Local-density approximation

Local-density approximations (LDA) are a class of approximations to the <a href="/facts/Exchange_interaction/orogl7qP">exchange</a>–<a href="/facts/Electron_correlation/1XpBFce2">correlation</a> (XC) energy <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> in <a href="/facts/Density_functional_theory/wwz4Gqdg">density functional theory</a> (DFT) that depend solely upon the value of the <a href="/facts/Electronic_density/L5yDsk6j">electronic density</a> at each point in space (and not, for example, derivatives of the density or the <a href="/facts/Kohn%25E2%2580%2593Sham_equations/mz2BklaC">Kohn–Sham orbitals</a>). Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the <a href="/facts/Homogeneous_electron_gas/X7kghINT">homogeneous electron gas</a> (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems (molecules and solids).
In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as

E
          
            
              x
              c
            
          
          
            
              L
              D
              A
            
          
        
        [
        ρ
        ]
        =
        ∫
        ρ
        (
        
          r
        
        )
        
          ϵ
          
            
              x
              c
            
          
        
        (
        ρ
        (
        
          r
        
        )
        )
         
        
          d
        
        
          r
        
         
        ,
      
    
    {\displaystyle E_{\rm {xc}}^{\mathrm {LDA} }[\rho ]=\int \rho (\mathbf {r} )\epsilon _{\rm {xc}}(\rho (\mathbf {r} ))\ \mathrm {d} \mathbf {r} \ ,}

where ρ is the <a href="/facts/Electronic_density/L5yDsk6j">electronic density</a> and єxc is the exchange-correlation energy per particle of a <a href="/facts/Homogeneous_electron_gas/X7kghINT">homogeneous electron gas</a> of charge density ρ. The exchange-correlation energy is decomposed into exchange and correlation terms linearly,

E
          
            
              x
              c
            
          
        
        =
        
          E
          
            
              x
            
          
        
        +
        
          E
          
            
              c
            
          
        
         
        ,
      
    
    {\displaystyle E_{\rm {xc}}=E_{\rm {x}}+E_{\rm {c}}\ ,}

so that separate expressions for Ex and Ec are sought. The exchange term takes on a simple analytic form for the HEG. Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for єc.
Local-density approximations are important in the construction of more sophisticated approximations to the exchange-correlation energy, such as <a href="/facts/Generalized_gradient_approximation/wwz4Gqdg">generalized gradient approximations</a> (GGA) or <a href="/facts/Hybrid_functional/ttzwSvHa">hybrid functionals</a>, as a desirable property of any approximate exchange-correlation functional is that it reproduce the exact results of the HEG for non-varying densities. As such, LDA's are often an explicit component of such functionals.
The local-density approximation was first introduced by <a href="/facts/Walter_Kohn/WCQ3DRlI">Walter Kohn</a> and <a href="/facts/Lu_Jeu_Sham/9nr2lC49">Lu Jeu Sham</a> in 1965.

Local-density approximation open-in-new

Local-density approximation