Density functional theory

Density functional theory is a totally different approach, where the system is characterised by its electron density, rather than a wave function. In this method, the following energy functional is minimised with respect to the density, $\rho$.

$$E[\rho(r)]=T - \sum_{K}Z\int\frac{\rho(r)}{r}dr + \frac{1}{2... ...ho(r_i)\rho(r_j)}{\vert r_i-r_j\vert}dr_idr_j + \xi_{xc}[\rho]$$ (1.8)

In similarity with the Hartree-Fock method, the first terms represent the kinetic energy, the electron-nucleus attraction, and the electron-electron repulsion, respectively. The last term, however, is a general functional for the exchange and correlation energy, not present in the Hartree-Fock method. The form of this functional, $\xi_{xc}$[$\rho$], is not known and many approximate functionals have been developed with varying accuracy. Today, hybrid functionals, such as the three-parameter B3LYP functional, seem to give the best agreement with experiments[17]. The success of the method have often been attributed to inclusion of the exact Hartree-Fock exchange and gradient corrections for exchange and correlation.

One of the problems with density functional theory is that there is no clear way to improve the accuracy since the exact functional is not known and thus, the development relies on intuition and comparison with experimental results. In contrast, it is clear how to improve the accuracy of the wave function approach - it is only computationally more demanding. Nevertheless, the method of choice throughout this thesis, has mostly been the B3LYP method. This method certainly has its limitations, but for transition-metal complexes, it gives unexpectedly reliable results for ground state properties and can be applied to systems well beyond the size limit of more accurate methods. It includes correlation and is much less basis set dependent than correlated wave function based methods.