The local density approximation (LDA) states that, for regions of a
material where the charge density is slowly varying, the exchange
correlation energy at that point can be considered the same as that
for a locally uniform electron gas of the same charge density (see
Figure 2.1). In this case we can write *E*_{xc} as

The spin polarised variation (local spin density approximation, or LSDA) replaces the spin averaged energy density in the above equation with the energy density for a polarised homogeneous electron gas.

Although this approximation is extremely simple, it is surprisingly
accurate, and forms the core of most modern DFT codes. It even works
reasonably well in systems where the charge density is rapidly
varying. However it tends to underpredict atomic ground state
energies and ionisation energies, while overpredicting binding
energies. It is also known to overly favour high spin state
structures. For these reasons there have been attempts to move beyond
the LDA, notably through the addition of *gradient corrections* to
incorporate longer range gradient effects [19]. However in
practise, although these improvements seem to give better total
energies the resultant structure is often worse, and at a greatly
increased computational cost. In general, the LDA is worse for small
molecules and improves with system size.

The only remaining problem is to find an approximate solution for the homogeneous electron gas exchange-correlation term, .There are several parameterised prescriptions for this, the one used in AIMPRO is based on the work by Ceperley and Alder [20].