Photodissociation of ${\text{Cr(CO)}}_4\text{bpy}$: A Non‐Adiabatic Dynamics Investigation

3.1. Potential Energy Scans

Potential energy curves of single‐reference methods are compared against high‐level MRCI/CASSCF(6,17) data in Figure 1a. All studied states are of $3d\to {\pi }_{\text{bpy}}^{\ast }$ MLCT character. The highest singlet $S_3$, denoted $2{\text{A}}_1$ as per its FC symmetry, is a $3d_{xz}\to {\pi }_{\text{bpy}}^{\ast }$ state. The lower $S_2$ and $S_1$ states, denoted $1{\text{B}}_1$ and $1{\text{B}}_2$, are of $3d_{z^2-x^2}\to {\pi }_{\text{bpy}}^{\ast }$ and $3d_{xy}\to {\pi }_{\text{bpy}}^{\ast }$ characters, respectively. See Figure S1 in ESI for orbital graphical representations. The high‐level MRCI/CASSCF(6,17) [43] curves feature a ground state barrier of ∼0.8 eV, and excited states show shallow minima. All single reference methods fail to qualitatively capture the ground state shape of PES. HF follows the reference ground state very closely, not showing any failure. UHF and UKS calculations of several functionals all give the spin operator value $⟨S^2⟩$ of 0.0 along the entire reaction coordinate, showing that a ground state dissociation produces two singlet products, and is of singlet character throughout. The ground state of CAM‐B3LYP has a significantly overestimated energy barrier of ∼1.9 eV. The transition energies at FC and dissociation‐limit geometries are in excellent agreement with MRCI/CASSCF(6,17), but intermediate geometries give overestimated energies of up to 0.5 eV (see Figure S2). Independently of this, the erroneous shape of the ground state renders all excited states non‐dissociative. The same PES calculations were repeated with all DFT functionals available in ORCA 5.0.3 (not shown) to investigate if this is a functional or method‐dependent problem. All have produced the same overestimated ground state energy barrier, showing that DFT is fundamentally unable to describe this bond‐breaking process and that a different method needs to be used. MP2 shows typical failure of describing dissociation processes, based on which ADC(2) calculations were not attempted. These considerations have led us to consider CASSCF as the method of choice.

Potential energy curves of SA4‐CASSCF(6,7) and SA12‐CASSCF(10,11)a are shown in Figure 1b. All CASSCF variants qualitatively reproduce the shape of reference curves. SA4‐CASSCF(6,7) and SA12‐CASSCF(10,11) a show minimum ground state geometries with bond lengths of 2.1 Å (1.9 Å for MRCI/CASSCF(6,17) and B3LYP) and excited state energy barriers: ∼0.1 eV for ${\text{S}}_1$ and ${\text{S}}_2$; $\sim 0.2\text{eV}$ for ${\text{S}}_3$ (0.1 eV and 0.02 eV for MRCI/CASSCF(6,17)). Interestingly, neither method explicitly includes orbitals or states involved in avoided crossings between $3d\to 3d_{x^2}$ and lower excited states, which are well established in literature to be the cause of quasi‐bound excited state shapes [37]. Reference MRCI/CASSCF(6,17) data does not include the relevant $3d_{x^2}$ orbital in its active space either, but this interaction is expected to be incorporated through the post‐CASSCF CISD calculation (although not for the ground state, which is just CASSCF(6,17)). Analysis of the CASSCF solution shows that, as the bond stretches, the ${\pi }_{\text{bpy}}^{\ast }$ orbital gains a contribution from the $3d_{x^2}$ basis function (see Table S4). CASSCF(10,11)b calculations (Figure S3a), which explicitly include the relevant $d_{x^2}$ orbital at FC geometry, show a much greater contribution of $3d_{x^2}$ basis function as the bond stretches, clearly showing state mixing. Points at which the contribution of this basis function increases considerably coincide with the local maxima and onsets of plateaus in CASSCF(10,11)b.

For SA4‐CASSCF(10,11)b, the ground state curve has a maximum at 3.0 Å and strongly underestimated energies beyond that point. $S_1$ and $S_2$ state curves remain nearly unchanged, while $S_3$ now features a gentle maximum at the same coordinate as the ground state. In SA7‐CASSCF(10,11)b, the ground state no longer has a local maximum, but all three excited states do. Simultaneously, they have significantly higher excitation energies throughout the entire curve, but more so near the FC region. The states $S_4$, $S_5$, and $S_6$ (not shown) are all of $3d\to 3{\text{d}}_{x^2}$ character at extended bond lengths (they are strongly mixed at the FC region) and all feature a local minimum at the same geometry as the maxima of lower states. This method clearly captures the avoided crossing but is, as a whole, of much worse quality. These characteristics are augmented further in SA12‐CASSCF(10,11)b, where all excited states are dissociative and have higher excitation energies. Our attempts at explicitly incorporating the avoided crossing have brought behavior that is more different from MRCI/CASSCF(6,17) and at a greater computational cost. The smaller active space, SA4‐CASSCF(6,7), produces the most quantitatively agreeable behavior, and so it was chosen to be used in further dynamic calculations.

To complete static studies, PES for triplet states were calculated at the SA3‐CASSCF(6,7) level of theory (see Figure S3d). Calculated triplets are the analogues to the excited states of SA4‐CASSCF(6,7). $T_1$ and $T_2$ are almost coincident in energy outside of the FC region. The character of triplet states, assigned based on contributing orbital transitions, is discontinuous around 2.00–2.75 Å. This suggests the existence of an accessible conical intersection seam between $1^3{\text{A}}_1$ and $1^3{\text{B}}_1$, or $T_3$ and $T_2$ states. This would enable quick depopulation of the ${\text{T}}_3$ state and support the observation of only two emissive triplet states [42]. The energy barrier of both $T_1$ and $T_2$ is under 0.1 eV, which is lower than for singlet states.

PES singlet curves of Cr–CO_eq were calculated using SA4‐CASSCF(6,7) (see Figure S4). Ground state in this coordinate has a higher energy barrier of 1.2 eV and excited state barriers of 0.7–1.0 eV. Furthermore, the ${\pi }_{\text{bpy}}^{\ast }$ orbital gains no contribution from the $3d_{x^2}$ orbital upon elongation of the bond, due to its perpendicular position to the ${\text{CO}}_{\text{eq}}$ ligand. This directionality of the avoided crossing is consistent with ligand selectivity of photodissociation. Simultaneously, the excited energy barriers are low enough to allow for equatorial dissociation if a complex is designed with an appropriate metal center and ligands, which both stabilize axial bonds and destabilize equatorial bonds.

From a simple electrostatic picture, Cr→bpy excitations lead to the depopulation of the metal center. This leaves less electron density for bonding between Cr and CO. Consequently, any Cr→bpy excitation would have a photodissociative effect, regardless of spin, which is in line with CO being a spectator ligand of the excitation process. Indeed, the lowering of excited state energy barriers is observed along both ${\text{CO}}_{\text{ax}}$ and ${\text{CO}}_{\text{eq}}$ dissociations, but the absence of $3d_{x^2}$ avoided crossing results in bound shapes of the excited states along ${\text{CO}}_{\text{eq}}$ dissociation pathway.

3.2. Electronic Dynamics

The evolution of Molecular Coulomb Hamiltonian (MCH) active state electronic populations is presented in Figure 2. The bright $S_3$ state starts decaying instantly and fits well with a monoexponential decay function with a lifetime of 86 fs (over 300 fs, $R^2=0.981$). This is in good agreement with the solvent‐independent experimental value of 96 fs [45]. Electronic population of $T_3$ and $T_2$ never exceeds 0.05 and 0.1, respectively. $T_1$ rises approximately linearly during 50–300 fs, reaching a plateau of ∼0.19. Calculating triplet state lifetimes would require much longer simulation times, but the existence of a $T_1$ plateau does indicate a much slower decay to $S_0$, in accordance with their measured picosecond lifetimes [42, 45]. The ground state population increases in a semi‐stepwise manner, with three flat regions at 0 fs, ∼120 fs, and ∼300 fs. The first two regions suggest that transitions to the ground state are favored at stretched bond geometries (discussed below), while the third is more likely due to the exhaustion of higher singlet state population and dephasing of coherent oscillations.

To complement electronic population data, the total count of hopping events and the net difference of downwards and upwards hops are presented in Table 1. The resulting approximate mechanism is shown in Figure 3. The main transfer of population goes through a cascade of $S_3\to S_2\to S_1\to S_0$, with branching points $S_2\to T_3$ and $S_1\to T_2$, and non‐negligible direct transfer of $S_3\to S_1$. $S_3\to S_2$ transitions appear to occur mainly at the very beginning of the simulation, suggesting that this is the only allowed transition in the FC region. The ‘middle’ states $S_2$, $S_1$, $T_3$, and $T_2$ are in dynamic equilibrium, owing to low energy separation, but with a net flow towards $S_1$ and $T_2$. From there, trajectories decay either via $S_1\to S_0$ or $T_2\to T_1$. Transitions to $S_0$ are mostly irreversible, though some back hops are observed. The $S_0$ state energy is frequently close to the ‘middle’ manifold during a trajectory, but once a transition to $S_0$ occurs, ground and excited state energies quickly diverge, preventing back hops. This is true even if a dissociating trajectory drops to the ground state, suggesting that this is due to a nuclear rearrangement of the complex, fully coordinated or not. Finally, there is a dynamic equilibrium between $T_2$ and $T_1$, which is in agreement with the observed two trapping triplet states [37, 42]. The lack of direct transfer $S_{1-3}\to T_1$ is in accordance with previous observations [45]. The apparent equilibrium between $T_1$ and $S_0$, as judged by the number of hops in Table 1, is due to low energy gaps and state mixing on trajectories in the $T_1$ state. The low net transfer into $T_3$ and $T_2$ states does not allow for a confident confirmation of the existence of a $T_3/T_2$ conical intersection, which was suggested by calculated PES.

3.3. Nuclear Dynamics

For each trajectory, the evolution of the two equivalent Cr–CO_ax bonds was tracked, and the one that has reached a greater value is shown in Figure 4a. The bond length of all Cr–CO_eq is shown in Figure S5. No dissociation of an equatorial ligand was observed, in accordance with literature [42]. For all trajectories, if the Cr–CO_ax bond length has passed 3.5 Å, it was not observed to drop back below this value, so it was chosen as the threshold after which a trajectory is considered dissociated. Upon excitation to the $S_3$ state, all trajectories begin stretching their Cr–CO bonds. For ${\text{CO}}_{\text{ax}}$ (${\text{CO}}_{\text{eq}}$), they begin at ∼2.02 (1.92) Å and oscillate between 2.0 (1.9) and 2.3 (2.1) Å with a period of ∼150 (80) fs. Upon reaching the first maximum of Cr–CO_ax, most trajectories proceed back down the oscillation pathway, while a small subset evolves towards longer bond lengths. Of this subset, some do not dissociate, instead taking a longer arc down, or hovering close to 3.0 Å for over 100 fs. The remaining other trajectories proceed beyond this avoided crossing region and dissociate.

At 400 fs, 9 out of 70 trajectories were observed to dissociate (7 dissociated trajectories were stopped before 400 fs), and all did so as a continuation of the upward motion of the Cr–CO_ax oscillation, giving a quantum yield of 13%. It is in rough agreement with experimental work, where it ranges from 1% to 10%, depending on experimental conditions [29, 39]. Seven trajectories dissociated during the first period, and two more in the second period. One additional trajectory dissociated beyond 400 fs, on the third period.

To investigate the importance of other motions, Figure 4b presents the time evolution of selected frequency‐mass scaled normal coordinates $\nu$. For each mode $i$, the mean displacement ${\mu }_i$ and standard deviation ${\sigma }_i$ (also referred to as activity) are presented in grey and blue, respectively. These parameters were averaged in time slices of 20 fs, up to 400 fs. Analysis of all modes is shown in Figure S6. The majority of normal modes show minimal temporal change. Those which vary most in time all involve carbonyl ligands: ${\nu }_{72}$ and ${\nu }_{73}$ are the antisymmetric and symmetric axial C‐O stretching modes, ${\nu }_{16}$ and ${\nu }_{17}$ are symmetric and antisymmetric Cr–CO_ax stretching modes, and ${\nu }_4$ involves _axOC‐Cr–CO_ax bending in the xz‐plane, with a lesser contribution of equatorial _eqOC‐Cr–CO_eq bending.

Several modes are set in motion in a coherent manner upon the initial electronic excitation: ${\nu }_{12}$ and ${\nu }_{14}$, which both have strong contributions from symmetric stretching of Cr‐N bonds; ${\nu }_{22}$, the symmetric stretch of Cr‐CO_eq; ${\nu }_{19}$, ${\nu }_{35}$, ${\nu }_{38}$, ${\nu }_{45}$, ${\nu }_{60}$, ${\nu }_{64}$, and ${\nu }_{66}$, which are all stretching modes of bipyridine bonds; ${\nu }_{27}$, which involves OC‐Cr‐CO and Cr‐C‐O bending of all carbonyl ligands. Of these, only modes ${\nu }_{22}$ and ${\nu }_{27}$ show a sizeable change in activity over time.

Several low‐frequency modes, in particular axial C‐Cr‐C bending ${\nu }_4$, increase in activity in a semi‐stepwise manner and not immediately upon excitation. The onset of this sharp increase happens at around 100 fs, which is the same period of time at which the Cr–CO_ax bond length was noted to diverge between dissociation, contraction, or hovering for an extended period. When considering dissociating trajectories alone (Insets of Figure S6), the activity of ${\nu }_{16}$ and ${\nu }_{17}$ is much greater, while the activity of ${\nu }_4$ is considerably reduced, though the mean displacement remains high in both. This behavior suggests a coupling between Cr–CO_ax stretching and _axOC‐Cr–CO_ax bending modes, which would act to inhibit dissociation.

These observations suggest a ballistic model of the photodissociation process. Looking again at PES of Figure 1, the $S_3$ state has a downward gradient away from the FC region at 2.02 Å, which is the center of the initial Wigner distribution. This slope gives the system kinetic energy towards dissociation. For most trajectories, this is not enough to overcome the energy barrier: The bond contracts back. A smaller subset will have kinetic energy that is approximately equal to the barrier. These trajectories will hover in the avoided crossing region plateau, on top of the energy barrier, before turning back after an extended period. Finally, those with even more energy will overcome the barrier with sufficient leftover kinetic energy to continue the motion towards dissociation on a flat surface. This is supported by the observation that the initial bond lengths of all dissociating trajectories are in the shorter half of the initial Wigner distribution. As time proceeds, for non‐dissociating trajectories, the excess energy within this oscillating normal mode is expected to equipartition into other modes of motion, making the dissociation less likely after subsequent periods. In other words, if a trajectory did not dissociate during the first period of bond oscillation (around 150 fs), it is less likely to dissociate later on. It is still possible as a probabilistic process, and three trajectories in total were observed to dissociate on second and third periods, but in significantly lesser numbers. It is also noted that solvent may prohibit dissociation after the first period of oscillation. A ballistic model is consistent with the ∼100 fs timescale of photodissociation [42, 45] and wavelength‐dependence of the quantum yield, as occupying a higher vibrational state will directly favor dissociation. The evolution of Cr–CO_eq (Figure S5) shows no dissociation. When compared with Cr–CO_ax, shorter periods of oscillation (80 vs. 150 fs) and shorter bond lengths at minimum geometry (1.92 vs. 2.02 Å) indicate a stronger bond, consistent with the bound shape of PES and higher energy barriers (Figure S4). Slight overestimation of the photodissociation quantum yield is a possible consequence of neglecting solvent effects. An explicit solvent could act as a physical barrier for the released molecule, shifting the energy barrier upwards, as well as damping vibrational motion, prohibiting dissociation beyond the first period of Cr–CO_ax oscillation.

3.4. Combined Nuclear and Electronic Dynamics

In this section, we combine the electronic and nuclear dynamic information, seen on the middle and bottom panels of Figure 4a. Seven out of ten dissociating trajectories are of triplet character at 3.5 Å. Five of these trajectories undergo intersystem crossing long before reaching the energy barrier at ∼3.0 Å; the remaining two do so in its vicinity. This is in contrast to the established understanding that triplet states are non‐dissociative. It is, however consistent with the calculated PES, wherein $T_1$ and $T_2$ have lower energy barriers of dissociation than all singlet states. Therefore, if a trajectory undergoes intersystem crossing early on, it will approach a lower energy barrier and be more likely to cross it. This supports the above‐explained mechanism, where the main determining factor is the energy barrier and not the occupied electronic and spin state.

A remaining question is how the nuclear conformation affects the probability of transitions. Cannizzo et al. [63] have noted a correlation between metal‐ligand vibrational frequencies and intersystem crossing time scales for analogous systems $\text{Re}(\text{L}){(\text{CO})}_3$(bpy) [L = Cl, Br, I]. This suggests that a distortion along the ligand‐metal stretching mode leads to an area with greater spin‐orbit coupling and favoured intersystem crossings. Similar observations were noted for the lifetimes of singlet MLCT states in Fe and Ru tris‐bpy complexes [8, 64]. In present simulations, the semi‐stepwise increase of $S_0$ population points towards a similar behavior.

To investigate the coupling between nuclear motions and electronic spin transitions, Figure 5a presents: (1) energy gap between states $S_1$ and $T_2$; (2) norm of the spin‐orbit coupling matrix element (SOCME), averaged over all trajectories as a function of normal mode displacement, for selected modes. A full plot over all modes is presented in Figure S7. We focus on the S ₁/T ₂ crossing, as it accounts for over half of all ISC hops (Table 1). Variation of SOCME was not observed to be large enough to indicate a favorable ISC at particular geometries. All states in the middle manifold ($S_{1-2}$ and $T_{1-3}$) were observed to evolve in parallel; that is, the pairwise energy gap remained near constant. For the S ₁/T ₂ pair, the energy gap was very close to degeneracy, as can be seen most clearly for mode ${\nu }_{17}$. A weak dependence of energy gap on nuclear displacement was noted for various bending modes involving Cr‐L bonds, with example modes ${\nu }_4$ and ${\nu }_{27}$ being shown. The symmetric Cr–CO_ax stretching mode does also show a weak dependence, and with the previously shown coupling with ${\nu }_4$, they would account for a weakly favoured ISC at extended Cr–CO_ax geometries. However, the near‐degeneracy of these two states remains as the dominant driving force for ISC. Figure 5b shows distributions of all hops and the initial Wigner distribution as a function of Cr–CO_ax bond length. For each trajectory, hops are shown against bond lengths of both ${\text{CO}}_{\text{ax}}$, with the one that has reached the farthest denoted ${\text{CO}}_1$; the other ${\text{CO}}_2$. ${\text{CO}}_2$ acts as a control distribution, approximating one that would statistically arise from vibrational motion alone. For ${\text{CO}}_1$ (${\text{CO}}_2$), the median of initial distribution is 1.99 Å while the distributions of hops have medians of 2.21, 2.24, and 2.27 Å (2.14, 2.12, and 2.14 Å) for $S\to S$, $T\to T$, and intersystem crossing hops, respectively. Comparison of ${\text{CO}}_1$ and ${\text{CO}}_2$ distributions shows that ISC hops are weakly favored at stretched bond geometries, in accordance with the above analysis of energy gap dependence.