  Many references exist in the density functional theory (DFT) literature to
the chemical potential of the electrons in an atom or a molecule. The origin of
this notion has been the identification of the Lagrange multiplier $\mu =
\partial E/\partial N$ in the Euler-Lagrange variational equation for the
ground state density as the chemical potential of the electrons. We first
discuss why the Lagrange multiplier in this case is an arbitrary constant and
therefore cannot be a physical characteristic of an atom or molecule. The
switching of the energy derivative ("chemical potential") from $-I$ to $-A$
when the electron number crosses the integer, called integer discontinuity or
derivative discontinuity, is not physical but only occurs when the nonphysical
noninteger electron systems and the corresponding energy and derivative
$\partial E/\partial N$ are chosen in a specific discontinuous way. The
question is discussed whether in fact the thermodynamical concept of a chemical
potential can be defined for the electrons in such few-electron systems as
atoms and molecules. The conclusion is that such systems lack important
characteristics of thermodynamic systems and do not afford the definition of a
chemical potential. They also cannot be considered as analogues of the open
systems of thermodynamics that can exchange particles with an environment (a
particles bath or other members of a Gibbsian ensemble). Thermodynamical
(statistical mechanical) concepts like chemical potential, open systems, grand
canonical ensemble etc. are not applicable to a few electron system like an
atom or molecule. A number of topics in DFT are critically reviewed in the
light of these findings: jumps in the Kohn-Sham potential when crossing an
integer number of electrons, the band gap problem, the
deviation-from-straight-lines error, the role of ensembles in DFT.
