Limitations

Despite the remarkable success of the LDA, its limitations mean that care must be taken in its application. For systems where the density varies slowly, the LDA tends to perform well, and chemical trends are well reproduced. In strongly correlated systems where an independent particle picture breaks down, the LDA is very inaccurate. The transition metal oxides XO (X=Fe,Mn,Ni) are all Mott insulators, but the LDA predicts that they are either semiconductors or metals. The LDA has been applied to high $T_{c}$ superconductors, but finds several to be metallic, when in reality they are insulating at 0K.

The LDA finds the wrong ground state for in many simpler cases. For example, the LDA finds the wrong ground state for the titanium atom. The LDA does not account for van der Waals bonding, and gives a very poor description of hydrogen bonding. These phenomena are essential for most of biochemistry: the structure of DNA of depends critically on hydrogen bonding, as do the changes in the structure of most molecules on solvation.

The success of the LDA has been shown by QMC calculations to result from a real-space cancellation of errors in the LDA exchange and correlation energies. The cancellation represents a difficulty when improvements to the LDA are attempted, as an improvement in only the exchange or correlation contributions may give worse results.

An obvious approach to improving the LDA is to include gradient corrections, by making $E_{xc}$ a functional of the density and its gradient:

$$E_{xc}[n({\bf r})] = \int d^3r \epsilon_{xc}(n({\bf r}))n({\b... ..._{xc}[n({\bf r}),\vert\nabla n({\bf r})\vert]d{\bf r} \;\;\; ,$$

where $F_{xc}$ is a correction chosen to satisfy one or several known limits for $E_{xc}$.

Clearly, there is no unique recipe for $F_{xc}$, and several dozen functionals have been proposed in the literature. They do not always represent a systematic improvement over the LDA and results must be carefully compared against experiment.The development of improved functionals is currently a very active area of research and although incremental improvements are likely, it is far from clear whether the research will be successful in providing the substantial increase in accuracy desired.