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LDA

LDA: Local Density Approximation
Definition:A method to approximate both the exchange and correlation energy in density functional methods
Explanation:One of the principle ideas of the density functional theory (→DFT) methods is to convert the many body problem into N single-particle equations in an effective field. As a consequence, electronic exchange and correlation interactions are not included and have to be considered separately. Exact functional for these electron–electron interactions is unknown except for the hypothetical free electron gas. For this theoretical model, the exchange energy can be calculated analytically, while the correlation energy can be derived from perturbation theory. Therefore, the exchange and correlation energy can be calculated as a function of gas density. LDA uses tabulated data of this function to approximate the electronic correlation and exchange energy of small volume elements of the crystal as the correlation and exchange energy of the free electron gas with the corresponding density. The total exchange and correlation energy of the crystal is then estimated as the sum over all volume elements. Despite such a crude simplification, the LDA method yields surprisingly good results. The LDA method can be improved as the generalized gradient method (→GGA) which additionally takes into account the density gradient.

The LDA is an approximation of the exchange-correlation functional where the latter is represented as a local functional of the electron density. Within the LDA, the electron gas is considered to be homogeneous, and the exchange-correlation energy, Exc[n], is an integral over all space with the exchange-correlation energy density, εxchom, at each point assumed to be the same as in a homogeneous electron gas with that density, which can be expressed as in the equation below. Therefore, the only necessary information is the exchange-correlation energy of the homogeneous gas as a function of density. The exchange energy of the homogeneous gas can be given analytically; the correlation energy can be calculated with a great accuracy with Monte-Carlo (→MC ) methods [1].

The equation below is obviously valid when the electronic densities vary slowly over space, but real atoms in solids violate this condition; thus, the LDA usually underestimates lattice constants and overerstimates bulk moduli of solids. As mentioned above, the more enhanced approximation is the GGA.
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SFB-Link:LDA methods are not used in the SFB since they are not reliable for a magnetic iron.
References:R. Dronskowski, Computational Chemistry of Solid State Materials, Wiley-VCH, Weinheim, 2005.
[1] R. M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, 2004