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Fig. 3.6 Bonding and antibonding orbitals of the hydrogen molecule (a), notations of the
distances between nuclei and electrons (b), and scheme of molecular orbitals (c) in the H 2
molecule
the wave function must be chosen. Practice shows that this can be done based on a
simplified Schr ¨ dinger equation, where the interaction between electrons is
neglected:
ðÞ ! H ðÞ 1
ðÞ ¼ H ðÞ þ H ðÞ ,
H 1
;
;
H ðÞ 1
ðÞ 1
ðÞ ¼ E ðÞ
ðÞ 1
ð Ψ
;
;
Ψ
ð :
;
It can be shown that this simplified equation can be partitioned into two ordinary
differential equations:
H ðÞˆ 1 ðÞ ¼ ʵ 1 ˆ 1 ðÞ ,
H ðÞˆ 2 ðÞ ¼ ʵ 2 ˆ 2 ðÞ ,
ðÞ 1
Ψ
ðÞ ¼ ˆ 1 ðÞˆ 2 ðÞ , E ¼ ʵ 1 þ ʵ 2 :
;
Consider solutions H (1) when the distance between the first electron and the
nucleus “ a ” is small and therefore r a 1 r b 1 . In this case in H (1) the term Ze 2 / r b 1 can
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