The Chemical Bond
http://www.catalysis.nl/~chembond/

Course Information
Course contents
Lecture Notes
Exercises
Contacting Information
What's New?
Homepage

Different interacting orbitals

Two different interacting orbitals is a more difficult problem than two identical interacting orbitals. The secular equation is given by

æ
ç
è
a1
b
b
a2
ö
÷
ø 		æ
ç
è
c1
c2
ö
÷
ø 		= e æ
ç
è
1
S
S
1
ö
÷
ø 		æ
ç
è
c1
c2
ö
÷
ø 		.
Without loss of generality we assume that a1 < a2. The way to determine the solutions is the same as for identical orbitals. The solutions are given by
e1 = 1
2(1-S2)
 é
ë
(a1+a2-bS)
-

Ö
 
(a2-a1)2 +4(a1S-b)(a2S-b)
 
 ù
û 		,
e2 = 1
2(1-S2)
 é
ë
(a1+a2-bS)
+

Ö
 
(a2-a1)2 +4(a1S-b)(a2S-b)
 
 ù
û 		.
This is quite an obscure expression. Although it's not always appropriate to neglect overlap, in general the non-diagonal elements of the overlap, but also of the Fock matrix, are small. So we can at least make an expansion in terms of S and b and retain only the lowest orders of non-vanishing terms. This leads to
e1
= a1- (a1S-b)2
a2-a1
 ,
e2
= a2+ (a2S-b)2
a2-a1
 .
The lower energy is e1, because a1 < a2. We can interpret the expression as meaning that the c1 orbital has shifted to a lower energy, because of the interaction with c2, and that the c2 has shifted to a higher energy. The latter shift is larger than the former. This can be seen from the average of the original and final levels.
1
2
 (e1+e2) = 1
2
 (a1+a2) +S é
ë 		(a1S-b)+(a2S-b) ù
û 		.
We have seen for two identical interacting orbitals that the last term is larger than zero. So we must have that the higher level shows a larger shift. We also see that these shifts are smaller when the difference between the original levels (a1 and a2) becomes larger. As a technical point note that the expansion of e1 and e2 is not valid if a2 = a1.

If we substitute the expression for e1 into the secular equation we obtain to lowest order in b and S

æ
ç
è
(b-a1S)2/(a2-a1)
b-a1S
b-a1S
a2-a1
ö
÷
ø 		æ
ç
è
c1
c2
ö
÷
ø 		= æ
ç
è
0
0
ö
÷
ø 		.
This gives us
c2 = c1 a1S-b
a2-a1
 ,
or |c2| < |c1|, because S and b are small. (More precise is that S and b must be so small that the numerator in the expression above is smaller than the difference between the energies of the interacting orbitals.) This means that the lower MO is similar to the c1. Similarly, we find upon substitution of the expression for e2 into the secular equation
æ
ç
è
a1-a2
b-a2S
b-a2S
(b-a2S)2/(a1-a2)
ö
÷
ø 		æ
ç
è
c1
c2
ö
÷
ø 		= æ
ç
è
0
0
ö
÷
ø 		,
which leads to
c1 = -c2 a2S-b
a2-a1
 ,
Consequently we have |c1| < |c2|, and the higher MO is similar to c2.

Qualitative MO theory.
Last updated: Thursday, March 20, 2003
© Dr. A.P.J. Jansen