LDA/LSDA

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. 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

Local Density and Local Spin Density Approximations (LDA/LSDA)... 

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)). 

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... 

Local-density approxmiation (LDA) comparison with GGA Local spm-deasity approxmiation (LSDA) Lmear-muffin-tm orbital (LMTO)... 

The initial implementation of DFT employed the so-called local density approximation, LDA (or, if we have separate a and [i spin, the local spin density approximation, LSDA). The basic assumption is that the density varies only slowly with distance -which it is locally constant. Another way of visualizing the concept of LDA is that we start with a homogeneous electron gas and subsequently localize the density around each external potential - each nucleus in a molecule or a solid. That the density is locally constant is indeed true for the intermediate densities, but not necessarily so in the high- and low-density regions. To correct for this, it was rec-... 

Fig. 3.6 Binding energy curves for the hydrogen molecule (lower panel). HF and HL are the Hartree-Fock and Heitler-London predictions, whereas LDA and LSDA are those for local density and local spin density approximations respectively. The upper panel gives the local magnetic moment within the LSDA self-consistent calculations. (After Gunnarsson and Lundquist (1976).)...

In our classification (7.2.3.4a-g) PBE is on rung three and TPSS is on rung four (these are rungs two and three if LDA and LSDA are collapsed into rung one). 

The LDA (or, in the case of radicals, the local spin density approximation, LSDA) exchange-correlation energy is generally expressed as... 

The LDA has been adopted in most DFT electronic stmcture calculations on sohds since the 1970s but was not considered accurate enough for quantum chemistry until the 1990s when refinements were made. Thus, even though the DFT formahsm is, in principle, exact, the many-body problem is stiU only solved approximately in the LDA scheme The LDA works best for metals. Band gaps tend to be underestimated. The local spin-density approximation (LSDA) is a generahzation of the LDA to account for electron spin ... 

For closed-shell systems LSDA is equal to LDA, and since this is the most common... 

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... 

The earliest class of DFT methods is known as local (electron) density approximation (LDA) methods in the case that the total electron density is decomposed into individual spin densities for +1 /2 and -1/2 spin we refer to these methods as local spin density approximation (LSDA) methods. In these methods the total molecular XC energy is evaluated by integration on a numerical grid of the electron density, and the energy is a function of only the specific value of the density at each point, hence the local density ... 

In addition to the OP formalism, several alternative schemes have been suggested in the past to account, within a relativistic band-structure calculation, for correlation effects not incorporated within the local approximation to SDFT (LSDA). For example, the LDA+U scheme has been applied to the compound CeSb (Antropov et al. 1995), a system that has a maximum Kerr-rotation angle of 90° (Pittini et al. 

Better results than with the LDA are obtained by an elaboration of the LDA in which electrons of a and spin in the uniform electron gas are assigned different spatial KS orbitals and from which different electron density functions and follow. This unrestricted LDA method (cf. UHF, section 5.2.3.6e) is called the local spin density approximation, LSDA, and has the advantages that it can handle systems with one or more unpaired electrons, like radicals, and systems in which electrons are becoming unpaired, such as molecules far from their equilibrium geometries even for ordinary molecules it appears to be more forgiving toward the use of (necessarily) inexact xcfunctionals [37], For species in which all the electrons are securely paired, the LSDA is equivalent to the LDA. Like and its functional derivative... 

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