According to Parr, the electronic chemical potential (μ) can be defined from the Lagrange multiplier [13] . This definition is exactly the same deduced by Pearson.

The electronic chemical potential μ can be calculated from the energies of the frontier molecular orbitals E_HOMO and E_LUMO as follows

Other parameters of global reactivity called global hardness (η) and overall softness (S) were introduced from the variation of energy from one stationary state to another. These parameters are given by:

With µ: chemical potential, ρ(r): electron density, n(r): external potential of the system.

The first partial derivative of μ with respect to N (the total number of electrons) is defined as the overall hardness η of the system [15] [16] . It is related to the quantity S which is the global softness of the system.

The global hardness η can also be calculated from the energies of the frontier molecular orbitals E_HOMO and E_LUMO as follows

From the chemical hardness, a further parameter of reactivity is determined in order to define the energy stability due to the charge transfer. Global electrophilicity index (ω) [17] which is linked to the chemical potential by the following relation

It should be noted that the nucleophilic index cannot be defined by a variational procedure, because there is no molecular electronic stabilization along the subtraction of the electron density of a molecule. In lack of a nucleophilic descriptor, Domingo et al. [18] proposed that the hypothesis that a weakly electrophilic molecule is systematically highly nucleophilic is true only for simple molecules. High values of nucleophilic correspond to low values of ionization potential and vice versa. Domingo et al. used the energy of the Highest Occupied Molecular Orbital (E_HOMO) obtained by the Kohn-Sham method [18] . The empirical (relative) nucleophilic index (N) [19] is defined as follows:

: Energy of tetracyanoethylene (TCE)

Here, tetracyanoethylene is taken as the reference in the nucleophilicity scale, because it presents lowest HOMO energy.

The maximum charge transfer () represents the maximum load proportion a system can acquire from its environment [20] . It is defined as follows:

Electrophilicity gap (Δω) between the reactants makes it possible to predict the polarity of the reaction [21] . This descriptor is defined by:

Fukui functions corresponding to the site k of a molecule is defined as the first derivative of the electronic density of a system with respect to the number of electrons N to a constant external potential [22] .

The condensed form of Fukui functions in a molecule with N electrons has been proposed by Yang et al. [23] by:

for nucleophilic attack (20)

for electrophilic attack (21)

for radical attack (22)

with

: Electronic population of the atom k in the neutral molecule

: Electronic population of the k atom in the anionic molecule

: Electron population of the k atom in the cationic molecule

The product of the global electrophilic index ω and the electrophilic Fukui index make it possible to identify the most electrophilic site. This site is identified by the determination of the local electrophilic index () [24] , defined as:

Similarly, the most nucleophilic site can be identified by the local nucleophilic index, [25] ; Defined as the product of the global nucleophilic index N and the nucleophilic Fukui index.

According to Domingo et al., The local electrophile and local nucleophilic index are reliable descriptors for the prediction of the most favorable nucleophilic-electrophilic interaction for the formation of a chemical bond between two atoms [26] . This chemical bond takes place between the nucleophilic site (characterized by the highest value) of the nucleophilic molecule and the electrophilic site (characterized by the greater value) of the electrophilic molecule.

The condensed Fukui Functions and the global softness S also make it possible to determine the local softness [27] , defined by:

The condensed local softnesses can be calculated from the following expressions:

where +, −, and 0 signs show nucleophilic, electrophilic, and radical attack, respectively.

According to the rule of Gazquez-Mendez [28] , Two chemical species interact through the atoms having equal or neighboring softnesses.

The values of the local reactivity descriptors, in this case the Fukui function (,), the local softness (,), the local electrophile power (), and the dual descriptors () of the reactivity have been determined. These values of the local descriptors on the atoms P₁, P₂, C₁ and C₂ of the reactants 1a, 1b and R₂ calculated according to the model of Domingo et al. With the Natural Population Atomic (NPA) [29] [30] [31] and Mulliken (MK) [31] approaches at B3LYP/6-311+G(d, p) are reported in Table 3.

On analysis of the values in Table 3, phosphines 1a and 1b have the highest values of the local nucleophilic indices. Similarly, the carbon C₁ of the compound R₂ has the highest value of the local electrophilic index (). This shows that the most favored interaction takes place between the P₁ atom of the compound 1a and the C₁ atom of the compound R₂ for the first reaction, and between the P₉ and C₁ atoms for the second reaction. Therefore, the formation of experimentally observed P₁-C₁ and P₂-C₁ bonds are correctly predicted by the Domingo model with the Mulliken and NPA approaches.

The prediction according to the Gazquez-Mendez model presents values of the Fukui function (,), local softness for reactants R₂ and local softness for reactants 1a or 1b. These values of the local descriptors on the atoms P₁, P₂, C₁ and C₂ of the reactants 1a, 1b and R₂ were calculated according to the Gazquez-Mendez model with the NPA population analyzes and MK at the B3LYP 6-311+G level (d, p) are given in Table 4.

Examination of the values in Table 4 indicates that the phosphines 1a, 1b and dichloroethylene R₂ have similar values of local softnesses (,) respectively on P₁, P₂ and C₁ atoms by the approach of Mulliken. This observation shows

that the most favored interaction takes place between the P₁ atom of the compound 1a and the C₁ atom of the dichloroethylene for the first reaction and the P₂ atom of the compound 1b and the C₁ atom of the dichloroethylene for the second reaction. Also in Table 4, with the NPA approach, the values of the local softnesses of the phosphines 1a and 1b on the atoms P₁ and P₂ are closer to the values of the local softnesses The C₁ atom of the compound R₂ compared to the C₂ atom, which also confirms the interaction between the phosphors of the various phosphines and the least substituted carbon.

In order to elucidate the reaction mechanism and compare the stability of the formed phosphines, the values of the reactant energies, the energies of the products, the energies of the transitions states and the activating energies are presented in Table 5 below.

From Table 5, it can be seen that the activation energy found by the reaction pathway give a value of 0.03765 kcal/mol. This value is inferior to what has been found by the reaction pathway through 2b (29.4925 kcal/mol). The phosphine 2a is more stable than that of 2b.

In order to show that the transition states (TS) are well connected to minima (reactants and products), the calculation of the Intrinsic Reactions Coordinates (IRC) and the curves of the reaction pathway, E = f (RC) are shown in Figures 3-5.

The evolution of the IRC curve (Figure 4), Indicates that the energy of the product obtained (−1892.770 Hartree) is greater than that of the reactant (−1892.772 Hartree) with thiophenyl as radical. This means that the compound (2b) obtained is kinetically unstable. The product 2a is therefore more stable than the product 2b, under the same conditions.

Figure 3 shows that the phosphorus carbon (P-C) bond is established in the transition state on the most hydrogenated C₁ carbon and the hydrogen of the phosphorus then on the C₂ carbon bonded to the chlorine. Similarly, the phosphorus carbon bond is established first on the most hydrogenated carbon during the formation of the product 3a. In the transition state, the hydrogen carbon bond is not yet established (Figure 5).

Reaction pathway therefore makes it possible to propose the following reaction mechanism Figure 6.

The products 3a and 3b have been obtained after the formation of the products 2a and 2b.

Thus, during the addition of 1a to R₂, respectively 1b to R₂, two reactions occur in succession:

This explains the stability of the products 3a and 3b obtained experimentally and the fact that they are major products [10] .

Products 3a and 3b are obtained in an excess of reactant (R₂), unlike product 2a and 2b which require less reactant.

The influence of the solvent on the getting of the product 2a was envisaged at B3LYP/6-31G (d, p). The results are shown in Table 6 below.

The study of the addition reaction envisaged in the solvents gives satisfactory results at the level of theory used. The activation barrier (E_a = 2.51 kcal / mol) is the lowest when the diethylene is used as a solvent. The diethylene favors the obtaining of a product kinetically more stable product. Consequently, the product 2a can be obtained optimally in diethylene. These theoretical predictions are in agreement with the experimental studies carried out (Figure 7).