Computational Chemistry

Step 4– Calculating the Fock matrix

(a) The one-electron matrices

From Eq.5.100

F¼TþVðHÞþVðHeÞþG¼HcoreþG ð 5 : 116 Þ

The one-electron matricesT,V(H) andV(He) (i.e.Hcore) follow immediately
from the one-electron integrals. The kinetic energy matrix is

T¼

T 11 T 12

T 21 T 22



¼

0 :6249 0: 2395

0 :2395 1: 1609



ð 5 : 117 Þ

T 11 is smaller thanT 22 , as the kinetic energy of an electron inf 1 (f(H)) is
smaller than that of an electron inf 2 (f(He)); this is expected since the larger
charge on the helium nucleus results in a larger kinetic energy for an electron in its
1 sorbital than for an electron in the hydrogen 1sorbital – classically speaking, the
electron must move faster to stay in orbit around the stronger-pulling He nucleus.
T 12 can be regarded as the kinetic energy of an electron in the H(1s)#He(1s)
overlap region.
The hydrogen potential energy matrix is

VðHÞ¼

V 11 ðHÞ V 12 ðHÞ

V 21 ðHÞ V 22 ðHÞ



¼

# 1 : 0300 # 0 : 4445

# 0 : 4445 # 0 : 6563



ð 5 : 118 Þ

All theV(H) values represent the attraction of an electron to the hydrogen
nucleus.V 11 (H) is the potential energy due to attraction of an electron inf 1 to
the hydrogen nucleus, andV 22 (H) is the potential energy due to attraction of an
electron inf 2 to the hydrogen nucleus. As expected, an electron inf 1 (f(H)) is
attracted to the H nucleus more strongly (the potential energy is more negative) than
is an electron inf 2 (f(He)).V 12 (H) can be regarded as the potential energy of
attraction to the hydrogen nucleus of an electron in the H(1s)#He(1s) overlap
region.
The helium potential energy matrix is

VðHeÞ¼

V 11 ðHeÞ V 12 ðHeÞ

V 21 ðHeÞ V 22 ðHeÞ



¼

# 1 : 2555 # 1 : 1110

# 1 : 1110 # 2 : 8076



ð 5 : 119 Þ

All theV(He) values represent the attraction of an electron to the helium nucleus.
V 11 (He), the potential energy of attraction of an electron inf(H) to the helium
nucleus, is of course less negative than the potential energy of attraction of an
electron inf(He) to this same nucleus.V 12 (He) can be taken as the potential energy
of attraction to the helium nucleus of an electron in the H(1s)#He(1s) overlap
region. An electron inf(He) is attracted to the helium nucleus more strongly than
an electron inf(H) is attracted to the hydrogen nucleus (#2.8076 inV(He)cf.
#1.0300 inV(H)), due to the greater nuclear charge of helium.

5.2 The Basic Principles of the ab initio Method 221