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Fig. 3.5 Scheme of the formation of the hydrogen atom
lowest one, with two electrons with opposite spins at each level. In the case of a
degenerate state, a single electron first occupies each component, which is subse-
quently filled by electrons with opposite spin.
This simple model of the electronic structure of atoms can also be used for
qualitative description of more complex systems. Consider two identical hydrogen
atoms that are far enough from each other, such that any interaction between them
can be neglected (Fig.
3.5
). Suppose that the atoms move toward each other. At
small distances between the nuclei of these atoms, their Coulomb potentials will
start to overlap, forming a single system. Simultaneously electron levels will be
changing as well. Let us estimate qualitatively the nature of these changes.
We introduce the notation for the hydrogen molecule (two nuclei at a distance
R
ab
from each other), as shown in Fig.
3.6
.
In this notation, the Schr¨dinger equation can be written as
e
2
r
12
,
H
1
ð Ψ
;
ðÞ
¼
E
1
;
Ψ
ð
,
H
1
1
;
ðÞ
¼
H ðÞþH ðÞþ
;
"
#
2
2
2
2
Ze
2
r
ai
Ze
2
r
bi
HðÞ
¼
ℏ
∂
x
i
þ
∂
y
i
þ
∂
,
i
¼ 1, 2
:
2
m
z
i
∂
∂
∂
It consists of three parts.
H
(1) and
H
(2) depend only on the coordinates of the
first and the second electrons, respectively, while the interaction between the
electrons is represented by a separate term. This interaction makes a significant
contribution to the calculated energy of the hydrogen molecule and cannot therefore
be neglected. At the same time, since the exact solution of the Schr¨dinger equation
is not known, necessitating the use of approximate methods, an analytical form of
