orbitals, and for wavefunction orbitals"ELUMOof M is said to correspond to the
electron affinity of M", not of M (see theionization energy and electron affinity
subsection above). So how do the results of calculations using the formula of Eq.7.33
compare with those using Eq.7.32? Table7.12gives values ofwcalculated using
QCISD(T)/6-311þG (Section 5.4.3) values ofIandA, which should give good
values of these latter two quantities, and compares thesewvalues with those from
HOMO/LUMO energies calculated by ab initio (MP2(FC)/6-31G) and by DFT
(B3LYP/6-31G*). For the two cations the agreement between the three ways of
calculatingwis good; for the other species it is erratic or bad, although the trends are
the same for the three methods within a given family (electronegativity increases from
anion to radical to cation). It seems likely that Eq.7.32is the sounder way to calculate
electronegativity. An exposition of the concept of electronegativity as the (negative)
average of the HOMO and LUMO energies, and the chemical potential ("w) as
lying at the midpoint of the HOMO/LUMO gap, has been given by Pearson [ 150 ].
Chemical hardness and softness are much newer ideas than electronegativity,
and they were quantified only fairly recently. Parr and Pearson (1983) proposed
to identify the curvature (i.e. the second derivative) of theEversusNgraph (e.g.
Fig.7.10) with hardness,[ 151 ]. This accords with the qualitative idea of hardness
as resistance to deformation, which itself accommodates the concept of a hard
molecule as resisting polarization – not being readily deformed in an electric field:
if we choose to define hardness as the curvature of theEversusNgraph, then
¼
@^2 E
@N^2
v¼
@m
@Nv¼"
@w
@Nv(7.34)
where m andw are introduced from Eqs.7.30 and 7.31. The hardness of a
species is then the amount by which its electronegativity – its ability to accept
Table 7.12 Electronegativity,w, and hardness,(cf. Fig.7.10). For each specieswandhave
been calculated in three ways: (1)From ionization energy (I) and electron affinity (A), usingw¼
½(I+A) and¼1/2(I-A).IandAwere calculated (QCISD(T)/6-311+G) as the energy
differences of the optimized-geometry species, i.e. adiabatic values. (2) From the MP2(FC)/6-
31G HOMO and LUMO, usingw¼"1/2(EHOMOþELUMO) and¼½(ELUMO"EHOMO). (3)
From the B3LYP/6-31G* Kohn–Sham HOMO and LUMO, as for (2). All the numbers refer to
units of eV
IAHOMO, MP2 LUMO, MP2 w: :
(HOMO, DFT) (LUMO, DFT) (I+A)/2, (I-A)/2,
HOMO/LUMO HOMO/LUMO
MP2, HOMO/
LUMO DFT
MP2, HOMO/
LUMO DFT
F+ 36 19.8 "37.6 ("30.0) "17.7 ("27.3) 27.9, 27.7, 28.7 8.1, 10.0, 1.4
F. 19.8 3 "19.5 ("14.5) "19.5 ("14.5) 11.4, 19.5, 14.5 8.4, 0, 0
F- 3 " 14 "2.1 (4.6) 42.1 (36.4) "5.5,"20.0,"20.5 8.4, 22.1, 15.9
HS+ 20.2 11.3 "20.3 ("16.8) "10.7 ("15.7) 15.8, 15.5, 16.3 4.5, 4.8, 0.6
HS. 11.3 1.7 "12.5 ("8.7) "12.5 ("8.7) 6.5, 12.5, 8.7 4.8, 0, 0
HS" 1.7 "6.4 "1.9 (1.3) 12.3 (8.4) "2.4,"5.2,"4.9 4.1, 7.1, 3.6
7.3 Applications of Density Functional Theory 501