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+hX. The value 1 is inserted for C-C bonds, a value kC-X for C-X bonds. For the majority of cases, the needed values for hX and kC-X can be taken from the table below.
hX C-X  kC-X
2.84  C-F  0.68
Cl  1.45  C-Cl  0.57
Br  1.16  C-Br  0.38
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

Ei  =  α - xi 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).
For the acrolein molecule in the ground state, the total energy of the four π-electrons is

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

### Eigenvectors

To any solution xi there is an eigenvector ci which is calculated as follows:

R=1 (hRS-EiδRS) ciR  =  0

If there is a bond R-S,
then hRSRS, otherwise hRS = 0
and αRS = 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 following electron configuration represents the ground state of the molecle:

occupancy:  Ψ12 Ψ22 Ψ30 Ψ40

The formula

qR  =  pRR  =  i bi ciR²

and

pRS  =  i bi ciR ciS

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

p-Matrix
0.8863
0.9342  1.0351
0.0  0.3479  0.7485
0.0  0.0  0.8909  1.3302

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

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

pRR = 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: QR = NR - pRR.

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

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

pRS = 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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