Quantum mechanics in chemistry

[Big-Daddy]

(Let's assume for the sake of this discussion that computational power were not an issue.)

What are the applications to chemistry of quantum mechanics? So far I have only seen the Schrodinger equation being solved for a system with a hydrogen nucleus and 1 electron. Is it possible in theory to find the molecular orbital set, and thus the molecular orbital diagram, for all energetically likely isomers of molecules with a set number and type of atoms and set number of electrons (directly from the information of which and how many atoms there are, and how many electrons there are and which atomic orbitals they originally belonged to)? How do we go about writing the equations to do this (solving them need not be demonstrated, nor the approximations you need to do so, although I don't mind if you care to show these things).

Given two molecules and conditions such as temperature and pressure, can you theoretically calculate a) whether they will react and, if so, what product they will form or b) given the product you expect to be formed, the rate constant for the reaction?

 

[mfb]

Is it possible in theory to find the molecular orbital set, and thus the molecular orbital diagram, for all energetically likely isomers of molecules with a set number and type of atoms and set number of electrons (directly from the information of which and how many atoms there are, and how many electrons there are and which atomic orbitals they originally belonged to)?

The first task is to identify the possible orbitals of the electrons - once you have that, just fill them with electrons. Finding those orbitals can be done approximately with computers, indeed, quantum chemistry does that all the time. Chemistry is "just" applied quantum mechanics.

How do we go about writing the equations to do this (solving them need not be demonstrated, nor the approximations you need to do so, although I don't mind if you care to show these things).

Just the Schroedinger equation.
It can be useful to use some approximations to simplify the calculations.

Given two molecules and conditions such as temperature and pressure, can you theoretically calculate a) whether they will react and, if so, what product they will form or b) given the product you expect to be formed, the rate constant for the reaction?

If computing power is not an issue: sure.

 

[epenguin]

Although you could say that in principle chemistry should be reducible to QM (I think one could argue but I won't) maybe the student's question is more asking an indication on what is the state of the art in actually doing that? Because many student textbooks will have done the H atom, and maybe the H₂⁺ molecule and aspects of the He atom, they leave it there and the rest of chemistry is covered in a qualitative, not to mention somewhat dogmatic, fashion. I don't know if this only indicates how long ago I did any, but for that again I may not be the only one here.

Clearly the H atom does throw a flood of light of principles on all atoms, but now good are we at calculating their properties, their spectra let alone their chemistry? Quite a wide question I realize.

For what I know a bit more about I would say that protein structure calculations and predictions are not just computational physics.

 

[mfb]

Protein folding and even multi-protein interactions can be estimated with computers (you can even contribute with your own PC). It is not the only method to evaluate the structure of proteins, sure.

Although you could say that in principle chemistry should be reducible to QM (I think one could argue but I won't)

Molecules are objects in our universe, so they can in principle be described with physics. That is a direct consequence of the scope of physics. And the laws for the interactions of electrons and nuclei are well understood, and described with quantum mechanics.

maybe the student's question is more asking an indication on what is the state of the art in actually doing that?

I mentioned that in my post.

 

[Big-Daddy]

The first task is to identify the possible orbitals of the electrons - once you have that, just fill them with electrons. Finding those orbitals can be done approximately with computers, indeed, quantum chemistry does that all the time. Chemistry is "just" applied quantum mechanics.
Just the Schroedinger equation.
It can be useful to use some approximations to simplify the calculations.

Let's look at this first then.

What is the required information we have to input into the Schrodinger equation? e.g. if I wanted to write the equation which will land me up with the structure of water and all of its isomers, alongside values of how energetically likely each isomer is (and thus how prevalent I should expect each to be), how would I specify that? Maybe if you take me through that example (you don't need to list every electron necessarily, I should get the gist!) or send me a link for that example or another one (which is a proper molecule, not just H or H₂⁺ or something). So we're talking about finding those orbitals here, i.e. we're going to find an equation/equations which when solved would yield the molecular orbital diagram(s) for the isomers of water, along with estimations of how energetically likely each isomer is?

And where do we start seeing the fact that the structure of the same substance in different phases may differ - e.g. PCl₅ in the gaseous phase has a different structure to what it has in the solid phase unit cell. Then there's complex ions, which exist only in solution or in double salts. How do we get explanations for these structures out of QM?

 

[Jorriss]

Electronic structure is very important in chemistry.

Most of theoretical chemistry outside of electronic structure is still mostly classical but classical molecular dynamics simulations are starting to involve more quantum corrections.

For example, path integral quantum mechanics is used to quantize nuclear motion which is very important in such chemical problems as proton transport.

 

[mfb]

@Big-Daddy: I think you'll have to look for a book or something similar for that level of detail.

 

[Big-Daddy]

Any suggestions? At least the name of the field in which this stuff can be found?

I would have looked into a book straight away but all the books I have stop after the H nucleus with 1 electron. They don't even begin to consider multiple electrons, much less large molecules.

If you do recommend a book or name the field, I'd appreciate being directed to the field which describes how to formulate the method for finding the structures of each isomer and how energetically likely each one is, in terms of writing the equations - i.e. one which is focused on the development of the method for writing the equations needed to specify the molecule, rather than the approximations or computational solution methods.

 

[cgk]

The field dealing with the electronic structure (and other properties) of molecules on quantum mechanical level is called ``Quantum Chemistry''. If you happen to have a university library around, you could look for introductory textbooks in this topic. If you want to get a detailed insight into how molecular properties are calculated, you could look up "Molecular electronic structure theory" by Helgaker, Jørgensen, and Olsen (but this is not an introductory textbook).

Probably the most important aspects to understand if you want to get into this topic are Hartree-Fock and second quantization, because everything else is based on that. And, basically, Hartree-Fock is the picture defining MO theory, and on which the ``conventional'' picture of chemical theory (including DFT, if you are honest) is based.

 

[ZapperZ]

Do people actually start with "Schrodinger equation" still in such type of calculation?

As cgk has mentioned, techniques such as density function theory are very common-place in tackling such problems, especially when they involved many-body interactions. This is one of the most obvious "application" of quantum mechanics in chemistry.

Zz.

 

[Jorriss]

Do people actually start with "Schrodinger equation" still in such type of calculation?

In a typical chemistry department quantum mechanics course, yes.

 

[cgk]

Do people actually start with "Schrodinger equation" still in such type of calculation?

Yes. At the very least, the Schrödinger equation is used to derive approximations. There is also a large field of electronic structure theory in which real many-body wave functions beyond mean field are routinely calculated (usually when DFT doesn't cut it). But also in other fields, like theoretical vibrational spectroscopy or reaction dynamics many approaches are based on wave function theory (even if they sometimes get lost in the process and replaced by other things, e.g., surface hopping or path integrals).

Chemistry is very quantum-y on the theoretical side.

 

[Big-Daddy]

And perhaps, while I wait for some books to arrive, you can humour me by answering an extension question.

Could quantum theory predict the properties of a mixture rather than a single pure substance? For instance, if I know that I have an aggregate mixture of certain liquids (in terms of solutions, and also in the broader sense, e.g. crude oil) - including the chemical structure of each liquid and the percentage composition by number of moles of each liquid in the mixture - could I predict the chemical properties of the mixture? Could I predict the physical properties of the mixture (e.g. viscosity, etc.)?

 

[mfb]

In theory: no problem.
In practice: looks like a hard problem.

 

[Jorriss]

And perhaps, while I wait for some books to arrive, you can humour me by answering an extension question.

Could quantum theory predict the properties of a mixture rather than a single pure substance? For instance, if I know that I have an aggregate mixture of certain liquids (in terms of solutions, and also in the broader sense, e.g. crude oil) - including the chemical structure of each liquid and the percentage composition by number of moles of each liquid in the mixture - could I predict the chemical properties of the mixture? Could I predict the physical properties of the mixture (e.g. viscosity, etc.)?

That problem is completely intractable using a full quantum treatment. It is possible to find bulk properties using classical mechanics with quantum mechanical corrections.

 

[Big-Daddy]

In theory: no problem.
In practice: looks like a hard problem.

Given that computing power is not an issue for my consideration?

That problem is completely intractable using a full quantum treatment. It is possible to find bulk properties using classical mechanics with quantum mechanical corrections.

Is the theoretical method to do it using QM laid out, and just difficult to actually process, or is it little known?

 

[mfb]

Given that computing power is not an issue for my consideration?

That's the big difference between theory and practice.
The required computation power for a full quantum-mechanical solution (based on the Schrödinger equation) grows really fast with the number of involved particles. With a real (and with every realistic) computer, you have to make some approximations.

 

[Big-Daddy]

That's the big difference between theory and practice.
The required computation power for a full quantum-mechanical solution (based on the Schrödinger equation) grows really fast with the number of involved particles. With a real (and with every realistic) computer, you have to make some approximations.

I just want to look into the method. Ways for actually getting a final answer out of it, I'll look up separately once I've grasped the possibilities in the field.

As I asked above, is the theoretical method to consider chemical and physical properties of aggregate mixtures using QM laid out, and just difficult to actually process, or is it unknown? Do such things even lie within the domain of QM?

 

[Jorriss]

Is the theoretical method to do it using QM laid out, and just difficult to actually process, or is it little known?

What do you mean by laid out?

In principle we have theories that can get us just about any level of accuracy from solving the schrodinger equation explicitly to a highly coarse grained, numerical, classical mechanics picture.

The issue is that the more complete pictures are almost always orders of magnitude more challenging to solve analytical, or orders of magnitude more time consuming to solve numerically.

In principle - completely in principle - you could solve for the properties of a mixture by solving the schrodinger equation for it.

 

[Big-Daddy]

In principle - completely in principle - you could solve for the properties of a mixture by solving the schrodinger equation for it.

And where I can learn how to specify the Schrodinger equation for a mixture? (If you would suggest looking into a book, please also suggest a name with it.)

I am assuming the activity of each component in the mixture, as well as what structures each component is, will be the defining factors for how we write the specific Schrodinger equation for the mixture.

 

[Jorriss]

And where I can learn how to specify the Schrodinger equation for a mixture? (If you would suggest looking into a book, please also suggest a name with it.)

I am assuming the activity of each component in the mixture, as well as what structures each component is, will be the defining factors for how we write the specific Schrodinger equation for the mixture.

It's not challenging at all to specify.

https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)#Many_particles

Nope, no activities or anything. Activities are far less fundamental than potentials. You just need to decide the potential between particles. Why are you interested in this specifically?

 

[Big-Daddy]

It's not challenging at all to specify.

https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)#Many_particles

Nope, no activities or anything. Activities are far less fundamental than potentials. You just need to decide the potential between particles.

OK, so what would the equation (i.e. full Hamiltonian, in this case) for a mixture of H₂ gas and He gas look like? I ask because, firstly, there doesn't seem to be provision for the electrons that belong directly to each of the atoms as individuals (we are just treating them as independent particles), and secondly, I want to see what constants need to be specified using non-QM tools, for a specific case. (You don't need to find the values but if you could point out which of the symbols in the Hamiltonian I actually need to find a value for before solving using the Schrodinger equation, I would be grateful.) And thirdly, it's very counter-intuitive and almost certainly wrong that a mixture of 25% H2 and 75% He gas has identical properties to a mixture of 75% H2 and 25% He gas. Unless you are saying the equation applies to the system microscopically?! (i.e. if there are 2 moles of each gas, we need to include some 8*10²⁴ particles in the summation you linked to, rather than just 7 (2 H electrons, 2 He electrons, 2 H nuclei, 1 He nucleus)?) Also, there aren't any Coulomb interactions?

The Wikipedia page you linked to is ok, except that it doesn't say how to write V as a function of vector distance apart r, which is very important to write an equation which can be solved in theory. Meanwhile, http://en.wikipedia.org/wiki/Molecular_Hamiltonian offers a different way of writing the Hamiltonian, but does not help me see how to either write V as a function of r, and already starts with an approximation (going straight to the Coulomb Hamiltonian rather than the full Hamiltonian) which immediately makes it difficult to see the relationship to the page you linked me to.

Can you do this example for the hydrogen-helium case I suggested?Why am I interested in this specifically? Well, I'm trying to attain an understanding of how much, and how, quantum mechanics covers chemistry as we understand it, in principle (i.e. if computational power can be considered a non-issue).

 

[Jorriss]

You just need to put a Coulomb potential between every particle and then, in principle, you have the correct Hamiltonian.

Why am I interested in this specifically? Well, I'm trying to attain an understanding of how much, and how, quantum mechanics covers chemistry as we understand it, in principle (i.e. if computational power can be considered a non-issue).

In principle, it covers virtually all of chemistry - except in those places where relativistic corrections are necessary.

 

[Big-Daddy]

You just need to put a Coulomb potential between every particle and then, in principle, you have the correct Hamiltonian.

Ok, let me see if I can find some examples on the Internet.

In principle, it covers virtually all of chemistry - except in those places where relativistic corrections are necessary.

Some things are looking a bit odd though. First of all, where are the temperature and pressure terms in the Schrodinger equation? I know my desired molecule can have different structures at different temperatures and pressures, particularly when it is in a different phase. Secondly, just inputting the particles involved will give me a single wave-function (in theory) and thus a single isomer of the molecule, whereas the molecule may have dozens of isomers which are reasonably energetically likely.

 

[Einstein Mcfly]

Ok, let me see if I can find some examples on the Internet.



Some things are looking a bit odd though. First of all, where are the temperature and pressure terms in the Schrodinger equation? I know my desired molecule can have different structures at different temperatures and pressures, particularly when it is in a different phase. Secondly, just inputting the particles involved will give me a single wave-function (in theory) and thus a single isomer of the molecule, whereas the molecule may have dozens of isomers which are reasonably energetically likely.

The SE will give you energy spectra that you can then along with statistical mechanics to get macroscopic properties like pressure.

As for isomers, what you specify in the SE are, as was mentioned above, potentials. These will depend on distances between nuclei and how many electrons there are and, when you're done, you'll have a set of solutions for each electron with an energy and a sense of where they're located. If you move the nuclei around, you'll change the potential and change the solutions for the electrons. Usually in quantum chemistry, you get rid of this issue by making the "Born-Oppenheimer approximation" that says that the nuclei are essentially fixed and you can compute the solutions for the electrons with that in mind. What you generally DO in quantum chemistry (as opposed to what you could do, in principle) is to look at a number of snapshots of nuclear positions (isomers) and see what their energy and Gibbs Free Energies are, see what barriers exist between them and then discuss the thermodynamics and kinetics of possible "reactions" that take you from one isomer to the other. This can be visualized in terms of an "energy surface".

Also, if you want to look at a way of doing quantum chemistry that takes things like resonance structures (kind of "electronic isomers") into account, look up Valence Bond Theory.

 

[DrDu]

These are all quite complicated topics and I suggest you to read a book or article on quantum chemistry and physical chemistry.

Just one point: The temperature does not enter the hamiltonian but is taken into account via a thermal ensemble where the occupation probability of an excited state of a molecule is given by Boltzmann factors exp (-E/kT), where E is the energy relative to the ground state.
Pressure effects, at least in gasses, are often calculated starting from a virial expansion, i.e. interactions of the molecules are considered according to the number of interacting molecules (2, 3, 4 etc.).

 

[Big-Daddy]

As for isomers, what you specify in the SE are, as was mentioned above, potentials. These will depend on distances between nuclei and how many electrons there are and, when you're done, you'll have a set of solutions for each electron with an energy and a sense of where they're located. If you move the nuclei around, you'll change the potential and change the solutions for the electrons. Usually in quantum chemistry, you get rid of this issue by making the "Born-Oppenheimer approximation" that says that the nuclei are essentially fixed and you can compute the solutions for the electrons with that in mind. What you generally DO in quantum chemistry (as opposed to what you could do, in principle) is to look at a number of snapshots of nuclear positions (isomers) and see what their energy and Gibbs Free Energies are, see what barriers exist between them and then discuss the thermodynamics and kinetics of possible "reactions" that take you from one isomer to the other. This can be visualized in terms of an "energy surface".

Thanks for the argument, very interesting and new way of considering what isomers actually are. Looking at this bit particularly:

As for isomers, what you specify in the SE are, as was mentioned above, potentials. These will depend on distances between nuclei and how many electrons there are and, when you're done, you'll have a set of solutions for each electron with an energy and a sense of where they're located. If you move the nuclei around, you'll change the potential and change the solutions for the electrons.

From the Schrodinger equation will in theory arise the wave-function solutions for this set of nuclei and electrons, I grasp, which I will then restrict with statistical mechanics to the correct wave-function that represents correctly the electrons in my molecule under these statistical mechanical conditions.

Does this wave-function refer to a single isomer? If not, how do we find the different "isomers" from the wave-function - because, after all, if one wave-function refers to more than one isomer, then there should be no concept of inter-conversion between isomers as the isomers would fundamentally be the same (which could be the case for resonance structures but not normal molecular isomers)?

 

[DrDu]

Most of quantum chemistry is built upon the Born Oppenheimer approximation as has been pointed out by Einstein Mcfly already.
http://en.wikipedia.org/wiki/Born_Oppenheimer
Concepts like molecular structure and isomers have no meaning at a fully quantum mechanical level but are emergent properties in the Born_Oppenheimer approximation which is the result of the vast difference of electronic and nuclear masses.

 

[Big-Daddy]

Concepts like molecular structure and isomers have no meaning at a fully quantum mechanical level but are emergent properties in the Born_Oppenheimer approximation which is the result of the vast difference of electronic and nuclear masses.

I've read the page you linked on the Born-Oppenheimer approximation. However, I'm still confused as to what you mean by "molecular structure and isomers have no meaning at a fully quantum mechanical level" - surely there must be something to distinguish isomers from each other at the most fundamental level, since they are indeed different species and we can isolate them separately from one another in the lab?

 

[Einstein Mcfly]

I've read the page you linked on the Born-Oppenheimer approximation. However, I'm still confused as to what you mean by "molecular structure and isomers have no meaning at a fully quantum mechanical level" - surely there must be something to distinguish isomers from each other at the most fundamental level, since they are indeed different species and we can isolate them separately from one another in the lab?

Each arrangement of nuclei for a given number of electrons will have its own wave functions (solutions to the SE for that particular potential). That wave function doesn't "know" anything about any other wave function based on any other potential (due to different coordinates for the nuclei or numbers of electrons). If you want to compare properties of isomers at the quantum mechanical level, you must do separate calculations for each system and then compare their properties. If one of them has a lower total energy than another (all else being equal), then you can say that this structure should be found more readily in nature than the higher energy one. It's not exactly that simple (there are factors like kinetics and further reactions and whatnot) but that's the general idea. Quantum mechanics and quantum chemistry don't solve everything all at once; you need to do quite a bit of work yourself to investigate the possibilities.

 

[DrDu]

I've read the page you linked on the Born-Oppenheimer approximation. However, I'm still confused as to what you mean by "molecular structure and isomers have no meaning at a fully quantum mechanical level" - surely there must be something to distinguish isomers from each other at the most fundamental level, since they are indeed different species and we can isolate them separately from one another in the lab?

Isomers can isomerize. On a most fundamental level, the eigenstates of the full hamiltonian are superpositions of different isomers.
The question why we really observe isomers is far from trivial. Probably it is due to interactions with neighbouring molecules, so it is a colligative effect. You may want to google for Hund's paradox.

 

[Einstein Mcfly]

Isomers can isomerize. On a most fundamental level, the eigenstates of the full hamiltonian are superpositions of different isomers.
The question why we really observe isomers is far from trivial. Probably it is due to interactions with neighbouring molecules, so it is a colligative effect. You may want to google for Hund's paradox.

Doesn't Hund's paradox just apply to chiral systems? I thought his question was more about non-degenerate isomers (boat versus chair and all that).

 

[DrDu]

Doesn't Hund's paradox just apply to chiral systems? I thought his question was more about non-degenerate isomers (boat versus chair and all that).

Yes, but as always, for degenerate states the consideration of superpositions is most relevant.