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?
[Physics] Theoretical treatment of Hydrogen bond
hydrogenmoleculesquantum mechanicsquantum-chemistry
Related Solutions
The divide is actually not between covalent and ionic, but rather a spectrum between localised and delocalised electrons. The history of all this is actually quite fascinating, and Phil Anderson in his book "More and Different" has a nice chapter on this. Essentially, around the time that people started doing quantum mechanics on molecules seriously, there were two schools of thinking which dominated.
On one side was Mott and more popularly, Hund and Pauli who thought of electrons as primarily attached to atoms and through electromagnetic interactions their motions/orbitals would be deformed and one gets molecules. This is the version usually taught in chemistry classes as with a few rules of thumb it is possible to qualitatively account for a vast range of behaviours.
On the other side was Slater with a dream of a machine which could simply compute the electronic structure by giving it the atoms and electrons. In this picture, the electrons are primarily thought of as delocalised over all the atoms, and through a rigorous procedure of perturbation theory one adds the effect of interactions between electrons and may achieve arbitrarily good precision.
The latter has the problem that the results are not intuitive --- there are no rules of thumb available and one is reduced to simply computing. The problem with the former is that to achieve high accuracy, the "rules of thumb" become exceedingly complex and are not really very easy to use or to compute with --- it lacks the simple regularity of the Slater dream machine. It is telling that essentially the latter has won, and nowadays it is routine to compute the electronic structure of quite large molecules (~1000 atoms) through brute-force (the technique is known as density functional theory, and there are commercial software available to do it).
In finite molecules one can actually show that in principle both approaches will work --- technically we speak of there being an adiabatic connection between the localised and delocalised states. The only practical difference is just how hard it is to carry out the calculations. However, in infinite molecules (e.g. solid crystals) this is not true, and there can be a proper phase transition between the two starting points. In that case, the localised approach corresponds to what is fancily called these days "strongly correlated systems" such as Mott insulators and magnetically ordered materials, and the delocalised approach are essentially metals (technical language: renormalises to be a Fermi liquid).
Nowadays there is a desire (from theoretical condensed matter physicists) to develop the localised approach again, as it may be possible to find some useful rules of thumb regarding magnetic materials, a prominent example of which are the high temperature superconductors.
To avoid re-treading old ground, this answer contains some previous literature that has been mentioned on this thread, as well as the surface layer obtainable via naive google searches:
- A. H. Wilson. The Ionised Hydrogen Molecule. Proc. Roy. Soc. Lond. Ser. A, Math. Phys. 118 no. 780, pp. 635-647 (1928).
- These lecture notes for CHEM-UA 127: Advanced General Chemistry I at NYU by M.E. Tuckerman. Contains the variable separation, and the claim that the problem is exactly solvable, without exhibiting that solution or providing any references.
- This CCL thread, linking to the following references:
- H. Eyring, G. Walter, and J. E. Kimball, Quantum Chemistry (Wiley, New York, 1946), pp. 201-203. Claims the problem is solvable and refers to Teller, Burrau, Hylleraas and Jaffe (below), all of which provide series solutions.
- E. Teller, Über das Waserstoffmolekülion, Z. Physik 61 no. 7-8, 458-480 (1930)
- G. Jaffé, Zur Theorie des Wasserstoffmolekülions, Z. Physik 87 no. 7-8, 535-544 (1934)
- E. A. Hylleraas, Über die Elektronenterme des Wasserstoffmoleküls, Z. Physik 71 no. 11-12, 739-763 (1931)
- Ø. Burrau, Berechnung des Energiewertes des Wasserstoffmolekel-Ions (H2+) im Normalzustand, Kgl. Danske Videnskab. Selskab. Mat.-Fys. 7, p. 1. (1927) (eprint)
- D. R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans. Roy. Soc. London 246, 215 (1953).
- H. Eyring, G. Walter, and J. E. Kimball, Quantum Chemistry (Wiley, New York, 1946), pp. 201-203. Claims the problem is solvable and refers to Teller, Burrau, Hylleraas and Jaffe (below), all of which provide series solutions.
From this list, the papers by Wilson, Teller, Jaffé, Hylleraas, Burrau and Bates contain derivations of the separation of variables as well a series solution for the resulting coupled equations, in which the quantization condition usually appears, if I understand correctly, as the requirement that the separation constant $\mu$ be a zero of a function defined by a continued fraction, as $$ f(\mu) = \mu+ \frac{ \frac{1\cdot2\lambda^2}{2\cdot3}}{1-\frac{\mu}{2\cdot 3} - \frac{ \frac{3\cdot4\lambda^2}{2\cdot3\cdot4\cdot5}}{1-\frac{\mu}{4\cdot 5} - \frac{ \frac{5\cdot6\lambda^2}{2\cdot3\cdot4\cdot5\cdot6\cdot7}}{1-\frac{\mu}{6\cdot 7} - \cdots } } } =0, $$ where $\lambda$ is essentially the energy eigenvalue.
I am extremely reluctant to call these series solutions as 'exact' or 'analytical', though of course this involves a personal judgement call. (As a contrast, I'm not that reluctant to call the Braak solution of the Rabi model an analytical solution, even though it shares many features with the ones in this reference list. To some extent, that's because it's more recent, so there hasn't been enough time to tell whether there's more connections to be made with those solutions, but intuitively they feel like they have more 'structure' around them.) However, maybe someone can come along with a review and simplified exposition of the series solutions, and make the case that the functions they define are as 'closed-form' as, say, the Bessel eigenfunctions of a cylindrical well?
Best Answer
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