Computational Chemistry

The angle brackets remind us that these energy terms are quantum-mechanical
average values or “expectation values”; each is a functional of the ground-state
electron density. Focussing first on the middle term, the one most easily dealt with:
the nucleus-electron potential energy is the sum over all 2nelectrons (as with our
treatment of ab initio theory, we will work with a closed-shell molecule which
perforce has an even number of electrons) of the potential corresponding to
attraction of an electron for all the nuclei A:

hiVNe ¼

X^2 n

i¼ 1

X

nuclei A

"

ZA

riA

¼

X^2 n

i¼ 1

vðriÞ (7.8)

ZA/riAis the potential energy due to interaction of electroniwith nucleus A at
the varying distancer;v(ri) is the external potential for the attraction of electronito
all the nuclei, and with it we can write the double summation more compactly. The
density functionrcan be introduced intoby using the fact [ 31 ] that

Z

c

X^2 n

i¼ 1

fðriÞcdt¼

Z

rðrÞfðrÞdr (7.9)

wheref(ri) is a function of the spatial coordinates of electroniof the system andC
is the total wavefunction (the integrations are over spatial and spin coordinatestau
on the left and spatial coordinates on the right). From Eqs.7.8and7.9, invoking the
concept of expectation value (Section 5.2.3.3)hiVNe ¼ cjV^Nejc
we get

hiVNe ¼

Z

r 0 ðrÞvðrÞdr (7.10)

So Eq.*7.7can be written

E 0 ¼hiT½r 0 Š þ

Z

r 0 ðrÞvðrÞdrþhiVee½r 0 Š (7.11)

The middle term is now a classical electrostatic attraction potential energy
expression. Unfortunately this equation for the energy cannot be used as it stands,
since we don’t know the kinetic and potential energy functionals in the energy
termshT[r 0 ]iandhVee[r 0 ]i.
To utilize Eq.7.11, Kohn and Sham introduced the idea of a fictitious reference
system of noninteracting electrons which give exactly the same electron density
distribution as the real system has. Addressing electronickineticenergy, let us
define the quantityDhT[r 0 ]i(don’t confuse Greek deltaD, an increment, with the
differential operator delr) as the deviation of the real electronic kinetic energy
from that of the reference system:

DhiT½r 0 Š hiT½r 0 Šrea"hiT½r 0 Šref

i:e:hiT½r 0 Š"hiT½r 0 Šref

(7.12)

452 7 Density Functional Calculations