  This work presents an alternative, general, and in-principle exact extension
of electronic Kohn-Sham density functional theory (KS-DFT) to the fully
quantum-mechanical molecular problem. Unlike in existing multi-component or
exact-factorization-based DFTs of electrons and nuclei, both nuclear and
electronic densities are mapped onto a fictitious electronically
non-interacting molecule (referred to as KS molecule), where the electrons
still interact with the nuclei. Moreover, in the present molecular KS-DFT, no
assumption is made about the mathematical form (exactly factorized or not) of
the molecular wavefunction. By expanding the KS molecular wavefunction \`a la
Born-Huang, we obtain a self-consistent set of "KS beyond Born-Oppenheimer"
electronic equations coupled to nuclear equations that describe nuclei
interacting among themselves and with non-interacting electrons. An exact
adiabatic connection formula is derived for the Hartree-exchange-correlation
energy of the electrons within the molecule and, on that basis, a practical
adiabatic density-functional approximation is proposed and discussed.
