Local-density approximation

Local-density approximations (LDA) are a class of approximations to the exchange-correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and not, for example, derivatives of the density or the Kohn-Sham orbitals). Many approaches can yield local approximations to the XC energy. Overwhelming, however, successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, and which then applied to realistic systems (molecules and solids).

In general, a local-density approximation for the exchange-correlation energy is written as

${\displaystyle E_{xc}^{\mathrm {LDA} }[\rho ]=\int \mathrm {d} \mathbf {r} \ \rho (\mathbf {r} )\epsilon _{xc}(\rho ),}$

where ρ is the electronic density and ε_xc, the exchange-correlation energy density, is a function of the density alone. The exchange-correlation energy is decomposed in to exchange and correlation terms linearly,

${\displaystyle E_{xc}=E_{x}+E_{c},\,}$

so that separate expressions for ε_x and ε_c 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 ε_c.

Homogeneous electron gas

Approximation for ε_xc depending only upon the density can be developed in numerous ways. The most successful approach is based on the homogeneous electron gas. This is constructed by placing N interacting electrons in to a volume, V, with a positive background charge keeping the system neutral. N and V are then taken to infinity in the manner that keeps the density (ρ = N / V) finite. This is a useful approximation as the total energy consists of contributions only from the kinetic energy and exchange-correlation energy, and that the wavefunction is expressible in terms of planewaves. In particular, for a constant density ρ, the exchange energy density is proportional to ρ^⅓.

Exchange functional

The exchange-energy density of a HEG is known analytically. The LDA for exchange employs this expression under the approximation that the exchange-energy in a system where the density in not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression^[1][2]

${\displaystyle E_{x}=\int \mathrm {d} \mathbf {r} \,\rho (\mathbf {r} )\left({\frac {3}{4}}\left({\frac {3}{\pi }}\right)^{1/3}\rho ^{1/3}(\mathbf {r} )\right)=C_{x}\int \mathrm {d} \mathbf {r} \ \rho ^{4/3}(\mathbf {r} ).}$ 

Correlation functional

Analytic expressions for the correlation energy of the HEG are not known except in the high- and low-density limits corresponding to infinitely-weak and infinitely-strong correlation. For a HEG with density ρ, the high-density limit of the correlation energy density is^[1]

${\displaystyle \epsilon _{c}=A\ln(r_{s})+B+r_{s}(C\ln(r_{s})+D),\,}$ 

and the low limit

${\displaystyle \epsilon _{c}={\frac {1}{2}}\left({\frac {g_{0}}{r_{s}}}+{\frac {g_{1}}{r_{s}^{3/2}}}+\dots \right),}$ 

where the Wigner-Seitz radius is related to the density as

${\displaystyle {\frac {4}{3}}\pi r_{s}^{3}={\frac {1}{\rho }}.}$ 

Accurate quantum Monte Carlo simulations for the energy of the HEG have been performed for several intermediate values of the density, in turn providing accurate values of the correlation energy density.^[3] The most popular LDA's to the correlation energy density interpolate these accurate values obtained from simulation while reproducing the exactly known limiting behavior. Various approaches, using different analytic forms for ε_c, have generated several LDA's for the correlation functional, including

• Vosko-Wilk-Nusair (VWN) ^[4]

• Perdew-Zunger (PZ81) ^[5]

• Cole-Perdew (CP) ^[6]

• Perdew-Wang (PW92) ^[7]

Predating these, and even the formal foundations of DFT itself, is the Wigner correlation functional obtained perturbatively from the HEG model.^[8]

References

1. Parr, Robert G (1994). Density-Functional Theory of Atoms and Molecules. Oxford: Oxford University Press. ISBN 978-0-19-509276-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
2. Dirac, P. A. M. (1930). "Note on exchange phenomena in the Thomas-Fermi atom". Proc. Cambridge Phil. Roy. Soc. 26: 376–385.
3. D. M. Ceperley and B. J. Alder (1980). "Ground State of the Electron Gas by a Stochastic Method" (abstract). Phys. Rev. Lett. 45: 566–569. doi:10.1103/PhysRevLett.45.566.
4. S. H. Vosko, L. Wilk and M. Nusair (1980). "Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis". Can. J. Phys. 58: 1200.
5. J. P. Perdew and A. Zunger (1981). "Self-interaction correction to density-functional approximations for many-electron systems" (abstract). Phys. Rev. B. 23: 5048. doi:10.1103/PhysRevB.23.5048.
6. L. A. Cole and J. P. Perdew (1982). "Calculated electron affinities of the elements" (abstract). Phys. Rev. A. 25: 1265. doi:10.1103/PhysRevA.25.1265.
7. John P. Perdew and Yue Wang (1992). "Accurate and simple analytic representation of the electron-gas correlation energy" (abstract). Phys. Rev. B. 45: 13244–13249. doi:10.1103/PhysRevB.45.13244.
8. E. Wigner (1934). "On the Interaction of Electrons in Metals" (abstract). Phys. Rev. 46: 1002–1011. doi:10.1103/PhysRev.46.1002.