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I would like to understand how the Hydrogen bond can be described through the Schroedinger equation. I don't need numerical methods that one uses them to simulate it, rather I need its treatment from theoretical point of view that can show also the probability of that electron will go around first and second atom, I searched the internet but couldn't find any treatment that shows what I mentioned. Can anyone offer an explanation?

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The molecule is clearly not an integrable - analytically solvable - system, so you shouldn't expect compact exact formulae without any numerical calculations. One may still say lots of things about approximate ways to imagine the wave function, the relative spins of the two electrons, and so on. –  Luboš Motl Oct 27 '12 at 14:40
    
@Luboš: Yes I know that, but I need a treatment that looks to extract some "behavioral" properties not the quantitative simulation it self... –  TMS Oct 27 '12 at 14:55
    
Well, if you realize it will cover the qualitative properties only, try e.g. hyperphysics.phy-astr.gsu.edu/hbase/molecule/hmol.html –  Luboš Motl Oct 27 '12 at 15:07
    
Nice illustrations, thx, anyway I need a more detailed treatment... as I read, London long time ago made some simple assumptions that brought him to calculate by using Schrodinger equation the distance in between atoms in $H_2$ molecule and bond energy, such a thing will be very helpful. –  TMS Oct 27 '12 at 15:22
    
It's better to just ask a direct question, such as "How does one understand the hydrogen bond using the Schroedinger equation?" In most cases (this included) you shouldn't just ask for references, it limits the kind of answers people can give. I've edited your question to show what I mean, but feel free to make further edits if you would like. –  David Z Oct 28 '12 at 1:54

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First, Hydrogen bond is not the bond in a Hydrogen molecule. A hydrogen bond is another kind of bond.

Second, chemical bonding cannot be described by the Schrödinger equation alone because this equation only describes isolated systems and an atom in a molecule is anything except isolated!

The Hydrogen molecule is trivial, there are only two atoms and are identical; therefore, the bond must be, more or less, that abstract 'line' connecting both nuclei, but the Schrödinger formalisms says little more. Where does start one atom and finish the other? At what separation distance the bond is broken? What happens for more complex molecules as cyclohexane? You solve the Schrödinger equation for the whole molecule but you do not get any bond. Is Carbon 1 bonded to Carbon 2? is to Carbon 4? Where does finish a Carbon atom and starts a Hydrogen atom? The Schrödinger equation cannot answer anything of this.

The traditional quantum chemical approach starts from the classical chemical theory, which already gives the bonds (classical chemical theory already says you that Carbon 1 in cyclohexene is only bonded to Carbons 2 and 6), and then uses that chemical information to rewrite the solutions to the Schrödinger equation (e.g. using localized orbitals) to mimic chemical bonding theory. But this is all a mess because you need a classical theory to interpret/rewrite quantum solutions for the whole molecule; moreover, the orbitals are not observable in this approach and atoms are not even defined.

The modern quantum chemical approach starts from Schwinger generalization of quantum mechanics to open systems. And uses this formalism to rigorously (and elegantly) define atoms and their bonds. This theory is the theory of atoms in molecules or AIM theory developed by Bader and coworkers. An atom is defined as a proper quantum open system. Another advantage is that AIM works with electron densities, which can be obtained by other methods (including experimental measurements) instead of working with unobservable wavefunctions.

Using AIM theory you can predict, in an ab initio fashion, that Carbon 1 in cyclohexene is only bonded to Carbons 2 and 6 without requiring a previous knowledge of classical chemical theory. The theory also gives a complete characterization of the kind of bonds in terms of a set of topological indices, and also gives atomic properties. It can be considered a proper quantum chemical theory.

Recently, it has been showed that AIM theory is related to the Bohm 'potential'. Concretely, it has been shown that the Bohm 'potential' gives, essentially, the same topology, symmetry, and chemical reactivity than the Bader Laplacian for $\mathrm{H}_2\mathrm{O}$ and other molecules. For an explanation of this close relation between Bader and Bohm approaches check the section 8 of this work

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I know this question is old but for anyone who comes across this post: Bader's theory and all of Bohm's potential theories are controversial and definitely not widely accepted by the quantum chemistry community. Schrodinger's equation can certainly describe hydrogen bonding. The only problem is that very accurate methods are computationally expensive. You can take a look at this article pubs.acs.org/doi/abs/10.1021/jz101245s where (well accepted) coupled cluster and perturbation theories predict H-bonded water clusters. –  Goku Feb 17 '13 at 6:29
    
@Goku Coupled cluster and perturbation theories are only numerical methods for solving the Schrödinger equation. The numerical solutions are next interpreted or analyzed using some chemical theory (Bader or otherwise) that defines atoms and bonds. Bader theory is rather standard in the mainstream chemical literature with standard computational chemistry suites such as Gaussian performing Bader analysis automatically. –  juanrga Feb 17 '13 at 16:44
    
@Goku Similar thoughts about Bohm theory, which is today established as one of the common interpretations of quantum theory. –  juanrga Feb 17 '13 at 16:44
    
@juannrga Ok, to be fair, I admit that Bader's charge analysis is rather standard and used by serious researchers. However, the reason for this is that there is no single satisfactory definition for charge of an atom in a molecule (that is why Gaussian also does, for example, Mulliken populations). But, except for this aspect of Bader's work, the rest is mostly Bader's own understanding of physics and it is not widely accepted. Also, wavefunction methods (which Bader seemed to dislike so much) predict densities that are in agreement with experimental observations. –  Goku Feb 17 '13 at 22:56
    
And with respect to Bohm theory, the only place I have seen this applied to chemistry in Jan C. Boeyens' controversial book "New Theories for Chemistry". His theory leads to results which contradict experiment (e.g. a T-shaped BH$_3$ molecule) and other parts of the book deal with controversial ideas such as a quasi-numerological approach to nuclear chemistry and an alternative special relativity theory. Anyway, we shouldn't start a debate here, but the OP should be aware of the different points of view.... –  Goku Feb 17 '13 at 23:02

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