2.3.1. Concerted Mechanism
The tri/tetra-methyl-substituted-tetroxane reactants are in a chair conformation (S-
c). The H atoms of the methyl groups in geminal position show a symmetric eclipsed conformation two to two with respect to the average plane of the molecule, in which all carbons are included (
Figure 1a). One H atom of each methyl group is in this plane, and two H atoms out of the plane of two geminal methyl groups are facing each other. The concerted mechanism has been calculated as one single step, where the transition state (TS) is called S-TS
X (
Table 2,
Figure 1a), with a transition vector (TV) of opening and closing the peroxide O-O bonds and opening C···O bonds to form the O
2 molecule and two molecules of acetaldehyde/acetone via the opening/closing of the C-O bonds. This TS and TV are depicted in
Figure 1a. The
Ea values are 66.3, 68.8, and 65.9 kcal/mol for TMT-axial (TMTax), TMT-equatorial (TMTeq), and ACDP, respectively, being lower than the less-substituted compounds of the series [
7,
8,
9]. The TMTax has lower
Ea than TMTeq probably due to the anomeric effect that stabilizes the axial isomer TS.
Taking into account all methyl derivatives, a correlation of the average values of
Ea as a function of the number of methyl groups has been performed (
Figure 1b). We observe that the
Ea decreases linearly with the increasing number of methyl groups, with a slope of −2.04 kcal/molCH
3 (
Figure 1b; in this figure, average values of axial and equatorial isomers of TMT are also considered). This slope is larger than that of the experimental
Ea as a function of the number of methyl group (Equation (3)). This fitting shows that the FDP
Ea for a concerted mechanism would be 73.8 kcal/mol, being considerably higher than the experimental value (29.3 kcal/mol) [
7]. Additionally, the value of
Ea = 65.9 kcal/mol for ACDP is the lowest of the series, which is much higher than our experimental value but is still high enough to avoid the concerted mechanism in front of the stepwise mechanism (vide supra).
The reaction product (S-P) (Branch S-I,
Scheme 1) shows an O
2 molecule in the singlet state surrounded by hydrogen bonds of two acetaldehyde/acetone molecules depending on if the reactant is TMT or ACDP. Reaction energy (
Er) is exothermic, being −31.1 and −28.0 kcal/mol for TMT axial and equatorial isomers, respectively, and −32.8 kcal/mol for ACDP. The series follows a linear equation as a function of the number of methyl groups:
Er = −17.1 − 4.0
n (R
2 = 0.98) (FDP is not included in this equation).
2.3.2. Stepwise Mechanism in the Singlet State
The stepwise mechanism is depicted in
Scheme 1 (Branch S-II). The mechanism has three steps: (i) the production of the open diradical structure (S-
o), (ii) the formation of acetone/acetaldehyde (S-
b′) and oxide-peroxide intermediates (S-
b), and (iii) the generation of the end products (S-
p), where the oxygen molecule is in the singlet state. Step (i) is common to all methyl derivatives of tetroxane. In Step (ii), only one possible reaction path is possible for ACDP, but in the case of TMT, two different secondary reaction pathways are possible (vide supra).
Let us start with the reactant, S-
c, which goes throughout one TS (S-TS
co) to the open diradical structure S-
o (
Figure 2). This S-TS
co has a much lower
Ea than the above S-TS
X. The
Ea for S-TS
co of TMT turns out to be 17.7 and 18.3 kcal/mol for axial and equatorial methyl position isomers, respectively (
Table 3 and
Table 4); and for ACDP
Ea = 18.5 kcal/mol (
Figure 2,
Table 5). Considering the
Ea as an average value (
Eaaver) of the axial and equatorial isomers, a function with the number (
n) of methyl groups of the different derivatives of the series can be expressed by a linear equation
Eaaver = 16.15 + 0.61
n, R
2 = 0.993. The most important distance in the TS, and so the simplified reaction coordinate, in the structure of S-TS
co, is the O···O distance, which, in the average of the axial and equatorial isomers (
dO···Oaver), also shows a linear function with the number of methyl groups:
dO···Oaver = 1.90 + 0.014
n, R
2 = 0.96. Indeed, the
Ea and
dO···O as a function of the number methyl groups show qualitative parallel trends, increasing the distance as increasing
Ea, but with a much lower slope of 0.014 Å/CH
3, indicating that the main coordinate of the TS has a small variation on the effect of substituents in the same reaction.
The first intermediate in the reaction is the diradical open structure, S-
o, with a distance between both O atoms of 3.22 Å and a spin density of −0.94 and 0.94 e
− (
Figure 2), indicating that both electrons have different spins, maintaining the identity of the singlet state. This intermediate gives endothermic reaction energy, increasing with the number (
n) of methyl groups, following also a linear function:
Erco = 9.0 + 1.22
n; R
2 = 0.992. The slope of
Erco is twice the slope of
Ea-TS
co, indicating that the methyl substitution effect is twice as large in the diradical open intermediates as in TS
co. Indeed, the linear equation relating both
Erco and
Ea (TS
co) is
Erco = −23.5 + 2.0
Ea (R
2 = 0.996). The O···O distance of S-
o follows the linear equation
dO···O = 3.519 − 0.074
n, R
2 = 0.997. Additionally, the reaction coordinate O···O of S-
o intermediate decreases with a slope of 4.6 Å with respect to this O···O distance in S-TS
co, following the linear equation
dO···Oo = 12.2 − 4.6
d(O···O)
TSco (R
2 = 0.99). All these effects agree with the Leffler–Hammond postulate [
16,
17].
The next step is the TS of a broken C-O bond, S-TS
ob′, which depends on either the two-methyl- or single-methyl-substituted C atom at TMT (
Figure 3). In the first case, acetone will be obtained as the product (S-
b′, Branch SII-ii of
Scheme 1), while in the second case, the mono-substituted carbon (R = H, CH
3 in
Scheme 1) will give acetaldehyde as the first product of reaction (S-
b′, Branch SII-i of
Scheme 1). Let us start with the mono-substituted carbon (Branch SII-i,
Scheme 1). In
Figure 3a the S-
o intermediate goes throughout S-TS
ob′ to S-
b′ (second intermediate product). The
Ea of S-TS
ob′ is 17.1 and 17.5 kcal/mol for axial and equatorial isomers with respect to the S-
o intermediate. In S-TS
ob′, the distinction between equatorial and axial isomers of CH
3 positions is not clear. Nonetheless, some slightly structural differences are found in the potential energy surface (PES) critical points. For this reason, average values are also taken. The transition vectors are mainly formed by the vibrations of C···O distance—which are 1.704 and 1.707 Å for axial and equatorial isomers, respectively—and the spin density is shared by all oxygen atoms. The product is an acetaldehyde molecule plus an oxide–peroxide diradical, S-
b′ (
Figure 3), whose
Er is exothermic with respect to the S-
o intermediate. However, it is slightly endothermic with respect to S-
c.
When acetone is left as the first product in TMT (Branch ii in
Scheme 1), the corresponding
Ea of TS
ob′ and
Erob′ are 15.4 and −13.9 kcal/mol, respectively, for the axial isomer and 15.4 and −12.7 kcal/mol, respectively, for the equatorial isomer (
Figure 3b). This reaction is more favorable than the above, where the aldehyde is left out, though the differences are not meaningful. ACDP yields acetone and a peroxyacetone diradical as reaction products in the second stage, giving 15.31 and −14.83 kcal/mol for
Ea and
Er, respectively.
Considering the Ea as an average value (Eaaver) of the axial and equatorial isomers and the two possible intermediates formed in this step of the different derivatives of the series, a linear function can be obtained. These Ea values low down to 1.3 kcal/molCH3, following the linear equation Ea(TSob′) = 20.6 − 1.3n (R2 = 0.93). The C···O distances have a linear function with respect to the number (n) of methyl groups: d(C···O) = 1.714 − 0.0013n (R2 = 0.99). The slope is quite small, indicating that this TS is quite constant with respect to the number of methyl groups. The average values of axial and equatorial isomers give an exothermicity of 2.5 kcal/molCH3 with respect to S-o, resulting from the linear equation Erob′ = −5.0 − 2.5n (R2 = 0.98). The correlation is low because the product’s disposition is not very regular. In general, the Ea of this reaction step decreases with the number of methyl groups, and Er is growing in exothermicity. These facts indicate the additivity of methyl groups in our methyl-tetroxane derivatives; that is, each methyl group contributes to decreasing the activation energy.
From now on, one of the products of the step (either acetaldehyde or acetone), S-
b′, is removed, and the oxide–peroxide diradical intermediate (S-
b) continues reacting. A second molecule of acetaldehyde or acetone and one oxygen molecule will be the products of this reaction step (
Table 3,
Table 4 and
Table 5). When S-
b is
·O-CH(CH
3)-O-O
·, TMT has the same
Ea of 11.6 kcal/mol (as average value). This value is similar to MFDP, and DMT [
8,
9]. The
Er of the acetaldehyde product (
Scheme 1, Branch SII-ii) is −29.01 and −28.64 kcal/mol with respect to
b for TMTax and TMTeq, respectively (
Figure 4a,
Table 4). However, when the acetone is the second product, S-
b is
·O-C(CH
3)
2-O-O
· (
Scheme 1, Branch SII-i), where the
Ea and
Er are 10.73 and −31.10 kcal/mol, respectively, for TMT and ACDP (
Figure 4b). Therefore, the reaction of TMT and ACDP in the singlet state is exothermic in the last step of reaction.
In the derivatives with three and four methyl groups, the largest TS is that of the first step—that is, S-TS
co, which could be the rate-limiting step. On the contrary, in FDP, MFDP, and DMT, the rate-limiting step looks like the second step, where S-TS
ob′ has the highest energy barrier [
7,
8,
9]. However, these energy differences are very small and both steps can be considered limiting step at experimental conditions. Our experimental
Ea for ACDP obtained in this work is 22.7 kcal/mol, which is consistent with our theoretical value of 18.5 kcal/mol of S-TS
co (
Table 5).
2.3.3. Thermolysis Reaction as Triplet State (Scheme 2)
For ACDP, a quasi-chair excited structure (T-qc) in the PES of the triplet state has been found at 14.9 kcal/mol above S-c. This structure is not as symmetric as S-c and has a dO···O = 2.82 Å, much larger than the other peroxide bonds (1.41 Å). The methyl groups are in the same conformation as S-c. This excited structure has a spin density of 0.942 e− on the two open oxygen atoms. The equivalent structure for TMT has not been found. The second structure found in the T-PES is the open diradical structure (T-o) close to S-o. This T-o presents an energy of 13.7 kcal/mol with respect to S-c and only −1.2 kcal/mol with respect to T-qc. Then, the energy of T-o is very close to T-qc, where the quasi-open structure of T-qc has a large dO···O distance similar to the open structure T-o. If a transition state exists from T-qc to T-o, we could not find it, but it should be with low Ea from T-o to T-qc.
The energy and structure of T-
o are very close to the S-
o, having a spin density of 0.939 e
− on each open oxygen of the peroxide bond, and an O···O distance of 3.222 Å for ACDP, and 3.286 and 3.330 Å for TMT axial and equatorial isomers, respectively (
Figure 5). The O···O distances decrease as the number (
n) of methyl groups following the linear equation
dO···O = 3.527 − 0.075
n (R
2 = 0.998). This equation and distances are very close to the S-
o-
dO···O distances, indicating the similar
o structures for the different derivatives and their very close variation with the number of methyl groups.
From T-
o a transition state T-TS
ob′ is also found. This new structure is the TS for breaking and forming the first acetone/acetaldehyde molecule. In TMT, different values are found depending on if either acetone or acetaldehyde is the first product in
b′. If acetaldehyde is the first product in
b′ (
Scheme 2, Branch i),
Ea is 16.98 and 15.13 kcal/mol for the axial and equatorial isomers, respectively (
Figure 5a). If acetone is the first product (T-
b′,
Scheme 2, Branch ii),
Ea is 15.20 kcal/mol for both axial and equatorial isomers (
Figure 5b). In ACDP,
Ea is 15.13 kcal/mol with respect to T-
o (
Figure 5b). The
Ea as a function of the number of methyl groups from FDP to DMT was previously calculated with a quadratic function [
9]. With the addition of the three and four substituted derivatives to the system, the quadratic function is
Ea = 25.3 − 6.2
n + 0.9
n2 (R
2 = 0.97), similar to that found in Reference [
9] (
Table 6,
Table 7 and
Table 8).
C···O distances for T-TS
ob′ are 1.708 Å for ACDP and 1.704 and 1.715 Å for TMT as average values of the position axial/equatorial isomers when acetaldehyde and acetone are the first products, respectively. These distances are not quite different from those of S-TS
ob′. TV (
Figure 5) is viewed as a vibration of the C···O breaking/forming the C···O bond. These distances vary with the number (
n) of methyl groups with a linear function: d
C···O = 1.714 − 0.0014
n (R
2 = 0.991). The slope of the function is very small, where the structure of the TS can be considered very stable as the number of methyl groups increases. This equation is very close to the d
C···O equation of the S-TS
ob′, which indicates an identity of structures in the TS
ob′ of the diradical part of reaction independently of the multiplicity of the state. Additionally, the
o diradicals also have very close structures for both multiplicity states, as already mentioned above. From this, we can extract that if both multiplicity states have equal reactants and TS’, the reaction path should be very close. The spin density of T-TS
ob′ is shared over all oxygen atoms in the structure (
Figure 5).
T-TS
ob′ goes to T-
b′, a super-molecular system with an acetone/acetaldehyde molecule and a peroxy-acetaldehyde/acetone diradical, with a C
acetone/acetaldehyde···O
peroxy distance of 3.956 and 3.473 Å for TMT
aver(ax-eq) and acetone as the first product of the reaction; 3.647 Å is the average of both position isomers when the acetaldehyde is the first product of the reaction. The reaction energy is −11.06 kcal/mol with respect to T-
o for TMT as an average value of both position isomers when acetaldehyde is the first product of reaction (
Figure 5a,
Table 6); and −13.36 kcal/mol for the equivalent value when acetone is the first product of the reaction (
Figure 5b,
Table 7). The
Er as a function of the number of methyl groups behaves decreasing with the linear function
Er = −5.5 − 2.9
n, R
2 = 0.93. The spin density of the triplet state is shared on the three oxygen atoms of the peroxy-acetone radical. From now on, the acetone molecule is removed from the super-molecular system and the critical point is optimized from the peroxy-acetone radical. Nonetheless, the energy of acetone molecule is constantly added up to the energy of the critical points in order to preserve the whole energy of the system. Following this diradical intermediate, T-
b, a T-TS
bp yields the final product (T-
p,
Figure 6). The
Ea is really the lowest of the T-PES and S-PES with values of 6.88–7.90 kcal/mol (
Table 6,
Table 7 and
Table 8). The
Ea as a function of the number of methyl groups is also linear:
Ea = 9.48 − 0.67
n, R
2 = 0.98. In the T-PES all
Eas low down with the increasing number of methyl groups, indicating the reaction rate is faster with the methyl substitution. Besides, this TS
bp is the lowest TS of the S- and T-PES. The last TS gives rise to the second acetaldehyde/acetone molecule plus the
3O
2 molecule. The TVs clearly show this formation (
Figure 6a,b). The spin density also is shared over the three oxygen atoms of the system. The highest
Ea on T-PES is that of T-TS
ob′, being the rate-limiting step of the reaction, which agrees with the other derivatives of the series.
Finally, the products are
3O
2 plus the second acetone/acetaldehyde molecule. The reaction energy with respect to T-
o is exothermic, with −58.2 kcal/mol for ACDP (
Erob′+
Erbp,
Table 8) and −54.2 and −53.9 kcal/mol for TMT
aver(ax-eq) for acetone (
Table 6) and acetaldehyde (
Table 7) as the second product, respectively. A linear function is also found for this step of reaction,
Erop = −41.4 − 4.3
n, R
2 = 0.998, describing the reaction energy of
op, giving an exothermicity of 4.3 kcal/molCH
3. Considering the first step of the reaction in singlet state for obtaining the biradical S-
o, the reaction energies of the reactionin triplet state with respect to reactant S-
c are −44.7 kcal/mol for ACDP and −41.4 and −40.7 kcal/mol for TMT
aver(ax-eq) for acetone and acetaldehyde as final products, respectively. If the reaction would proceed in singlet state PES, the
Er of the equivalent critical point for S-
p, with respect to S-
c is −24.0 kcal/mol and −29.9 kcal/mol for TMT
aver(ax-eq) for acetone and acetaldehyde as final products, respectively and −32.4 kcal/mol with respect to S-
c for ACDP (
Table 3,
Table 4 and
Table 5). This indicates that the reaction energies on T-PES are much more exothermic than in the S-PES, as expected based on the fact that the singlet molecular oxygen is more unstable than triplet molecular oxygen. The results in the T-PES are more in agreement with the experimental reaction energies of the diperoxide compounds.
The influence of the methyl groups in the thermolysis reaction of tetroxane derivatives increases the exothermicity of the reaction, and they change the rate-limiting step from the second step to the first step, although the differences in Ea are very small, and both TSs could be viewed as limiting steps at experimental conditions; additionally, they also change the geometry of intermediates and transition states in an approximately linear way. The singlet and triplet states present very close structures at the different critical point intermediates of the reaction. These similarities of structures indicate a possible nonadiabatic transition from the singlet to triplet states in such a way that reaction would be more exothermic if this transition of different multiplicity states is produced, which would agree with the high exothermicity of the tetroxanes. Therefore, it is worth providing general details on the intersystem crossing process, as described in the next section.