  Recently, the progression toward more exact density functional theory has
been questioned, implying a need for more formal ways to systematically measure
progress, i.e. a path. Here I use the Hohenberg-Kohn theorems and the
definition of normality by Burke et al. to define a path toward exactness and
straying from the path by separating errors in \r{ho} and E[\r{ho}]. A
consistent path toward exactness involves minimizing both errors. Second, a
suitably diverse test set of trial densities \r{ho}' can be used to estimate
the significance of errors in \r{ho} without knowing the exact densities which
are often computationally inaccessible. To illustrate this, the systems
previously studied by Medvedev et al., the first ionization energies of atoms
with Z = 1 to 10, the ionization energy of water, and the bond dissociation
energies of five diatomic molecules were investigated and benchmarked against
CCSD(T)/aug-cc-pV5Z. A test set of four functionals of distinct designs was
used: B3LYP, PBE, M06, and S-VWN. For atomic cations regardless of charge and
compactness up to Z = 10, the energy effects of variations in \r{ho} are < 4
kJ/mol (chemical accuracy) defined here as normal, even though these four
functionals ranked very differently in the previous test. Thus, the off-path
behavior for such cations is energy-wise insignificant and in fact,
indeterminate because of noise from other errors. An interesting oscillating
behavior in the density sensitivity is observed vs. Z, explained by orbital
occupation effects. Finally, it is shown that even large normal problems such
as the Co-C bond energy of cobalamins can use simpler (e.g. PBE) trial
densities to drastically speed up computation by loss of a few kJ/mol in
accuracy.
