Computational Chemistry

at various places altered? This function measures changes in electron density that
accompany chemical reactions, and has been used to try to rationalize and predict
the variation of reactivity from site to site in a molecule.
Electronegativity, hardness and softness, and the Fukui function will now be
explained quantitatively. These concepts can be analyzed using wavefunction
theory, but are often treated in connection with DFT, perhaps because much of
the underlying theory was formulated in this context [ 145 ]. Consider the effect
on the energy of a molecule, atom or ion, of adding electrons. Figure7.10shows
how the energy of a fluorine cation F+changes as one and then another electron
is added, giving a radical F.and then an anion F". The number of electronsNwe
can add to F+is integral, 1, 2,...(Nis taken here as 0 for F+, and is thus 1 for the
radical and 2 for the anion), but mathematically we can consider adding continuous
electronic chargeN; the line through the three points is then a continuous curve and
we can examine (∂E/∂N)Z,the derivative ofEwith respect toNat constant nuclear
charge. In 1876 Josiah Willard Gibbs published his theoretical studies of the effect
on the energy of a system of a change in its composition. The derivativem¼(∂E/
∂n)T,p, is the change in energy caused by an infinitesimal change in the number of
molesn. This derivative is called the chemical potential. HereEis Gibbs free
energyGand temperature and pressure are constant; the chemical potential can

N = number of electrons added

E = energy

(hartrees)

–98.00000

–100.00000

–99.00000

0

F. –99.57169

F– – 99.68061

F+ –98.84358

1 2

Fig. 7.10 Change of energy (for Fþ,Fland F") as electrons are added to a species. The energies
were calculated at the QCISD(T)/6-311þG* level. The slope of the curve at any point (first
derivative) is the electronic chemical potential, and the negative of the slope the electronegativity,
of the species at that point. The curvature at any point (second derivative) is the hardness of the
species). See too Table7.12

498 7 Density Functional Calculations