Various functionals are known in density functional theory to represent the total ground-state energy of a many electron system. i) The orbital-dependent functional introduced by Kohn and Sham [1] proved to be the basis for the universal and very successful computational approach in modelling of complex molecules and complex materials despite the fact that one its components (exchange-correlation energy) must be approximated. Such calculations scale nominally as N**3 with the size of the system although many efforts have been made on approaching linear scaling by using special algorithms. ii) The formalism using the orbital-free functional as defined originally in Hohenberg-Kohn theorems [2] could lead to the linear scaling but, due t o the unknown form of its kinetic energy component, practical calculations are only possible for some systems. iii) Representing the total energy as the multi-functional E[rho1,rho2,...,rhoM] epending explicitly on several electron densities of the subsys tems making up the whole investigated system (rho=rho1+rho2+...+rhoM) offers yet another alternative to the Kohn-Sham route to determine the ground-state properties of large polyatomic systems [3]. The multi-functional based formulation of density functional theory was also used to demonstrate that the exact description of the electronic state of an embedded molecule can be derived from Kohn-Sham-like equations applying the first-principles based and orbital-free embedding potential [4]. The embedding calculations allow one to reach linear or even better scaling by introducing additional controllable approximations concerning the electron density of the total system.

The following issues relevant to the E[rho1,rho2,...,rhoM]-based formalism will be addressed: a) its exact functionals limit, b) the recent developmen ts concerning approximating E[rho1,rho2,...,rhoM] [5], c) the possible uses of this formalism in various types of calculations and the corresponding scaling behaviour, d) some recent applications in the studies of complex systems.

[1] W. Kohn and L. J. Sham,

Phys. Rev. vol 140 (1965) A1133

[2] P. Hohenberg and W. Kohn,

Phys. Rev. vol. 136 (1964) B864 < /dd>

[3] P. Cortona,

Phys. Rev. B vol. 44 (1991) 8454

[4] T.A. Wesolowski and A. Warshel

J. Phys. Chem. vol. 97 (1993) 8050

[5] T.A. Wesolowski and F. Tran,

J. Chem. Phys. vol. 118 (2003) 2072