First, we want to demonstrate the most simple manner of how Hückel (Erich Hückel, German physicist and theoretical chemist, 18961980) calculated molecular orbitals. Anyone truely interested in the method is invited to study Hückel's treatment of the acrolein molecule. We are already familiar with the LCAO approximation. It states, in general:
Σ_{k}c_{k}(H_{ik}  E · S_{ik}) = 0 
We now
In consequence, we employ the following secular equation to describe a diatomic molecule:
(α  E) · c_{1}
+ β · c_{2} = 0
β · c_{1} + (α  E) · c_{2} = 0
We only recieve non trivial (i.e. c_{i}≠0) solutions in cases where the secular determinant becomes zero:

αE  β  = 0  
β  αE 
Division by β and denoting (αE)/β as x yields
x  1  = 0  
1  x 
The determinant is resolved (multiplication of the elements in both diagonals, subtraction of the product obtained in the secondary diagonal from the product in obtained in the main diagonal)
x^{2}1 = 0 <=> x_{1/2} = ± 1  
or  E_{1} = α + β and E_{2} = α  β 
In the case of the molecule H_{2}, we obtain E_{total} = 2 (α+β) as state of minimal energy. Two electrons are within this state. The measured reduction of energy in the molecule H_{2} is E = − 0.174 a.u.. According to the mathematical treatment presented here, this equals 2β, therefore
For the H_{2}^{+} ion, E_{total} = α + β, we expect an energy reduction of
In analogy, the πbond in ethen H_{2}C=CH_{2} is to be described. Each carbon atom contributes one electron to the πsystem, therefore a total energy of E_{total} = 2α + 2β is obtained. In the excited state π → π* of ethen, one electron is 2β above the ground state. Together with spectroscopic measurements, these considerations lead to β = 75 kJ/mol, a value that, in turn, allows to predict the bonding energy.
For the cation H_{3}^{+}, we get the following secular determinants and energy terms:
linear H_{3}^{+}  circular H_{3}^{+}  



Solutions  
x_{1} = 0  E_{1} = α  x_{1/2} = 1  E_{1/2} = α  β  
x_{2/3} = ±√2  E_{2/3} = α ± β  x_{3} = 2  E_{3} = α + 2β  
E_{total} = 2(α + β√2) = 2α + 2.83β  E_{total} = 2α + 4β  
Fig. 1: Representation of the orbitals related the energy value α+b√2 (above, highest energy), α (middle) and αβ√2 (below, lowest energy). 
I  √2·c_{1} + 1·c_{2} + 0·c_{3} = 0 
II  1·c_{1} + √2·c_{2} + 1·c_{3} = 0 
III  0·c_{1} + 1·c_{2} + √2·c_{3} = 0 
Equation I establishes the relation c_{2} = √2·c_{1}. Inserted in equation II, we get c_{3} = −c_{1}−√2·c_{2}
= −c_{1} + 2·c_{1} = c_{1}. Thus, for the highest energy level, we find the orbital
π:  Ψ = c_{1} (Φ_{1}  √2·Φ_{2} + Φ_{3}) 
Normalization of the wave function yields the value of the coefficient c_{1}. Note that
1 = c_{1}² (1² + √2² + 1²) ⇒ c_{1} = ^{1}/_{2}
The secular determinant of butadien (C_{4}H_{6}, CH_{2}=CH−CH=CH_{2}) leads to an equation of the forth degree. The zeros of this equation are:

The zeros of this equation are:
x_{1} = − ½ − ½ √5 = −1.618 x_{2} = + ½ − ½ √5 = −0.618 x_{3} = − ½ + ½ √5 = +0.618 x_{4} = + ½ + ½ √5 = +1.618 
The single electron energies of the four LCAO molecular orbitals result from equation E = α  x β
E_{1} = α + 1.618 β
E_{2} = α + 0.618 β E_{3} = α − 0.618 β E_{4} = α − 1.618 β 
Four electrons need to be placed. The total energy of the molecule in the ground state is the sum of orbital energies occupied by electrons:
E_{total} = 2 (α + 1.618β) + 2 (α + 0.618β) = 4α + 4.472 β
If we compare this result with the 2·(2α + 2β) for two ethen molecules, we come to the conclusion that the energy of butadien is by an amount of [4α4.472 β]  [4α+4β] = 0.472 β below the energy of two πbonds. An energy difference of this kind, between a description with located conjugated πbonds and molecular orbitals obtained in an approach like Hueckel's is called delocalization energy.
Now we are able to estimate the energies of many systems quite easily. To obtain the respective wave functions, we determine the eigenvectors.
To each solution x_{i} of the secular equation, there is an eigenvector c_{i} which is calculated as follows: We insert the eigenvalue x_{i} into the secular equation and recieve three independent equations (one of the four is linear dependent ). We solve this system of equations. The value for the last coefficient emerges when the wave function is normalized. The method is exemplified for the first eigenvalue x_{1} = 1.618).
I  x_{1}c_{1} + 1·c_{2} + 0·c_{3} + 0·c_{4} = 0 
II  1·c_{1} + x_{1}c_{2} + 1·c_{3} + 0·c_{4} = 0 
III  0·c_{1} + 1·c_{2} + x_{1}c_{3} + 1·c_{4} = 0 
IV  0·c_{1} + 0·c_{2} + 1·c_{3} + x_{1}c_{4} = 0 
Equation I states:
c_{2} = −x_{1}c_{1} = 1.618 c_{1}
Inserted in equation II:
c_{3} = −c_{1}− x_{1}c_{2} = −c_{1} + 1.618 · 1.618 c_{1} = +1.618 c_{1}
Inserted in equation III:
c_{4} = −c_{2}− x_{1}c_{3} = −1.618 c_{1} + 1.618 · 1.618 c_{1} = c_{1}
The wave function 1π of the respective molecular orbital is a linear combination of the four porbitals Φ of the carbon atoms. In an analogous way, the wave functions for orbitals of higher energy Ψ_{2}, Ψ_{3} and Ψ_{4} are recieved by inserting the eigenvalues into the secular equations.
1π:  Ψ_{1} = c_{1}(Φ_{1} + 1.618Φ_{2} + 1.618Φ_{3} + Φ_{4}) 
2π:  Ψ_{2} = c_{1}(1.618Φ_{1} + Φ_{2} − Φ_{3} − 1.618Φ_{4}) 
3π^{*}:  Ψ_{3} = c_{1}(1.618Φ_{1} − Φ_{2} − Φ_{3} + 1.618Φ_{4}) 
4π^{*}:  Ψ_{4} = c_{1}(Φ_{1}− 1.618Φ_{2} + 1.618Φ_{3 } − Φ_{4}) 
Again, normalization yields coefficient c_{1}. Note that Φ_{i}·Φ_{k} = δ_{ik}:
1 = c_{1}² (1.618²
+ 1² + 1² + 1.618²)
c_{1} = 0.372
The following table provides an overview on the recieved coefficients for the butadien molecule for the four eigenvalues. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are denoted.
MO 1  MO 2
HOMO 
MO 3
LUMO 
MO 4  

eigenvalue  1.618  0.618  +0.618  +1.618 
atom 1  +0.372  +0.601  +0.601  +0.372 
atom 2  +0.607  +0.372  0.372  0.601 
atom 3  +0.601  0.372  0.372  +0.601 
atom 4  +0.372  0.601  +0.601  0.372 
4π^{*}  throughout antibonding  
3π^{*}  1  2; 3  4 bonding
2  3 antibonding 

2π  1  2; 3  4 bonding
2  3 antibonding 

1π  throughout (1234) bonding  
Fig. 2: Representation of the molecular orbitals obtained with Hueckel's approach to the butadien molecule. The hatched area shall represent positive values of the wave function 
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