Microscopic Modelling of Correlated Low-dimensional Systems

Chapter 2: Method 13
In magnetic systems or, in general, in systems where open electronic shells are involved,
better approximations to the exchange-correlation functional can be obtained by introducing
the two spins densities, ρ↑(r) and ρ↓(r), such that ρ(r) = ρ↑(r) + ρ↓(r), and (ζ(r) =
ρ↑(r) − ρ↓(r))/ρ(r) is the magnetization density. The non-interacting kinetic energy splits
trivially into spin-up and a spin-down contributions, and the external and Hartree potentials
depend on the fully density ρ(r), but the approximate exchange-correlation functional will
depend on both spin densities independently, Exc = Exc[ρ↑(r), ρ↓(r)]. The density, given
by Eq. (2.16) contains a double summation, over the spin states and over the number of
electrons in each spin state. These latter have to be determined according to the single-
particle eigenvalues, by asking for the lowest N = N↑ + N↓ to be occupied.
The equivalent of the LDA in spin-polarized systems is the local spin density approximation
(LSDA), which basically consists of replacing the exchange-correlation energy density with
a spin-polarized expression:
E LSDA
�
xc [ρ↑(r), ρ↓(r)] =
[ρ↑(r) + ρ↓(r)]ɛ h xc[ρ↑(r), ρ↓(r)]dr (2.18)
obtained for instance, by interpolating between the fully-polarized and fully-unpolarized
exchange-correlation energy densities using an appropriate expression that depends on ζ(r).
To extend the local density approximation to systems with more significant non-
homogeneous densities, several techniques have been proposed. The most successful one
is the generalized gradient approximation (GGA), where the real Exc[ρ(r)] is expressed as a
functional of the density ρ(r) and its gradient ∇ρ(r):
where Fxc is a correction.
�
Exc[ρ] =
�
ρ(r)ɛxc(ρ(r)) dr +
Fxc[ρ, |∇ρ|] dr (2.19)
The GGA formalism gives a better description of inhomogeneous systems, like transition
metals, and it significantly improves the binding and atomic energies, it improves bond
lengths, angles, predicting good results also in the cases where LDA fails. It improves ener-
getics, geometries and dynamical properties of water, ice and water clusters. GGA accounts
specifically for density gradients that are neglected in pure LDA. For the GGA calculations
performed in this work, the Perdew-Burke-Ernzerhof (PBE96) [93] parameterization for
the exchange-correlation functional was used. This functional improves the description of
hydrogen-bonded systems.