Computational Chemistry

@E

@N



v

¼

EðM"Þ"EðMþÞ

2 " 0

¼

"ðIþAÞ

2

i.e., using Eq.7.31

w¼

IþA

2

(7.32)

To use this formula one can employ experimental or calculated adiabatic (or
vertical, if the species from removal or addition of an electron are not stationary
points) values ofIandA. This same formula (Eq.7.32) forwwas elegantly derived
by Mulliken (1934) [ 149 ] using only the definitions ofIandA. Consider the
reactions

XþY!XþþY"

and

XþY!X"þYþ

If X and Y have the same electronegativity then the energy changes of the two
reactions are equal, since X and Y have the same proclivities for gaining and for
losing electrons, i.e.

IðXÞ"AðYÞ¼IðYÞ"AðXÞ

i:e:ðIþAÞ for X¼ðIþAÞ for Y

So it makes sense to define electronegativity asIþA; the factor of ½(Eq.7.32)
was said by Mulliken to be “probably better for some purposes” (perhaps he meant
to makewthe arithmetic mean ofIandA, an easily-grasped concept).
Electronegativity has also been expressed in terms of orbital energies, by takingI
as the negative of the HOMO energy andAas the negative of the LUMO energy
[ 150 ]. This gives

w¼

"ðEHOMOþELUMOÞ

2

(7.33)

This expression has the advantage over Eq.7.32that one needs only the HOMO
and LUMO energies of the species, which are provided by a one-pot calculation
(i.e. by what is operationally a single calculation), but to use Eq.7.32one needs the
ionization energy and electron affinity, the rigorous calculation of which demands
the energies of M, Mþ, and M"; cf. the Fukui functions for SCN"later in this
section. How good is Eq.7.33?I¼"EHOMOis a fairly good approximation for the
orbitals of wavefunction theory, but not for the Kohn–Sham orbitals of current
DFT, andA¼"ELUMOis only a very rough approximation for the Kohn–Sham

500 7 Density Functional Calculations