2.1.1. Polymer Model Selection

Constructing representative polymer fragments is a critical step in modeling polymer transformations, since model size effects, steric hindrance, and electronic delocalization strongly influence calculated energetics. We recommend short polymer fragments that reproduce the local electronic and steric environment of the reaction center while keeping the calculations computationally efficient [Reference Papanikolaou, Wu, Huber and Mavrikakis49, Reference Ezendu, Soyemi and Szilvási98, Reference Ezendu and Szilvási99, Reference Wu, Papanikolaou and Cheng100]. Each new polymer system nevertheless requires fragment-size validation because chain-length effects, stereochemistry, and conformational constraints can alter the reaction energetics. Researchers can perform that validation by increasing fragment length until activation barriers and reaction energies no longer change significantly. To avoid artifacts from chain ends, we embed the reacting monomer unit within the fragment and include enough neighboring monomer units to capture inductive and conjugative interactions that significantly influence local reactivity.

Polymer fragments can adopt multiple conformations and distinct stereochemical configurations, such as cis (Z) or trans (E) isomers and isotactic or syndiotactic chain arrangement. When experimental or computational precedent exists, literature reports can guide the choice of conformation or isomer. In the absence of such guidance, researchers can compare activation barriers across candidate isomers for a key reaction step and select the configuration with the lowest energy pathway while considering reaction center location. End-chain and mid-chain environments impose different steric and backbone constraints and therefore alter the accessible conformational space. To avoid chain-end artifacts, we embed the reacting monomer unit within the fragment and include enough neighboring monomer units to capture inductive and conjugative interactions that significantly influence local reactivity [Reference Ezendu, Soyemi and Szilvási98, Reference Ezendu and Szilvási99]. Researchers can address the reduced flexibility imposed by mid-chain environments by introducing strain through incremental increases in the distance between the terminal carbon atoms of the fragment and then evaluating the resulting changes in Gibbs energies and activation barriers [Reference Lugo, Edeleva, Van Steenberge and Sabbe101].

Polymer systems with additional complexity, such as branching, tacticity variations, or chain defects, require additional models that incorporate these features. A common strategy is to modify the repeat/monomer unit at or near the reaction center to represent a structural irregularity, such as replacing a regular monomer with one containing a missing substituent, an added functional group, or an altered stereochemical orientation [Reference Ezendu, Soyemi and Szilvási98]. Embedding the modified monomer within the polymer fragment allows for a direct comparison of reaction energetics and mechanistic pathways between a fragment with a structural defect and its defect-free counterpart. For example, defect-containing fragments have been shown to change the course of reactions along the polymer chain, and such comparisons demonstrate that structural defects can shift activation barriers and modify accessible mechanistic pathways [Reference Ezendu, Soyemi and Szilvási98, Reference Ezendu and Szilvási99].

Good Practices and Recommendations

• • Use polymer fragments that balance chemical realism with computational efficiency; 3–5 monomer units are often sufficient.

• • Place the reacting monomer away from chain termini and use appropriate capping groups to prevent artificial polarization.

• • Preserve inductive and conjugative influences from adjacent units when defining chain ends to preserve local electronic effects.

• • Evaluate relevant conformers or stereoisomers; compare activation barriers if no precedent study exists.

• • Incorporate branching, tacticity, or structural defects into additional models when these features may influence reactivity.

2.1.2. Choice of DFT Method

Selecting an exchange–correlation functional is a nontrivial choice in DFT studies, since functionals vary in their ability to describe different classes of reactions, and no single option is universally reliable. Hundreds of functionals exist [Reference Mardirossian and Head-Gordon102], organized according to Jacob’s ladder [Reference Perdew and Schmidt103], and it is generally believed that higher rungs offer improved accuracy at higher computational cost [Reference Perdew and Schmidt103]. For polymer transformations that require accurate reaction barriers and reaction energies, the functional must reliably describe bond-breaking and bond-forming events, often in extended, conjugated, or sterically constrained environments. Current computational resources permit geometry optimizations with functionals at rung 3 or higher for polymer fragments, including meta-generalized gradient approximation (GGA) and hybrid DFT functionals [Reference Mardirossian and Head-Gordon102]. Functionals such as PBE0 [Reference Perdew, Burke and Ernzerhof104, Reference Adamo and Barone105], B3LYP [Reference Becke106, Reference Chai and Head-Gordon107], M06-2X [Reference Zhao and Truhlar108], and ωB97X-D [Reference Chai and Head-Gordon107, Reference Chai and Head-Gordon109, Reference Zheng, Zhao and Truhlar110] are frequently evaluated, as they have shown consistent performance for barrier calculation relative to HL ab initio methods such as canonical coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) [Reference Riplinger, Sandhoefer, Hansen and Neese93, Reference Bartlett and Musiał111, Reference Raghavachari, Trucks, Pople and Head-Gordon112]. The chemical composition of the full system, including the polymer backbone, substituents, catalyst, and solvent, should guide functional choice because some functionals describe main-group chemistry more reliably, whereas others better capture transition-metal or heavy p-block reactivity. Regardless of the specific choice, most DFT functionals share systematic limitations, particularly self-interaction error (SIE) [Reference Perdew and Zunger113, Reference Mori-Sánchez, Cohen and Yang114, Reference Bao, Gagliardi and Truhlar115, Reference Cohen, Mori-Sánchez and Yang116] and the inadequate treatment of long-range correlation [Reference Cohen, Mori-Sánchez and Yang116]. Self-interaction error arises from the spurious or unphysical interaction of an electron with its own charge density and can strongly affect charged or delocalized states [Reference Perdew and Zunger113, Reference Mori-Sánchez, Cohen and Yang114, Reference Bao, Gagliardi and Truhlar115, Reference Cohen, Mori-Sánchez and Yang116]. Global and range-separated hybrid functionals incorporate a fraction of exact Hartree–Fock exchange, which helps reduce SIE and improves barrier heights and charge localization, particularly for anionic or radical intermediates [Reference Mardirossian and Head-Gordon102, Reference Cohen, Mori-Sánchez and Yang116]. Long-range correlation effects arise from dispersion (London) interactions between distant parts of the polymer or between the polymer and its environment and can significantly alter relative stabilities and barriers [Reference Grimme117, Reference Becke and Johnson118]. Incorporating dispersion corrections is therefore essential because noncovalent interactions can influence both conformational preference and reaction energetics [Reference Chai and Head-Gordon109, Reference Grimme, Antony, Ehrlich and Krieg119]. Dispersion correction methods such as D3 (with zero or BJ damping) [Reference Becke and Johnson118, Reference Grimme, Antony, Ehrlich and Krieg119, Reference Becke and Johnson120], D4 [Reference Caldeweyher, Ehlert and Hansen121], MBD [Reference Tkatchenko, DiStasio, Car and Scheffler122, Reference Tkatchenko and Scheffler123], VV10 [Reference Vydrov and Van Voorhis124, Reference Liang, Feng, Hait and Head-Gordon125], or XDM [Reference Becke and Johnson120, Reference Johnson and Becke126, Reference Johnson and Becke127] can be added to most functionals [Reference Grimme, Hansen, Brandenburg and Bannwarth128], while others, such as ωB97X-D, include dispersion treatment intrinsically. We recommend applying dispersion corrections consistently across reactants, intermediates, and transition states so that their relative energies remain comparable and are not biased by inconsistent dispersion treatment.

Once a suitable functional has been chosen, the next step is to select a basis set that can describe bond breaking and formation along the reaction coordinate and yield reliable reaction and activation energies. A well-chosen basis set includes polarization functions to describe bond breaking and formation, diffuse functions to treat anions and other loosely bound electrons, and sufficient valence functions to capture electron correlation accurately. Increasing the size and completeness of a basis set generally improves the description of the electron density but also increases computational cost. Minimal and small split-valence basis sets lack sufficient functions per atom to reliably describe bond rearrangements, which increase basis set incompleteness error (BSIE) and basis set superposition error (BSSE) [Reference Helgaker, Jorgensen and Olsen129, Reference Bursch, Mewes, Hansen and Grimme130]. Basis set incompleteness error occurs when the basis set is too small to accurately describe the electronic structure of the polymer fragment, while BSSE arises from overlapping basis functions between interacting fragments because the basis set is too small [Reference Helgaker, Jorgensen and Olsen129, Reference Bursch, Mewes, Hansen and Grimme130, Reference Van Mourik, Wilson, Peterson, Woon, Dunning and Sabin131, Reference Balabin132]. Both errors can distort bond lengths, angles, and dihedrals, shift reaction and barrier energies, and artificially stabilize polymer intermediates and transition states [Reference Van Mourik, Wilson, Peterson, Woon, Dunning and Sabin131, Reference Balabin132]. To minimize these errors, larger basis sets such as quadruple-ζ(Z) can significantly decrease basis set errors but may be impractical for routine geometry optimizations. In practice, a cost-effective polarized basis set can be used for geometry optimizations, with larger basis sets such as triple- or quadruple-ζ(Z) applied for single-point energy calculations when resources permit. Common Gaussian-type basis sets include those from the Pople [Reference Ditchfield, Hehre and Pople133, Reference Hehre, Ditchfield and Pople134], Dunning [Reference Dunning135, Reference Wilson, van Mourik and Dunning136], Ahlrichs [Reference Weigend, Häser, Patzelt and Ahlrichs137, Reference Weigend and Ahlrichs138], and Jensen [Reference Jensen139, Reference Jensen140] families, each providing a systematic hierarchy that increases the level of theory and calculation accuracy [Reference Dunning135, Reference Jensen141]. We recommend the Ahlrichs basis set family, as they are consistently available for most elements on the periodic table. We find that diffuse functions are generally unnecessary for neutral covalent polymers but are important for systems with significant negative charge density or for polarizability calculations. For heavy atoms beyond the fourth period, one should combine scalar-relativistic Hamiltonians (such as DKH [Reference Reiher142], ZORA [Reference Pollak and Weigend143, Reference Centoducatte, de Oliveira, Jorge and Camiletti144], or X2C [Reference Kutzelnigg and Liu145, Reference Saue146]) with compatible scalar-relativistic basis sets (for example, SARC [Reference Pantazis, Chen, Landis and Neese147], ANO [Reference Almlöf and Taylor148, Reference Widmark, Zobel, Vysotskiy, Tsuchiya and Veryazov149], or def2 variants [Reference Pollak and Weigend143]) to capture relativistic effects that can significantly influence bond lengths, bond strengths, and reactivity.

Good Practices and Recommendations

• • Survey commonly used functionals for the reaction type of interest and then evaluate their accuracy through benchmarking (see Section 2.1.4).

• • Include dispersion correction in all calculations.

• • Evaluate basis sets on the chosen polymer fragment by checking convergence of geometry and reaction energetics to ensure the basis set is large enough to capture bond-breaking and bond-formation while remaining computationally efficient.

• • Always use diffuse functions when modeling anions, highly polar transition states, or polarizability.

• • Apply effective core potentials or scalar-relativistic corrections for heavy atoms (N > 36).

2.1.3. Solvation Model

Solvent effects significantly influence the energetics of polymer transformations since changes in polarity, dielectric constant, or hydrogen-bonding environment can stabilize or destabilize reactants, intermediates, and transition states. Neglecting solvation can thus lead to substantial errors in calculated barriers and reaction energies, and sometimes result in qualitatively incorrect electronic structures. In practice, the solvent can be treated with continuum (implicit) models, microsolvation clusters, or fully explicit solvent molecules, depending on the level of detail and computational cost one is willing to accept [Reference Harvey, Himo, Maseras and Perrin150, Reference Tomasi151]. Explicit solvation is computationally demanding for polymer transformations because it requires sampling polymer fragments with many explicit solvent molecules. Therefore, the MUSIK framework uses implicit solvation models as the standard approach, balancing accuracy and efficiency. Implicit or continuum models like SMD [Reference Aleksandr, Marenich and Truhlar152], CPCM [Reference Barone and Cossi153, Reference Takano and Houk154], or COSMO-RS [Reference Klamt155, Reference Klamt and Schüürmann156] capture bulk dielectric properties and non-electrostatic effects, providing an efficient way to include solvent effects. However, because implicit solvation models average the solvent response, they may underestimate activation barriers in systems where strong localized solute–solvent interactions are important [Reference Harvey, Himo, Maseras and Perrin150, Reference Ryu, Park and Kim157]. The effect is even more pronounced when dispersion corrections are neglected [Reference Chai and Head-Gordon109, Reference Edeleva, Van Steenberge, Sabbe and D’hooge158]. While continuum models work well for many polymers with nonpolar backbones [Reference Ezendu, Soyemi and Szilvási98, Reference Ezendu and Szilvási99], systems with polar functional groups, hydrogen-bonding motifs, or intermediates with partial charge separation can be more sensitive to solvation. In such cases, implicit models generally predict the correct qualitative trends, but further benchmarking against higher-level calculations or explicit-solvent cluster models may be necessary for precise reaction energetics.

Good Practices and Recommendations

• • Choose the solvation model and solvent parameters to reflect the experimental medium or the intended reaction environment.

• • Apply the solvation model consistently across reactants, intermediates, and transition states to avoid systematic shifts in relative energetics.

2.1.4. Benchmarking and Selecting Computational Methods

Benchmarking establishes the reliability of candidate DFT methods by comparing their calculated energetics with a HL reference. The goal is to identify a functional–basis set combination that reproduces barrier heights and reaction energies with sufficient accuracy for polymer transformations, while remaining computationally tractable.

Selecting an appropriate reference method is essential for reliable benchmarking [Reference Mata and Suhm159]. Canonical CCSD(T) [Reference Riplinger, Sandhoefer, Hansen and Neese93, Reference Raghavachari, Trucks, Pople and Head-Gordon112] serves as the gold standard for small molecules, but it is prohibitively expensive for larger systems because of its computational cost [Reference Liakos, Sparta, Kesharwani, Martin and Neese96, Reference Bartlett and Musiał111]. Local CC methods such as DLPNO–CCSD(T) [Reference Riplinger and Neese95, Reference Riplinger, Pinski, Becker, Valeev and Neese97] or LNO-CCSD(T) [Reference Rolik, Szegedy, Ladjánszki, Ladóczki and Kállay94, Reference Nagy, Samu and Kállay160] provide a practical alternative, offering accuracy close to canonical CCSD(T) while reducing the formal scaling to nearly linear with system size. High-level CC approaches, both canonical and local variants, are implemented in widely used software packages such as ORCA [Reference Neese161, Reference Neese162], MRCC [Reference Kállay, Nagy and Mester163, Reference Mester, Nagy and Csóka164], and MOLPRO [Reference Werner, Knowles, Knizia, Manby and Schütz165, Reference Werner, Knowles and Manby166], with more recent interfaces also available in PySCF [Reference Sun, Berkelbach and Blunt167, Reference Sun, Zhang and Banerjee168].

We suggest benchmarking as follows: First, we optimize the geometries of representative polymer fragments with candidate DFT functionals in widely available software such as Gaussian [Reference Frisch, Trucks and Schlegel169], ORCA, Q-Chem [Reference Kong, White and Krylov170, Reference Shao, Gan and Epifanovsky171], or NWChem [Reference Mejia-Rodriguez, Aprà and Autschbach172, Reference Valiev, Bylaska and Govind173]. Next, we compute single-point reference energies at the HL using these DFT-optimized geometries and compare reaction energies and barrier heights using DFT against the HL reference values using error metrics such as mean absolute error, root-mean-square error, or maximum deviation. Then, we select the DFT method that gives reaction and barrier energies in closest agreement with the HL reference across all benchmarked steps. In parallel, we also benchmark the structure of representative intermediate(s) or transition state(s) across the tested DFT functionals and basis sets. We recommend comparing key reaction coordinate bond lengths, angles, or dihedrals across the tested DFT methods; if these metrics do not change significantly, the chosen basis set is adequate for HL single-point calculations. Accurate geometries are essential because differences in bond lengths, angles, or torsions lead directly to errors in the HL energies, reducing the reliability of the benchmark. It should be noted that basis-set errors, including BSIE and BSSE (see Section 2.1.2), reduce accuracy at every stage of benchmarking, from geometry optimization and transition-state searches up to HL single-point energy calculations. Therefore, to minimize these errors, use at least triple-ζ basis sets for single-point energy calculations, include diffuse functions for weakly bound regions, and check basis-set convergence for systems with multiple interacting fragments. For consistency, apply the same solvation model and dispersion correction to all species and calculations to maintain comparability and prevent systematic errors in relative energies.

Once a suitable computational method is established, we recommend reiterating the model benchmark and validating the fragment size separately to confirm that the model captures the steric and electronic influences of the polymer environment. One can perform model validation by calculating reaction energetics, either electronic energies or Gibbs free energies, for progressively longer chain fragments until the values converge. Convergence demonstrates that the selected computational setup is likely transferable across system sizes.

Good Practices and Recommendations

• • Benchmark DFT methods systematically by varying the type of functionals, basis sets, dispersion corrections, and solvation models to identify combinations that yield reliable energetics.

• • Keep geometries and solvation models consistent between DFT and HL reference calculations so that differences arise only from the electronic structure method.

• • Use local CC methods as the HL reference method.

• • For more accurate reference energies, apply at least triple-ζ basis sets to minimize BSIE and BSSE.

• • Reiterate model and method benchmarking to validate that the model and method consistently produce accurate results

2.3.1. DFT-Level Energies

At the DFT level, energy barriers are obtained by calculating the electronic energy differences between the transition state or intermediate and the sum of the reactant energies as follows:

(1) $$ {\varDelta E}_{DFT}^X={E}_{DFT}^X-\sum {E}_{reactants, DFT} $$

where the superscript X refers to either the transition state (‡) or an intermediate or product (INT). $ {E}_{DFT} $ represents the total self-consistent electronic energy, typically including dispersion correction.

The Gibbs free energy at the DFT level combines electronic, zero-point, thermal, and entropic contributions:

(2) $$ {G}_{DFT}={E}_{DFT}+{ZPE}_{DFT}+{E}_{Thermal}-T{S}_{DFT} $$

where $ {ZPE}_{DFT} $ is the zero-point vibrational energy, $ {E}_{Thermal} $ includes the thermal energy contributions of translational, rotational, and vibrational degrees of freedom, as well as the $ {k}_BT $ term, T is the temperature (298 K), and $ {S}_{DFT} $ is the entropy term [Reference Frisch, Trucks and Schlegel169, Reference Frisch, Trucks and Schlegel187].

The Gibbs free energy barrier for a step, similar to Equation 1, is calculated as:

(3) $$ \varDelta {G}_{DFT}^X={G}_{DFT}^X-\sum {G}_{reactants, DFT} $$

Kinetic parameters such as the activation energy and the pre-exponential factor are obtained by substituting the electronic and Gibbs free energies from Equations 1 to 3 into the Arrhenius–Eyring rate constant relationship as shown below:

(4) $$ k=A{e}^{-\frac{E_a}{RT}}=\frac{\kappa {k}_BT}{h}{e}^{-\frac{\varDelta {G_{DFT}}^{\ddagger }}{RT}} $$

where A is the pre-exponential factor, $ {E}_a $ is the activation energy, R is the universal gas constant, T is the temperature, $ {k}_BT $ is the Boltzmann constant, h is Planck’s constant, and κ is the transmission coefficient (usually taken as unity). Most studies apply classical transition-state theory ( κ  = 1), which assumes that every trajectory crossing the transition state proceeds directly to products without recrossing and that quantum tunneling through the barrier is negligible. In cases where these assumptions do not hold, a more accurate approach is to include dynamical effects that modify the rate constant through a transmission coefficient that can vary with temperature. We recommend applying methods like canonical variational transition-state theory to capture recrossing and the small-curvature tunneling approximation to describe tunneling, obtaining temperature-dependent transmission coefficients that reveal how these effects alter the overall rate relative to the classical estimate [Reference Truhlar and Garrett188, Reference Bao and Truhlar189, Reference Olajide and Szilvási190]. We recommend that k and A must be checked to ensure that their units remain consistent with the reaction molecularity and the standard-state convention used in Equation 4.

The activation energy at the DFT level is evaluated as:

(5) $$ {E}_a=\Delta {E}_{DFT}^{\ddagger }+\Delta {ZPE}_{DFT}^{\ddagger }+{\Delta E}_{Thermal}^{\ddagger } $$

where $ {ZPE}_{DFT} $ is the zero-point vibrational energy, $ {E}_{Thermal} $ includes the thermal energy contributions of translational, rotational, and vibrational degrees of freedom as well as the $ {k}_BT $ term.

The Arrhenius–Eyring expression in Equation 4 contains both the activation energy and the Gibbs free energy barrier. Substituting the activation energy from Equation 5 and the Gibbs free energy definition from Equation 3 allows the separation of the entropy contribution. The pre-exponential factor is then obtained as:

(6) $$ A=\frac{{\kappa k}_BT}{h}{e}^{\frac{{\Delta S}_{DFT}^{\ddagger }}{R}} $$

where Δ $ {S}_{DFT} $ is the entropy change between the transition state and the reactants.

2.3.2. Thermochemical Corrections

Accurate modeling in agreement with experiments requires the proper treatment of temperature, pressure, and entropy effects. Neglecting thermodynamic corrections introduces non-systematic errors in free energies, which in turn alter the rate parameters supplied to kMC simulations. In the MUSIK framework, vibrational analyses at the quantum-chemical level provide zero-point energies, thermal contributions, and entropic terms. Post-processing tools such as Shermo [Reference Lu and Chen191], GoodVibes [Reference Luchini, Alegre-Requena, Funes-Ardoiz and Paton192], or Arkane [Reference Dana, Johnson and Allen193] facilitate vibrational analysis and allow the extraction of zero-point, thermal, and entropic contributions directly from DFT calculations. Applying such corrections uniformly across all species ensures that the relative free energy remains comparable to experimental conditions. Free energies are typically referenced to the gas-phase standard state (1 atm) by most quantum-chemical software packages, which should be converted to the 1-M solution standard state when plastic transformations occur in solution. The need for concentration correction becomes critical when the reaction stoichiometry involves a net change in the number of moles, because the standard-state term no longer cancels between reactants and products. The corrected free energy is then:

(7) $$ {\varDelta G}_{DFT}^X=\varDelta {G}_{DFT}^X+\mathrm{RT}\;\ln\;\frac{c^{\Delta v}=1\;\mathrm{M}}{c^{\Delta v}={c}_{ref}} $$

Here, Δν is the change in the number of moles from reactants to the transition state or intermediate and $ {c}_{ref}=\frac{p}{RT} $ is the molar concentration for the gas-phase standard state, where p is the pressure (1 atm), R is the universal gas constant, and T is the reference temperature. For example, a bimolecular association reaction (Δν = −1) at 298 K corresponds to a correction of approximately −1.89 kcal mol⁻¹ or 7.92 kJ mol⁻¹.

2.3.3. High-Level (HL) Single-Point Energies

To achieve near-chemical accuracy in calculated energetics, the MUSIK framework improves upon DFT-level results by applying HL single-point electronic structure methods using local CCSD(T) methods that we find as an affordable balance between accuracy and computational cost. For each reactant, transition state, and intermediate, we calculate HL single-point energies on the DFT-optimized geometry of the lowest-energy conformer.

The activation energy at a higher level of theory is evaluated analogously to the DFT expression in Equation 5:

(8) $$ {E}_a=\Delta {E}_{HL}^{\ddagger }+\Delta {ZPE}_{DFT}^{\ddagger }+{\Delta E}_{Thermal}^{\ddagger } $$

where $ \Delta {E}_{HL}^{\ddagger 0} $ is the CC electronic energy of the transition states. The association/dissociation steps without an internal energy barrier require estimating an effective activation free energy that can be done by applying the translational-entropy penalty for association steps [Reference Ezendu and Szilvási99]. To quantify the translational entropy, we utilize the simplified form of the Sackur–Tetrode equation [Reference Ryu, Park and Kim157, Reference Tetrode194], which separates the contributions of mass and temperature under ideal gas conditions:

(9) $$ {S}_{trans}=\frac{3}{2}R\ln M+\frac{5}{2}R\ln T+2.315 $$

where M represents the molar mass, T is the temperature, and R is the ideal gas constant. Consequently, for a 2 → 1 association under the 1-M standard state at 298 K, the translational entropic penalty typically falls in the 40–50 kJ mol⁻¹ range.

The HL Gibbs free energy is obtained by adding DFT-derived thermal corrections to the HL single-point energy:

(10) $$ {G}_{HL}={E}_{HL}+\left({G}_{DFT}-{E}_{DFT}\right) $$

The final HL Gibbs free energies of intermediates and transition states with correction to the solution-phase standard state are then written as:

(11) $$ \varDelta {G}_{HL}^X={G}_{HL}^X-\sum {G}_{reactants, HL} $$

We note that forward barriers are evaluated for every elementary step. However, reverse barriers can be calculated and included when their magnitude is small or when the reverse process occurs on a timescale comparable to the forward reaction. A forward barrier corresponds to the Gibbs free energy difference between a transition state and the preceding intermediate or reactant, while a reverse barrier corresponds to the Gibbs free energy difference between the same transition state and the subsequent intermediate or product.

Good Practices and Recommendations

• • Use near-linear scaling local CCSD(T) methods to obtain accurate HL energies.

• • Archive all computational data, including optimized DFT geometries, HL single-point calculations, and thermochemical corrections, in accessible repositories to ensure transparency and reproducibility.

2.4.1. Definition of the Polymer Lattice

For modeling polymer transformation, the kMC lattice is naturally described by a one-dimensional topology whereby each lattice site corresponds to a backbone carbon atom, a repeat unit, or a reactive segment. However, additional considerations such as branched or cross-linked polymers may require two- or three-dimensional topologies to capture connectivity across multiple chains, similar to lattice definitions common in kMC modeling of surface reactions [Reference Stamatakis and Vlachos196, Reference Pineda and Stamatakis209, Reference Andersen, Panosetti and Reuter210, Reference Kunz, Kuhn and Deutschmann213, Reference Stamatakis216]. In MUSIK, boundary conditions are defined explicitly, where periodic boundaries connect the chain ends to mimic an infinitely repeating polymer, and open boundaries leave chain ends free to participate in termination or end-group reactions. Once the lattice topology and boundary of the polymer lattice are established, we identify which part (site) of the polymer lattice can participate in an elementary step. We assign site types to each site to capture distinct chemical environments or identity, such as backbone, end group, repeat unit, defect, or branch. Site types determine which reaction templates apply locally and enable simplification by grouping chemically equivalent sites under one label, minimizing the event catalog. Next, we define connectivity to identify which sites act as neighbors and can react together. Within our MUSIK framework, we specify site type and connectivity and label the site type consistently to ensure that the lattice represents both polymer structure and the local environments within the polymer that govern reactivity. Additionally, we recommend setting lattice length to reflect experimental molecular weight: convert molecular weight to a count of repeat units and use that count as the number of lattice sites.

2.4.2. Mapping of Reaction Mechanism onto the Lattice

Mapping quantum-chemical reaction mechanisms onto the polymer lattice defines the elementary events that drive the stochastic evolution of the system. The mapping translates results from small, representative fragments into lattice-level rules that capture bond breaking, bond formation, or chain rearrangements. In MUSIK, we represent each elementary step and its energetics consistently on the chosen lattice. We recommend selecting a set of lattice sites that is large enough to include all positions that influence a given elementary step yet small enough to avoid slow simulations and then confirming through numerical convergence that expanding the site size no longer changes the mapped event energetics or kinetics significantly. Next, we map the site types introduced in Section 2.4.1 onto the selected sites so that every elementary step can be expressed with the chosen site-type scheme. Section 3.2.2 provides detailed, step-by-step examples of the mapping procedure. Finally, we assign each mapped event a kinetic parameter derived from HL quantum mechanical calculations. Typical kMC inputs are the activation energy E_a, pre-exponential factor A, and rate constant 𝑘 (see Equation 4). For rate calculations, we recommend following the guidelines in Section 2.3 and deriving forward and reverse rates from the same quantum-chemical data so that microscopic reversibility holds. Some kMC engines allow explicit input of forward and reverse parameters, whereas others enforce thermodynamic consistency automatically through the equilibrium relation in Equation 13:

(13) $$ \frac{k_{fwd}}{k_{rev}}={\exp}^{\left(-\frac{{\varDelta G}^o}{RT}\right)} $$

where $ {k}_{fwd} $ and $ {k}_{rev} $ are the forward and reverse rate constants, $ {\varDelta G}^o $ is the standard-state Gibbs free energy difference between products and reactants, R is the gas constant, and T is the reaction temperature.

2.4.3. Lateral Interactions

In lattice-based kMC, lateral interactions account for how energetics change when species occupy neighboring lattice sites. While originally studied in the context of adsorbates on solid surfaces [Reference Stamatakis and Vlachos196, Reference Pineda and Stamatakis209, Reference Andersen, Panosetti and Reuter210, Reference Nielsen, d’Avezac, Hetherington and Stamatakis217, Reference Li and Grabow218], the same principles apply to polymers when the presence or absence of neighboring reactive groups, defects, or substituents on one repeat unit affects the reactivity of adjacent ones [Reference Ezendu, Soyemi and Szilvási98, Reference Ezendu and Szilvási99]. Such effects may stabilize or destabilize a transition state, alter reaction barriers, or shift selectivity depending on local structure. For instance, we showed that the formation of unsaturated bonds or the presence of substituents on one repeat unit can alter the barrier for transformations occurring on neighboring units, either lowering or raising the barriers depending on the local environment [Reference Ezendu, Soyemi and Szilvási98].

In frameworks that include lateral interactions, local environments modify site energetics by assigning corrections to base reaction energies or rate constants according to the configuration of neighboring sites. The most common approach is the cluster expansion Hamiltonian (CEH) [Reference Stamatakis and Vlachos196, Reference Pineda and Stamatakis209, Reference Andersen, Panosetti and Reuter210, Reference Nielsen, d’Avezac, Hetherington and Stamatakis217, Reference Sanchez, Ducastelle and Gratias219, Reference Benson, Yadavalli and Stamatakis220], which expresses interaction energies as contributions from pairwise or higher-order site clusters. For polymer systems, first-nearest-neighbor interactions usually capture short-range effects. Higher-order terms become necessary when strongly directional interactions extend beyond adjacent sites, such as hydrogen-bond chains, extended π–π stacking, or long-range dipole–dipole coupling that influence multiple repeat units. Detailed theory and algorithms for CEH can be found in the original references [Reference Stamatakis and Vlachos196, Reference Sanchez, Ducastelle and Gratias219, Reference Benson, Yadavalli and Stamatakis220]. Within the MUSIK framework, we adopt this standard treatment by passing environment-dependent corrections to the kMC engine rather than redefining the interaction model. Users supply the base reaction energetics from Sections 2.1–2.3 and indicate which neighbor shells can influence a given event, and the underlying kMC solver then applies the corresponding cluster-based adjustments at run time according to the local configuration.

Good Practices and Recommendations

• • Enumerate all possible site combinations for each event; account for symmetry and boundaries.

• • Choose the event span and neighborhood radius just large enough to capture chemistry without slowing the simulation.

• • Keep site types and labels consistent, enumerate relevant site combinations for every reaction to avoid bias, and keep site-type labels and site labels consistent across inputs and outputs.

• • Apply lateral-interaction corrections only to event instances that meet the defined neighborhood features and state which terms are corrected.

• • Maintain reproducibility by providing scripts, inputs, and instructions (README) that recreate at least one figure end-to-end.