How do we *know* the Schrodinger equation for H2+ can't be solved?

[AxiomOfChoice]

In all the introductions to the Born-Oppenheimer approximation I've seen, they make the following claim:

"If you write out the stationary Schrodinger equation for the simplest molecule -- H[itex]_2^+[/itex] -- even it cannot be solved analytically, so we are forced to make an approximation."

But how do we know it can't be solved analytically? Is that something that can be proved, or is it just the case that no one has been clever enough to figure out the analytical solution?

My question probably stems from a misunderstanding of what the word "analytical" means. I'd love for someone to clear this up for me!

 

[Nugatory]

But how do we know it can't be solved analytically? Is that something that can be proved, or is it just the case that no one has been clever enough to figure out the analytical solution?

It's the latter. The first few google hits for "hydrogen molecule schrodinger equation" will give you a feel for just how hairy this differential equation is.

 

[Bill_K]

But how do we know it can't be solved analytically? Is that something that can be proved, or is it just the case that no one has been clever enough to figure out the analytical solution?

Given an ordinary differential equation, you can always "solve it analytically" by inventing a new special function!

However, what can be proved is that the Schrodinger equation for this case is not separable, i.e. it cannot be broken down into a set of ordinary differential equations.

 

[dextercioby]

The H_2+ molecular ion is a 3-body problem, just like the Helium atom. The PDE is not separable.

 

[AxiomOfChoice]

The H_2+ molecular ion is a 3-body problem, just like the Helium atom. The PDE is not separable.

This seems to imply that the only PDEs that can be solved analytically are separable PDEs. Is that really the case?

Again, I think the real point at issue here is what, exactly, it means to say that a PDE can be solved analytically. Does it just mean: "There exists a standard set of techniques that, when applied to the equation, yield its solution, possibly after grinding through some integrals and algebra"? In other words, if someone got fantastically lucky and just happened to correctly guess a solution of the PDE, we couldn't say that the equation had been solved analytically. This seems like a plausible way to parse it.

 

[AxiomOfChoice]

Given an ordinary differential equation, you can always "solve it analytically" by inventing a new special function!

This is an interesting "way out"! But do I take it to mean that, if I have some partial differential equation in the variables [itex]x_1,x_2,\ldots, x_n[/itex], then I can say, "Let the solution to this PDE be [itex]f(x_1,x_2,\ldots, x_n)[/itex]," and I can therefore claim to have solved the PDE analytically? That doesn't seem right...

 

[ModusPwnd]

Thats what we do with trigonometric functions and the harmonic oscillator.

 

[cgk]

As long as you can give an algorithm to compute a solution to any given required degree of accuracy (in principle), you can always say "let f(x1,x2,...) be the limit of the following algorithm..." and define this as the solution of the problem.

If you are honest, this is not very different from what we do with trigonometric functions, special functions, or other mathematical function definitions, just as Bill_K said. In some sense the only difference between a problem which can be "solved analytically" and a problem which cannot is that for the former case it is often (not always!) easier in practice to actually compute the solution, because the corresponding functions are already implemented in standard programs or libraries. But even that is not always the case. If your "analytic solution" happens to be some funky hypergeometric function... good luck.

 

[WannabeNewton]

That doesn't seem right...

This is what we have existence and uniqueness theorems for-they guarantee analytic solution(s) under the right conditions. Whether or not we can actually solve for the guaranteed solutions in closed form is an entirely different story. In nearly all cases we cannot solve in closed form-including the helium atom. We simply do not have the techniques for it.

 

[AxiomOfChoice]

We simply do not have the techniques for it.

Ok. But this doesn't mean such techniques will never be found or are somehow incapable of being found, right?

 

[DrDu]

I am not sure what you mean with "all introductions to the Born Oppenheimer equation". Once you invoke the Born Oppenheimer approximation, the electronic problem becomes a separable differential equation in elliptic coordinates, though its solution still involves some numeric effort.

 

[f95toli]

Ok. But this doesn't mean such techniques will never be found or are somehow incapable of being found, right?

But then the question is WHY you would like to have an analytical solution?
Even when we do have analytical solutions for PDEs they are rarely particularly useful; the reason being that they tend to involve -as has already been mentioned- e.g. infinite sums over complicated special functions which at minmum will require access to tables to actually evaluate (and if the sum is infinite you obviously have to truncate it); in practice we just use numerical methods on computers instead because it is faster.

Don't get me wrong, sometimes there are good reasons to have analytical formulas even if you are not actually going to use them to calculate something; but if the formulas are too complicated and cumbersome this too becomes pointless.

Note also that it is quite often obvious from the numerical solution to a problem that you will never be able to find a useful analytical solution; the behaviour is simply too complicated.

 

[Nugatory]

Ok. But this doesn't mean such techniques will never be found or are somehow incapable of being found, right?

Never say never :)
This isn't like the halting problem or trisecting an angle, where we can prove that there is no solution.

At the same time, you shouldn't get too hung up on exact solutions. Even in classical mechanics, which has been around for thee-plus centuries...

We have a fairly easy exact solution for the trajectory of a thrown ball, if we neglect air resistance; this is high school physics. But you'll be several years into a college-level physics/engineering curriculum before you'll be able to solve the problem if we include air resistance - and that's still assuming that the ball is an ideal smooth sphere. And if we want to include the effects of irregularities such as seams and stitching in the surface of the ball... That problem is more complex than the hydrogen molecule, and no more solved. That doesn't stop us from building airplanes that fly and spaceships that survive reentry and land where we expect.

 

[AxiomOfChoice]

I am not sure what you mean with "all introductions to the Born Oppenheimer equation". Once you invoke the Born Oppenheimer approximation, the electronic problem becomes a separable differential equation in elliptic coordinates, though its solution still involves some numeric effort.

Yes, and I am quite familiar with the Born-Oppenheimer-to-elliptic-coordinates approach to solving the H[itex]_2^+[/itex] molecule. But this occurs only after the B-O approximation is invoked, and the approximation is (in my experience) always motivated by the observation that the full stationary Schrodinger equation -- before we "clamp down" the nuclei -- cannot be solved analytically.

 

[atyy]

Is AxiomOfChoice asking whether there anything like Abel's proof for the insolubility of the quintic or http://math.stackexchange.com/quest...e-that-a-function-has-no-closed-form-integral?

 

[AxiomOfChoice]

Is AxiomOfChoice asking whether there anything like Abel's proof for the insolubility of the quintic or http://math.stackexchange.com/quest...e-that-a-function-has-no-closed-form-integral?

Yes, this is kind of what I'm getting at. If a textbook says "there is no analytic solution to this PDE," I take that to mean that someone has proved there is no analytic solution.

If it is only a matter of no one knowing how to arrive at the solution at present - regardless of how long that's been the case - then it would seem "there is no analytic solution" is a poor choice of words.

 

[DrDu]

Personally, I have never seen it as a motivation for the BO approximation that the full problem cannot be solved analytically. Even after the BO approximation, there are no molecules where the electronic problem could be solved analytically, maybe besides H2+. However for the electronic problem usually the Hartree Fock approximation or LDA DFT is a good zeroth order starting point onto which more sofisticated methods can build up.
I have come about some papers where the full nuclear + electronic problem is solved without invoking the BO approximation, usually using a basis expansion e.g. into hyperspherical harmonics.

 

[AxiomOfChoice]

I have come about some papers where the full nuclear + electronic problem is solved without invoking the BO approximation, usually using a basis expansion e.g. into hyperspherical harmonics.

That's very interesting. Can you share some of them?

 

[DrDu]

I don't remember them, but Google Scholar spits up some:
http://link.springer.com/article/10.1007/BF01114921
http://arizona.openrepository.com/arizona/bitstream/10150/186023/1/azu_td_9307683_sip1_m.pdf
http://neon.phy.vanderbilt.edu/docs/publications/bubin_cr_113_36_2013.pdf

 

[D H]

If it is only a matter of no one knowing how to arrive at the solution at present - regardless of how long that's been the case - then it would seem "there is no analytic solution" is a poor choice of words.

When you see "there is no analytic solution ..." you should be reading that as meaning "there does not exist a solution that uses only a finite set of elementary operations on a finite set of elementary functions". That's too long, so mathematicians just say "there is no analytic solution" for short.

For example, consider ##\frac {df}{dx} = \frac{\sin x}{x}## (or ##\int \frac{\sin x}{x}\,dx##) This does not have an analytic solution in the elementary functions. Provably so, in fact. This particular differential equation (and the corresponding integral) come up very often, and it is fairly easy to evaluate numerically. So it's given a special name, the sine integral. The sine integral is one of many non-elementary special functions.

As Bill_K mentioned, given any ##f(x)##, one can always denote ##F(x) = \int_a^x f(t)\,dt## as a new special function and voila! problem solved. Well, not really. You have to have some way of evaluate that integral numerically, for one thing. For another, it needs to pop up in lots of different places to deserve that "special" label.

As to how you would go about proving that no analytic solution exists for some differential equation (remember what that means), you would need to invoke Liouville's theorem or use differential Galois theory.

 

[williamsn]

Essentially, in electronic structure theory, any system with at most 2 electrons can be solved exactly. If you want to attempt an analytical solution, be my guest, but I'll use GAUSSIAN and get the same answer weeks to months (aka ~$10,000 for a grad student) sooner. But there is a definite limit in some sense in that a 3e- system, such as a Lithium atom, cannot be exactly solved even numerically, due to electron-electron correlation that comes in thanks to Pauli and Hund. H2+ can probably be solved analytically, but it's more useful to introduce BO and PT at that stage and build up HF, CC, DFT, etc for larger systems from a trivial base case.

 

[AxiomOfChoice]

Essentially, in electronic structure theory, any system with at most 2 electrons can be solved exactly.

Ok. So this brings up a few questions: (1) What's the exact solution for H2+, a system with at most 2 electrons? (2) How is the exact solution different from the analytic solution? Or are you using those two words interchangeably?

 

[AxiomOfChoice]

When you see "there is no analytic solution ..." you should be reading that as meaning "there does not exist a solution that uses only a finite set of elementary operations on a finite set of elementary functions". That's too long, so mathematicians just say "there is no analytic solution" for short.

For example, consider ##\frac {df}{dx} = \frac{\sin x}{x}## (or ##\int \frac{\sin x}{x}\,dx##) This does not have an analytic solution in the elementary functions. Provably so, in fact. This particular differential equation (and the corresponding integral) come up very often, and it is fairly easy to evaluate numerically. So it's given a special name, the sine integral. The sine integral is one of many non-elementary special functions.

As Bill_K mentioned, given any ##f(x)##, one can always denote ##F(x) = \int_a^x f(t)\,dt## as a new special function and voila! problem solved. Well, not really. You have to have some way of evaluate that integral numerically, for one thing. For another, it needs to pop up in lots of different places to deserve that "special" label.

As to how you would go about proving that no analytic solution exists for some differential equation (remember what that means), you would need to invoke Liouville's theorem or use differential Galois theory.

This was a very helpful response. Thanks! Although I do have one clarifying question: what sorts of operations fall under the heading of "elementary operations"? Also, in what sorts of places -- other than, say, introductions to differential Galois theory -- do we encounter the equation [itex]x\ f'(x) = \sin x[/itex]?

 

[Bill_K]

Essentially, in electronic structure theory, any system with at most 2 electrons can be solved exactly.

This would come as a surprise to Bethe and Salpeter, authors of Quantum Mechanics of One- and Two-Electron Atoms.

Approximations have to be used to solve the wave equation for He-like atoms. It is convenient to use different approximations under different circumstances, depending on whether the quantum number n or ℓ is large or small, whether Z is large or small, whether an accurate energy eigenvalue or a relatively simple wavefunction is more important, and so on...

 

[williamsn]

My bad Bill, I meant 1 electron (Li is just much harder to solve than He due to exchange interactions). But, being a 1 electron system, H2+ is exactly numerically solvable. There is no closed form analytical solution, but it can be expressed as linear combination of hydrogenic orbitals giving bonding/antibonding interactions. It should look like a morse potential for the bonding combination and a decaying exponential for the anti bonding (as functions of internuclear distance).

 

[AxiomOfChoice]

But, being a 1 electron system, H2+ is exactly numerically solvable.

What does it mean to be exactly numerically solvable? Forgive me for being so pedantic, but it seems that people are saying a lot of conflicting things in this thread, and I'm wondering if it isn't just an issue of defining terms.

There is no closed form analytical solution, but it can be expressed as linear combination of hydrogenic orbitals giving bonding/antibonding interactions. It should look like a morse potential for the bonding combination and a decaying exponential for the anti bonding (as functions of internuclear distance).

Would you be kind enough to provide a source? If you can link to something I can get access to, that would be marvelous. Thanks!

 

[williamsn]

By exact, I mean that you don't need to introduce any approximations like variational theory or a self consistent field in order to solve the problem. For this problem, you will wind up with a few nasty integrals that do not have a closed form solution. An easy example of what I mean is a transcendental equation. For example, the solution for the energies of a particle bound in a finite square of width d is sqrt(E/(V-E)) = tan(d*sqrt(2m(V-E))). You would need to numerically solve for E, but your answer is exact up to the error inherent in your computation.

For H2+, Pauling/Wilson's Intro to Quantum does an excellent job (unfortunately, I think the original papers are in German). In Chapter 12 section 1, they treat the ion variationally, getting an approximate solution that was obtained numerically. In the next section, they show that the equation is actually separable in confocal elliptical coords and give the results of numerical solutions to the integrals that yield the energy eigenvalues as a function of internuclear distance, which are exact up to the accuracy of their numerical integration scheme.

 

[jonjacson]

Does someone have the mathematical demonstration step by step on the non solvability of that equation?