Tutorial 7        Kimball Tutorial 8      Tutorial 9    

Content: 1) s,p,d,.. functions;  2) Bonding clouds without nucleus;  3) Overlap of core- with valence-clouds; 4) π-clouds; 5) Introduction to the repository of “how to derive a quantitative Lewis structure with Kimball's electron clouds from an experimental molecular structure”

3d, 4d transition metal compounds;  Ionic radii, Electronegativity are all topics being soon cast into Tutorial form. Meanwhile you’ll find many discussions on this site: Electronegativity, 3d-metalcomplexes, 4d-metals, Ne-cores
ES 05 Sept 2017//2002//1982

1) s-, p-, d-functions

We have been using these functions in Tutorial7 without introducing them properly:
We copy from Tut1,2 Kimball’s transformation of the H-atom ground state:


where R is the expectation value for the distance of the electron from the proton, determined over its whole spatial extent from r=0 to r → ∞ . This equation is true for H[1s], and R=3/2 a0. For the p-, d- and higher spherical functions such an “average” radius is harder to imagine, because, as single components, e.g. px,py,pz, they do not show spherical symmetry. Only the whole group of the three p or five d functions has this symmetry. For this the mathematicians have determined an average radius in atomic units a0 (e.g. Margenau & Murphy, The Mathematics of Physics and Chemistry, Van Nostrand 1943, 1955, p.127):  


Since for a certain principle quantum number n, ℓ ≤ n-1, we have for n=1 ⟨r1s⟩ = R = 3/2 a0 as above. For n=2 one gets s and p functions with ℓ=0, ℓ=1 resp. Hence ⟨r2s⟩ = 6, ⟨r2p⟩ = 5 a0, and for n=3 and d functions, ℓ=2, ⟨r3d⟩ = 10.5 a0.
Except for the spherically symmetric ns functions these values cannot serve as Kimball’s radii. We compute for the He[1s2s] state ⟨r2s⟩ = 6, and for the lobes of He[2p] ⟨r2p⟩ = 4, which can easily be read off the diagram below. The diameter for a He[3d] lobe is a bit below 13. The exact number is 12.728 = 9√2 as derived and computed in E3d2p.

And here is the summary for the He excited states. Note the different scale for the projection of the 1s3d state:

2) Bonding clouds without nucleus

This is the third type of Kimball cloud, after the X-H and the lone pair cloud. Examples are the H3C●CH3 or C●OH clouds. Contrary to the lone pairs, we have sound experimental guidance, because we know the bond distances C-C, C-OH, and most other X-Y “single” bonds, see the extensive discussion in Li-Li . We get a bonding cloud diameter from the bond length minus the two adjacent core cloud radii. The latter are well known from the pertinent noble gas ions, e.g. Li(+),...,C(4+),..., Mg(2+),..., Cl(7+), or Br(7+). They are easily computed and vary only slightly from one molecule to the next. The reason is, that the electron density in the small cores is very high, owing to the charges of “heavy” nuclei, compressing the 2 electrons. Hence, the cores are only slightly perturbed by different bonding environments and their radii can safely be tabulated.
Here, you find a step-by-step computation of ethane

3) Overlap of core- with valence-clouds

For more complex molecules there is a simple procedure: Just draw a structural formula and join all the atomsymbols with one stroke = Lewis electron pair. Replace all atom symbols by their [He]- or [Ne]- core clouds and the bonding strokes by a bonding cloud. Then look at the points where a double or triple bond should be placed and follow the procedure under 4) of this Tutorial. Much of this routine has been automatized and built into the program used for computing real life molecules as can be seen in any example of the library.
The argument for setting up a Kimball structure with core clouds touching valence clouds is summarized here.

4) π-clouds

How do we model the double or triple bond? You get a glimpse into some controversy and a full story on how these concepts can be handled in ethene.

5) Quantitative Lewis model from an experimental structure

Some 200 organic molecules from about 4 to over 1000 atoms are presented on this site. In quantum chemical parlance these are all “single point” computations, meaning: The energy and other properties are for the structures given, without any optimization and without adding vibrational zeropoint energies - Re structures, interpreted by Kimball’s procedures. Hence, the accuracy of the nuclear coordinates taken from various sources, referenced in the computations, has not been questioned and is of unknown quality. Optimizing the thus derived molecular properties needs a good Hessian matrix of force constants. This has not been developed yet for the Kimball model.

Take e.g. a rather small molecule of 31 nuclei with several challenging features: Lorazepam, a benzodiazepine, often prescribed as sleeping pill (and named by a phantasy name, e.g. “Temesta™” by Pfizer). If you have a large screen you should now open the link, go through the series of steps described here and identify those on the Lorazepam computation. You can also printout either this short tutorial or the long computation and follow the arguments on screen and print.

1) Input: Two formats can be selected. If the molecule is available we choose the Wolfram ChemData repository whose sources are referenced. It usually also presents a 3D model, a conventional projected structural and a stoichiometric formula.- Just write the name of the molecule, you wish to compute, into logs="name of molecule" of any template molecule you find on the list. If Wolfram's ChemData repository knows that name, you are already done. Unless the molecule has peculier features, not yet provided for, the program runs smoothly to its end, overwriting the content of the template. If Wolfram does not know the name, you have to use the other input variant and provide logs="name of molecule" and a text file with the lines: Atomsymbol, x, y, z The coordinates are assumed to be in Å. If not, you have to provide the correct multiplier from your units to atomic units in the line "t=...". The location of the coordinate file is input under
pos=ReadList["Drive:\\ Directory \\ name.txt", Word, WordSeparators -> {" "}];
as shown on the Lorazepam template.

2) Analyze the “atomic” constituents, as a chemist would do it: Sorting the atom symbols into an array with C atoms first, H atoms last (this is not the usual sort of chemists, but very convenient for the computation!). Then selecting nuclear charge, core cloud charge, valence electrons and full Lewis shell of a “saturated” atom from a file “Spec”, you can view.

3) Analyze Lewis structure: Determine the distance matrix, compute the nuclear repulsion energy, select bonded nuclei with a distance criterion, matrix Dij; get the radii of single bond clouds X-Y by subtracting the core radii of X and Y from the bondlength and dividing the clouddiameter → Lij.

4) Determine Lewis’ properties: Number of σ bonds, additional π bonds required, number of lone pairs

5) Compute the zeropoint kinetic energy of core and bonding clouds

6) Determine connectivity matrix by localizing the bonded neighbors of every nucleus. With this in hand, you can now

7) localize positions of π clouds and put them into the molecular skeleton as “pz” clouds, radius from gauge molecules given

8) Localize position, size, orientation and number of (equivalent) lone pairs: This is the trickiest part of the whole program. Most of the algorithms developed is in several subroutines you can download and inspect.

9) Collect info on σ bonding clouds, determine locations of their centers between the adjacent cores, add the computed locations of π-bonds and lone pairs; then plot everything (stereo!) and tabulate number and categories, like σ, π-bonds, lone pairs, of what has been drawn for a check.

10) Now collect coordinates of cores, all type of bonding and lone pairs into one array, associate charges and determine their sum - just a control that nothing has been overlooked.

11) Determine Vee, the electron-electron repulsion energy, Vne, the electron-nuclear attraction energy, Vnn, check the nuclear repulsion energy, already determined with the original coordinates, and all contributions of electron-electron repulsion in the same cloud, and the attraction energy of protons, eccentric in their X-H cloud. Finally we have to add the kinetic energy of the π-clouds and lone pairs to that of the σ-skeleton, determined earlier, to get

12) the final results of energy components, the Virial and the Politzer ratio and a full Hellmann-Feynman force analysis.

The nice thing about all this: You obtain almost everything, a simple quantum chemical computation gets you, in just a few seconds! and: The mathematics to do this is transparent and shown in detail. Except for a few matrix operations it is just basic mathematics, no calculus, no integrals.

If you look at some of these molecular computations with Mathematica you are able to change everything and test your own ideas, perhaps improving what I have tried to show you or falsifying statements you may find erroneous.

Created with the Wolfram Language