Electron configuration based on Hückel's mathematical treatment of acrolein

With acrolein as example, Hückel's method to calculate molecular orbitals and the results are explained. (chapter based on a work by Dr. Horst Bögel).

The acrolein molecule (CH₂ = CH - CH = O)

First, we build a secular determinant which describes the structure of acrolein. The atomic parameter are found within the diagonal, for a carbon atom, x is inserted, for heteroatoms X the sum x+h_X. The value 1 is inserted for C-C bonds, a value k_C-X for C-X bonds. For the majority of cases, the needed values for h_X and k_C-X can be taken from the table below.

X	hX		C-X	kC-X
F	2.84		C-F	0.68
Cl	1.45		C-Cl	0.57
Br	1.16		C-Br	0.38
J	0.78		C-J	0.19
:O	2.06		C-O:	1.31
:O-CH3	1.96		C=O	1.93
.O	1.18		C-N:	1.30
:N	1.47		C=N.	1.06
.N	0.83		C-CH3	0.18
:CH3	0.88		N-O	1.95
from: G. DERFLINGER, H. LISCHKA: Mh. Chemie 100 (1969) 1003

 
 

Acrolein's secular determinant is

x   1.0   0.0   0.0   1.0   x   1.0   0.0   0.0   1.0   x   1.93   0.0   0.0   1.93  x+1.18

Transformation of this determinant yields an equation of the fourth degree:

x⁴ + 1.18 x³ - 5.725x² - 2.36x + 3.725 = 0

The zeros of this equation or, respectively, the eigenvalues of the determinant are

x1 = -2.7654

x2 = -1.0207

x3 = 0.6880

x4 = 1.9182

The four eigenvalues and the following equation yields the energy levels of the molecular orbitals

E_i  =  α - x_i β_-

These levels are shown in the diagram here. Empiric values are α= -11 eV and β= -2.5 eV. The total energy of the molecule equals the sum of the electrons' energies in the respective molecular orbitals. The total energy of the π-electrons is E  =  ∑i=1 bi⋅Ei where bi represents the occupancy of orbitals with the values 0 (empty), 1 and 2 (pair of electrons).

E = 2(α+2.7654β)+2(α+1.0207β) = 4α+7.5722β = -25.0695 eV

Eigenvectors

∑_R=1 (h_RS-E_iδ_RS) c_iR  =  0

If there is a bond R-S,
then h_RS =β_RS, otherwise h_RS = 0
and α_RS = 1 for R=S, otherwise _RS = 0.

The following table contains the results:

	MO 1	MO 2  HOMO	MO 3  LUMO	MO 4
eigenvalue	-2.7654	-1.0207	0.6880	1.9182
atom 1	0.0919	-0.6593	-0.6990	-0.2613
atom 2	0.2542	-0.6730	0.4809	0.5012
atom 3	0.6111	-0.0276	0.3682	-0.7002
atom 4	0.7439	0.3341	-0.3804	0.4362

The obtained molecular orbitals are depicted below:

MO Ψ12
MO Ψ22 (HOMO)
MO Ψ32 (LUMO)
MO Ψ42

Bond order, atomic charge, π-electron density

The formula

q_R  =  p_RR  =  ∑_i b_i c_iR²

p_RS  =  ∑_i b_i c_iR c_iS

allow to establish a p-matrix that, in turn, serves to construct a molecule diagram.
 

The values within diagonal of the matrix indicate the electron density (p_RR) near the respective atoms.

Example: density of electrons near carbon atom C₍₂₎

p_RR = 2 * (0.2542)² + 2 * (0.6730)² + 0 * (0.4809)² + 0 * (0.5012)² = 1.0351

The formal charge of any atom is: Q_R = N_R - p_RR.

Besides the diagonal of the matrix, the bond order appears.

Example: bond order for the π-bond C=O between carbon and oxygen

p_RS = 2 * (0.6111 * 0.7439) + 2 * (0.0276 * -0.3341) + 0 * (0.3682 * -0.3804) + 0 * (-0.7002 * 0.4362) = 0.8909

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