The Fulminic Acid-Acetylene Cycloaddition is Not Concerted
Raymond A. Firestone
330 W 72 St., Apt 10A, New York, NY 10023
Revised June 15, 2017
Recently, two important papers appeared from different groups, both calculating by B3LYP/6-31G the reaction coordinate (RC) for the title reaction. Both concluded that concert obtains, although the formation of the first (C-C) bond was completed before the second bond (C-O) even began to form. The molecule after the first, but before the second, bond could only be a diradical. Why then concert? Because, said group 2, the interbond interval was so short, ~13 fs. The present manuscript rebuts concert on these grounds: (1) diradicals like this have often (≥62x) been intercepted, proving their independent existence; (2) there is no experimental evidence that the 13 fs interval is correct; (3) concert fails to explain why acetylenic dipolarophiles are not more reactive than ethylenic ones; (4) the Born-Oppenheimer principle says that electronic motion is immensely faster than nuclear motion, and that no Woodward-Hoffmann transition state (TS) exists on this RC; (5) in thermal reactions, RRKM requires activated molecules to undergo intramolecular vibrational redistribution (IVR) before bonding changes can occur, which takes ~500-1000 fs; (6) experimental measurement of bond formation times are 3 ns (Au-Au); in SN2 and carbonyl additions (C-C), ≥10 ns; even with photochemical activation (C-C), ~20 ps. All these facts are incompatible with concert, and fit diradicals.
The concerted mechanism for the Diels-Alder reaction (DA) has held sway since 1935 . 28 years later, Huisgen recognized 1,3-dipolar cycloadditions (1,3-DC) as a new class of reactions that appear to proceed in the same way, i.e., concerted . In 1937, Littman suggested on experimental grounds that the diradical mechanism might be a better one for the DA , and he has had a number of followers: Kistiakowsky , Walling  and Firestone , among others.
The advent of the Woodward-Hoffmann Rules in 1965  seemed to end the discussion, for a symmetry-allowed concerted cycloaddition must be much faster than a stepwise one since the rate-determining step for concert has two more bonding electrons than that for diradicals, and also because a concerted transition state (TS) for either DA or 1,3-DC possesses some aromatic stabilization lacking in the stepwise TS.
The two bonding electrons present in the concerted TS but absent in the diradical provide ~80 Kcal/mol extra stabilization for concert, reduced to ~40-50 after adjustment for radical stabilization energies . This gives rise to an enormous factor, about 1030 in rate, that conclusively ensures that any given cycloaddition must be either concerted or stepwise, but never both mechanisms side by side.
The difference between the observed Ea and that for diradicals, subsequently named energy of concert (Econ), tells us what that correct mechanism is. If Econ is large, it is concerted; if small, stepwise. Huisgen calculated from experimental data that Econ must be 37-58 Kcal/mol , and Firestone estimated, also from experimental data, ~43 Kcal/mol. So the two opposite camps agree on this, if not much else. Econs ranging from 12-30 Kcal/mol have also been derived from quantum mechanical calculations (QMC), e.g., 12.4, [10, 11], but these are suspect because they vary so widely and also because calculations unsupported by experiment are hypotheses, not facts. Although Econ is derived from one experimental number and another that is perforce only estimated, it is surprisingly easy to determine it precisely .
There are now 42 experimentally verified examples of DA and 1,3-DC that are unquestionably stepwise-diradical, and 46 more whose Econs are much too small for concert . These are 88 out of thousands of examples, the rest of which are generally said to be concerted. However, proving concert experimentally is no simple matter, requiring mass balances of well over 99%. How high must they be? The most precise in the literature  are 99.997% and 99.94%, but even these give rise to minimum Econs of only 3.5 and 1.7 Kcal/mol . Concert is still possible here, for no loss in stereospecificity could be detected, but it is certainly not proven.
The Current Scene
In the past decade, two major research groups have analyzed reaction (1) by quantum mechanical calculations (QMC) using the B3LYP/6-31G method. Both pronounced this particular (hypothetical) 1,3-DC to be concerted:
(1) HC=N=O + acetylene → isoxazole
However, perusal of their data reveals that in both studies the C-C and C-O bonds are calculated to form sequentially, not simultaneously. That means that they cannot be concerted. In the first , the new C-C bond forms to completion before the C-O bond even begins to form. The only possible structure for the interim species is a diradical.
The second paper , however, depicts the same scenario – stepwise – but declares it a concerted cycloaddition because the time gap between formation of the two new bonds in reaction (1) is calculated to be only 13-15 fs, i.e., less than one vibration-time. Other cycloadditions within a group of 18 related cases had calculated time gaps ranging from 0 to 15 fs. A subsequent paper described a similar scenario for DA [15, 16].
Could this  apply to all 1,3-DCs and DAs? No, because in most of the 88 examples in ref.  a diradical was intercepted in some way: rotation within the dienophile (15 examples), intramolecular H transfer (26), loss of a small molecule (3), capture by a free-radical trap (1), or consequences of singlet-triplet interconversion by means of a nearby heavy atom (17). There is an 18th highly probable example . Interception proves the independent existence of diradicals for long enough – much longer than 15 fs – to be diverted to another pathway. Other examples exist in which concert is impossible because Econ is too small, or zero, sometimes even less than zero which makes concert absurd . Therefore, the scenario in ref  cannot be true in these 88 cases. It is also weak because this QM method “has systematic errors for hetero systems.” 
Another drawback in the fs argument is that it has no experimental support. No one has actually measured the time lapse between bonds 1 and 2. Without that it’s a hypothesis and no more. However, in some other reactions people have experimentally determined the time to form bonds, which turns out to be much longer than 13-15 fs (v.i.).
Furthermore, even the hypothesis is on shaky ground. Refs.  and  are not the last word in these calculations. A third group of theoreticians, using the same method, reached the opposite conclusion, i.e., that these cycloadditions are stepwise-diradical . This makes it even more imperative to settle the question with hard facts .
In reaction (1), the authors recognized a problem in that the calculated Eas were the same for both ethylenic and acetylenic dipolarophiles. There was no kinetic advantage for acetylenes even though the cost of opening a triple bond is much less than that of a double bond. Furthermore, there was no kinetic advantage when the cycloadduct was aromatic . Parallel results were reported for cycloaddition of hydrazoic acid to ethylene and acetylene, and formaldimine and HCN . This problem was also remarked elsewhere . These results had been long known experimentally, with the same comment .
This conflicts with concert  , for it is easier by ~10 Kcal/mol to break an acetylenic than an ethylenic pi-bond. In addition, when the cycloadduct is aromatic, it should have a further ~10 Kcal/mol lowering of the Ea for concert, where there is only a single TS whose properties are derived from both the reactants and the product.
If we assume that only half of each of the two 10 Kcal/mol advantages for the acetylene is present in the Ea, acetylenes should be >10,000,000x more reactive than olefins. But they are not. Half of each advantage is reasonable since the TS in one typical 1,3-DC is about halfway down the RC [18, 24].
Why does this problem not apply to diradicals? Because here there are two sequential TSs, the first rate-determining. At this TS (virtually identical with the diradical), the entire acetylenic pi-bond is consumed, favoring it over an ethylene by the full 10 Kcal/mol. However, this diradical contains a vinylic radical which has 10 Kcal/mol more strain energy than a saturated one. So they cancel exactly . As for aromaticity, it plays no role here because its inception is delayed to the second TS, after the rate-determining first TS. So the diradical pathway double- and triple-bonded dipolarophiles should both react at similar rates, aromatic or not, which they do.
Clearly the acetylene-ethylene question in 1,3-DCs contradicts concert. This question, first discussed in 1968 , is also recognized in ref. , but without explanation.
Sequence of Bond Formation
Now consider the species that exists immediately after completion of the first bond but before inception of the second in ref ’s scheme.
The first bond has a normal length, but the gap between the C and the O not yet joined together must be longer than a normal bond; otherwise it would already be the second bond. The as-yet-nonexistent second bond cannot be conflated with the C-O bond in the product because the Born-Oppenheimer approximation  teaches that the lightweight electrons, whose mass is 1/29440 that of an O nucleus and 1/23920 of a CH unit, move so much faster than O atoms or CH units that they assume new equilibrium positions essentially instantaneously at every new point in space traversed by the heavier atoms as they move, relatively slowly, to their new bonding positions. The electrons follow, and cannot precede, the heavy atoms. At this instant – this very fs – the molecule is a diradical. The electrons cannot move into position for bond 2 during formation of bond 1, because the heavy atoms aren’t there yet , so there must be two successive TSs. At no time is there an aromatic Woodward-Hoffmann TS, and concert is ruled out. The time interval between formation of bonds 1 and 2 must be long enough to permit interception of the diradical (v.s.).
Time Required for Bond Formation
How long does it take to form the second bond? In 1922 Lindemann pointed out that since collisions are the means for energizing molecules to cross the activation barrier in either direction, bond formation or cleavage cannot occur within a time as short as a single vibration (1013/sec), for if it did, collisions (1010/sec) would be rate-determining, which would make all such reactions second order, not first . But first order reactions do occur. This is known as the Unimolecular Anomaly. Collisions reversibly activate molecules to high vibrational states, and these are what go ahead to the product, but it takes a lot longer than 10-13 sec (100 fs).
Lindemann’s insight strongly influenced the evolution of chemical kinetics, and modern RRKM theory  holds that intramolecular vibrational energy redistribution (IVR) in molecules whose energy is ~ Ea is always completed before further changes in bonding can occur (ergodic behavior). IVR is normally ~0.5-1 ps , i.e. 500-1000 fs, so bond formation must take longer than this. Large molecules (>3-4 atoms) obey RRKM [29, 30], i.e., long reaction times relative to IVR. Thus reaction times as short as 13 fs cannot be correct.
Nonergodic Behavior – Reaction Times Faster Than IVR
This is indeed seen , but only under special conditions: a very weak breaking bond (e.g. van der Waals dimers), or pressures >100 atm , or with ultrashort activation pulses to energies far above Ea . None of these conditions applies to reaction (1), Normally, it is collisions, not hν, that elevate reactants to energies ~ Ea, but never far above Ea.
Time Delay in Unimolecular Reactions
Consider an intramolecular 1,3-DC in detail. The reactant is vibrationally activated by collision. If subsequent events occurred in a few fs, the overall transformation would be second order: first order in both reactant and solvent. Unimolecular kinetics can exist only if the first vibrationally activated reactants live long enough for most of them to be deactivated by another collision while a few other activated molecules eventually go on to product. That’s tens of ps, not fs, so concert is ruled out.
Time Delay in Bimolecular Reactions
I chose an intramolecular 1,3-DC to begin discussion of time delay in order to utilize the Unimolecular Anomaly. Many examples now exist of bimolecular reactions where energetic, but very short, interaction times proceed at very low rates. Second-order reactions are not bound by the Anomaly, but it is likely that they also involve collisional activation, followed by reaction of vibrationally activated molecules. Some examples:
In 1968 Bauer reported the pericyclic reaction (2)
(2) H2 + ND3 ↔ HD + ND2H
at 700o in a shock tube . As with unimolecular reactions, the exchange does not take place directly via translational energy (ET), for the kinetic expression is first order in ND3 and in Ar, but zero order in H2. Clearly ND3 is vibrationally activated by collision with Ar, and then reacts with cold H2. There must be a time delay between the two encounters comparable with the delay in unimolecular reactions between collisional activation to a vibrationally activated state with subsequent rearrangement. Bauer and his associates reported a number of examples like this.
Similarly, high ET was unable to induce reaction between vibrationally cold HI + DI, but if one partner had high EV reaction took place .
D + H2 reaction slows down as ET rises above Ea because the reaction time is too short . Similarly, with Li + HF (EV = 0) in crossed beams, the rate rises sharply as ET goes down .
In all these cases, short reaction times are less productive than long ones. Fs reaction times are much too short. Nowadays there are many more examples.
Experimental Measurement of Bond Formation Times
A recent publication describes the measurement of the time it takes to form a single Au-Au bond between two Au(CN)2 moieties. It takes 3 ns, i.e., 3×106 fs . Gold is not a special case, for similar times are also typical with small organic molecules.
When bond formation in a simple SN2 is done with excess vibrational energy (EV), bonds form faster than IVR (non-RRKM). Even then, however, reaction times are ~20 ps  or tens of ps , about a thousand times longer than 13 fs. In other SN2 and carbonyl additions where RRKM is obeyed and IVR is complete, reaction times are ≥10 ns .
A bond-breaking reaction, cyclohexadiene → 1,3,5-hexatriene, was initiated by 267 nm irradiation and tracked by x-ray scatter, finding an 80 fs bond-breaking time. The huge excess of energy instantaneously absorbed ensured nonergodic behavior, so had purely thermal activation been used, the reaction time would have surely been longer than IVR, i.e., >>80 fs .
The diradical formed thermally from fulminic acid + acetylene has no excess EV and, therefore, cannot proceed to cycloadduct faster than IVR (~0.5 – 1 ps). Furthermore, it is not necessarily formed in a perfect conformation to make the second bond. It must first alter atomic positions, which also takes time. (We know that spin-paired diradicals often rotate, even as much as 180°, before forming the second bond ). Therefore, formation of a second covalent bond cannot occur in only 13 fs.
Two quantum chemistry groups have calculated the mechanism of fulminic acid-acetylene cycloaddition, proposing concert although their own data show a stepwise mechanism with a short-lived diradical intermediate. A third group takes the opposite position, casting doubt on all QM conclusions. However, group 2 espouses concert because of the very short (13-15 fs) lifetime of the diradical, a calculated, but not experimentally supported, time interval.
But a diradical is still a diradical regardless of its lifetime. The Born-Oppenheimer Principle holds that electrons move so much faster than atoms that formation of each of the two new bonds is an independent event.
The concerted hypothesis is also disproved by numerous experiments showing that the intermediate diradicals can be intercepted and, therefore, must exist independently, as well as others where Econ is too low to permit concert. In Science, experiment, not theory, is paramount.
Furthermore, the calculated 13-15 fs delay is much too short to comply with the Lindemann principle and RRKM, and several experimental measurements of bonding times range from thousands to millions of fs.
Finally, one may ask why molecules in symmetry-allowed cycloadditions follow the higher-Ea diradical pathway. The simplest answer is that something else not yet recognized prevents concert. Discovery of the orbital symmetry rules does not mean that there are no other sets of rules. More on this anon.
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