also be defined with respect to internal energyUor Helmholtz free energyA
(Section 5.5.2.1a) [ 146 ]. By analogy, (∂E/∂N)Z, the change in energy with respect
to number of added electrons at constant nuclear charge, is theelectronicchemical
potential (or in an understood context just the chemical potential) of an atom. For a
molecule the differentiation is at constant nuclear framework, the charges and their
positions being constant, i.e. constant external potential,v(Section7.2.3.1). So for
an atom, ion or molecule
m¼@E
@N
v(7.30)
The electronic chemical potential of a molecular (including atomic or ionic)
species, according to Eq.7.30, is the infinitesimal change in energy when electronic
charge is added to it. Figure7.10suggests that the energy will drop when charge
is added to a species, at least as far as common charges (from aboutþ3 to"1)
go, and indeed, even for fluorine’s electronegative antithesis, lithium, the energy
drops along the sequence Liþ, Li., Li"(QCISD(T)/6-311þG* gives energies of
"7.23584,"7.43203,"7.45448h, respectively). Now, since one feels intuitively
that the more electronegative a species, the more its energy should drop when it
acquires electrons, we suspect that there should be a link between the chemical
potential and electronegativity. If we choose for convenience to tag most electro-
negativities with positive values, then since (∂E/∂N)vis negative we might define
the electronegativitywas the negative of the electronic chemical potential:
w¼"m¼"@E
@N
v(7.31)
From this viewpoint the electronegativity of a species is the drop in energy when
an infinitesimal amount (infinitesimal so that it remains the same species) of
electronic charge enters it. It is a measure of how hospitable an atom or ion, or a
group or an atom in a molecule (Section 5.5.4), is to the ingress of electronic charge,
which fits in with our intuitive concept of electronegativity.
This definition of electronegativity was given in 1961 [ 147 ] and later (1978)
discussed in the context of DFT [ 148 ]. Equation7.31could be used to calculate
electronegativity by fitting an empirical curve to calculated energies for, e.g. Mþ,M
and M", and calculating the slope (gradient, first derivative) at the point of interest;
however, the equation can be used to derive a simple approximate formula for
electronegativity using a three-point approximation. For consecutive species M+,M
and M"(constant nuclear framework), let the energies beE(Mþ),E(M), andE
(M"). Then by definition
E(Mþ)"E(M)¼I, the ionization energy of M
andE(M)"E(M")¼A, the electron affinity of M
Adding:E(M+)"E(M")¼IþA
So approximating the derivative at the point corresponding to M as the change in
EwhenNgoes from 0 to 2, divided by this change in electron number, we get
7.3 Applications of Density Functional Theory 499