Computational Chemistry

Comparing the above P’s with Eqs.5.136we see that convergence to within our one

part in 1,000 criterion has not occurred: the largest change in the density matrix is |

(0.1996#0.1885)/0.1885|¼0.059, which is above 0.001, so the SCF procedure is

repeated.

Three more SCF cycles were carried out; the results of the “zeroth cycle” (the

initial guess) and the five cycles are summarized in Table5.1. Only with the fifth

cycle has convergence been achieved, i.e. have the changes in all the density

matrix elements fallen below one part in 1,000 (the largest change is inP 11 ,

|(0.2020#0.2019)/0.2019|¼0.0005<0.001). In actual practice, a convergence

criterion of from about one part in 10^4 to 1 in 10^8 is used, depending on the program

and the particular kind of calculation. The coefficients and the density matrix

elements change smoothly, although the energy levels andEtotalHF show some oscil-

lation. To reduce the number of steps needed to achieve convergence, programs

sometimes extrapolate the density matrix, i.e. estimate the finalPvalues and use

these estimates to initiate the final few SCF cycles.

Often the main result from a Hartree–Fock (i.e. an SCF) calculation is the energy

of the molecule (the calculation of energy may be subsumed into a geometry

optimization, which is really the task of finding the minimum-energy geometry).

The STO-1G energy of HHeþwith an internuclear distance of 0.800 A ̊ may be

calculated from our results:

The electronic energy is

EHF¼

Xn

i¼ 1

eiþ

1

2

Xm

r¼ 1

Xm

s¼ 1

PrsHcorers ð 5 : 147 ¼ 5 : 90 Þ

The internuclear repulsion energy is

VNN¼

X

allm;v

ZmZv

rmv

ð 5 : 148 ¼ 5 : 92 Þ

and the total internal energy of the molecule at 0 K (except for zero point energy –

Section 5.2.3.6.4) is

Table 5.1 Results of initial guess and SCF cycles on HHeþat bond length 0.800 A ̊using the
STO-1G basis set. Energies (e 1 ,e 2 , andEtotalHF) are in hartrees
Initial guess
(zeroth
cycle)

First cycle Second cycle Third cycle Fourth cycle Fifth cycle

e 1 ,e 2 ,– #1.4027,
#0.0756

#1.4447,

#0.1062

#1.4466,

#0.1054

#1.4473,

#0.1056

#1.4470,
#0.1051
c 11 ,c 21 0.249,
0.867

0.3070,

0.8100

0.3159,

0.8034

0.3175,

0.8022

0.3177,

0.8021

0.3178,
0.8020
c 12 ,c 22 – 1.1145,
#0.8247

1.1120,

#0.8319

1.1115,

#0.8323

1.1115,

#0.8325

1.1114,
#0.8325
P 11 0.1240 0.1885 0.1996 0.2010 0.2019 0.2020
P 12 0.4318 0.4973 0.5076 0.5094 0.5097 0.5097
P 22 1.5034 1.3122 1.2909 1.2870 1.2867 1.2864
EtotalHF – #2.3992 #2.4419 #2.4428 #2.4443 #2.4438

228 5 Ab initio Calculations