Gas-phase reactivity of [Ca(formamide)]²⁺ complex: an example of different dynamical behaviours

Abstract

In the present contribution, we have summarized our recent work on the comprehension of [Ca(formamide)]²⁺ complex gas-phase unimolecular dissociation. By using different theoretical approaches, we were able to revise the original (and typical for such kind of problems) understanding given in terms of stationary points on the potential energy surface, which did not provide a satisfactory explanation of the experimentally observed reactivity. In particular, we point out how non-statistical and non-intrinsic reaction coordinate mechanisms are of fundamental importance.

This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.

1. Introduction

Gas-phase reactivity is a long-standing topic of physical chemistry. Since its origins, experimental and theoretical approaches were used to understand reaction pathways, kinetics and dynamics. Doubly charged cations are of particular interest because they can undergo two types of fragmentations: (i) neutral loss, in which a doubly charged and a neutral species are produced and (ii) Coulomb explosion, in which due to an electronic rearrangement leading to two separate positive charges, the molecule is destabilized and the formation of two singly charged ions is rapidly observed.

The possibility of formation of stable doubly charged species in the gas phase is a long-standing problem, mainly when they are formed by the association of a doubly charged transition metal ion, M²⁺, and an organic base (B), since these species usually are either themochemically or kinetically unstable [1]. The system undergoes a spontaneous deprotonation of the base leading to the monocation [M(B-H)]⁺, the species experimentally observed [2–5]. However, this is not the case with alkaline-earth dications such as Ca²⁺ since [CaB]²⁺ species are stable and detectable in the gas phase. Thus, doubly charged molecular ions like [Ca(urea)]²⁺ or [Ca(formamide)]²⁺ are formed, and can be mass-selected to investigate their unimolecular reactivity [6–14]. Indeed, a paradigmatic example is provided by [Ca(formamide)]²⁺ clusters whose collision-induced dissociation (CID) was studied by mass spectrometry coupled with static quantum chemistry calculations by Eizaguirre et al. [15]. In such study, the reactivity was elucidated by carrying out a rather complete survey of the topology of the corresponding potential energy surface (PES). This system constitutes, however, a prime example of the incomplete picture that the topology of the PES offers for the rationalization of the observed reactivity.

As a matter of fact, the neutral loss of formamide yielding Ca²⁺ is the most energetically demanding reaction, over all the other possible products. This simple fact does not allow one to explain the most obvious experimental outcome clearly showing that the dominant reactive channel is that associated precisely with loss of formamide, yielding Ca²⁺. This apparent contradiction between the topology of the PES and experimental evidence moved us to re-explore the unimolecular reactivity of [Ca(formamide)]²⁺ ion through a deep investigation of its dynamics on the reactive PES in terms of kinetics [16] and dynamics approaches [17,18]. In this paper, we will review these studies.

This paper is organized as follows: the different theoretical methods used are listed in §2, followed by the different results obtained (§3), while §4 provides concluding remarks.

2. Theoretical methods

To obtain a comprehensive picture of unimolecular dissociation of [Ca(formamide)]²⁺ complex, we have employed and coupled different theoretical methods that we briefly review here. (i) Rice–Ramsperger–Kassel–Marcus (RRKM) calculations, to obtain rate constants in the statistical limit. (ii) Explicit collision dynamics between the doubly charged cation and an inert gas. These simulations provided two pieces of information: the fast reactivity (on femtosecond time scale) and the amount of transferred energy. (iii) Post-transition state (post-TS) dynamics to investigate whether the evolution on the known PES follows the intrinsic reaction coordinate (IRC) or not, if there are bifurcations and what is the role of rotational activation. These calculations were done by employing Gaussian09 [19] quantum chemistry package for the location of stationary points, IRC, frequencies and for energy and forces in the chemical dynamics, VENUS [20] for dynamics (coupled with Gaussian09) and the Zhu and Hase code [21] for RRKM rate constant calculations.

(a). Rate constants

Unimolecular dissociation rate constants can be obtained by means of RRKM theory [22]. This theory, which is based on the statistical population of phase space during the reaction, is very powerful as it allows obtaining rate constants from the geometries, energy and vibrational frequencies of reactants and transition states (TSs). Once they are located, the microcanonical rate constant for a unimolecular reaction (that in the present case is direct fragmentation and/or isomerization due to energy deposited by collision(s)) reads

2.1

where is the sum of states of the TS, ρ(E) is the density of states of reactants, σ is the symmetry factor and h is the Planck constant. Often only vibrational density and sum of states are taken into account, and they can be easily obtained with the direct count method as proposed by Bayer & Swinehart [23]. This approach is computationally accessible also for relatively large molecules, although in the past to speed up calculations the semi-classical formula proposed by Whitten & Rabinovitch [24,25] was often used. Results are generally very close to the exact direct count method.

As we have mentioned, vibrational sum and density of states are generally enough to correctly describe the reactivity; this means that rotational activation can be disregarded. While often vibrational activation is enough in unimolecular dissociation kinetic modelling of CID [26,27], in some cases it was shown that the rotation activation is important [28–31]. Anyhow, in order for rotational activation to have an important role in k(E) value, reactants and TSs should have very different moments of inertia (and geometries), as schematically shown in figure 1.

Figure 1.

Schematic of the effect of rotational energy on reaction barrier heights, and therefore on rate constants. Solid line is the electronic PES without rotational energy, dashed is with rotational energy when the reactant state has less rotational energy than the TS.

Rotational energy for a symmetric top (I_x = I_y) is

2.2

where , , J = 0, 1, 2, … and K = 0, ±1, ± 2, …, ± J.

For an ‘almost’ symmetric top (where I_x ∼ I_y), the approximation

2.3

is generally employed. The quantum number J is a constant of motion and therefore is always adiabatic, i.e. there is no energy exchange. On the other hand, the quantum number K can be treated as an active rotor—it allows energy exchange between vibrational and rotational modes—or as an adiabatic rotor. The different treatment of K quantum number results in different RRKM rate constant formula [21]. In the case of K adiabatic, the rate constant depends on energy, E, and J and K quantum number specific values:

2.4

where is the rotational energy of the TS and E_rot is the same for the minimum energy structure. In the case of K active, the rate constant can be obtained by two approaches. In the first, the density and sum of states are calculated by summing over the contributions of all possible K values, leading to

2.5

In the second approach, the density and sum of states are directly obtained by convolutions between the densities and the sum of states for the internal (vibrational) degrees of freedom and the active external rotation (the rotation corresponding to K quantum number), such that

2.6

and

2.7

where is the active energy of the TS and is the active energy of the energized reactant. J and K quantum numbers are obtained from the rotational energy of the molecule. Then the repartition between J and K can be done following three limit cases:

• (1) all the rotational energy is given in the (x,y)-plane, thus K = 0 and

• (2) the rotational energy is equally distributed among the three axes, such that and ; and

• (3) all the rotational energy is placed along the z-axis, such that

Eizaguirre et al. [15] computed all the PES associated with [Ca(formamide)]²⁺ unimolecular reactivity. Therefore, it is possible to obtain the associated rate constants using these data. Anyhow, in the case of fragmentations, often the last step does not correspond to a tight TS (i.e. a saddle point geometry). In this case, we likely have a loose TS that can be located as a minimum in the sum of states along the reaction coordinate. A good example is the formamide neutral loss, which implies the elongation of the Ca²⁺–O bond until its rupture, leading to Ca²⁺ and HCONH₂. Locating a loose TS is not trivial, as one has first to scan the reaction coordinate, minimizing all the other degrees of freedom, verify that the reaction really proceeds as we suppose it should (i.e. the system does not take another pathway), project the vibrational frequencies along the reaction coordinate and verify that we have 3N – 7 real frequencies. This is a tedious procedure and, unfortunately, not always successful, though necessary when the rate limiting step corresponds to the loose TS. Formamide neutral loss directly proceeds from the reactants, making necessary to locate that loose TS. It is worth noting that this procedure leads to an energy-dependent TS geometry. A schematic representation of loose TS characterization as a function of energy is given in figure 2.

Figure 2.

Schematic showing how the minimum of the sum of states varies as a function of the internal energy of the molecule along the reaction coordinate (a). The position of the loose TS varies consequently on the PES since it depends on its internal energy (b).

(b). Collisional dynamics

In CID experiments, the selected ion is accelerated with a given energy in the laboratory framework (E_lab) and sent to a collision chamber where an inert gas is present (generally Ar or N₂). Simulations are done by colliding the molecular ion (the selected species) with an Ar atom or an N₂ molecule.

Both reactivity and energy transfer (comparing results for the same ion with the same relative collision energy) seem to be lower using N₂ as projectile than using Ar [31]. Therefore, we use Ar in our simulations, even though the experiments were performed using N₂. The only difference expected is a higher energy transfer with Ar, and consequently a higher reactivity, but not any difference in the reaction mechanisms after the molecule has been energized by collision.

The reference system is the centre-of-mass (COM), such that the total energy and total angular momentum are conserved. The collision energy in the COM framework (E_COM) can be easily obtained from E_lab (which is set in the experiments) by employing the simple relationship [32]

2.8

where m_i is the mass of the ion and m_g is the mass of the atom (or molecule) of the gas. We tuned the collision energy, mimicking experimentally breakdown curves.

The following set-up is used for simulations. Normal mode energies are sampled using a Boltzmann distribution for a given temperature:

2.9

where ν_i is the vibrational frequency of the ith mode, n_i is the corresponding vibrational quantum number, k_b is the Boltzmann constant and T_vib the vibrational temperature (T_vib = 300 K in current simulations). The resulting normal-mode energies are partitioned between kinetic and potential energy by choosing a random phase for each normal mode. Rotational energy and angular momentum for the ion are selected by assuming separability of vibrational and rotational motion. Thus, initial rotational conditions are obtained by assuming a thermal partitioning of RT/2 about each internal rotational axis (we assumed T = 300 K also for rotation). Afterward, vibrational and rotational energies are transformed into Cartesian coordinates and momenta following algorithms implemented in VENUS [20]. Random orientations in Euler angles between the (rigid body) ion and projectile (Ar atom) are sampled in order to account for the random directions of the Ar−[Ca(formamide)]²⁺ collisions. Then the ion-projectile relative energy is set, and possible impact parameters, b, are considered. Here, b parameter is sampled between zero and b_max = 3.0 Å in the present case, depending on the size of the molecule.

Time propagation is achieved solving Newton equations of motion numerically, using the velocity Verlet algorithm [33] with a time step of 0.2 fs. The initial ion−Ar distance is 8.0 Å, and trajectories are stopped at a 100 Å ion−Ar distance, to ensure no interactions between Ar and ion. This corresponds to a total integration time of about 2.5 ps per trajectory. A trajectory was also stopped if a reactive channel was identified. In that case, a criterion distance of 7.0 Å was used to guarantee no interactions between fragments. For each case, 300 trajectories were computed to correctly describe the process under study.

[Ca(formamide)]²⁺ complex is treated at density functional theory level while the interaction with Ar is modelled by using an analytical function proposed by Meroueh and Hase for collisions between Ar and protonated polyglycines [34] and used recently in CID simulations of different systems [30,35–40]. The analytical potential is purely repulsive

2.10

since A, B and C are positive (r_ij is the distance between Ar and each j atom of the ion). Parameters were obtained by fitting QCISD(T)/6-31++G(d,p) interaction potential. For [Ca(formamide)]²⁺ electronic structure theory, we used G96LYP and BLYP functionals with the 6-31G(d) basis set. We previously assessed the level of theory in order to better reproduce CCSD(T) barriers [16].

(c). Post-transition state dynamics

Once TSs are identified and the amount of transferred energy obtained, it is possible to use dynamics simulations to study how the system evolves from the TS towards the different reaction channels. Thus, in some cases it is meaningful to initialize trajectories at a TS with initial conditions for either a constant temperature in accord with TS theory or at constant energy for a unimolecular reaction, in accord with RRKM. This sort of dynamics also allows us to explore the non-IRC unimolecular reactivity of the ion, including bifurcations.

For statistical unimolecular decomposition at constant energy, there is a microcanonical ensemble of states for the dissociating molecule and every vibrational/rotational state at the TS with energy in the range 0 to E − E₀ has equal probability of being populated [41,42]. Here, E is the energy of the unimolecular reactant above its zero-point vibrational energy (ZPVE) level and E₀ is the difference between the TS and reactant ZPVE levels; that is, the quantum threshold. The total TS energy is the sum . The vibrational and reaction coordinate energies at a TS, in excess of the ZPVE, were added using microcanonical quasi-classical normal-mode sampling. In this case, there are only 3N – 7 vibrational degrees of freedom (nonlinear polyatomic molecule) instead of 3N – 6 for the minima. The new degree of freedom at the TS is the reaction coordinate translation. Thus, the phase for each vibrational mode is selected at random, while the reaction coordinate energy is added to its momenta. When rotational excitation is considered, one-third of the total rotational energy (E_ROT) is added about each rotational axis.

The direct dynamics trajectories were initiated at five different TSs on the [Ca(formamide)]²⁺ PES. For each of them, we ran a set of trajectories (a) for which the initial structure has no excess energy (E_MIN). In this case, the energy of the system corresponds to the potential energy difference between the reactants and the TS, plus the ZPVE of the TS. Then, a second set (b) for which the vibrational degrees of freedom of the TS were excited (E_VIB), and a third set (c) for which the energy was used to excite external rotation (E_ROT). A schematic representation is shown in figure 3.

Figure 3.

Schematic explanation of the post-TS dynamics. Trajectories are generated at the TS, following both forward and backward directions with respect to the reaction coordinate. These trajectories can have different internal energy, partitioned between vibrational and rotational energy as schematically drawn. Finally, the forward reaction can take multiple pathways (in particular cases bifurcations) leading to products that are different from what the TS should provide following the IRC.

For vibrational excitation, option (b), two different energies were used. For one, 35 kcal mol⁻¹ was added in excess of the ZPVE to each of the five TSs. For the other, 120 kcal mol⁻¹ was added relative to the [Ca(formamide)]²⁺ potential energy minimum with ZPVE. Thus, for this case, the initial TS energy depends on the TS considered. For option (c), rotational excitations of 35 and 64 kcal mol⁻¹ were added to the TSs. These excitation energies were chosen according to the amount of transferred energy obtained from the collision dynamics simulations of [Ca(formamide)]²⁺ CID [17]. The time step used for numerical integration is 0.2 fs using, as above, velocity Verlet algorithm. Trajectories were ended after 2 ps or when the separation between reaction products exceeded 7 Å. A total of 50 trajectories were computed for each set of trajectories; thus, a total of 250 trajectories were computed for each of the five TSs considered.

3. Results

(a). Experimental results and potential energy surface topology

The fragments produced by means of CID of [Ca(formamide)]²⁺ complexes with N₂ can be classified in two main groups: (i) monocations arising from Coulomb explosion processes, namely [Ca(OH)]⁺, [HCNH]⁺, [Ca(NH₂)]⁺ and HCO⁺ and (ii) dications, namely Ca²⁺, [Ca(OH₂)]²⁺, [Ca–NH₃]²⁺ and [C,O,Ca]²⁺, produced in alternative neutral loss reactions. Figure 4 shows a schematic summary of those products.

Figure 4.

Summary of the products obtained by electrospray tandem mass spectrometry (ESI-MS/MS) of [Ca(formamide)]²⁺ complexes [15]. Corresponding m/z values are shown. Ca is in dark yellow, O in red, C in dark grey, N in blue and H in light grey. The squares mark Coulomb explosion products.

A detailed analysis of the PES at B3LYP/cc-pWCVTZ level of theory allowed one to establish the mechanisms behind the aforementioned fragmentation process [15,16]. [Ca(formamide)]²⁺ complexes undergo a charge separation reaction yielding [Ca(NH₂)]⁺ + HCO⁺. However, the formation of [Ca(OH)]⁺ + [HCNH]⁺, which are the most stable products, originates in the complexes involving formimidic acid (its structure can be seen in the int2 of figure 5), produced by a previous isomerization of formamide. These [Ca(formimidic acid)]²⁺ complexes are also the origin of a multistep mechanism which leads to a very stable local minimum, in which the metal dication is simultaneously bound to HCN and to H₂O. The dissociation of this complex finally leads to the loss of H₂O or HCN, both experimentally observed, although the former is slightly more stable. Finally, it is important to emphasize that the CID spectrum obtained at 11 eV (laboratory frame) shows that the dominant process corresponds to the appearance of Ca²⁺, that is, to the loss of the intact formamide molecule, which is apparently in clear contrast, as mentioned in the Introduction, with the fact that the whole PES of [Ca(formamide)]²⁺ system is lower in energy than the formamide + Ca²⁺ reaction channel. The knowledge of the critical point of the PES was, then, the first point to investigate whether unimolecular dissociation kinetics can give some explanation on the reactivity more than the mere analysis of energetics, in particular, related to the observation that the Ca²⁺ product shows a high intensity. In the next subsection, we then report the kinetic results.

Figure 5.

PES at G96LYP/6-31G(d) level of theory based on the one computed by Eizaguirre et al. [15] using B3LYP/cc-pwCVTZ. The names of the rate constants for the five possible pathways starting from the minima are shown within squares. The energies (in kcal mol⁻¹) of the different stationary points with respect to min1 are shown in parentheses. From int2 a multistep mechanism (not shown) will eventually lead to Coulomb explosion products, [CaOH]⁺ + [HCNH]⁺, or HCN neutral loss yielding [Ca(H₂O)]²⁺.

(b). Statistical unimolecular dissociation

A simplified picture of the re-computed PES at the G96LYP/6-31G(d) level of theory (used in the dynamics simulations) is shown in figure 5.

Then, we can obtain, per each elementary event, the RRKM rate constants (see §2a). For this, one essential parameter is the internal energy of the reactant ion. This value is, in principle, not known just from the experimental conditions: in fact the collision energy does not correspond to the activation energy since only a fraction of the former is given to the ion in the inelastic scattering process. A simple way of evaluating the transferred energy is to perform chemical dynamics simulation of the ion + Ar (the projectile in the present case) collision. From such simulations, we obtain, for the non-reactive trajectories, the energy transferred to the ion in the collision. We can then use this information to study the statistical dissociation processes using the microcanonical rate constants, k_i(E), previously obtained.

The direction taken by the system from the minimum with the energy given by the collision can be crucial to determine the final reaction pathway. We used RRKM theory to investigate the kinetics of the pathways directly accessible from min1. Figure 6 displays the corresponding rate constants as a function of the internal energy for these five pathways along with the vibrational energy distribution obtained from dynamics simulations. Indeed, taking into account the amount of transferred energy, the pathway labelled k_E is not energetically accessible. The reaction labelled k_D, leading to int2, should be considered carefully. On the one hand, the reverse reaction, rate constant k_−D, is several orders of magnitude faster than the forward reaction, k_D. Thus, in principle, this pathway could be discarded. However, it is necessary to take into account the form of the PES beyond the stationary point int2; in the case where the subsequent reaction from int2 would be faster than k_−D, depopulating the intermediate forward would be faster than going back to min1. Yet, a careful study of the PES beyond int2 (see details—that were skipped in figure 5—in fig. 9 of [15]) shows that the paths from int2 to the products [Ca(H₂O)]²⁺+ HNC and [CaOH]⁺ + [HCNH]⁺ imply many different minima and TSs. Therefore, it would be a slower pathway than the ones directly connecting the reactant to products. We discarded then this pathway in favour of the three remaining ones:

• (1) The neutral loss of formamide. This process occurs through loose TS and for which the rate constant is k_A.

• (2) The Coulomb explosion corresponding to the formation of [Ca(NH₂)]⁺ + [HCO]⁺. It occurs via a tight transition state (TS_C) and the corresponding rate constant is k_C.

• (3) The CO neutral loss and formation of [Ca(NH₃)]²⁺ ion. This last process occurs via a tight TS (TS_B) with the corresponding rate constant, k_B.

Figure 6.

Vibrational energy distribution (histograms in white/blue) obtained from non-reactive trajectories after collisional dynamics (E_coll = 230 kcal mol^–1). We also show RRKM rate constants as a function of internal vibrational energy for the five pathways starting at min1. In black and red are the rate constants for pathways leading directly to products from the reactant while in grey are the rate constants for pathways that imply different intermediates before reaching the products [15]. Dashed lines correspond to backward reactions. The blue vertical dashed line marks the values of the different rate constants for a selected value of internal energy (E_int = 84 kcal mol⁻¹).

Reactions (2) and (3) differ only in the position of the proton: in (2) it stays on C atom, and [HCO]⁺ ion leaves; in (3) the H is transferred to NH₂ group leading to neutral CO.

Combining the energy dependence of unimolecular rate constants with the vibrational activation of the ion after collisions we can understand many aspects of chemical reactivity. First, at low energies (less than 70 kcal mol⁻¹) only k_A is active and fragmentations leading to neutral loss of formamide dominate. Note that we have non-zero values of k_A also for energies smaller than 84 kcal mol⁻¹ (that is the energy to form Ca²⁺ + formamide). This occurs because we have a loose TS and the rate constant corresponds to the depopulation of the reactants that is defined by the location of the TS via variational TS theory (figure 2), which has, by definition, an energy that is smaller than the dissociation energy. Physically for energies less than 84 kcal mol⁻¹, we do not have formation of this product. Increasing energy, k_B and k_C also start playing a role. However, reaction corresponding to k_A is faster up to very high energies (higher than what should be obtained by low-energy CID experiments). For vibrational energies between 70 and 110 kcal mol^–1 (so in the energy range that the ion should get by collisions at typical experimental conditions), the channel leading to neutral loss (k_B) is faster than that leading to the two monocationic fragment formation (k_C). TS_B does not directly lead to fragments, but to an intermediate that can convert back to reactants with a rate constant, k_−B. However, the backward reaction is slower at higher energies and of the same order of magnitude for the smallest energies. Furthermore, since the barrier connecting the intermediate with the final products (see TS_F in figure 5) is very low, one can almost disregard this last step of the PES mechanism and consider that, once the TS_B is passed, the [Ca(NH₃)]²⁺ + CO products are obtained straight. Considering the kinetics of a subset of the available reactive space we can conclude that formamide neutral loss is favoured. Even if the final product energy is higher than the TS leading to other products, it is kinetically favoured due to the softness of the potential energy landscape (and this because it is a simple loose TS that needs just a Ca–O elongation to lead to products far enough to dissociate).

From the kinetic analysis of the PES we have thus a first explanation as to why the Ca²⁺ + formamide is observed with higher intensity than other products. A full statistical description considering all the different channels would be described by RRKM master equation procedure or a kinetic Monte Carlo using the full set of rate constants. Here, we focused our attention on the statistical processes concerning the first activation steps. Note that in a similar context, it was shown that while the dissociation starting from the minimum is statistical, this is not always the case when starting from high-energy structures [43]. In the following, we used two dynamical approaches to understand how the other products can be formed: collisional simulations (§3c) and post-TS dynamics (§3d).

(c). Direct fragmentation pathways

When performing collisional dynamics, a fraction of trajectories are able to react in the simulation time scale. From simulations, we obtained the following reactions: (1) formamide neutral loss, (2) CaNH₂⁺ + HCO⁺ and (3) CaNH₃²⁺ + CO. In table 1, we report the percentage obtained for each reaction pathway as a function of collision energy and method used.

Table 1.

Percentage of reactive versus non-reactive trajectories and collisional dynamics products obtained from simulations at three different collision energies (in kcal mol^–1).

collision energy	180	230	280
non-reactives (%)	76	63	60
reactives (%)	24	37	40
Ca2+ + formamide (%)	100	96	98
[Ca(NH2)]+ + [HCO]+ (%)	—	3	2
[Ca(NH3)]2+ + CO (%)	—	1	—

We should note that we obtained as dominant product the Ca²⁺ + formamide neutral loss and very few Coulomb explosions. The other product observed in very few trajectories is [Ca(NH₃)]²⁺ + CO. These products are in agreement with the kinetic picture arising from RRKM rate constants, but, as we will see, in some cases they proceed through different mechanisms.

From collisional simulations, we may retrieve not only branching ratios, but also kinetic information. In figure 7, we report the reaction time as a function of internal energy as obtained from chemical dynamics. These times can be compared with RRKM rate constants (here in terms of half-life times, t_1/2(E) = −(ln 0.5)/k(E)). For Ca²⁺ + formamide reaction the collisional times differ significantly from the RRKM predicted times, in particular, at low energy. A similar picture holds for the other neutral loss, but here we should stress that the number of reactive trajectories is not enough to make any conclusion. On the other hand, for the Coulomb explosion reaction, there are only two points that match with the predicted RRKM reaction times. However, in general, both sets of times differ significantly, indicating that those trajectories neither follow a direct fragmentation mechanism nor react through a full intramolecular vibrational energy redistribution (IVR) mechanism, which is assumed in the RRKM theory. This suggests that the actual mechanism is between these two limiting cases; i.e. the energy is distributed within the internal degrees of freedom of the molecule, but the reaction takes place before a complete IVR could be achieved.

Figure 7.

Reaction time versus transferred energy obtained from chemical dynamics simulations (dots, squares and triangles) and half-life times (t_1/2) predicted by RRKM (solid lines). Both were obtained using the G96LYP/6-31G(d) level of theory. Results are shown for trajectories yielding formamide neutral loss (a), k_A, CO neutral loss (b), k_B, and Coulomb explosion yielding [Ca(NH₂)]⁺ + [HCO]⁺ (c), k_C. (Online version in colour.)

Collision simulations have shown that products other than Ca²⁺ + formamide can be obtained, and in particular [Ca(NH₂)]⁺ + [HCO]⁺ and [Ca(NH₃)]²⁺ + CO. On the other hand, some more products observed in experiments are not obtained in the simulation time length. From collision results, we were able, anyhow, to know the final internal energies of the non-reacting ions and use them to understand post-TS dynamics from which we understood the formation of the missing products. Next section will show the principal results of post-TS dynamics.

(d). Post-transition state dynamics

When computing PES, i.e. locating minima and TSs, to explain reactivity it is normally assumed that each individual TS leads to one, and only one, energy well (corresponding to the final products or a reaction intermediate). This is for instance one of the main assumptions of transition state theory.

In a topological analysis of the PES, each TS is verified by means of IRC calculations [44]. Once a TS is found, the confirmation that it is indeed the desired TS is done when the IRC calculation (forward and backward) actually connects reactants with the expected products. The IRC is a minimum energy path along the PES in which one follows the gradient on the electronic energy with an initial velocity vector parallel to the imaginary mode at the TS and orthogonal to all other modes. It is assumed that the orthogonal modes are equilibrated and have the minimum energy. A dynamics, on the other hand, has finite velocity along all the modes, such that it is not obliged to follow the IRC. In the case of simple PES, one can observe just an oscillation along the IRC, such that it can be considered as a good approximation of the dynamical pathways. In more complex PES, on the other hand, where a deviation from the IRC can result in a totally different reaction pathway, more subtle effects come into play. Dynamical factors become fundamental to understand reactivity. Reaction dynamics that do not follow the IRC are referred to as non-IRC dynamics.

An interesting and particular case of non-IRC behaviour appears when a PES bifurcation is encountered. The direct consequence is that one single TS may lead to two (or more) different products [45,46]. A bifurcation is related to the presence of a valley-ridge inflection point [47]. There exists methods based on the topological analysis of the PES to locate those points; however, they are computationally intensive and prohibitive for large molecules [48–51]. In this regard, post-TS simulations [52] are a good alternative to point out the presence of a bifurcation since they provide detailed information concerning post-TS dynamics [53–59].

To complete the study of [Ca(formamide)]²⁺ reactivity, we have thus performed a set of post-TS dynamics using as starting points the TSs directly connected with the minimum energy structure. To ensure that some of the non-reactive trajectories have enough internal energy to reach the TS after the collision, we resorted to the vibrational and rotational energy distributions (obtained from CID simulations). As shown in figure 8, some of these trajectories have enough internal energy to eventually dissociate. Furthermore, in figure 8 it can be seen that the TSs' excess energy is not only vibrational, but in some instances there is a non-negligible rotational activation. We have therefore also analysed the effect of rotational activation on such post-TS dynamics.

Figure 8.

Vibrational (in orange) and rotational (in white/red) energy distributions for the non-reactive trajectories at three collision energies. Vertical lines mark the energy for the different TSs that can be reached from min1. Lines in bold correspond to the three accessible pathways from min1. The red line identifies the Coulomb explosion reaction.

Direct collisional dynamics are on short time scales (few picoseconds) and consequently, even if some trajectories have enough energy to react, they will not do so during the simulated time. We have then assumed that these trajectories can, in a long time scale, statistically reach the TS with the rovibrational excess energy acquired in the collision with Ar. This may be seen as a shortcut allowing us to sample a larger area of the PES and thus study phenomena that cannot be directly observed by collision dynamics. In contrast with RRKM analysis, we will have now information about the PES after crossing the TS.

Particularly interesting are results starting from three TSs: TS_B, which, in principle, connects the initial structure with [Ca(NH₃)]²⁺ + CO products; TS_C, which should lead to the Coulomb explosion [CaNH₂]⁺ + HCO⁺ products; and TS_D, which should eventually provide either Coulomb explosion products [CaOH]⁺ + [HCNH]⁺ or HCN neutral loss yielding [Ca(H₂O]²⁺. Results are summarized in figure 9, where we show how starting from each TS we can have trajectories leading to multiple products. In particular:

• (1) When initiating trajectories at TS_B we obtain both the IRC product [Ca(NH₃₎]²⁺ + CO and the non-IRC product CaNH₂⁺ + HCO⁺. This is an example of non-IRC dynamics, showing that when the TS is excited, the path leading to other products is opened. However, the final product distribution does not depend on how the energy is injected (i.e. we have the same results when the TS is excited with the same amount of energy on vibrational degrees of freedom or on external rotation, see details in [18]).

• (2) When starting from TS_C we obtain again the two pathways mentioned in point (1). In this case, the fact that trajectories initiated with less energy than ZPVE also follow both pathways strongly suggests the presence of a bifurcation. Furthermore, for this TS, the way of exciting the system (vibrational versus rotational) has a drastic effect on the final product distribution.

• (3) When starting from TS_D a complex dynamics is also observed. Both products (i) [Ca(H₂O)]²⁺ + HNC and (ii) [CaOH]⁺ + [HCNH]⁺ are obtained. For both Eizaguirre et al. [15] computed a multistep mechanism leading to these products, confirming all the TSs and intermediates by IRC calculations. However, the trajectories leading to these products do not follow the computed PES. Thus, again we have non-IRC dynamics. Here, as the excitation energy on the starting TS structure increases, the product distribution changes. However, there are no significant differences regarding where this energy is initially located (vibration or rotation).

Figure 9.

Example of post-TS trajectories initiated at three different TSs. For each initial structure, we show the time evolution of characteristic distances for an ensemble of trajectories (a different colour for each individual trajectory). Note that the printed distances may vary from one TS to another.

Post-TS dynamics provided us the full set of mechanisms responsible for the experimental formation of different products. In particular, now also the products that are formed through TS_D are obtained and the corresponding pathway is much shorter than that supposed to occur from PES analysis (and also from a kinetic study on this PES).

4. Concluding remarks

The most relevant conclusion of our re-visiting the CID reactivity of [Ca(formamide)]²⁺ molecular ions is that only a part of the observed reactivity can be accounted for from a survey of the topology of the PES, and from a kinetic treatment that only takes into consideration statistically driven processes. In fact, it seems that a complete picture can be only obtained when direct collision and post-TS dynamics are accounted for. Indeed, some reaction paths are only open if one does not impose a statistical behaviour to the system, that is, if one allows for non-IRC processes to occur. This is precisely the case when the flux bifurcates on the downhill pathway towards products, after crossing a given TS. Rather interestingly, non-IRC dynamics appear systematically linked to the competition between the formation of two monocations (Coulomb explosion) or the loss of a neutral yielding a smaller doubly charged cation. The possibility that one single TS may lead to two different products implies the appearance of a new kind of selectivity that is controlled after the TS is crossed, and not before as usually assumed for kinetic control. In this way, reaction selectivity is controlled by dynamical factors, and in some specific cases may depend on where the excitation energy is initially located. Indeed, although in most cases the product distribution is similar regardless of where the excitation energy comes from, rotational or vibrational activation of the TS, in some cases, we showed that rotational excitation may have a key role in determining the reaction pathway.

The collision-induced fragmentation of [Ca(formamide)]²⁺ ion provides a prototypical example showing how even a relatively small system can give rise to complicated reaction mechanisms that can only be rationalized when dynamical effects, such as bifurcations or non-IRC dynamics, are taken into account. We believe that a similar approach, which combines different theoretical methods, may be quite useful to further account for unimolecular reactivity, especially for those cases when the use of one single technique fails to do so.

Also importantly, this analysis still leaves, as an open question, whether this non-IRC/bifurcated character is a systematic feature of the post-TS dynamics important to explain unimolecular dissociation of this class of systems or whether it is just a peculiarity of the specific [Ca(formamide)]²⁺ complex investigated here. This will be the subject of future work.

Authors' contributions

A.M.-S. carried out the molecular dynamics calculations and participated in the design of the study. R.S. conceived and designed the study. M.Y. coordinated the study. All authors helped draft the manuscript and gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the projects TQ2015-63997-C2, CTQ2013-43698-P, FOTOCARBON-CM S2013/MIT-2841, and by the COST Action CM1204. Also partial support was provided by ANR DynBioReact (grant no. ANR-14-CE06-0029-01).