Computational Chemistry

electrons – decreases when an infinitesimal amount of electronic charge is added to

it. Intuitively, a hard molecule is like a rigid container that does not yield as

electrons are forced in, so the pressure, analogous to the electron density, inside

builds up, resisting the ingress of more electrons. A soft molecule may be likened to

a balloon that can expand as it acquires electrons, so that the ability to accept still

more electrons is not so seriously compromised. Softness is logically the reciprocal

of hardness:

s¼

1



(7.35)

and qualitatively, of course, it is the opposite in all ways.

To approximate hardness byIandA(cf. the approximation of electronegativity

by Eq.7.32), we approximate theE¼f(N) curve (cf. Fig.7.10) by a general

quadratic (since itlookslike a quadratic):

E¼aN^2 þbNþc

@^2 E

@N^2

¼ 2 a

We will now let M denote any atom or molecule, and M+and M"the species

formed by removal or addition of an electron from M.

E(M) corresponds toN¼1 andE(M") corresponds toN¼2, so substituting into

our quadratic equation

EðÞ¼M a 12

'(

þbðÞþ 1 c¼aþbþc

and

EðÞ¼M" a 22

'(

þbðÞþ 2 c¼ 4 aþ 2 bþc

and so

2 a¼c+E(M")" 2 E(M)

SinceE(0)¼E(M+)¼a(0^2 )+b(0) +c¼c,

2 a¼EðÞþMþ EðÞ"M" 2 EðÞ¼M ½Š"EðÞ"Mþ EðÞM ½Š¼EðÞ"M EðÞM" I"A

i:e:¼

@^2 E

@N^2



v

¼I"A (7.36)

Actually, the hardness is commonlydefinedashalfthe curvature of theEversus

Ngraph, giving

¼

1

2

@^2 E

@N^2



v

¼

I"A

2

(7.37)

502 7 Density Functional Calculations