How to calculate the rate constants for nonradiative transitions between the MS components of spin multiplets?

Predicting the rates of spin-dependent processes characterised by nonradiative transitions between electronic states with different spin multiplicities is important for understanding the mechanisms of many photochemical and catalytic reactions. To calculate these rates, it is necessary to define the spin state representation and the couplings between these states that drives the interstate transitions. In this work, we describe three different approaches to calculating the spin–orbit coupling (SOC), transition probabilities, and rate constants between the MS components of the electronic states with different spin multiplicities. We implemented these approaches in our nonadiabatic statistical theory (NAST) software package, which predicts the transition probabilities and rate constants of spin-dependent processes using information obtained from electronic structure calculations. We discuss the advantage and drawbacks of each approach and, as an example, calculate the rate constants for transitions between the spin states in the active site model of the protein rubredoxin. GRAPHICAL ABSTRACT


Introduction
Spin-dependent phenomena in chemistry, physics and biology have attracted considerable interest for a long time [1][2][3][4][5][6]. These phenomena often depend on the spin-orbit coupling (SOC), an interaction between the intrinsic magnetic moments of electrons and their orbital angular momentums [7,8]. Compared to other spin interactions, such as the spin-spin, orbit-orbit, and Zeeman interactions, the SOC is usually the strongest. Examples of the SOC-driven processes include the spin-forbidden reaction for a ring-closure of triplet cyclopentane-1,3diyl [9,10], the intersystem crossing in the photoactivated CONTACT  Mn-based complexes leading to the singlet oxygen generation with applications in photodynamic therapy [11,12], and the quantum tunnelling intersystem crossing in the photoactivated thiophosgene molecule [13][14][15][16][17]. Moreover, the SOC-mediated transitions between the electronic states with different spin multiplicities are believed to play a role in the biological electron transfer in metalloproteins [18,19]. The SOC effects are also central to the electron spin dynamics in quantum dots [20], coherent state spin qubits [21] and molecular magnets [22].
In Section 2, we introduce NAST and three different approaches to calculate the transition probabilities and rate constants from SOCMEs using the LZ and WC formulas. The same section discusses the applicability of the LZ and WC probabilities for the introduced approaches. In Section 3, we demonstrate how the three approaches work by calculating the transition probabilities and rate constants between the spin states in the active site model of the protein rubredoxin. We discuss the contributions of the M S -specific probabilities and rate constants to the corresponding effective state-averaged quantities. We conclude with the discussion of the advantages and drawbacks of each approach and outline the future directions in the NAST development.

Transition probabilities and rates between electronic states with different spin multiplicities
The microcanonical rate constant k(E) expressed as a function of the rovibrational energy E can be written as: The integral represents the convolution of the rovibrational density of states ρ X and the interstate transition probability P at the MECP geometry, while ε ⊥ is the energy partitioned in the reaction coordinate. The prefactor contains the rovibrational density of states ρ R , the symmetry numbers σ R and σ X , the number of chiral MECP isomers γ X , and the Planck constant h. The subscripts R and X indicate the properties of the reactant and MECP structures, respectively. A temperaturedependent (canonical) rate constant can be calculated by averaging the microcanonical constant of Equation (1) over the internal energy Boltzmann distribution [31].
The interstate transition probability can be calculated with the double-passage Landau-Zener (LZ) or weak coupling (WC) formulas using the SOC between spin states.
In the above equations, | g| = |g 1 − g 2 | andḡ = (|g 1 ||g 2 |) 1/2 are defined in terms of the two gradients g 1 and g 2 of the crossing PESs at MECP, μ ⊥ is the reduced mass for the motion along the reaction coordinate, is the reduced Planck constant, and Ai is the Airy function. The MECP energy relative to the energy of reactant, E X , can be redefined as E X → E X + ZPE X − ZPE R to account for the zero-point energy (ZPE). Equations (1)-(3) can be used to calculate the transition probabilities and rate constants between the spin states in several different representations by defining SOC for a specific representation.

Effective SOC approach
The commonly used effective spin-orbit coupling (SOC eff ) is obtained as the root-mean-square of the SOCME between all M S components of the low-spin (LS) and high-spin (HS) states [27] (LS and HS refers to the spin quantum number S): In Equation (4), S and S are the spin quantum numbers for LS and HS states, M S and M S are the corresponding magnetic quantum numbers, andĤ SO is the spin-orbit operator.
For the often-encountered case of a singlet-triplet crossing, the sum of the non-relativistic HamiltonianĤ 0 and the spin-orbit HamiltonianĤ SO in the spin-diabatic M S basis is represented by the following 4 × 4 matrix: The diagonal terms, E S = 0, 0|Ĥ 0 |0, 0 and E T = 1, M S |Ĥ 0 |1, M S with M S = −1, 0, +1 are the energies of the singlet and triplet states. The M S = 0 component of the singlet state couples to all three M S components of the triplet state. The non-zero SOCMEs are z = 0, 0|Ĥ SO |1, −1 , ib = 0, 0|Ĥ SO |1, 0 and z * = 0, 0|Ĥ SO |1, +1 , where z * is the complex conjugate of z, and b is real. Therefore, there are only two independent coupling parameters (ICPs).
Here, we consider a linear crossing model with E S = r and E T = −r, where r is a reaction coordinate ( Figure 1(a)). The spin-adiabatic states, shown in Figure 1(b), are obtained by diagonalising the Hamiltonian matrix in Equation (5). The direct energy gap between the two non-degenerate spin-adiabatic states E 1 and E 4 is equal to twice the effective SOC defined by Equation (4). This observation provides a strong justification for calculating the SOC between singlet and triplet states using Equation (4). However, as we show below, if the multiplicity of LS state is higher than one, there are multiple energy gaps between adiabatic states, and the physical meaning of the effective SOC value defined by Equation (4) is not clear [32]. One may assume that multiple energy gaps in the adiabatic representation correspond to SOCMEs between the individual M S components of the spin states. In the next two subsections, we show that this assumption is not correct.

M S -specific approach
In the M S -specific approach, the interstate transition probabilities, P M S ,M S , are evaluated using the individual SOCMEs [31]. Because the transition probabilities are calculated between spin-diabatic states, the same energy gradients and Hessian matrices are used for all M S components of the same spin multiplet. By replacing the SOC value in Equations (2)  , which reduces the computational cost. For example, in the case of a singlet-triplet crossing, the SOCME values z and b replace the SOC in the probability formulas. The obtained transition probabilities can be employed to calculate the microcanonical rate constants between individual M S components of the crossing multiplets using Equation (1). This M S -specific approach can be used to evaluate the contributions of the rate constants between each pair of M S components into the rate constant obtained with the state-averaged approach. Moreover, the M S -specific approach is needed to model the spin-dependent kinetics in an external magnetic field, which breaks the degeneracy of the M S components of the same spin multiplet and therefore requires to treat transitions between each pair of M S components separately.

Intermediate approach
The intermediate approach for computing SOC is similar to the effective approach defined by Equation (4), with the exception that the summation performed only over the M S component of HS state.
Note that the notation SOC |±M S | indicates that the values for the −M S and +M S components of LS state are equal. For the singlet-triplet crossing, the intermediate and effective approaches are identical. However, as demonstrated below, for the crossings between states with higher multiplicities, the intermediate approach provides couplings that correspond to the energy gaps between spinadiabatic states.
For example, the Hamiltonian for a doublet-quartet crossing in the spin-diabatic basis can be written in the form of 6 × 6 matrix, In Equation (7), E D and E Q are the energies of the doublet and quartet states, and the ICPs are defined as  = (|z 1 | 2 + b 2 + |z 2 | 2 ) 1/2 . As shown in Figure 2, these two equal couplings correspond to half of the energy gaps between the two pairs of adiabatic states (E 1 = E 2 and E 5 = E 6 ). In contrast, the effective approach gives only one coupling value, SOC eff = 2SOC 1 2 .
As an alternative, the intermediate SOC can be defined as the sum over the M S components of LS state. However, in this case, the number of distinct non-zero coupling is larger than the number of energy gaps in the adiabatic representation. Therefore, in this work, we adopt the definition of Equation (6).
For a triplet-quintet crossing, the spin-diabatic Hamiltonian is The  between the three pairs of spin-adiabatic states shown in Figure 3. The intermediate approach is applicable to the crossings between the pairs of spin states of any multiplicities.
The diagram in Figure 4 shows the SOC values used in the intermediate approach for the crossings between the spin states up to sextet. In Supporting Information, we consider a quartet-sextet crossing with four complex (z 1 , z 2 , z 3 and z 4 ) and two imaginary (ib 1 and ib 2 ) ICPs, producing two pairs of equal couplings that match two gaps between the spin-adiabatic states ( Figure S1).

Applicability of the weak-coupling and Landau-Zenner probabilities
From Equations (4) and (6), it is obvious that the effective SOC is a sum of the intermediate couplings: Because the WC transition probability is proportional to the square of SOC, Equation (9) implies that the WC effective probability between two spin multiplets is equal to the sum of intermediate transition probabilities: In contrast, the LZ transition probability expression contains SOC in the argument of an exponential function, and the relation similar to Equation (10) is not valid. However, for a small argument, the power series of exponential function can be truncated as exp(x) ≈ 1 + x, leading to the approximate relation between the effective and intermediate LZ transition probabilities: Moreover, similar equations are valid for the effective probability expressed in terms of the M S -specific transition probabilities. The applicability of both the WC and LZ probability expressions are limited by the magnitude of SOC, which must not be too large [32]. The M Sspecific and intermediate approaches extend the applicability range of the WC and LZ expressions because the corresponding couplings are smaller than SOC eff .

Results and discussion
In this section, we apply the three approaches for calculating the transition probabilities and rate constants discussed in the previous section to the active site model of the small iron-sulfur protein rubredoxin. In our earlier studies of the [Fe(SCH 3 ) 4 ] n (n = 0, 1-and 2-) rubredoxin active site model [19,44], we found several crossings between electronic states with different spin multiplicities. Here, we focus on the triplet-quintet and quartet-sextet crossings in the active site model with n = 2-and 1-, respectively. The M S -specific SOCME values obtained from the first-order perturbation theory at the CASCI/def2-TZVP level of theory are reported in Table 1 [19]. The effective and intermediate SOC values were calculated from SOCMEs using Equations (4) and (6), respectively. All other electronic structure calculations including the single-point energy, geometry optimisation, energy gradient, and Hessian computations were performed at the 7i z 4 = 0.9 + 4.0i ib 1 = 7.5i ib 2 = 9.2i PBE/def2-TZVP level of theory with the GAMESS suite of programme [45]. The transition probabilities for the triplet-quintet spin states of [Fe(SCH 3 ) 4 ] 2− and the quartet-sextet spin states of [Fe(SCH 3 ) 4 ] 1− calculated with the LZ and WC formulas are shown in Figure 5 and S2, respectively. The tunnelling regions are shown as purple boxes. For the triplet-quintet crossing, the effective transition probability is labelled as P eff , the intermediate probabilities are labelled as P 0 and P 1 , and the M S -specific probabilities are labelled as P 1,2 , P 1,1 , P 1,0 , P 0,1 , and P 0,0 . As stated earlier, the probabilities and rate constants are equal for transitions from ±M S state components (Table S1); therefore, only one of these probabilities is plotted. This degeneracy is accounted for by the factor of two in front of the probabilities in the legends of Figures 5 and S2 (Table 1), while the gradient differences and the geometric mean of the gradients are similar [19]. Therefore, for the LZ transition probabilities calculated with the effective and intermediate approaches, the deviation from Equation (11) is expected to be more noticeable for the triplet-quintet crossing in accordance with the theory. In addition, as shown in Figure 5(a,c), the sums of the intermediate and M S -specific probabilities (red lines) can reach unphysical values greater than unity. As the internal energy increases, the deviation from Equation (11) becomes smaller. In contrast, than using WC equation, the effective probability is exactly equal to the sums of the intermediate and M S -specific probabilities, as demonstrated in Figure 5(b,d). The approximation of Equation (11) works better for the [Fe(SCH 3 ) 4 ] 1− complex with a weaker SOC, as evidenced by the nearly perfect agreements between the LZ effective probability and the sums of the intermediate and M S -specific probabilities ( Figure  S2). For the triplet-quintet crossing, the transition probabilities P 0,0 and P 1,1 , which correlate with the large SOCME values b 1 and b 2 (Table 1), make the most significant contributions to the overall transition probability. For the quartet-sextet crossing, the transition probabilities P 1/2,1/2 and P 3/2,3/2 obtained from the b 2 and b 1 SOCMEs make the largest contribution to the overall transition probability. The MECPs are located at 43.5 and 25.6 kJ/mol above the reactants for the complexes with n = 2-and 1-, respectively. The WC formula predicts a non-zero tunnelling probability at the energies below the MECPs and probability oscillations due to the interference between the reaction paths at the energies above MECPs, as illustrated in Figure 5(b,d) and S2b,d. Figure 6 displays the microcanonical rate constants for the transitions between each pair of the spin-states in [Fe(SCH 3 ) 4 ] 2− calculated from the corresponding probabilities ( Figure 5). The rate constants for [Fe(SCH 3 ) 4 ] 1− are shown in Figure S3. As expected, the rate constants calculated using the LZ transition probabilities are zero  in the tunnelling regions shown in purple. The WC transition probability formula accounts for quantum tunnelling and predicts non-zero values for the rate constants in these regions. For the rate constants calculated using the LZ transition probabilities, there is a noticeable difference between the effective rate constant and the sum over the intermediate rate constants (Figure 6(a)). The same is true for the sum over M S -specific rate constants (Figure 6(c)). In contrast, the sum over the intermediate rate constants and the sum over M S -specific rate constants obtained using the WC probabilities are exactly equal to the effective rate constant (Figure 6(b,d)). This is consistent with the form of the rate constant defined by Equation (1), which reduces to the sum of the M Sspecific rate constants, if the overall transition probability can be written as a sum of the M S -specific probabilities (Equation (11)). As clearly seen from Figure 6(c,d), the rate constants of the transitions between different M S components of the spin multiplets can differ by several orders of magnitudes. Predicting these M S -specific rate constants is important for emerging experiments targeting intersystem crossings between individual components of spin multiplets [46].

Conclusion
We discussed three approaches for predicting the kinetics of the spin-dependent processes with nonadiabatic statistical theory using different representations for the spin-diabatic electronic states. In addition to the previously used effective approach and the M S -specific approach, we introduced a new intermediate approach.
In the effective approach, the calculated transition probability and rate constant correspond to an effective transition between two spin multiplets. In the M S -specific approach, the probabilities and rate constants are calculated for transitions between all pairs of the M S components of two interacting spin multiplets. This approach is useful to describe situations than the transition rates between different M S components are significantly different from each other, including intersystem crossings in an external magnet field. In the intermediate approach, the SOC is computed by summing over the M S components of the high-spin state only. For the singlet-triplet crossing, the intermediate and effective approaches are identical. However, for the crossings between states with higher spin multiplicities, only the intermediate approach produces the spin-orbit couplings that correspond to the energy gaps between spin-adiabatic states. To demonstrate the differences between the three approaches, we applied them to the transitions between the electronic states with different spin multiplicities in the active site model of the protein rubredoxin.
Because the computational costs of three approaches are similar, the choice of a specific approach depends on the modelled experimental conditions. We expect that the M S -specific approach will be useful for studying the kinetics of spin-dependent processes in the presence of an external magnetic field, while the intermediate approach will help to compare the results of the chemical kinetics and dynamics studies performed using the spindiabatic and spin-adiabatic representations of electronic states. We plan to expand these three approaches for calculating the transition probabilities and rate constants, so they can be used with the more accurate Zhu-Nakamura transition probability equations, which account for the shape of the crossing potentials and are not limited by the strength of interstate coupling.