====== Results of the PMO Theory Approach ======

The //PMO// theory remains the best tool to discuss the reactivity of various chemical reactions. Our reaction is a suitable case for its application and, moreover, in π-approximation. The interaction energy of the PMO theory, also called "charge transfer" (CT) energy, has a well-known form:

{{:reactivity:radical_addition:pmo-1.png?direct&300|}} 
<html><span class="rfloat"><br><b>(eq. 1)</b></span></html>

where the summing is done over a set of spin orbitals and //γ// is the resonance integral on the atomic orbitals (//AO//) of reactive centers //r// and //s//.

Using RHF π-orbitals of isolated reagents we calculated CT energies (//E<sup>CT</sup>//) of all series in consideration (see, e.g., Tables V and VI). Figure 4 shows that there is some symbasis between them and the activation barriers of corresponding reactions. Addition to a substituted position (α-addition) gives less satisfactory correlations, that are obviously caused by a greater steric effect (deformation of reaction center). 

So, //E<sup>CT</sup>// could also be taken for another factor of reactivity. But, there is one inconvenience with it: it is not an ordinary index of reactivity; it is the whole set of energies of MO\|s and their coefficients of the AO\|s of the reaction center. It is no wonder that when using the //PMO// theory researchers frequently truncate orbital sets to //HOMO// and //LUMO// to avoid the summing in **Eq. (1)**; this method is known as the frontier molecular orbitals (//FMO//) approach.

Although //FMO// cannot be approved as a true approximation (see [[#Footnote|Footnote]]), it has an obvious advantage: the coefficients and energies of FMO may be used as indices of reactivity.

The use of //localized molecular orbitals// can serve as another approach. For example, using a projective procedure we managed to build a pair of //localized MOs// of a monomer that combined ≈95% of π-density both of occupied and unoccupied shells. It is a pity that even 5% is too great an error for such sums as **Eq. (1)**. So, a desire arises to "correct" the energies (or coefficients) of such localized orbitals in such a way that they would give **exact** values of the sums in **Eq. (1)**. As a result, we would get some //fictitious// but //effective// orbitals.

Now, we are going to propose an [[.:altpmo|alternative notation]] of the //PMO// **Eq. (1)**, that gives such effective MO\|s.

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**Table V.** //HOO<sup>•</sup>// addition to vinyl monomers.
//E<sub>a</sub>//, kcal/mol; //E<sup>CT</sup>//, (γ<sup>2</sup>/eV); //N// и //G//, eV.

^  \\ Monomer  ^  β-attack  ^^^^  α-attack  ^^^^
^  :::  ^  //E<sub>a</sub>//  ^  //E<sup>CT</sup>//  ^  //N//  ^  //G//  ^  //E<sub>a</sub>//  ^  //E<sup>CT</sup>//  ^  //N//  ^  //G//  ^
| //Eth//  |  26.4  |  −0.286  |  1.304  |  14.1  |  ||||
| //Prop//  |  25.8  |  −0.294  |  1.147  |  13.7  |  31.0  |  −0.285  |  1.460  |  14.2  |
| //AN//  |  26.0  |  −0.287  |  −1.740  |  14.1  |  30.3  |  −0.271  |  0.284  |  14.8  |
| //MA//  |  26.8  |  −0.284  |  −2.187  |  14.4  |  31.3  |  −0.270  |  0.726  |  14.8  |
| //VA//  |  23.9  |  −0.307  |  2.011  |  13.3  |  26.8  |  −0.278  |  0.738  |  14.4  |
| //VCl//  |  26.5  |  −0.284  |  −0.355  |  14.1  |  30.2  |  −0.273  |  0.233  |  14.7  |
| //VCl2//  |  25.7  |  −0.289  |  −1.625  |  14.0  |  32.0  |  −0.266  |  −0.459  |  15.1  |
| //St//  |  23.0  |  −0.311  |  1.371  |  13.0  |  ||||
| //ViNaph//  |  22.7  |  −0.313  |  1.379  |  12.9  |  ||||
| //2ViPy//  |  23.3  |  −0.303  |  0.788  |  13.3  |  ||||
| //4ViPy//  |  23.4  |  −0.299  |  0.125  |  13.4  |  ||||
| //Bu2ene//  |  22.6  |  −0.310  |  1.395  |  13.1  |  ||||
| //Hex3ene//  |  21.8  |  −0.315  |  1.400  |  12.9  |  ||||

**Table VI.**
//CH<sub>3</sub><sup>•</sup>// addition to vinyl monomers.
^  \\ Monomer  ^  β-attack  ^^^^  α-attack  ^^^^
^  :::  ^  //E<sub>a</sub>//  ^  //E<sup>CT</sup>//  ^  //N//  ^  //G//  ^  //E<sub>a</sub>//  ^  //E<sup>CT</sup>//  ^  //N//  ^  //G//  ^
| //Eth//  |  13.9  |  −0.349  |  −0.735  |  11.5  |  ||||
| //Prop//  |  13.6  |  −0.359  |  −0.960  |  11.2  |  18.1  |  −0.348  |  −0.675  |  11.5  |
| //St//  |  11.4  |  −0.378  |  −1.054  |  10.7  |  −0.344  |  −0.766  |  11.7  |
| //AN//  |  13.0  |  −0.391  |  −3.405  |  11.3  |  17.0  |  −0.355  |  −1.714  |  11.5  |
| //MA//  |  13.4  |  −0.388  |  −3.738  |  11.5  |  18.0  |  −0.349  |  −1.342  |  11.6  |
| //VA//  |  12.1  |  −0.370  |  −0.385  |  10.8  |  16.7  |  −0.346  |  −1.353  |  11.7  |
| //VCl//  |  13.3  |  −0.369  |  −2.216  |  11.3  |  16.9  |  −0.354  |  −1.716  |  11.5  |
| //VCl2//  |  12.6  |  −0.396  |  −3.343  |  11.1  |  19.6  |  −0.363  |  −2.394  |  11.5  |

**Figure 4.** Relationships between activation energies and CT energy of the //PMO// method, //HOO// addition. •, β-attack; ○, α-attack. 1, //Eth//; 2, //Prop//; 3, //St//; 4, //AN//; 5, //MA//; 6, //VA//; 7, //VCl//; 8, //VCl2//.

{{:reactivity:radical_addition:pmo-2.png?direct&300|}}

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=== Footnote: ===
In our case the //FMO// approach failed entirely, for example, styrene having the most reactive //C=C// bond (both experimental and calculational data) appears to be completely inert from the point of view of the //FMO// approach. Obviously, this is due to truncation of many π-orbitals in this case, not to the different electronic structure of styrene.

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