Electron microscopes cannot clearly depict the exact shape and structure of atoms and molecules, even though it does show a vague, cloudy image. In my AP chemistry class, I learned that the bond angle of some molecules is 109.5 degrees. How is this bond angle determined so precisely, if the bonds cannot be accurately observed through a microscope?
Molecules – How Are Bond Angles Determined?
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I wish there was an easy answer, but this is actually somewhat complicated, and to some extent is more art than science. There are several simple models that are used to predict molecular geometry, one of the most common is the VESPR model. Based on this model, one can begin to perform calculations of energy associated with different vibrational modes of the molecule. It is the comparison of calculated vibrational modes using the non-relativistic Schrodinger equation to those values observed in spectroscopic data that verify that the model is correct. It turns out rather surprisingly that many simple molecules can have a structure determined by simple models such as VESPR. However, as the molecules become increasingly complex, numerical simulation techniques are required, especially in the cases of complex proteins.
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
Best Answer
The positions of atoms respective to each other in a crystal lattice (solid) can be determined by X-ray crystallography. From these positions bond lengths and bond angles can also be calculated accurately.
Probably the most memorable case of solving the geometrical structure of a molecule was Franklin and Gosling's X-ray crystallography of DNA, information later used by Watson and Crick to solve the mystery of DNA's structure.
For many simple (binary) compounds molecular shapes and bond angles can also be determined theoretically (see link).