Computational Chemistry

P¼

P 11 P 12 P 13 ..

.

P 1 m

P 21 P 22 P 23 ((( P 2 m

... ... ... ...

Pm 1 Pm 2 Pm 3 ((( Pmm

0 B B B B @

1 C C C C A

ð 5 : 80 Þ

where the density matrix elements are

Ptu¼ 2

Xn

j¼ 1

c$tjcuj t¼ 1 ; 2 ;...;m and u¼ 1 ; 2 ;...;m $ð 5 : 81 Þ

(sometimesPis defined as∑c*c). From Eqs.5.78and5.81:

Frs¼Hcorers ð 1 Þþ

Xm

t¼ 1

Xm

u¼ 1

Ptu ðrsjtuÞ#

1

2

ðrujtsÞ



$ð 5 : 82 Þ

Equation5.82,aslightmodificationofEq.5.78,isthekeyequationin
calculating the ab initio Fock matrix (you need memorize this equation only to
the extent that the Fock matrix element consists ofHcore,P,andthetwo-electron
integrals). Each density matrix elementPturepresents the coefficientscfor a
particular pair of basis functionsftandfu,summedoveralltheoccupiedMO’s
ci(i¼1, 2,...,n). We use the density matrix here just as a convenient way to
express the Fock matrix elements, and to formulate the calculation of properties
arising from electron distribution (Section 5.5.4), although there is far more to the
density matrix concept [ 27 ]. Equation5.82enables the MO wavefunctionsc
(which are linear combinations of thec’s andf’s) and their energy levelseto
be calculated by iterative diagonalization of the Fock matrix.
Equation5.17(E¼ 2 ∑Hþ∑∑(2J#K)) gives one expression for the molecular
electronic energyE. If we wish to calculateEfrom the energy levels, we must note
that in the HF methodEis not simply twice the sum of the energies of then
occupied energy levels, i.e. it is not the sum of the one-electron energies (as we take
it to be in the simple and extended H€uckel methods). This is because the MO energy
level valueerepresents the energy of one electronsubject to interaction with all the
other electrons. The energy of an electron is thus its kinetic energy plus its
electron–nuclear attractive potential energy (Hcore), plus, courtesy of theJandK
integrals (Section 5.2.3.5and Eqs. (5.48)–(5.50¼5.83¼5.50)), the potential
energy from repulsion of all the other electrons:

ei¼Hiicoreþ

Xn

j¼ 1

ð 2 Jijð 1 Þ#Kijð 1 ÞÞð 5 : 83 ¼ 5 : 50 Þ

If we add the energies of electron 1 and electron 2, say, we are adding, besides
the kinetic energies of these electrons, the repulsion energy of electron 1 on electron
2, 3, 4,..., and the repulsion energy of electron 2 on electron 1, 3, 4,...– in other

210 5 Ab initio Calculations