Application of Hückel Molecular Orbital Theory (HMO) on Hetero-Conjugated Molecule, 3-Aminopropenal

Very often the application of quantum mechanics into chemistry represents a challenging task for chemistry students. However, this can be very usual exercise, and we have shown the easiness on the molecule of 3-aminopropenal, which is an interesting example because of the existence of the conjugation system consisting of the carbonyl group, alkenyl system and the lone electronic pair on nitrogen without any symmetry. Coefficients obtained using the Hückel secular determinant were 2.0484, -1.7328, -0.7827, +0.4892 and +1.5747.


1.Introduction
There is worldwide problem on the brain drain and insufficient interest in natural sciences, particularly physics and chemistry, which youngsters see as "waste of time and energy". Common trend in teaching is present through the application of different teaching methods, and popularization of natural sciences [1][2][3].
Covalent bond formation between two or more atoms in the molecule can be explained using valence-structural theory (VS method) and molecular-orbital method (MO-method). MO method investigates the molecule as an entity, thus on the same way as an atom, with the difference of the motion of electrons under the influence of more nuclei, i.e. in molecular orbitals.
Schrodinger equation (1) and its modified form (2) are used for the calculation of the energy of the electron in molecular orbitals. ^= (1) (- In equations (1) and (2), is a wave function, E is the energy of the molecular orbital, ^i s a quantum mechanical operator, which represents the total energy of the system. Usually, it is scary for students, but it only represents the group of the operations written in a shorter form that must be performed over the function ψ: Unfortunately, total solutions of the wave equation are today only possible for the simplest case of the hydrogen atom.
In case of the molecule with n electrons and η nuclei with the charge Zµ, the equation for Hamiltonian becomes: (3) The equation (3) can be simplified by the introduction of Born-Oppenheimer approximation and by the omission of terms related completely to nuclei.

Materials and methods
The inability of the accurate solution of the wave equation for multiatomic molecules gave different approximate approaches. One of the best is a linear combination of atomic orbitals (LCAO) [4,5], and Huckel [6] posted a theory suitable for the determination of the electronic structure of olefinic and aromatic hydrocarbons (Hückel molecular orbital theory (HMO theory)) [7]. Hückel introduced approximations into modified wave equation, and they are: where Hii is a matrix of elements which appear on the diagonal of the determinant. These terms represent the energy of the interactions of electrons in the isolated 2pz orbital with its own nucleus [8].
Integral of the resonance (β) represents the energy of the interaction of electrons of two nuclei at the same time, made by the overlapping of 2pz AO on the observed atoms [8].

Hij=∫
= or 0 (5) In this case, the interaction should be quite similar for all ij pairs because all σ-bond distances are approximately the same and all 2pz AO share the same node plane. Therefore, it is not incorrect to put approximation that these Hij terms are equal to the same value β. If centers i and j are not directly connected, the internuclear distance should be longer and the interaction energy represented with Hij should be significantly smaller. Then, the approximation is performed to make these terms zero in order to simplify the calculation [8].
Overlapping integrals (Sij) represent the measure of the overlapping of atomic p orbitals.
Introducing all three approximations, secular equations and determinants are becoming simpler and easier for solving, even for the more complex systems: If we develop the determinant, we are getting n th level of the secular polynomial PS which possesses real roots of the shape: (α-εj)=mjβ (j=1, 2, 3, …n) (7) and n connected energy values: εj=α-mjβ, (8) which are permitted energy values for MO or so-called Hückel eigenvalues. If a molecule contains a heteroatom (O, N, S), the correction must be introduced for α and β integrals related to the carbon atom [9,10]: (10) In the equations (9) and (10), α0 is a standard value, β0 is a standard bond integral, hx is the empirical constant without unit, which is characteristic for X, and depends on the difference in electronegativity of X and carbon [9,10]. One of the ways to get the parameter hx is the use of Pauling value for the electronegativity (χ): hx=k(χx-χC); it is the common practice to get k very close to one (Table 1S). kij is being taken as one for all normal C-C bonds in polyenes.
Coefficients Cji are fractions of atomic orbitals in molecular MO: and for each of these Ψj (or eigenfunctions) it exists the appropriate root mj (or xj) which is related to the energy of the electron in that particular MO. For each Ψj there is a set of secular equations from which it is possible to get the secular determinant. The problem is now the incorporation of each root into the corresponding set of secular equations, and the solving of the set of coefficients Cji for each Ψj. This gives us a description of the behavior (spatial distribution) of electrons in each MO. They can be obtained from secular equations: If we know energies of molecular orbitals (E), we can calculate total energies of the electrons, resonance energies, etc., and use coefficients Cji molecular-orbital indexes.
There are several very useful parameters: Electronic density (q) represents the probability of finding the electrons on particular atoms in the molecular orbital, and it can be calculated according to the equation (12): where ni is the number of electrons in MO, Cji 2 electronic density on atoms, and i occupied orbital. Electronic charge (ξ) is the charge of the atom and represents the difference of formal charge (Z) [11] and electronic density (q):

ξ=Z-q, Q=1 for a neutral carbon atom Z=(number of the group in the Periodic table)-(number of common electrons/2+number of lone electrons) Z(O)=1, Z(N)=1, Z(C)=1
ρ and N bond order ρ is a π bond order which determines the fraction of the electronic density which belongs to a particular bond. ρXY=∑ In the equation (13), X and Y are neighboring atoms, C-coefficient, i-bonded orbital, ni-number of atoms in the bonded orbital. N is the total bond order and represents the sum of σ and π bonds. NXY=1+ρXY, where 1 stands for σ bond, ρXY is π bond order, X and Y are neighboring atoms. Index of free valence (F) represents the reactivity of the center: where σ-bond orders are excluded from the consideration because it is assumed that any carbon in π system is sp 2 hybridized and completely σ-bonded to three other atoms. Regarding reactivity, this is understandable because molecules which contain π usually react firstly through electrons the most available or electrons of the highest energy, i.e. π-electrons. Nmax can be obtained by the observation of the central atom in the hypothetic molecule of trimethylmethane with a value √3 or 1.732 and represents the maximum number of π bonds. Nr=∑ -total sum of bond order around the observed center FX=1.732-∑

3.Results and discussions
Semiempirical HMO theory was applied for the determination of E, ψ and MO indexes of the acyclic conjugated compound represented with the structural formulae: Firstly, Huckel secular determinant was established: By the development of the determinant, we are getting polynomial which solutions are zeros of the polynomial (x1…n) ( Table 1).

Figure 1. Diagram of the energy levels for the investigated molecule
Filling of the molecular orbitals (MO) with electrons is almost the same as the filling of atomic orbitals and the same principles are followed.
On the basis of the obtained parameters, the reactivity of compounds can be calculated and the correlation with some properties of the molecule (l, µ, etc.).

Conclusions
By the application of semi-empirical HMO theory, approximative values of energies of molecular orbitals were found, explicit values of wave functions were presented, and MO indexes were calculated in an acyclic conjugated compound with two heteroatoms (O, N) in positions α and ω. Calculated theoretical parameters can be used in correlation with experimental data of the series of similar molecules.