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Content: Chemical Structure, 2nd part: “Must a Molecule have a Shape?”, R.G. Woolley, J.Am.Chem.Soc. 1978, 100(4),1073
How does Kimball’s model deal with the “classical” notion of molecular structure?
ES 18 August 2017/30 Dec 2016//24 Mar 2015//2002//1982

“Must a Molecule have a Shape ?”

This picture of an anharmonic oscillator can be found at www.chemsoft.ch/chemed/linbox0.htm, Potential 5, together with the Pascal program, p.8-9 to calculate the quantum mechanical eigenvalues and eigenfunctions = wavefunctions of the anharmonic oscillator, and several other potentials, which you may click.>

This picture shows the nuclear wavefunction belonging to the 8th eigenvalue of the anharmonic oscillator above.

What is the size of the zero point vibrational energy, e.g. in methane ? 23.9 kcal/mol = 100 kJ/mol. If we look at a small protein, Crambin with 46 amino acid residues, this sums up to about kcal/mol. If this energy was allowed to percolate through the molecule it could easily break some 100 C-C or C-H bonds! Hence, no “shape” would be stable. Fortunately, zero point energy is localised to a high degree and unavailable for transport elsewhere (this article describes a patent(!) for making use of zero point energy, getting loose!).

You see now Woolley’s argument: If all nuclei in molecules and solids jiggle around some percents of their distances to neighbors, how could we describe structure or talk about shape? Experimentally, there is no difficulty. Look at a recent X-ray crystallographically determined skeleton of a molecule. There is a “thermal ellipsoid” at the points of highest electron density.

Stereo from: OR TEP, Carroll K. Johnson, ORNL-3794,1965

 The centers of the thermal ellipsoids define the position of nuclei in the measured “shape”, the “Re” (actually Ro, see below) points. The ellipsoids become smaller, the lower the temperature. Near 0 Kelvin, perhaps at 4 K in liquid He, the ellipsoid shrinks to the extent of the average zero point motion of every nucleus. Very accurate X-ray structures are, therefore, measured near 0 K.- BTW, the electronic kinetic energy, e.g. in a Kimball free cloud, is practically not dependent on temperature. The electrons move so fast around that several 100 to some 1000 K do not measurably change their kinetic energy! The zero point energy makes them already very hot, several 104 K, if it was thermal energy.It is important to note, that I have only picked a small part of Woolleys' paper. There are more fundamental topics, e.g. connected to vibronic coupling and radiationless relaxation of excited states, which are beyond this simple tutorial. Another field are "floppy molecules", i.e. non rigid molecular systems which do not have a "permanent" structure even at low or ambient temperatures. Examples are Li3, CH5+, PF5, C5H10 and a vast number of others. A basic non rigid system is NH3, where the "umbrella vibration" happens at ambient temperature, preventing the separation of enantiomers for a substituted NABC, unless the ligands are so unwieldy that a large energy barrier locks the molecule in one of its chiral species. We might go into this in a separate tutorial.For the Kimball model and for most sophisticated quantum chemical computations we need a simplification: This is a fixed nuclear skeleton, characterized by the distances of those “Re” points, explained above. It defines what we mean by the notion “chemical structure” or “shape” and obtains the attribute Re-structure. It is connected to the fundamental Born-Oppenheimer approximation which postulates separation of electronic and nuclear motion. Hence, we learn that the chemists molecular “shape” is just an approximation. We handle it in macroscopic models on a screen or as Dreiding and plastic models in real life and can successfully interpret chemical experiments. Nature has done us a big favor by making electrons so tiny, or nuclei so heavy, in comparison! Otherwise there would be no molecular shape and we could not philosophize about it, because we would not exist (this is no endorsement of the "Anthropic principle", however). I’ve written Re sometimes as “Re”, above. There are other structural measures than the bond length: Angles between bond directions, involving three nuclei, and dihedral angles ϑ, the angles between two planes spanned by  four nuclei, also, a triangle and a 4th nucleus not in that plane (or in that plane, then ϑ =180°, otherwise different from 180°). We discuss ammonia again, where a ∠HNH = 106.67° has been given and characterized as ∠e. NH3 has a symmetric vibration where the H nuclei just breathe in the way of inflating/deflating an equilateral triangle (and the N nucleus vibrates slightly perpendicular to it, the soft beginning of the umbrella vibration!). This has an equilibrium size and an ∠HNH with it. That is the ∠e angle, where “e” signifies equilibrium, as above. There exists an ∠o angle, similar to Ro for bondlength. These measures belong to the structure in the first vibrational level with zero point energy only, hence named “0” by the spectroscopists. Because the anharmonic, real oscillator, has an asymmetrical potential curve as in the picture above, the Ro value is a bit larger than the Re value. You’ll find e.g. for NH3 an ∠oHNH of 107.78°, 1.1° larger than the ∠e value.- I mention all these details because chemistry texts seldom give complete definitions for the data they present in tables. Check ammonia!I close this tutorial with emphasizing, that the Re(∠e, ϑe)-structures, we deal about in theory, lectures and text books, are NOT observable! The Ro(∠o,ϑo)-structures can be measured with spectroscopic or X-ray crystallographic tools. The Re structure pertains to the minimum energy potential “surface” (also called “Born-Oppenheimer surface”) which quantum chemistry programs deliver, often without explicit reference to the Born-Oppenheimer approximation. It is computed from the measured or calculated points of the potential curve.-Just helping: For a diatomic molecule we have a potential curve, for a triatomic system this is a plane, for a multinuclear molecule, finally, a “hypersurface” of many dimensions, hard to imagine, but computable and often illustrated in the form of cuts, e.g. along a reaction path. Every text book has such cuts to discuss “transition states” and intermediates on an energy scale.

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