Dynamics of CO₂ activation by gas-phase transition metal ions: the importance of intersystem crossing

Abstract

The activation of CO₂ at isolated transition-metal centers represents a prototypical problem for understanding elementary steps relevant to single-atom catalysis. Fundamental knowledge of such systems can be acquired by investigating gas phase reactions between transition-metal ions and molecules. Because open-shell transition-metal species often possess multiple accessible spin states, their reactions with CO₂ can proceed along competing spin-changing and spin-conserving pathways. Understanding how spin–orbit coupling influences these competing pathways therefore calls for a direct comparison between scattering experiments and multi-state dynamical simulations. In this Perspective, we summarize our combined experimental and theoretical investigations of the Ta⁺, Nb⁺, and Zr⁺ + CO₂ reactions. Crossed-beam velocity map imaging provides energy- and angle-resolved differential cross sections, while trajectory surface-hopping simulations on first-principles based full-dimensional multi-spin potential energy surfaces enable a dynamical treatment of intersystem crossing and spin-conserving channels on an equal footing. In all three systems, intersystem crossing competes with the spin-conserving channel for the control of the overall reaction dynamics and kinetics. These reactions all proceed predominantly via an indirect mechanism, as evidenced by the nearly isotropic differential cross sections consistent with long-lived complex formation, and the respective product energy distributions indicate substantial energy deposition into internal modes. Despite their similar potential-energy topographies, Ta⁺ and Nb⁺ + CO₂ reactions are dominated by spin-changing pathways at all energies investigated, whereas in the Zr⁺ system the spin-conserving channel becomes competitive. This difference arises from the markedly different magnitudes of the spin–orbit coupling, which determines the efficiency of intersystem crossing and thereby the balance between the two pathways.

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1 Introduction

Carbon dioxide is a major greenhouse gas that has important implications for Earth's climate. There has been much recent effort towards its sequestration¹ and conversion into value-added chemicals.² The main difficulty in the latter chemical approach is the activation of this inert molecule. Current processes typically involve transition-metal catalysts.³ As a result, a thorough mechanistic understanding of transition-metal mediated activation of CO₂ is of great importance.

An important recent development in heterogeneous catalysis is the emergence of single atom catalysts (SACs), which typically involve atomically dispersed transition-metal species on oxide surfaces.^4–6 The SACs not only improve atom efficiency, but also introduce new mechanisms and control of catalytic activity. It is now established that the environment of these supported atomic species, such as coordination numbers, has a substantial impact on the activity of SACs.⁷ So far, however, studies have focused on geometric and energetic factors. Only limited attention has been given to explore the influence of the spin states of the transition-metal active sites.^8,9

In homogeneous catalysis, it is well established that ligands strongly influence the spin states of the central open-shell transition-metal cation, which in turn show vastly different reactivities.¹⁰ Several recent studies have indeed demonstrated that catalytic activity of SACs can be modulated by an external magnetic field.^11,12 These observations clearly point to the importance of the electronic spin of transition-metal SACs in catalysis, but a systematic understanding of spin chemistry has not been achieved, largely due to difficulties in controlling the spin state of SACs on support surfaces.

A reductionist approach is to first understand the reactivity of transition-metal cations in the gas phase.^13,14 The absence of ligands and/or solvent allows the extraction of the most important principles that control reactivity. Several pioneering studies have been reported on gaseous transition-metal cation activation of CO₂, but they have until recently been restricted to kinetics,^15–28 or detection of activated intermediates by spectroscopy.^29–32

Because of d electrons, open-shell transition-metal cations typically have multiple low-lying spin states. It has been pointed out that the lowest energy spin state is often less reactive than its higher energy counterparts.^14,33–36 However, a reaction under thermal conditions can still proceed via a barrierless pathway facilitated by intersystem crossing (ISC). This phenomenon is exemplified by the prototypical reaction between FeO⁺ and H₂.^37–41 Experimental evidence indicated that its rate coefficient has a negative temperature dependence, signalling a barrierless reaction pathway. However, the magnitude of the rate coefficient is significantly less than the Langevin limit, suggestive of a kinetic bottleneck. These seemingly contradicting observations can be readily rationalized by a “two-state reactivity” (TSR) model,^42,43 in which the barrierless pathway is facilitated by nonadiabatic ISC between two spin states.^44,45 In the meantime, the inefficiency of the reaction is caused by the nominally spin forbidden nature of the process. Obviously, the TSR mechanism depends critically on the strength of the spin–orbit coupling (SOC) of the transition metal.⁴⁶

Recently, the RPTU group has leveraged the velocity map imaging (VMI) technique⁴⁷ to study ion–molecule reactions involving transition-metal cations.^48–51 This technique allows the determination of much more detailed dynamic information beyond kinetics. Such information includes energy and angle differential cross sections, providing unprecedented details of the reaction dynamics. Perhaps most importantly, knowledge about the long-range attraction and the role played by spin sheds valuable light on the molecular level processes of this type of reactions. The energy and angle differential cross sections encode information about the rearrangement of atoms during the chemical transformation and how energy is utilized and (re)distributed among the product degrees of freedom. These experiments are complemented by full-dimensional theoretical investigations of the reaction dynamics (and kinetics) based on first principles potential energy surfaces (PESs), carried out by the UNM team.^51–53 The combined experiment-theory endeavour leads to an in-depth understanding of the microscopic reaction mechanism, particularly how the SOC influences the reactivity and dynamics in an unexpected way.

In this Perspective, we summarize our recent work on the following reactions, in which CO₂ is activated by three transition-metal cations (tantalum, niobium and zirconium):

⁵Ta⁺ + ¹CO₂ → ³TaO⁺ + ¹CO ΔE = −1.987 eV (1)

→ ⁵TaO⁺ + ¹CO ΔE = 1.543 eV (2)

⁵Nb⁺ + ¹CO₂ → ³NbO⁺ + ¹CO ΔE = −1.572 eV (3)

→ ⁵NbO⁺ + ¹CO ΔE = 2.061 eV (4)

⁴Zr⁺ + ¹CO₂ → ²ZrO⁺ + ¹CO ΔE = −1.990 eV (5)

→ ⁴ZrO⁺ + ¹CO ΔE = 1.510 eV (6)

Reactions (1), (3) and (5) represent the ISC channels, while reactions (2), (4) and (6) represent the spin conserving channels (SCC). Given their different energetics and SOC strengths (329.8, 127.2, and 38.0 cm⁻¹ for Ta⁺, Nb⁺, and Zr⁺), the three systems present an interesting prototype for understanding the role of ISC in transition-metal activation of small molecules.

2 Methods

2.1 Experiment

We recorded energy and angle differential cross sections of transition-metal ion molecule reactions using a combination of crossed beams and 3D velocity map imaging.^47,54,55 A general description of the experiment is given here, as details have been published previously.^48,49,51,55 We prepare the ion beam using a home-built laser vaporization source.^56,57 The second harmonic of a Nd:YAG laser (532 nm, 20 Hz, ≈ 4 mJ per pulse (5 × 10⁷ W cm⁻²)) is focused onto a rotating metal foil. A synchronized helium pulse (≈ 8 bar He, 40 µs) extracts the plasma plume perpendicular to the laser axis toward the interaction region of the spectrometer. The ion beam is confined to an expansion channel in which the ions undergo on the order of 10⁴–10⁵ collisions with helium before entering the high vacuum region of the source chamber (≈5 × 10⁶–1 ×10⁵ mbar during operation). We have no direct control over the electronic state distribution within the ion beam but assume only significant contributions from the electronic ground state and first electronically excited state to be present because the next electronically excited state is too far up in energy and statistically unfavourable⁵⁸ in all three cases. We estimate an upper limit of 20% for the contribution of the first electronically excited state.⁴⁸ The source is operated such that only M⁺ cations contribute to the ion beam and cluster formation is suppressed. The M⁺ beam is transferred into the interaction region of the velocity map imaging stack^47,59 using a set of deflectors and Einzel-lenses. Subsequently, it is characterized by 2D velocity map imaging to adjust energy and angle. A potential bias to the source region relative to that of the imaging stack can be used to adjust the collision energy. The CO₂ molecular beam is prepared using a second home-built piezo valve⁶⁰ and expanding pure carbon dioxide. Similar to the ion beam, the molecular beam is characterized by 2D velocity map imaging after non-resonant electron impact ionization. The 1σ-error of the collision energy due to velocity and angular spread of both input beams is between 65 to 200 meV for the presented experiments according to Gaussian error propagation.⁴⁷ The two reactant beams are crossed at a relative angle of ≈150° and product ions MO⁺ are perpendicularly extracted towards a position and time sensitive detector. Product ions hit a mircochannel plate phosphor screen combination and a CCD-camera records the position of impact. A photo multiplier tube simultaneously records the arrival time. The 3D Newton sphere can be reconstructed from the experimental data by making use of the cylindrical symmetry in the center-of-mass frame of the collision. This allows us to recover the velocity along z-direction (time-of-flight axis) perpendicular from the scattering plane.⁴⁷

The data analysis follows the method established by Wester and co-workers.^47,55,59 The 3D Newton sphere is again projected onto a 2D representation to allow for comparison to the conventional representation of differential cross sections. Integrating the original product ion velocity distribution over energy (velocity) or scattering angle yields 1D-histograms. Kinetic energy distributions are calculated directly from experimental velocities while internal energy distributions are calculated making use of energy conservation under the experimental single collision conditions. The reaction enthalpy is a necessary input for the energy balance and taken either from literature or calculations. Kinematic cut-offs, which give the maximum possible product ion velocity taking energy and momentum conservation into account, are calculated using the same energy balance.

2.2 Theory

The reactions of transition-metal cations with CO₂ involve open-shell electronic configurations and often proceed through multiple spin manifolds. In many cases, the spin state correlating with the ground-state reactants features a significant barrier, while a lower-lying reaction pathway exists in another spin state. The reaction therefore proceeds through ISC between different spin states, rather than along a single spin-adiabatic pathway. Under such circumstances, a quantitative description of the reaction kinetics cannot rely solely on stationary-point energetics on a single PES. Instead, it requires a unified treatment of the electronic structure, global PESs for multiple spin states, the interstate couplings that enable spin transitions, and nonadiabatic dynamics simulations that capture the competition between these pathways.

In our recent studies on the Ta⁺, Nb⁺, and Zr⁺ + CO₂ reactions,^51–53 essentially the same theoretical approach was employed, enabling direct comparisons between experiment and theory across this series of transition-metal ions.

2.2.1 Electronic structure calculations. For these transition-metal systems, high-level multireference configuration interaction (MRCI) or coupled-cluster (CC) calculations provide reliable energetics at stationary points but are computationally prohibitive for constructing global PESs. Therefore, density functional theory (DFT) was adopted as a practical compromise between accuracy and computational cost. In the Ta⁺ and Nb⁺ + CO₂ reactions, the B3LYP functional^61,62 with aug-cc-pVDZ basis set for C and O, ECP60MDF_AVDZ⁶³ for Ta, and ECP28MDF_AVDZ⁶⁴ for Nb was employed (denoted as B3LYP/DZ), while the Zr⁺ system used a range-separated hybrid functional (ωB97XD)⁶⁵ with the same basis sets as Nb⁺ + CO₂ system (denoted as ωB97XD/DZ), based on extensive benchmarking. Benchmark calculations show that the chosen DFT methods can qualitatively describe the three systems for key stationary points, the energy differences between DFT and CC are typically within about 0.3 eV, while the computational cost is dramatically reduced.

For the description of ISC, SOCs were evaluated using MRCI calculations based on CASSCF wave functions, with the same basis set employed in the DFT calculations for the three systems. The active space of four electrons in eight orbitals (4e, 8o) was used for the Ta⁺ and Nb⁺ + CO₂ systems, whereas an active space of 9 electrons in 8 orbitals (9e, 8o) was chosen for the Zr⁺ + CO₂ system. The MRCI SOC matrix elements were computed at minimum energy crossing points (MECPs), and the root-mean-square of their sum⁶⁶ was used in our dynamical calculations.

2.2.2 Potential energy surfaces. Because ion–molecule reactions are strongly influenced by long-range electrostatic interactions, the global PES for each spin state was constructed by combining the short-range (SR) DFT potential and a long-range (LR) term:

V = sV_SR + (1 − s)V_LR, (7)

where V_SR and V_LR denote SR and LR PESs, respectively. The two terms are smoothly connected through a switch function,

s = {1 − tanh[3(R − R₀)]}/2, (8)

where R is the distance between transition-metal ion and the center-of-mass (COM) of CO₂ and R₀ is a parameter that defines the switching region. The values of R₀ are set to 5.0, 5.5, and 5.0 Å for the Ta⁺, Nb⁺, and Zr⁺ + CO₂ systems, respectively.

The SR PES for each spin state of the three systems was developed by fitting approximately 50 000 data points sampled via an active learning scheme using the permutation invariant polynomial-neural network (PIP-NN) approach,^67,68 which incorporates the permutation symmetry of identical atoms through PIPs embedded in the NN input layer. For each of the three systems, the energies of the sampled data points were computed at their respective DFT levels, as discussed above.

The LR term in the entrance channel is written as

V_LR = V_CO2 + V_ES, (9)

where V_CO2 denotes the PES of the isolated CO₂ molecule and V_ES accounts for the electrostatic interaction between the ion and CO₂. The electrostatic term is given byin which the 1/R³ and 1/R⁴ contributions correspond to the charge-quadrupole and charge-induced dipole interactions, respectively. Here ϑ is the angle between the CO₂ quadrupolar axis (the OCO axis) and the intermolecular vector R, and q = +1 represents ionic charge. The CO₂ quadrupole moment Θ and its polarizability parameters α and α_‖ − α_⊥ are taken as −4.27 Debye Å, 2.63 ų, and 2.10 ų, respectively, according to literature values.^69,70 The LR term for an excited spin state is obtained by shifting the lowest spin state term according to the calculated excitation energy of the transition-metal ion.

2.2.3 Trajectory surface hopping. The nonadiabatic reaction dynamics were investigated using Tully's fewest switches surface hopping (FSSH) method⁷¹ on the coupled multi-state PESs. In this mixed quantum-classical framework, nuclear motion is propagated classically on an adiabatic surface, while the electronic state populations evolve quantum mechanically along the trajectory. Transitions between spin states are governed by hopping probabilities derived from the SOC, and when a hop occurs, the nuclear momenta are adjusted to conserve the total energy. The SOC constants employed in the FSSH simulations are 329.8, 127.2, and 38.0 cm⁻¹ for Ta⁺ (between quintet and triplet), Nb⁺ (between quintet and triplet), and Zr⁺ (between quartet and doublet) + CO₂ systems, respectively.

The reactive integral cross section (ICS), σ_r(E_rel), was calculated according to

σ_r(E_rel) = πb_max²P_r(E_rel), (11)

where P_r(E_rel) represents the reaction probability at collision energy E_rel. P_r(E_rel) is given by the ratio between the number of reactive trajectories N_r and the total number of trajectories N_tot at the given collision energy E_rel. b_max is the maximum reactive impact parameter. For each trajectory, the impact parameter (b) is sampled as b = b_maxζ^1/2 with ζ being a uniformly distributed random number between 0 and 1.

The differential cross section (DCS) is then obtained from

where Ω denotes the scattering solid angle and P_r(θ) is the normalized angular distribution of the reactive trajectories in the scattering angle (θ), defined as

Here, v⃑_i and v⃑_f are the initial velocity vector of the incident M⁺ and the final velocity vector of product MO⁺, respectively. Forward scattering (θ = 0°) corresponds to a direct rebound of product MO⁺, whereas backward scattering (θ = 180°) indicates that the product MO⁺ moves in the same direction as the incident M⁺.

3 Potential energy surfaces

The reaction pathways with the corresponding stationary points for the Ta⁺, Nb⁺, and Zr⁺ + CO₂ reactions are shown in Fig. 1. The energies were computed at the B3LYP/DZ level for the Ta⁺ and Nb⁺ + CO₂ systems and at the ωB97XD/DZ level for the Zr⁺ + CO₂ system. In the reactant asymptotic regions, the ground electronic states for Ta⁺, Nb⁺, and Zr⁺ are quintet (⁵F), quintet (⁵D), and quartet (⁴F), respectively. Each ion also possesses a low-lying first excited electronic state, namely ³F for Ta⁺, ³P for Nb⁺, and ²D for Zr⁺, located 0.387, 0.640, and 0.419 eV above their corresponding ground state at the respective DFT levels. These values are in good agreement with the experimental excitation energies of 0.428, 0.688, and 0.450 eV.⁵⁸ It should be noted that the experimental values correspond to spin–orbit averaged energies, whereas SOC effects were not explicitly included in our DFT calculations.

Fig. 1 Schematic reaction paths for the reaction M⁺ + CO₂ → MO⁺ + CO (M = Ta (red), Nb (blue) and Zr (black)) for the respective spin states for the free cation M⁺ and the free MO⁺. Energies (ZPE-corrected, in eV) were calculated at the B3LYP/DZ level for M = Ta and Nb and at the ωB97XD/DZ level for M = Zr. Note that no ZPE is included for the MECP. Adapted with permission from ref. 52 (Copyright 2024 American Chemical Society); ref. 53 (Copyright 2024 American Chemical Society); and adapted with permission from ref. 51.

Overall, the three systems follow similar reaction pathways. As illustrated in Fig. 1, along the ground-state PES, the M⁺ ion first approaches CO₂ and forms a reactant complex (RC) [M⁺CO₂], primarily stabilized by long-range electrostatic interactions. The RC lies 0.902, 0.980, and 0.949 eV below the separated reactants for Ta⁺, Nb⁺, and Zr⁺, respectively. On the PES correlating with the electronic ground state of M⁺ + CO₂, the formation of MO⁺ + CO is endothermic, with reaction energies of 1.543, 2.061, and 1.510 eV for the three systems at their respective DFT levels. There is also a very high saddle point (SP), 0.640, 0.898, and 0.631 eV for Ta⁺, Nb⁺, and Zr⁺, respectively, on the electronic ground state. Such sizable barriers and pronounced endothermicity render an adiabatic reactive channel unlikely at low collision energies.

However, the reactivity is enabled via ISC to a different spin state, on which the barriers are substantially reduced and even become submerged for Ta⁺ and Zr⁺. The corresponding barrier heights are −0.278, 0.005, and −0.358 eV for Ta⁺, Nb⁺, and Zr⁺ + CO₂, respectively. Specifically, on the PES correlating with the first excited spin state of M⁺ + CO₂, the SP appears earlier along the reaction coordinate than the crossing seam. Consequently, once ISC occurs, the system can evolve toward products without surmounting additional barriers. This crossing feature suggests that the overall reactivity is governed primarily by ISC, rather than by the intrinsic ground-state potential barriers. Such a mechanism accounts for the efficient reactivity observed experimentally. A more detailed discussion of this mechanism is provided in the following sections. Indeed, measurements show that these reactions proceed efficiently at room temperature and at low collision energies, with rate coefficients on the same order of magnitude as the Langevin collision limit.^22,26,49 If the reaction proceeds to the ground-state product channel, on the other hand, it is exothermic by 1.987, 1.572 and 1.990 for the Ta⁺, Nb⁺, and Zr⁺ systems, respectively. These values are close to the available experimental values of 1.646 ± 0.125 eV for the Ta⁺ system and 2.310 ± 0.11 eV for the Zr⁺ system. A detailed comparison between DFT results and higher-level benchmark calculations can be found in the literature^51–53 and is therefore not repeated here.

4 Energy and mechanistic insight

Experimental energy and angle DCSs for Ta⁺ + CO₂ (reactions (1) and (2)) from crossed beam experiments give insights into the underlying microscopic mechanisms (see Fig. 2(a)–(c)).⁴⁸ The experimental product ion velocity distributions for the TaO⁺ product ions have been recorded at three different relative collision energies E_rel = 1.0, 1.4, 2.0 eV, with the highest one well above the SCC channel (reaction (2), Fig. 1) which is not expected to contribute due to the established multi-state character of the reaction.^17,26 The TaO⁺ ions are dominantly scattered isotropically around the COM (Fig. 2(a)–(c)). The isotropic scattering is a signature of indirect dynamics, i.e., a complex mediated mechanism.^72,73 An interaction complex is formed that lives for several rotational periods and subsequently dissociates statistically. Products are scattered into all scattering angles θ. The fully isotropic patterns as seen in Fig. 2(a)–(c) further hint at low impact parameter collisions associated with randomization of angular momenta between orbital angular momentum of the collision and rotational angular momentum of the products (MO⁺ and CO) due to the no longer strict orbital angular momentum conservation.⁷²

Fig. 2 Experimental energy and angle differential cross sections for Ta⁺ + CO₂. Product ion TaO⁺ velocity distributions (a)–(c) for the reaction Ta⁺ + CO₂ → TaO⁺ + CO for three different relative collision energies E_rel = 1.0, 1.4, and 2.0 eV. The two superimposed circles indicate the kinematic cut-offs for the two lowest electronic states of the tantalum cation (the ground state ⁵Ta⁺: green circle, the first electronically excited state ³Ta⁺: orange circle). Histograms are normalized such that the bin with highest intensity is set to 1 to allow for a comparable visual impression. A simplified Newton diagram is shown at the top illustrating the relative orientation of the reactant beams and the scattering angle θ. At a scattering angle close to 180° incomplete background subtraction leads to residual ion beam signal being visible. (d) Product ion kinetic energy distributions E_kin (TaO⁺). Evaluated scattering range θ = 0°–165° as indicated by the pink dashed area in (a)–(c). Adapted with permission from ref. 48.

The product ions are scattered well away from the kinematic cut-offs, which are represented by rings superimposed onto the velocity distributions: ⁵Ta⁺ (green) and ³Ta⁺ (orange). ³Ta⁺ is the first electronically excited state of Ta⁺ ≈ 0.4 eV above the ground state.⁵⁸ Low product kinetic energies mean significant partitioning of energy into internal degrees of freedom of TaO⁺ and/or CO. Upon increasing the collision energy by 1 eV, the mean product kinetic energy release (sum of kinetic energy of TaO⁺ and CO) does not change within experimental errors. The direct comparison of the 1D histograms is given in Fig. 2(d). This indicates that additional collision energy is mostly partitioned into internal excitation. We assume this to take place in the pre-reaction well. The RC is dynamically trapped behind a barrier,^74,75 which we refer to as bottleneck. Once this bottleneck is passed, products are quickly formed and the energy released during product formation is dominantly partitioned into kinetic energy. In summary, the indirect dynamics and the near constant product kinetic energy release are surprising because the reaction is considered to be very efficient, is highly exothermic, and involves only four atoms, which results in few internal degrees of freedom to store energy in.

Dynamics for the reaction with niobium (reactions (3) and (4)) and zirconium (reactions (5) and (6)) have also been investigated.^49,51 A comparison of all three metals at a similar collision energy of about 2.0 eV is given in Fig. 3(a)–(c). The same dynamics are observed for all three elements: dominantly indirect with a high degree of internal excitation and near constant product kinetic energy release. A comparison of experiment and theory now allows us to gain an even deeper insight into the underlying dynamics. Fig. 3(d)–(f) shows the results from our FSSH simulations. These also show low product ion velocities and dominant indirect dynamics.^52,53 The good qualitative agreement, which is even better for the lower collision energies not shown here, allows us to extract details not revealed by the experiment alone. Deviations seen for tantalum, for which the highest collision energy is shown, are possibly due to the approximations in the treatment of the spin–orbit manifold or from high-energy electronic states that are not included in the current model. The kinematic cut-offs in Fig. 3 now refer to the respective ISC (orange) and the SCC (white) channels. While for tantalum and niobium the ISC channel is the dominant pathway irrespective of the collision energy, the FSSH simulations revealed that the SCC channel becomes dominant for zirconium once it becomes energetically feasible.⁵¹ The simulated DCSs for both channels can thus be given separately (Fig. 3(f) and (g)). Direct deconvolution of ISC and SCC is not possible for the experimental data (Fig. 3c) but they are not inconsistent with the simulations. Please note the intensity distribution in the backward hemisphere with respect to the kinematic cut-off of the SCC.

Fig. 3 Differential cross sections for Ta⁺/Nb⁺/Zr⁺ + CO₂ at a relative collision energy E_rel ≈ 2 eV. The upper row (a)–(c) shows the experimental data in direct comparison to results from simulations (d)–(f) for the ISC pathway and additionally to the SCC for zirconium (g). The superimposed circles give the kinematic cut-offs for MO⁺ ions formed via ISC (orange) or SCC (white). The bottom row (h)–(j) shows a comparison of the integrated angular distribution for the DCS shown in (a)–(g) (experiment black and simulation blue). For niobium, only the scattering range θ = 0°–143° is shown because of signal-to-noise due to incomplete background subtraction in (b). The complete angular range is shown in (i) but the area omitted in (b) indicated by the grey dashed area. In case of zirconium the total distribution (ISC + SCC) is shown for the simulation. Adapted with permission from ref. 48; ref. 49; ref. 52 (Copyright 2024 American Chemical Society); ref. 53 (Copyright 2024 American Chemical Society); and ref. 51.

These multi-state reactions have two candidates as possible bottlenecks: ISC or the “classical transition state SP” (see Fig. 1). Both potential bottlenecks are submerged with respect to the free reactants Ta⁺ + CO₂ and trap the reaction in the same local minimum (RC). Fig. 4a illustrates the theoretically calculated distribution of quintet-to-triplet transitions at E_rel = 2.0 eV for the Ta⁺ + CO₂ reaction, comparing nonreactive trajectories with those forming triplet products. Among nonreactive trajectories, quintet-to-triplet hopping is rare: only about 3.2% of trajectories exhibit at least one such transition. Note that nonreactive trajectories that make one transition from the quintet to triplet would have to make the reverse transition to return to the quintet reactant channel. In contrast, for trajectories yielding triplet products, most reactive events involve a single quintet-to-triplet transition, accounting for about 79.0% of the triplet-producing trajectories. The remaining reactive trajectories undergo 2–10 transitions, but the frequency drops rapidly with increasing numbers of hops.

Fig. 4 (a) Histogram of ISC events (quintet → triplet) observed in the nonreactive channel (black) and ⁵Ta⁺ + CO₂ → ³TaO⁺ + CO (red) reactive channel E_rel = 2.0 eV. (b) Energy profiles of the quintet and triplet states along the reaction coordinate defined by the imaginary mode of the ³SP, with the remaining normal coordinates constrained. (c) Statistical distributions of the C–O bond length at the seam-crossing geometries sampled from reactive trajectories at E_rel = 2.0 eV. The dashed line indicates the C–O distance at ³SP structure. Adapted with permission from ref. 52 (Copyright 2024, American Chemical Society).

Taken together, these statistics indicate that ISC is the rate-limiting step at low collision energies. If crossing the barrier on a given spin state PES were the bottleneck instead, one would expect many trajectories to undergo multiple spin transitions before reaching the product channel. This behaviour contrasts sharply with the FeO⁺ + H₂ reaction, where barrier crossing, rather than spin-state change, controls the reactivity at low collision energies.⁴⁵ The key distinction between that reaction and the systems discussed here lies in the position of the crossing seam. As shown in Fig. 1, the crossing seam between the ground state and first-excited state in the M⁺ + CO₂ reactions is located on the product side of the first SP. This feature is further illustrated in Fig. 4b, which presents one-dimensional cuts of the quintet and triplet PESs along the reaction coordinate associated with the ³SP for the Ta⁺ + CO₂ system. Consequently, once ISC occurs, the system can proceed directly toward the triplet product channel without surmounting additional barriers. This interpretation is supported by the distribution of the C–O bond distances at the quintet-to-triplet hopping geometries shown in Fig. 4c. In contrast, for the FeO⁺ + H₂ → Fe⁺ + H₂O reaction, the sextet–quartet crossing seam lies before the ⁴SP, making barrier crossing unavoidable and therefore rate-limiting.⁴⁵

5 Competition between ISC and SCC channels

5.1 Branching ratio

A quantitate representation of the branching ratios for the ISC (full symbols) and SCC (open symbols) for all three elements has been extracted from simulations and is shown in Fig. 5. The energy required for the endothermic SCC to open is given for Ta⁺ (red), Nb⁺ (blue) and Zr⁺ (black) by the arrows at the top of Fig. 5. Only for zirconium the SCC contributes within the investigated energy range and becomes dominant for E_rel > 2.2 eV. The experimental collision energies are highlighted by the larger symbols to distinguish them from other data points obtained from simulations. The latter were used to confirm that the lack of observation of the SCC for tantalum and niobium was not due to a threshold effect, especially for niobium.

Fig. 5 Competition between ISC and SCC. Branching ratio of ISC (closed symbols) and SCC (open symbols) for Zr⁺ (black circles), Nb⁺ (blue diamonds) and Ta⁺ (red squares) reactions as a function of relative collision energy E_rel for the title reaction.^51–53 Larger symbols refer to energies at which experimental DCS are available. The arrows on the upper energy axis indicate the minimum energy required for the endothermic SCC. The GIB results for Zr⁺ and Nb⁺ are also included for comparison, in which the ISC and SCC contributions were estimated using a combination of exothermic and endothermic models fitted to the experimental data.^20,22

It is interesting to note that there is independent evidence for the SCC pathway of Zr⁺ and Nb⁺ in the guided ion beam (GIB) experiments by Sievers and Armentrout.^20,22 The GIB data shown in Fig. 5 are qualitatively consistent with our theoretical predictions, although quantitative discrepancies exist between the experimental results and the FSSH simulations. Please note the experimental data of Sievers and Armentrout were a fit to the overall cross section, and there are significant uncertainties concerning the relative contributions of the two channels, particularly in the transition region. In the experiment, the SCC channel for the Zr⁺ + CO₂ reaction opens at E_rel = 0.75 eV, and its contributions increase rapidly with increasing collisional energy. In contrast, for the Nb⁺ + CO₂ reaction, the SCC channel opens at about 2.0 eV and grows more gradually. As a result, the ISC channel remains dominant even at relatively high collision energies (up to 3.3 eV).

The contrasting product branching patterns among the three systems can be clearly attributed to differences in their SOC strengths. In the vicinity of the crossing seam, the calculated SOC for the Zr⁺ + CO₂ system amounts to only 38.0 cm⁻¹, markedly weaker than the values of 329.8 and 127.2 cm⁻¹ obtained for the Ta⁺ and Nb⁺ + CO₂ systems, respectively. Such a reduced coupling diminishes the efficiency of spin-state transitions in the zirconium case. Table 1 summarizes the calculated lifetime of the RC and PC at different collision energies. At low collision energies, the RC lifetimes in Zr⁺ + CO₂ reaction (e.g., 1.09 ps at E_rel = 2.2 eV) is considerably longer than those in the Ta⁺ (0.31 ps) and Nb⁺ (0.39 ps) + CO₂ reactions, reflecting its relatively low spin-state transition efficiency. The low transition probability consequently enhances the population transfer into the SCC pathway.

In addition, the lifetime of the respective RCs for the ISC is in all cases by at least a factor of three larger than that for the PC [MO⁺CO]. This again confirms that the dynamic trapping and energy redistribution takes place on the reactant side of the bottleneck and swiftly moves through the PC despite it being the deepest minimum on the PES (see Fig. 1). In case of the SCC for Zr⁺ + CO₂, the lifetimes of the RC and PC both show a similar energy dependence, as expected for a classic endothermic over-the-barrier reaction.

5.2 Energy partitioning: (non)-statistical?

The energy flow during a reaction is one of the main objectives in dynamics studies. So far, we have distinguished between the ISC and SCC channels, i.e., electronic excitation of the MO⁺ product ion. At low relative collision energies of a couple of eV, that is a somewhat special case often associated with reactions that allow for multi-state reactivity.^51,76,77 Much more common is rotational and vibrational excitation of molecular products. Fig. 6 shows the calculated mean amount of energy partitioned into translation of MO⁺ and CO (grey symbols) and into rotations (orange (MO⁺) and green (CO) symbols) and vibration (red (MO⁺) and blue (CO) symbols) of each molecule in absolute values (eV). Despite the indirect mechanism and the generally high degree of ro-vibrational excitation of CO and especially MO⁺, the energy partitioning is not statistical for the ISC pathway. In each case, little to no additional collision energy is partitioned into translation and for each case, a preferential mode exists into which most of the additional (collision) energy is partitioned into.

Fig. 6 Product energy distributions for the reactions with (a) Ta⁺,⁵² (b) Nb⁺,⁵³ and (c) and (d) Zr⁺ (ref. 51) into the products MO⁺ + CO in absolute energy scales (eV) from the FSSH simulations. The energy is partitioned into translation of MO⁺ + CO (grey circles), vibration (MO⁺ red squares, CO blue triangle) and rotation (MO⁺ orange open square, CO green open triangle). In case of zirconium the ISC (c) and SCC (d) are each shown. Dashed lines are to guide the eye. Despite the indirect mechanism, only the SCC for zirconium is showing statistical energy partitioning among all degrees of freedom. All other reactions have either a preferred channel that most energy is partitioned into or that takes up most of the additional collision energy.

Upon comparing the amount of energy partitioned into MO⁺ and CO, it becomes apparent that on average more energy is partitioned into MO⁺ (red and orange symbols) compared to CO (blue and green symbols). CO receives similar amounts of energy for all three systems. Additional collision energy is preferably partitioned into MO⁺. While rotation shows the highest excitation for tantalum, additional collision energy is partitioned into rotation and vibration alike (Fig. 6a).⁵² On the other hand, ZrO⁺ and NbO⁺ vibration picks up almost all of the additional collision energy (Fig. 6b and c).^51,53 The significant rotational excitation, in all three cases, is in agreement with the dynamic signature seen in the product ion velocity distributions and redistribution of angular momenta from the collision into product rotational momentum. Nevertheless, the dominant character of the rotations in TaO⁺ was unexpected prior to simulations.^51,52 On the other hand, the dominance the ZrO⁺ stretching vibration is striking for zirconium.⁵¹ The energy partitioning for the ISC pathway and SCC can be compared for zirconium. Here, the differences between the pathways becomes obvious. The energy partitioning for the SCC is statistical, i.e., each degree of freedom picks up similar amounts of the additional collision energy. In Fig. 6d, this is illustrated by the same slopes indicated by the dashed lines. Even though the absolute amount of energy partitioned into translation is higher for the exothermic ISC channels, its relative contribution to the energy budget is lower than for the endothermic SCC.

The energies given in Fig. 6 are the mean values extracted from product vibrational and rotational state distributions.^51–53 A closer look at the distributions reveals similarities between the three transition-metal elements for MO⁺ and differences of MO⁺ to CO (Table 2). In all cases, the vibrational and rotational state distributions for CO are much narrower compared to respective distributions for MO⁺. Generally, the CO vibrational state distributions peak at low v (v = 1–2). The rotational distributions for CO show a common envelop shape with a mean J of ≈40–50 (see Table 2). The distributions for MO⁺ are much broader with a mean rotational excitation J̄ exceeding 100 with populations extending up to J = 250. The rotational distribution is inverted which is the most obvious for TaO⁺. Vibrational state distributions extend as far as v = 30 for the MO⁺ stretching vibration. While the distributions peak at low v for TaO⁺ and NbO⁺ and subsequently decay. The distributions show a much less steep decay for ²ZrO⁺. Here, the vibration picks up the additional collision energy. The three reactions show similarities but also distinct differences, especially with respect to the mean energy partitioning into translation, vibration and rotation but also on the resulting state distributions.

To better understand this, we applied the sudden vector projection (SVP) model.⁷⁸ The SVP model is based on the assumption that the energy disposed into a product mode is correlated to its projection onto the reaction coordinate. We can make use of the model because the calculated structures of the SP and the MECP are very similar and the reaction swiftly proceeds to products once the ISC happens, i.e., the PCs’ lifetimes are very short (see Table 1). The SVP agrees reasonably well with the FSSH results, predicting that a larger fraction of the energy is partitioned into product translation than into other degrees of freedom (Ta: 0.591, Nb: 0.391, Zr: 0.567). SVP slightly overestimates the CO rotation which might be due to the MECP being slightly on the product side. Please note that the SVP is about energy partitioning upon product formation, i.e., once the bottleneck is passed. Taking this to mind, the SVP values for translation make even more sense with respect to the kinetic energy release seen in experiment and FSSH simulations.

Table 1 Calculated lifetimes of the pre-reaction complex (RC) and the post-reaction (PC) complex along the reaction pathway for the relative collision energies used in the crossed beam experiments (data for Ta⁺ + CO₂ from ref. 52, for Nb⁺ + CO₂ from ref. 53 and for Zr⁺ + CO₂ from ref. 51)

M^+		Erel/eV	Lifetime/ps
			RC [M^+CO2]	PC [MO^+CO]
Ta	ISC	1.0	0.36	0.10
		1.4	0.33	0.10
		2.0	0.33	0.08
		2.2	0.31	0.08
Nb	ISC	1.3	1.30	0.20
		1.8	0.56	0.13
		2.2	0.39	0.08
Zr	ISC	1.7	1.36	0.47
		2.2	1.09	0.34
		3.3	0.48	0.24
	SCC	1.7	1.17	1.65
		2.2	0.87	0.59
		3.3	0.36	0.24

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Table 2 Vibrational and rotational quanta MO⁺ and CO and the mean vibrational and rotational state and SVP values. Please note that because the endothermicity of the quartet channel, the barrier is not expected to govern the product energy distribution. Instead, the dynamics are controlled by the exit channel. As a result, SVP fails to yield accurate predictions (data for Ta⁺ + CO₂ from ref. 49 and 52, for Nb⁺ + CO₂ from ref. 49 and 53 and for Zr⁺ + CO₂ from ref. 51)

	Erel/eV	Vibration						Rotation
		MO^+			CO			MO^+			CO
		ν/cm^−1	SVP	v̄ (vmax)	ν/cm^−1	SVP	v̄ (vmax)	B/cm^−1	SVP	J̄ (Jmax)	B/cm^−1	SVP	J̄ (Jmax)
^3TaO^+ + CO	1.0	1081	0.301	5(20)	2187	0.071	1(10)	0.34	0.339	140(236)	1.92	0.428	38(89)
	1.4			6(24)			1(15)			147(245)			39(104)
	2.0			7(31)			1(15)			154(262)			42(111)
^3NbO^+ + CO	1.3	1072	0.273	4(21)	2187	0.051	0(6)	0.45	0.265	122(222)	1.92	0.391	46(92)
	2.2			8(32)			1(11)			120(250)			48(104)
^2ZrO^+ + CO	1.7	1023	0.224	8(31)	2187	0.066	1(9)	0.43	0.439	131(254)	1.92	0.576	48(102)
	2.3			11(35)			1(7)			131(269)			46(104)
	3.3			15(35)			2(11)			136(284)			44(110)
^4ZrO^+ + CO	1.7	712	0.035	0(1)	2187	0.030	0(0)	0.33	0.731	35(76)	1.92	0.372	12(28)
	2.3			1(6)			0(2)			61(143)			19(49)
	3.3			3(13)			0(6)			91(209)			26(85)

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6 Conclusions

Crossed-beam velocity map imaging experiments were employed to obtain energy- and angle-resolved product velocity images for the Ta⁺, Nb⁺, and Zr⁺ + CO₂ reactions. In parallel, nonadiabatic dynamics were investigated using trajectory surface-hopping on globally accurate full-dimensional multi-state PESs.

The measured DCSs provide direct evidence that all three reactions are dominated by indirect mechanisms involving long-lived entrance complexes. Experimentally, a substantial portion of the available energy is partitioned into internal excitation of the products and the translational energy of the products does not shift with increasing collision energy. Complementary trajectory analyses reveal that most of the available energy is dispersed into internal degrees of freedom of the products, with significant vibrational and rotational excitation of the metal oxide cation.

A central mechanistic outcome of the combined experimental and theoretical analysis is that ISC constitutes the rate-determining step in all three systems. The overall reactivity and product branching are therefore controlled by the efficiency of ISC. Although the potential energy topographies are closely related, the three systems differ in the relative importance of spin-changing and spin-conserving pathways. For Ta⁺ and Nb⁺, the reactions proceed predominantly through ISC channels under the investigated conditions. In contrast, in the Zr⁺ system the spin-conserving channel (SCC) becomes competitive once energetically accessible, resulting in distinctly different branching ratios. These trends are consistently rationalized by the markedly different SOC strengths: the larger SOC in Ta⁺ and Nb⁺ enhances ISC efficiency and thereby promotes spin-changing pathways, whereas the substantially smaller SOC in Zr⁺ reduces ISC efficiency and permits competition from the spin-conserving channel.

We further note that the combined experiment-theory approach discussed here is applicable to many other systems, as illustrated by our recent work on the Ta⁺ + CH₄ reaction.⁷⁶ These detailed information helps to further our understanding of transition-metal reactivity towards small molecules and might help to elucidate the spin control in SAC catalysis.

Author contributions

J. M. and H. G. prepared the manuscript with contributions from all authors. Contributions to the original studies featured here are given in the respective publications.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this Perspective has been included as part of the five original publications, which are the basis for the submitted manuscript. The original work is cited appropriately in the text and the respective figure captions. Additional data in numerical format is available from the authors upon request.^{48,49,51–53}

Acknowledgements

Work at RPTU was supported by the Deutsche Forschungsgemeinschaft DFG (Project No. 50027921). Work at UNM was supported by Air Force Office of Scientific Research (Grant No. FA9550-22-1-0350 to H. G.). We thank all co-workers who participated in the crossed beam experiment (Maximilian E. Huber, Martin Wedele and Maurice Birk). Special thanks go to Prof. Milan Ončák as well as to Shaun G. Ard, Nicholas S. Shuman and Albert A. Viggiano. J. M. thanks Björn Bastian and Prof. Roland Wester for the use of the data analysis package for the crossed beam data.

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Footnote

† These authors contributed equally.

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