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Not a physicist or Chemist, just interested in QM and it's applications.

I've been reading lately about Quantum Chemistry and it occurred to me that since we can model electron orbitals in QM and solve for the shape of the orbitals that result (the S, P, D, F etc orbitals), which explains why neutral atoms can form bonds together (valence electron wave function spreading out into a lower energy state between atoms, the difference going into kinetic energy).

So, it occurs to me that we should be able to represent chemical reactions as some kind of interaction between groups of atoms. So, is this realistically (computationally) possible? And if so, how complex of reactions can we represent in QM? Are we struggling to model H2 or the 4H + O2 => 2H2O reaction, or are computers at the point where we can model the methylation of Benzene or ATP synthesis?

Bonus points if anyone can link to visualisation videos of reactions that demonstrate their Quantum nature!

thanks!

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    $\begingroup$ Up to 2009, quantum mechanics can simulate 1 million atoms sciencedirect.com/science/article/pii/S0009261409005041, I guess for reaction it will involve more complicated details, nevertheless the range may be similar $\endgroup$ – user26143 Oct 21 '13 at 8:24
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    $\begingroup$ Well it depends very much on the sort of accuracy you require. If you want the energy levels and full wavefunctions to machine precision I'm told the current practical limit is about 6 or 7 particles, which is very hard to beat due to the exponentiality of Hilbert space. Jim Mitroy's group at CDU are pros at this. The techniques for larger numbers of particles rely on certain approximation schemes. A quantum chemist would know what schemes are useful for what practical purposes, such as DFT as mentioned by @user26143. $\endgroup$ – Michael Brown Oct 21 '13 at 8:38
  • $\begingroup$ hey, what! I went to CDU! I saw Jim Mitroy around, but I didn't do engineering so I never had him as a lecturer. Small World! Until now I had no idea that he was a QM researcher, let alone world-class. Amazing. $\endgroup$ – matao Oct 21 '13 at 9:05
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    $\begingroup$ I would be wary of comparing the performance of a metal simulation (as in the "1 mil atom" sim link above) with chemistry. I'm not a quantum chemist but I think in the metal case the atom locations are fixed and the electron density is evaluated, while what the OP asks would have to involve both electron orbital simulation and molecular/atom dynamics. This combination is called QM/MM (or QMMM) and used to simulate for example enzyme active sites together with the movement of the sites, but I don't have any performance figures. $\endgroup$ – BjornW Oct 21 '13 at 9:26
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    $\begingroup$ Another problem is whether MM belongs to QM as OP asked $\endgroup$ – user26143 Oct 21 '13 at 9:36
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Yes, it is possible to model chemical reactions with QM. In quantum chemistry, one often solves for the Schrodinger equation of the molecular Hamiltonian assuming the Born-Oppenheimer approximation. Within this approximation, the energy is a function of the coordinates of the atomic nuclei. Thus, to model a reaction you can follow the lowest energy path from the atomic coordinates of the reactants to those of the products.

Now, how complex can the reactions be and still be computationally feasible to model? That depends on two things: (1) how much accuracy do you need? (2) what kind of effects are involved in the reaction? For example, with modern computers and using Density Functional Theory (DFT), it is perfectly feasible to model the methylation of benzene (or even much larger systems), but I wouldn't expect an accuracy better than around 4 kcal/mol in the energies. However, I do not know of any method that is able to give an accurate description of the dissociation of the Cr$_2$ dimer, even though larger systems can be modelled accurately with a variety of methods. The reason for this is that the effects of electronic correlation are very strong in Cr$_2$, and these effects are inherently difficult and computationally costly to take into account.

If you want the best possible result (i.e. taking into account all of the correlation effects), you need to use a method called Full Configuration Interaction (FCI). In FCI, you minimize the energy of a wavefunction which is a linear combination of all the possible configurations (Slater determinants) that the system can take. This is the most general wavefunction possible and thus, by the variational principle, when you minimize its energy you get the exact numerical result for the Schrodinger equation for a given basis set (i.e. the functions that you use to represent your orbitals). However, the time complexity of this method is roughly $O(M!)$, where $M$ is the number of basis functions, and the result is really exact only when $M$ tends to infinty. In practice, $M$ only needs to be very large to converge but this is still so costly than we can only afford to do FCI for systems with no more than around 10 electrons. This is why there are so many approximation methods in quantum chemistry; so that one can use a method appropriate for size of the system and the accuracy desired. DFT is extremely popular because it is relatively cheap and accurate (formally, $O(M^4)$ and average errors of about 4 kcal/mol in energies as noted above). Other good methods that are in between FCI and DFT in accuracy and cost are coupled cluster methods ($\sim O(M^7)$) and CASPT2 (combinatorial cost), which is a kind of FCI in a subset of electrons and orbitals and adds a second order perturbation theory correction on top of that.

With respect to examples of reactions, you should probably check for the Woodward-Hoffman rules (http://en.wikipedia.org/wiki/Woodward%E2%80%93Hoffmann_rules). These are some simple rules based on QM and molecular orbital theory which can predict the outcome of a great deal of reactions that could not be explained in any other way.

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  • $\begingroup$ Maybe you could take a look at this question I'm interested in. Do these techniques have anything to say there? $\endgroup$ – WetSavannaAnimal Oct 22 '13 at 11:05
  • $\begingroup$ @WetSavannaAnimalakaRodVance I posted an answer to the question in your link. You can model any kind of absorbance/emission spectra rigurously with any method that yields a wavefunction. All methods I mentioned yield wavefunctions, except DFT which gives densities. You can, however, use TDDFT (en.wikipedia.org/wiki/Time-dependent_density_functional_theory) or $\Delta$SCF (pubs.acs.org/doi/abs/10.1021/ct400356k) to calculate spectra with DFT. $\endgroup$ – Goku Oct 23 '13 at 3:55

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