Ab Initio methods for chemical reactions?

[jonjacson]

Well, I know this is not the "Chemistry" subforum but the question is all about quantum physics.

When you study the Schrödinger equation you can compute the time evolution of the wave function, see what energy levels are possible etc. You can calculate the spectra of atoms describing them as a many body quantum system.

My question is, if I just use the Schrödinger equation, Would it be possible to get the products of a chemichal reaction?

I mean, if I "plug" into the quantum mechanics equation the wave function of two systems made of different atoms, Will the equations tell me how the systems react with each other? Will the equations tell me what are the final products?

Any books explaining this?

edit:

I just found this:

https://www.pnas.org/content/111/1/15

So the answer is yes but just to the simplest of them?

 

[Vanadium 50]

Well, yes and no.

Consider Benzene. It has 12 nuclei and 42 electrons, so you have 53 coupled differential equations, plus the Fermi-Diract and Bose-Einstein constraints. So it's possible, but not practical.

 

[A. Neumaier]

Will the equations tell me how the systems react with each other? Will the equations tell me what are the final products?

Yes. This is commonly done in the Born-Oppenheimer approximation.
Apart from spectroscopy and laser-induced chemistry, quantum chemistry works under the assumptions that at fixed nucleus positions, the electrons are in their ground state. Due to the Born-Oppenheimer approximation, this allows one to consider the nuclei as point particles collectively moving on an ab-initio potential energy surface determined by the electronic ground state calculations, which can be done in various ways, for large molecules commonly by DFT. The nuclei can now be treated classically (molecular dynamics; unbounded motion defines chemical reactions) or quantum mechanically (scattering theory; the continuous spectrum defines reactions).

Consider Benzene. It has 12 nuclei and 42 electrons, so you have 53 coupled differential equations, plus the Fermi-Diract and Bose-Einstein constraints. So it's possible, but not practical.

Benzene is still a very small case and is quite accurately tractable by ab initio methods.

 

[f95toli]

You need to be a bit careful here. When you are talking about chemical reactions you are talking about a dynamical problem. Reaction pathways are frequently decided by differences in energy and these differences can be much, much smaller than the absolute energy of the levels involved.
This means that not only do you need to be able to calculate the energies involved, you need to be able to do so very accurately. This in turn means that many approximation methods that work reasonably well for a static problems can not be used for reactions.

This is incidentally one of the reasons there are many potential applications for even quite small quantum computers (~150-200 qubits) in quantum chemistry; as long the calculations can be done with high enough accuracy it would allow to study chemical reactions that are completely intractable using a classical computer; and many of the molecules people want to study are not actually very large (caffeine is a famous example).

 

[A. Neumaier]

not only do you need to be able to calculate the energies involved, you need to be able to do so very accurately. This in turn means that many approximation methods that work reasonably well for a static problems can not be used for reactions.

In principle yes. Fortunately, errors of ab initio methods are to a large extent systematic, so differences to the minimal energy are often more accurate than the absolute energies. In any case, appropriate ab initio methods for getting potential energy surfaces are appropriately applied in practice to quite large molecules.

 

[Vanadium 50]

approximation

I don't think he's talking about approximations.

 

[A. Neumaier]

I don't think he's talking about approximations.

Virtually all of quantum mechanic's use consists of approximations. Exact results in quantum mecharics are very rare and even then apply to idealized situations only (e.g., the hydrogen atom without relativistic corrections).

 

[Vanadium 50]

irtually all of quantum mechanic's use consists in approximations.

True. But that seems not to be what the OP is asking about.

 

[A. Neumaier]

True. But that seems not to be what the OP is asking about.

Oh, in principle, if we could do the exact computations, no approximation would be needed, and the conclusions I gave would still hold. They would just not be computable and not give insight. insight in nontrivial quantum mechnaics usually requires simplification, and approximation is one of the ways to achieve that.

 

[Vanadium 50]

True. But that seems not to be what the OP is asking about.

 

[A. Neumaier]

True. But that seems not to be what the OP is asking about.

So what is he asking about if it is neither answerable through approximation nor an in principle question?

 

[jonjacson]

(scattering theory; the continuous spectrum defines reactions).

Can you recommend me any books talking about this?

You need to be a bit careful here. When you are talking about chemical reactions you are talking about a dynamical problem. Reaction pathways are frequently decided by differences in energy and these differences can be much, much smaller than the absolute energy of the levels involved.
This means that not only do you need to be able to calculate the energies involved, you need to be able to do so very accurately. This in turn means that many approximation methods that work reasonably well for a static problems can not be used for reactions.
.

Thanks, that is very interesting.

 

[A. Neumaier]

Can you recommend me any books talking about this?

The simplest of all reactions, AB to A+B, is, in the center of mass frame, and expressed in terms of the distance between A and B, just the case of an anharmonic oscillator with a potential that vanishes at infinity , is infinite at zero, and has a local minimizer with positive energy. The spectrum is continuous, and the eigenvectors of the Schroedinger equation describe the scattering states, i.e., how an initial unstable compound state turns into a superposition of states at large distance. From this you get decay rates. In general you have multi-particle scattering on a complicated potential energy surface; see, e.g., https://en.wikipedia.org/wiki/Transition-state_theory

There is an extensive book on scattering theory by Newton, which has all the details of multi-particle scattering from a physics point of view.

 

[DrDu]

Benzene is still a very small case and is quite accurately tractable by ab initio methods.

That's not true as far as dynamical calculations are concerned, not even in BO approximation.

 

[A. Neumaier]

That's not true as far as dynamical calculations are concerned, not even in BO approximation.

The label ab initio is traditionally not only given to the B/O approximation but to all approximation methods that do no use experimental input.

 

[DrDu]

Of course. What I mean is that propagating the Born Oppenheimer wavefunction on a single potential energy surface is a fromidable task even for molecules involving only 4 or five atoms. The article, the OP is citing in PNAS considers the reaction of only 3 atoms on one potential energy surface.
Benzene is definitely not small in this respect.

 

[A. Neumaier]

Of course. What I mean is that propagating the Born Oppenheimer wavefunction on a single potential energy surface is a fromidable task even for molecules involving only 4 or five atoms. The article, the OP is citing in PNAS considers the reaction of only 3 atoms on one potential energy surface.
Benzene is definitely not small in this respect.

Yes, But for doing classical molecular dynamics on the PES determined ab initio using the B/O approximation, Benzene is small. This still counts as ab initio. Much larger molecules can be handled in this way; see, e.g., the 2011 paper Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations by J. Behler.

 

[f95toli]

Yes, But for doing classical molecular dynamics on the PES determined ab initio using the B/O approximation, Benzene is small. This still counts as ab initio. Much larger molecules can be handled in this way; see, e.g., the 2011 paper Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations by J. Behler.

But this does not change the fact that many, many important chemical reactions can NOT be simulated (or even understood) using a classical computer. Doing calculations with chemical accuracy ( typically 1.6e-3 hartree) is currently impossible for even relatively small molecules.

This is one of the reasons why quantum computers are interesting for quantum chemistry.
See e.g.
https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.011021
which also briefly discussed the "classical" problem

 

[A. Neumaier]

Doing calculations with chemical accuracy ( typically 1.6e-3 hartree) is currently impossible for even relatively small molecules.

This is one of the reasons why quantum computers are interesting for quantum chemistry.

Doing calculations with chemical accuracy is currently much more impossible with quantum computers even for much tinier molecules - the paper you cite only treats $H_2$! And it is very inaccurate, producing spurious energy levels...

 

[f95toli]

Doing calculations with chemical accuracy is currently much more impossible with quantum computers even for much tinier molecules - the paper you cite only treats $H_2$! And it is very inaccurate, producing spurious energy levels...

True, I don't think I've stated otherwise.
What I said was that a QC could potentially solve problems that are always going to be intractable on a classical computer.
But yes, it is going to take a few years to get there. You need about ~150 well-behaved qubits to solve interesting problems, at the moment Google is has a 72 qubit processor (IBM is at 49), but rumor has it that the qubits are misbehaving.

 

[A. Neumaier]

Of course. What I mean is that propagating the Born Oppenheimer wavefunction on a single potential energy surface is a formidable task even for molecules involving only 4 or five atoms. [...] Benzene is definitely not small in this respect.

But also not large. You need to look not at an arbitrary paper but at the state of the art.

MCTDH is a public software system for solving time-dependent Schrödinger equations. It says in the Brief description that ''we have recently used MCTDH to study the multi-dimensional Henon-Heiles Hamiltonian including up to 32 degrees of freedom''. Benzene has $N=12$ hence $3N-6=30$ degrees of freedom (after separation of the center of mass motion and rotational degrees of freedom).

The successful treatment of benzene already in 1998/99 is explicitly mentioned.