1.1 Variation Method (Molecular Orbital Linear
Combination of Atomic Orbitals, MO-LCAO)
By using the variation method, the hypothesis on W n is that it consists in a linear
combinations of atomic orbitals (AOs)
W n ¼ R r c nr u r
r indicates the rth atom
The hypothesis on H is absent
For a biatomic bielectronic system,
W ¼ C 1 u 1 þ C 2 u 2 E ¼ E C 1 C 2
ð
Þ
E ¼
Z
W
à HWds=W
Ã
Wds
¼
Z
C 1 u 1 þ C 2 u 2
ð
Þ
à H C 1 u 1 þ C 2 u 2
ð
Þ ds= C 1 u 1 þ C 2 u 2
ð
Þ
à C 1 u 1 þ C 2 u 2 Þds
ð
¼
R
C 1 u 1
à HC 1 u 1 þ C 1 u 1
à HC 2 u 2 þ C 2 u 2
à HC 1 u 1 þ C 2 u 2
à HC 2 u 2
½
Š =
C 1 u 1
à C 1 u 1 þ C 1 u 1
à C 2 u 2 þ C 2 u 2
à C 1 u 1 þ C 2 u 2
à C 2 u 2
½
Š
if
R u i
à Hu j ds ¼ H ij and
R u i
Ã
u j ds ¼ S ij H 21 ¼ H 12 S 21 ¼ S 12
H ii = Coulomb integral
H ij = resonance integral
S ij = overlap integral
E ¼ C 1
2 H 11 þ 2C 1 C 2 H 12 þ C 2
2 H 22 = C 1
2 S 11 þ 2C 1 C 2
2 S 12 þ C 2
2 S 22
The energies have to be minimum varying C 1 and C 2 (variation condition)
dE=dc 1 ¼ 0
dE=dc 2 ¼ 0
ð1:AÞ
(1.A) is a two equation linear system, to derive E 1,2 and C 1,2 values.
E 1,2 are depending on the Coulomb, and on the resonance and overlap integrals,
which are difficult to be calculated. The system (1.A) becomes
C 1 H 11 À E S 11
ð
Þ þ C 2 H 12 À E S 12
ð
Þ¼0
C 1 H 21 À E S 21
ð
Þ þ C 2 H 22 À E S 22
ð
Þ¼0
One solution is C 1 ¼ C 2 ¼ 0 W ¼ 0
The other ones may be obtained from the following determinant:
2
1 The Electronic Structure Determination