# Is water really $H_2O$? On a comment concerning the quantum mechanical description of water made by Hilary Putnam

I'm not sure I would have phrased it exactly that way, but I think his statement is by-and-large defensible. The crux of the issue is that, in liquid water, there is no sharp line between intramolecular and intermolecular H-O bonding. The intramolecular H-O bond is closer than the intermolecular H-O hydrogen bond, but it's a difference of degree not kind, and in molecular simulations you'll frequently see smooth transitions where an H gradually detaches from one O while attaching to another O.

A neat demonstration of this fact is when you mix D2O with H2O; in a very short amount of time the liquid will be 50% HDO molecules!

So in the mess of liquid water, you can't neatly & perfectly split up all the atoms into H2O triplets. And if you try to define some criterion like "if an H+O are closer than distance X, I'll say they're part of the same molecule", you'll find that your criterion is kinda arbitrary, and more importantly you'll end up dividing the liquid into a mixture of H2O, OH, H3O, H3O2, H4O2, H5O2, etc. "molecules" at any given instant of time.

[One shouldn't take this too far though--at any instant, there are *plenty* of unambiguous tightly-bonded H2O triplets.]

(I don't have any good references off the top of my head, but I used to do research in the physical chemistry of water, and these are my vague memories from reading papers and attending technical lectures back then.)

I think it's tosh. A very small fraction of liquid $\rm H_2O$ will spontaneously decompose and recombine with very small amount of $\rm H_3O^+ $ (hydronium ion) and $\rm OH^-$ (hydroxyl) ions. Weak hydrogen bonds may form between neighboring water molecules, but that doesn't really count as $\rm H_4O_2$ or higher order molecules.

While I have a great deal of respect for the study of philosophy, I think they can go far amiss when they try to invoke technicalities of the sciences in their arguments. Feynman's experience when discussing the nature of "essential objects" in a philosophy seminar is relevant.

I am not sure I will call this an "answer" so much as a lengthy comment that nonetheless points to a plausible answer. I think I have an idea as to what is going on here, based on intuition I realized from an earlier discussion I had once on this forum.

The trick is that *what's taught in those chem books is in some strict sense a 'lie'* when it comes to things like this involving quantum mechanics. To understand what we mean, we should consider a rather simpler example, namely that of the question of *atomic orbitals* and atoms heavier than hydrogen. As we all know, the usual set of orbitals $1s$, $2s$, $2p$, etc. are all cute names for the energy eigenstates of hydrogen or a hydrogen-like atom, described by a solution to the Schrodinger equation with a Hamiltonian (in position basis)

$$\hat{H} = \frac{\hat{p}^2}{2m_e} - \frac{1}{4\pi \epsilon_0} \frac{e^2}{r^2}$$

where as usual $\hat{p} = -i\hbar \nabla$ is the momentum operator, $e$ is the charge on the electron and $m_e$ is the electron mass. The first term is the kinetic term, the second is the Coulomb's law term for the potential about the nucleus. The solutions to this Schrodinger equation are the orbitals, and are good enough 3-dimensional wave functions $\psi(x, y, z)$ (or perhaps since we used $r$, $\psi(r, \theta, \phi)$).

However, when you go up one electron to Helium (He), you get a wave function that involves *six* coordinates - 3 for each electron: $\psi(r_1, \theta_1, \phi_1, r_2, \theta_2, \phi_2)$, and moreover the Hamiltonian contains a cross potential term for the repulsive interaction between the two electrons. When one tries to solve this, one finds out that the random variables for the individual electron positions are not statistically independent, that is, each electron cannot be described on its own as simply having a wave function that does not depend on the position of the other electron: depending on where one is measured to be, the probability to measure the other will be different. (Another way to say this is there is a non-zero correlation. Technically you have to use a somewhat modified notion of "independence" as the electrons are indistinguishable fermions, than the ordinary notion in classical statistics you'd see outside quantum mechanics or in other fields of science beyond physics, where "independent" means an *antisymmetric combination*) This is *entanglement*, and it means that in the most honest description we cannot speak of separate orbitals at all but must think about the holistic 6-dimensional wave function for the atom. Now this doesn't mean that you can't talk of the "atom" at all as a physical entity - you can describe roughly a shape by taking an "electron localization function" that is the probability to find *ANY* electron in some volume of space, and this is the "shape of the atom" in some sense (it looks like a blob), but *on their own* the electrons cannot honestly be assigned to separate orbitals, in other words it makes in the strictest sense no sense to talk of "both electrons are in the $1s$ orbital" etc. because those are simply not solutions of the Schrodinger equation.

Nonetheless, because we cannot solve this *three-body problem* exactly and analytically, we try to *approximate* it using various fictitious constructs, and one such way to do that *IS* to try and use a product of the hydrogen orbitals, suitably adjusted with various types of correction, and this works to a reasonable extent, such that the more corrections and bells and whistles you add to it the more accurate you can get it. You could think of these as being other "models" of the atom and different "pictures" beyond the orbital picture, but in the "exact" case it's none of these, it's a statistically correlated electron blob which you have to describe using a 6-dimensional wave for the actual probabilities.

So I am commenting that perhaps the same is what applies for water as well. Given two interacting water molecules, you can write up a gigantic Hamiltonian with six nuclei and 20 electrons and myriad cross terms such that the wave function for the whole thing is a 26-particle, 78-dimensional monstrosity at any given time, with none of its component particles truly statistically independent, just another blob effectively. Some states or *approximate solutions* of this may "look like" for some suitable definition (e.g. like the ELF above) two "separate" water molecules (that is, $2\ \mathrm{H}_2 \mathrm{O}$), others may look like them more joined together (that is, $\mathrm{H}_4 \mathrm{O}_2$), and a more "real life" configuration could be better approximated by summing together these in a superposition, but even that may not necessarily be *the* most accurate setup. If a 3 body problem is hard, a 26-body one is insane, much less a roughly $10^{24}$-body one for a full cup of water.

(FWIW if you want to talk about the fullest quantum solution for a *hydrogen* atom you will want to include in the wave function a freely-varying position for the nucleus. If you do this the solutions to Schrodinger's equation are a product of the usual electron-orbital wave functions and a free-particle wave function for the nucleus, and the latter is what allows say, a hydrogen atom to diffract through a two-slit matter-wave setup. Good luck finding this in a textbook talking the solution of the hydrogen atom btw. The physical reasoning is relatively simple: there is a probability distribution on the nucleus's position. Once you've found the nucleus by one measurement though, then the probability of the electron will be obtained by an orbital centered around that position at which you saw it.)

As said though I don't know if this is exactly what is going on, but it seems to be right. The original discussion I had with orbitals was at - pay attention to Emilio Pisanty's answer:

What does an orbital mean in atoms with multiple electrons? What do the orbitals of Helium look like?

and it makes sense the original asker cannot find information because in truth it is hard to find information even on the case of orbitals by searching terms like "orbitals are fake" or "orbitals are a fiction", you get only a literal handful of results - which suggests it's not generally something most are aware of, or at least this specific phrase is not used to describe the situation.

As I said I don't know if nor do I expect the above to be exactly the "truth", I'm just putting this out more to suggest the general direction of what's going on. Basically you have the true solution which is none of these things, really, but you can approximate it to various better and better extents with more complex approximative models that look like at one level separate individual molecules, then another as superpositions of multi-molecule forms, then another as something even more complex, etc. and also likewise with the detailing in the individual atoms themselves. FWIW the same kind of thing is also done with classical mechanics as well, e.g. describing orbits in a complex planetary system. You can to a first order treat them as separate elliptical orbits, then if you want to add more detail you can add things like precession modeled as certain (simplified) kinds of planet-planet interactions, etc. The "truest" solution is they're none of this stuff, but to get closer to it you can add in various details - it's a general feature of how physics is done.