# Reaction rate estimation for relatively complex reactions

I am looking for a summary of your favorite methods to estimate a reaction rate constant for a “complex” polyatomic reaction (requiring a cell of ~100 atoms) from ab initio methods (the one I have in mind currently is degradation of a surface layer of perovskite by superoxide, following:

$${\small \ce{4CH_3NH_3PbI_3^* + O_2^- \rightarrow 4PbI_2 + 2I2 + 2H_2O + 4CH_3NH_2}}\tag{1}$$

The most basic approach I know of would be to perform a DFT based NEB (nudged elastic band) calculation to identify the saddle point, compute the energy barrier $$E_a$$ in the ground state, and use a Boltzmann-like relationship $$\nu(T)=\nu_0.e^{-\beta E_a}$$ to speculate a characteristic reaction time at finite temperature. This approach has several weaknesses, in order of importance for me:

• For a “complex” reaction, finding the actual saddle point with a simple NEB seems like wishful thinking to me: there could be many reaction pathways, steps, etc… which makes it challenging to set up the NEB properly. The cost of testing many potential paths would also quickly become prohibitive  since a single NEB on a relatively large system is already pretty expensive (at least at my scale)
• The term $$\nu_0$$ is quite a big unknown. I’ve seen people use “educated guesses” to plug a reasonable value into it (like the frequency of oscillations in the crystal as an attempt frequency), but it seems quite hand-wavy and could easily be way off, which begs the question of knowing whether such a guess-based result has much significance

• It seems like a fairly harsh approximation to encapsulate all effects of thermalization in a Boltzmann law.

I have seen more sophisticated methods that build upon this and bring more accuracy, like the semiclassical transition state theory (SCTST), and basically alleviate the 2 last points, but still requires to find the saddle point, and leaves me with my issue #1, which is that it might be unpractical for more complex reactions (correct me if I'm wrong and if there are reliable methods to find this saddle point)

In summary:

What are the schemes available to compute reaction rates at finite temperatures for relatively big systems and multi-molecular reactions? I am particularly interested in accuracy vs complexity trade-off (i.e. I am interested in having a “not-so-precise-but-kinda-reasonable” result if it makes the calculations more affordable)

• Regarding this specific reaction, I'm afraid that no existing computational method may give a reliable estimate of the empirical reaction rate, given that the formed PbI2 will probably cover up the remaining perovskite. One can only hope to estimate the reaction rate in the first microsecond or so, where the surface of the perovskite can still be considered fresh Commented Sep 24, 2021 at 8:37
• @wzkchem5 Yes, you are right, and this is what I am interested in (that's what I tried to mean by "surface reaction", i.e. only what happens to the first exposed layer) Commented Sep 24, 2021 at 8:39
• I think ab-initio molecular dynamics could give you some useful information about the dynamics of the reaction, although I don't think a reliable reaction rate could be obtained for something complex as this. Commented Sep 24, 2021 at 13:48
• You can try fragmentation methods like in pubs.acs.org/doi/abs/10.1021/acs.jpcb.5b08446 and transition state theory. Other possibilities are ONION, QM/MM and embedding schemes. Commented Feb 14, 2022 at 19:28
• Was the suggestion by @AntoniodeOliveira-Filho helpful? Also Antonio, if you could expand that comment into an answer it would be very helpful for the site! I'm trying to clear up the unanswered queue :) Commented Feb 24, 2022 at 3:42

Relatively big systems can be treated with fragmentation methods that allow one to reduce the cost of the overall computation by using the most accurate and/or expensive approach on the most relevant part of the system and leaving the rest with cheaper and/or less accurate approximations. This allows one to find for an optimal trade-off between cost and accuracy as the user can choose the way that the system will be fragmented and the level of theory used in each part of the system. If the resulting model is cheap enough for geometry optimizations and vibrational frequencies calculations (not all degrees of freedom have to be optimized to cut costs), rate constants can be calculated using convencional transition state theory, i.e., the Eyring equation $$k = \frac{k_\mathrm{B}T}{hc^\circ}e^{-\Delta^\ddagger G/RT}$$

Here is a couple of fragmentation methods and examples of their use (there are many others)

## Fragment Molecular Orbital (FMO)

The fragment molecular orbital method (FMO) can be used to compute very large molecular systems by dividing them into fragments and performing ab initio or density functional calculations of fragments and their dimers, whereby the Coulomb field from the whole system is included. The FMO code is very efficiently parallelized utilizing the generalized distributed data interface.

The FMO method was used, for example, to investigate the adsorption of toluene and phenol on a faujasite zeolite, with good accuracy, reproducing ab initio results.

1. K. Kitaura, E. Ikeo, T. Asada, T. Nakano, M. Uebayasi, Fragment molecular orbital method: an approximate computational method for large molecules. Chem. Phys. Lett. 313, 701–706 (1999).

2. S. Tanaka, Y. Mochizuki, Y. Komeiji, Y. Okiyama, K. Fukuzawa, Electron-correlated fragment-molecular-orbital calculations for biomolecular and nano systems. Phys. Chem. Chem. Phys. 16, 10310–10344 (2014).

3. D. G. Fedorov, J. H. Jensen, R. C. Deka, K. Kitaura, Covalent bond fragmentation suitable to describe solids in the fragment molecular orbital method. J. Phys. Chem. A 112, 11808–11816 (2008).

## Surface Integrated Molecular Orbital/Molecular Mechanics (SIMOMM)

Surface Integrated Molecular Orbital/Molecular Mechanics (SIMOMM) is a QM/MM (quantum mechanics/molecular mechanics) optimization scheme for modeling surfaces that is based on the IMOMM (Integrated Molecular Orbital/Molecular Mechanics) approach of Maseras and Morokuma.

A combination of the surface integrated molecular orbital/molecular mechanics (SIMOMM) and fragment molecular orbital (FMO) methods was used to investigate the aldol reaction catalyzed by an amine-substituted mesoporous silica nanoparticle (amine-MSN) surface using a large molecular cluster model ($$\ce{Si392O958C6NH361}$$).

1. J. R. Shoemaker, L. W. Burggraf, M. S. Gordon, SIMOMM: An integrated molecular orbital/molecular mechanics optimization scheme for surfaces. J. Phys. Chem. A 103, 3245–3251 (1999).

2. F. Maseras, K. Morokuma, IMOMM: A new integratedab initio + molecular mechanics geometry optimization scheme of equilibrium structures and transition states. J. Comput. Chem. 16, 1170–1179 (1995).

3. A. P. de Lima Batista, F. Zahariev, I. I. Slowing, A. A. C. Braga, F. R. Ornellas, M. S. Gordon, Silanol-assisted carbinolamine formation in an Amine-functionalized mesoporous silica surface: Theoretical investigation by fragmentation methods. J. Phys. Chem. B 120, 1660–1669 (2016).

## WF-in-DFT Embedding

WF-in-DFT is a projection-based quantum embedding for electronic structure, which provides a formally exact method for density functional theory (DFT) embedding. The method also provides a rigorous and accurate approach for describing a small part of a chemical system at the level of a correlated wavefunction (WF) method while the remainder of the system is described at the level of DFT.

This method was used to study the oxidation potentials of neat ethylene carbonate, neat dimethyl carbonate, and 1:1 mixtures of ethylene carbonate and dimethyl carbonate in a embedding scheme with a CCSD(T) description of the oxidized molecule, a DFT description of the surrounding molecules, and a molecular mechanics (MM) description of more distant molecules.

1. S. J. R. Lee, M. Welborn, F. R. Manby, T. F. Miller III, Projection-based wavefunction-in-DFT embedding. Acc. Chem. Res. 52, 1359–1368 (2019).

2. T. A. Barnes, J. W. Kaminski, O. Borodin, T. F. Miller III, Ab initio characterization of the electrochemical stability and solvation properties of condensed-phase ethylene carbonate and dimethyl carbonate mixtures. J. Phys. Chem. C 119, 3865–3880 (2015).