  The relation between the derivative of the energy with respect to occupation
number and the orbital energy, $\partial E/\partial n_i = \epsilon_i$, was
first introduced by Slater for approximate total energy expressions such as
Hartree-Fock and exchange-only LDA, and his derivation holds for hybrid
functionals as well. We argue that Janak's extension of this relation to
(exact) Kohn-Sham density functional theory is not valid. The reason is the
nonexistence of systems with noninteger electron number, and therefore of the
derivative of the total energy with respect to electron number, $\partial
E/\partial N$. How to handle the lack of a defined derivative $\partial
E/\partial N$ at the integer point, is demonstrated using the Lagrange
multiplier technique to enforce constraints. The well-known straight-line
behavior of the energy as derived from statistical physical considerations
[J.P. Perdew, R. G. Parr, M. Levy and J.J. Balduz, Phys. Rev. Lett. 49, 1691
(1982)] for the average energy of a molecule in a macroscopic sample ("dilute
gas") as a function of average electron number is not a property of a single
molecule at $T=0$. One may choose to represent the energy of a molecule in the
nonphysical domain of noninteger densities by a straight-line functional, but
the arbitrariness of this choice precludes the drawing of physical conclusions
from it.
