Density Functional Theory (DFT) has provide so far the most 
succesful tools for realistic calculations of the properties of complex quantum system. We investigate the fundamental problems related to DFT by developing new methods based on the role of metric spaces in quantum mechanics. In our previous works the metric properties of ground state wavefunction and desity were explored for the non-magnetic and magnetic quantum systems. Currently we are developing a new method for the investigation of the metric properties for the exchange functional by exploiting the special features of the exchange x-hole functional. A better understanding of this mapping is conducive to the possibility of improving the quality of the exchange-correlation functional currently used in DFT.
