initio methods do. 

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... 

A second calculation was done for a two-layer tubule using density functional theory in the local density approximation to establish the optimum interlayer distance between an inner (5,5) armchair tubule and an outer armchair (10,10) tubule. The result of this calculation yielded a 3.39 A interlayer separation... 

The electronic properties of single-walled carbon nanotubes have been studied theoretically using different methods[4-12. It is found that if n — wr is a multiple of 3, the nanotube will be metallic otherwise, it wiU exhibit a semiconducting behavior. Calculations on a 2D array of identical armchair nanotubes with parallel tube axes within the local density approximation framework indicate that a crystal with a hexagonal packing of the tubes is most stable, and that intertubule interactions render the system semiconducting with a zero energy gap[35]. 

According to many authors (e.g. Handy, 1993), the local density approximation (LDA) is not adequate for useful predictions in computational chemistry. 

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

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