Multicomponent density-functional theory for electrons and nuclei

We present a general multi-component density functional theory in which electrons and nuclei are treated completely quantum mechanically, without the use of a Born-Oppenheimer approximation. The two fundamental quantities in terms of which our theory is formulated are the nuclear N-body density and the electron density expressed in coordinates referring to the nuclear framework. For these two densities coupled Kohn-Sham equations are derived and the electron-nuclear correlation functional is analyzed in detail. The formalism is tested on the hydrogen molecule $H_2$ and its positive ion $H_2^+$ using several approximations for the electron-nuclear correlation functional.