Theoretical calculations are made by using the software called Gaussian 03 [21] .

The different molecular structures have been optimized using the Functional Density Theory (DFT) [22] [23] [24] . The level of calculation used is B3LYP associated with bases 6-311G (d) and 6-311+G (d). The calculations are carried out gaseous phase and solution. N, N-Dimethylformamide (DMF) being one of the solvents used for the synthesis of these quinolines [18] , the calculations in solution are thus carried out in DMF. The Polarized Continuum Model (PCM) [25] [26] is taken as a solvation model. The optimization of the molecular geometries is followed by calculations of vibration frequencies at the same level of the theory in order to verify that the structures obtained are minimal.

Knowledge of variations in energy contributions to internal energy which occurs between the products and the reagents helps to characterize energetically a chemical reaction. For a given energy parameter X, its variation is determined according to the following relation (1):

Δ r X = ∑ ​ X ( products ) − ∑ ​ X ( reactants ) (1)

The thermodynamic determined quantities in this study are the reaction enthalpies ( Δ r H ), the reaction free enthalpies ( Δ r G ) and the equilibrium constants (K_eq). These quantities are calculated by using relations (2) and (3) below:

Δ r G = ∑ ​ Δ f G ( Products ) − ∑ ​ Δ f G ( Reactants ) (2)

Δ r H = ∑ ​ Δ f H ( Products ) − ∑ ​ Δ f H ( Reactants ) (3)

The equilibrium constant denoted K_eq characterizes the equilibrium state of a chemical reaction. The value of the equilibrium constant depends solely on the considered chemical reaction and the temperature. In this work, the equilibrium constants are determined at 298 K from the following relation:

K e q = exp ( − Δ r G ( T ) / R T )

The Frontier Molecular Orbital (FMO) theory allows to understand the chemical reactivity. In particular, the prediction of the acidity of the different tautomers is made from the calculation of the deprotonation energies. The study of the reactivity of these molecules is carried out by taking into account the FMO [27] . This theory indicates the ease with which a molecule can be transformed or can react with other molecules. It gives a prediction of the excitation properties of a molecule. As a result, it constitutes quantum parameters for the determination of molecular reactivity [28] [29] . The smaller the energy gap between the HOMO and LUMO orbitals, the easier the molecule can be excited. Similarly a large gap HOMO-LUMO indicates a high stability of the studied molecule which implies its low reactivity during chemical reactions. The condensed Fukui functions are obtained using the Yang and Mortier procedure [30] [31] [32] based on the finite difference method. The different values of local descriptors are calculated from equations (4) and (5) below:

f k + = q k ( N + 1 ) − q k ( N ) (4)

f k − = q k ( N ) − q k ( N − 1 ) (5)

where: q k ( N ) : Electronic population of the atom k in the neutral form; q k ( N + 1 ) : Electronic population of the atom k in the anionic form; q k ( N − 1 ) : Electronic population of the atom k in the cationic form.

We can gather all the information contained in the two Fukui functions in a single function called dual descriptor. This function is represented by expression (6) which is below.

Δ f = f k + − f k − (6)

where Δf is a function which represent the reactivity of the molecule, it is positive in the electrophilic areas and negative in the nucleophilic ones [33] .

The chemical potential μ measures the tendency of the electron cloud to escape from the molecule. It is a global property of the molecular system which is equal to the slope of the total electronic energy E. This energy is a function of the number of electrons N with an external potential V (r) constant. The studied parameter corresponds to the opposite of the electronegativity χ as defined by Pauling and Mulliken [34] [35] . This parameter is determined by the relation below (7)

μ = ( ∂ E ∂ N ) = − χ (7)

μ can be expressed as a function of the ionization potential IP and the electronic affinity EA and it is calculated by using the following relation (8).

μ = − IP + EA 2 = − χ (8)

The hardness ɳ η is determined from the first derivative of the chemical potential. And it must be noted that the softness S is the inverse of the hardness [36] [37] . They are given by the following relation (9)

η = ( ∂ μ ∂ N ) = ( ∂ 2 E ∂ E 2 ) = 1 S (9)

According to the theory of acids and bases, developed by Pearson [38] , these quantities can be expressed as a function of both the Ionization Potentials (IP) and Electronic Affinity (EA). It permits to determine the relation (10)

η = 1 S = ( IP − EA ) / 2 (10)

In order to characterize other electronic behaviors of the systems, we have interested ourselves to other descriptors such as the electrophile index ω which evaluate the ability of a molecule to generate an electron transfer. It is calculated according to the following formula [39] .

ω = μ 2 2 η = χ 2 2 η (11)

These descriptors can be calculated in a very simple way in the framework of Koopmans’ approximation [40] according to which:

IP = − E HOMO andEA = − E LUMO

E HOMO and E LUMO are the energies of the HOMO and LUMO orbitals.

De protonation energies are widely used for the prediction of Brönsted acidity in gaseous and aqueous phases [41] [42] [43] [44] . The deprotonation reaction is defined by the following relationship (12).

A H → A − + H + (12)

The Gibbs energy Δ H 0 298 of the protonic abstraction reaction is as follows (13):

Δ G 0 298 = Δ H 0 298 − T Δ S 0 298 (13)

The contribution of entropy is given by the expression (14):

− T Δ S 0 298 = − T [ S ( A − ) + S ( H + ) − S ( A H ) ] (14)

For a temperature of T = 298 K at standard pressure, the second term T S ( H + ) = 32.5 Kj .mol -1 gives the expression (15) [42] [43] .

Δ G 0 298 = Δ H 0 298 − T [ S ( A − ) − S ( A H ) ] − 32.5 (15)

which leads to the following relation (16):

Δ G 0 298 = G ( A − ) − G ( A H ) − 32.5 (16)

Previous theoretical work on the prediction of acidity has shown that gradient-corrected functional and hybrid functionals such as B3LYP associated with bases 6-311+G(d) and 6-311+G(3df, 3pd) lead to results in good agreement with the experimental data [45] [46] [47] [48] . This motivates our choice of the level of calculation implemented in this work.

The calculated thermodynamic parameters are free enthalpies and enthalpies of formation. From these thermodynamic quantities, the reaction enthalpies and the equilibrium constants are determined. The order of relative stability of the tautomers is established from the enthalpy energies of formation of the different compounds. The values of the thermodynamic quantities calculated in gaseous phase and in DMF solution are gathered in Table 1.

Keq1 and Keq2 are respectively equilibrium constants in directions 1 and 2

(Figure 1). The values of the equilibrium constants mentioned in Table 1 are approximately equal to 1 which indicates that the reaction is in its equilibrium domain. Experimental data have shown that the probable existence of a tautomer relationship could explain the failure of tetrahydroquinoline to be obtained from 5,8-dimethoxy-quinolin-4-one by Reetz conditions [17] . Equilibrium constancy calculations confirm this hypothesis. They highlight a tautomeric equilibrium relationship between different tautomers. For two given tautomers, the results show that the ketone forms (Q-4-one, DMQ, DBrQ) have the lowest formation enthalpy values (Table 1). These are therefore more stable than the enol forms. In the case of 5,8-dimethoxy-quinolin-4-one, the enthalpies of formation of tautomers have very similar values. Both ketone and enol forms have a very similar stability. The reaction enthalpies are all positive which means that the equilibrium reaction is an endothermic one.

Gibbs free energies of deprotonation ( Δ G 0 298 ) are calculated in the gas phase at B3LYP/6-311+G (d). The values of this thermodynamic quantity are grouped together in Table 5.

The grouped data in Table 5 show an increase in the free enthalpy of the deportation reaction with the methoxyl group. This energy decreases with bromine.

According to these data, we can establish the increasing order of the studied molecules by taking into account the free energy of deprotonation ( Δ G 0 298 ) at 298 K: This order is: ΔG[DBrQ (O¹¹-H)] < ΔG[DBrQ (N¹-H)] < ΔG[Q-4-one (O¹¹-H)] < ΔG[Q-4-one (N¹-H)] < ΔG[DMQ (O¹¹-H)] < ΔG[DMQ (N¹-H)]. Bromine contributes to reducing the free energies of deprotonation of the nitrogen and oxygen atoms while the methoxyl group increases the same thermodynamic parameter on the same atoms. These results indicate that a deprotonation of the N¹-H nitrogen in the case of brominated quinolin-4-one will be easier than that of the nitrogen (N¹-H) of the methoxylated derivative. Work has shown that there is an inverse relationship between the amplitude of ΔG and the strength of the acid. The lower the value of ΔG, the more acidic the site is [41] [42] [43] [44] . In this way, the nitrogen atoms N¹-H and oxygen O¹¹-H in the brominated compounds are more acidic. These sites are less acidic in the methoxylated derivatives. Through the analysis of these free energies of deprotonation, we can establish the following decreasing order of studied molecules according to acidity: DBrQ (O¹¹-H) > DBrQ (N¹-H) > Q-4-one (O¹¹-H)> Q-4-one (N¹-H)] > DMQ (O¹¹-H)> DMQ (N¹-H). For a given quinoline, the deprotonation energy of the hydroxyl (O¹¹-H) of the enol form is lower than that of the nitrogen (N¹-H) of the ketone form.