ABSTRACT
For i = 1 and n = 2 (Diatomic molecule ethylene)
ψ1 = C11 φ1 + C12 φ2 (7.3)
The energy of this MO is obtained with the help of Eq. (7.1). Thus, H ψ1 = E ψ1 is multiplied by ψ1 (ψ1 is a real function) on both sides and integrated with respect to dτ (dx, dy, dz). Then expression for E is:
E = ψ ψ τ
ψ ψ τ 1 1
H d
(7.4)
E = (C C ) H (C C ) d
ϕ ϕ ϕ ϕ
ϕ ϕ
+ +
+ ∫ τ
2 dτ∫
or E = C H d 2 C C H dÄ C H d
ϕ ϕ ϕ ϕ ϕ ϕτ τ+ + ∫∫∫ 1 2
12 2ϕ ϕ ϕ ϕ2 2
1 2 d 2 C C d C d11 12 1 2τ τ τ+ + ∫∫∫
= C H 2 C C H C H C S 2 C C S C S 11 2
+ + + +
(7.5)
where Hij = Hji = ∫ φi H φj dτ, i.e., H12 = H21 = ∫ φ1 H φ2 dτ = ∫ φ2 H φ1 dτ;
H11 = ∫ φ1 H φ1 dτ; H22 = ∫ φ2 H φ2 dτ
Sij = ∫ φi φj dτ
S12 = ∫ φ1 φ2 dτ
2 dτ.
