Theoretical treatment of Hydrogen bond? 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?
 A: 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
A: The following references contain relatively complete treatments of hydrogen-bonding between two oxygens -O-H...O- using differential equations based on the quantum harmonic oscillator: 


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*Self-Consistent Einstein Model and Theory of Anharmonic Surface Vibration. I, Progress of Theoretical Physics, Vol. 58, No. 3, September 1977 by Takeo MATSUBARA and Kenji KAMIYA. 

*ISOTOPE EFFECT OF HYDROGEN BOND VIBRATIONS IN A THREE-DIMENSIONAL MODEL, Theoretical and Experimental Chemistry, VoL 35, No. 6, 1999 by S. S. Rozhkov, E. A. Shadchin and S. P. Sirenko

*Dahl, J. P., & Springborg, M. (1988). The Morse oscillator in position space, momentum space, and phase space. Journal of Chemical Physics, 88(7), 4535-4547.

*PROTON TRANSFER AND COHERENT PHENOMENA IN MOLECULAR STRUCTURES WITH HYDROGEN BONDS, Advances in Chemical Physics, Volume 125 by V. V. KRASNOHOLOVETS, P. M. TOMCHUK, and S. P. LUKYANETS.


All evolve into a fair amount of numerical modelling, however, the underlying Schrodinger equation can be appreciated in all of these works.
P.S. If anyone knows of a good paper concerning a hydrogen bond within a Nemethy Helix please post citation.
