Tutorial 6             Kimball Tutorial 6a            Tutorial 7

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 ?”

I like to address the provocative statement of R.G.Woolley. For most chemists his question sounds absolutely absurd, since for them “shape” is the most chemical attribute of matter. How could e.g. the genetic code survive decades, or millenia of years, if it was not preserved in the “shape” of a stable double helix of complementary nucleotide strands (of course, it is recreated at every cell division, but the pattern=shape prevails)? How could chemists correctly synthesize molecules with several chiral centers by cleverly making use of tiny differences of reaction velocities, if the “shape” would not hold them from loosing the target into a statistical mix of dozens of enantiomers (did you ever read the synthesis of VitaminB12 with 9 chiral centers by Albert Eschenmoser and Robert B. Woodward, finished 1972, a classic)?

Ok. Woolley deals with the fact, that quantum chemistry (usually) handles only electrons by Schrödinger’s differential equation, whereas it assumes that the nuclei and their positions (= shape) can be treated “classically”. They are not variables but simple parameters on the way to solve the equation. A stable molecule is found, as described earlier, by changing the position parameters in small steps and solving that differential equation (of second order) for every step. The size and directions of the steps are governed by sophisticated minimization algorithms. An approximation is “converged”, if the energy change per step has fallen below a predefined limit. This procedure means, that the electronic and nuclear motions are treated separately from each other (called “adiabatic” separation); i.e. the nuclei are held fixed while the electron distribution is calculated for that shape. If this is not yet near the equilibrium, there are still forces active to move the nuclei. Very near the equilibrium the forces vanish and the energy has a minimum.-
For a long time, it seemed hopeless to try for the full solution of an electronic-nuclear wave equation, where both types of particle move around. Advanced methodology and especially the phenomenal power of modern computers has changed this. Gamess-US, the program mentioned at the beginning of tutorial5, now has a modul for solving approximately such a full problem for molecules not too large. This is important because modern spectroscopy has reached a timescale of sub-femtoseconds, <10-15 sec, for observing elementary steps of chemical reactions, where, naturally, electronic AND nuclear motions can no more be separated (“nonadiabatic”). Forming a product in a chemical reaction means motion of nuclei to a new molecular entity and building appropriate new electronic distributions on the fly. However, all this is far removed for our tutorial of a simple model but should be part of our perspective.

I told you earlier that in atoms, molecules and solid lattices nuclei have a “probability” cloud of positions which is for the proton in the H-atom about 2000 times smaller than the electron cloud, and for the same quantum mechanical cause. This is well known for the vibrations of nuclei “clamped” to a certain position in a molecule, the left-over mode of motion when a nucleus has lost its “freedom” by engaging in a bond (a dog on a leash can only waggle his tail). In the picture, below, we look at a diatomic molecule. The red curve is called the potential [energy] curve. It traces the energy of the bond as function of the bond distance. One nucleus is held fixed at the origin, left, the other moves away from it to the right. The equilibrium distance is at the lowest energy point, designated as Re. Far out to the right, the molecule dissociates into two atoms. Left of Re we push the nuclei together which produces a steep increase in energy and a strong repulsion - the bonding electron cloud is compressed and the nuclei repel each other electrostatically.-
If that’s clear now, we move to the most interesting part. If we try to measure the bond distance, we do not find the molecule uniquely at Re, although this marks the most stable point of the potential curve. One finds an uncertainty of bond length depicted approximately by the width of the first blue horizontal line from bottom, the zeroth level of the vibration. The nuclei vibrate against each other with an energy from the Re point to that level. This is the zero point kinetic energy of the vibration which does not vanish even if we freeze the molecule to almost zero Kelvin (hence the name!). It has been predicted by the Heisenberg uncertainty relation. Its existence is the experimental proof for the probability cloud of the nuclei, "clamped" in a molecular skeleton. If we heat the system away from 0 Kelvin, the oscillator gains frequency and amplitude: It goes up the ladder of the blue steps, and quantum mechanics allows only those discrete energy levels. The particular potential curve of the picture below carries 11 levels of a bonded molecule. Heating above the 11th level the molecule dissociates thermally. Instead of using heat we could excite a vibrational mode from the lowest level by infrared radiation of narrow wavelength, perhaps from a tunable laser, and populate one level after the other (this is a simplified picture! we neglect rotation). This is done in spectroscopy, from which we have learned all about this ladder system and, arguing backwards from experiment, the potential energy curve and the zero point energy! Naturally, this all is now well understood and a precise theory allows the accurate prediction of thousands of energy levels in more complex molecules (more than 10’000 have been observed for CH4 and are completely understood through the work of Martin Quack and coworkers, ETH Zürich, 2010).


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 K_tutorial6a_12.png 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.

OR TEP of Glucose
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.   

Created with the Wolfram Language