| are happy with Bent bonds (pun intended), enjoy them! [So did Linus Pauling, while the MO-pioneers Friedrich Hund and Robert S. Mulliken, did not approve of them. Their canonical orbitals are nonlocal. Later, transformations by Boys or Edmiston-Rüdenberg led to localized orbitals, an operation leaving the total energy unchanged. These make double bonds with (proper orbital) "bananas". However, another localization scheme by Pipek-Mezey conserves the σ-π separation (and total energy). Recently, Björn O. Roos has shown, that at high precision the "double bond" concept looses its innocence (e.g. Int.J.Quant.Chem. 18(1980)175-189 first published 19 JUN 2009). But this all is for proper quantum chemical orbitals, not for 3D spherical charge clouds! It shows that there is no profit from fighting over mental images that keep the world invariant (see Frank Jensen, Introduction to Computational Chemistry, Wiley N.Y., 2nd ed. 2008, p.304-9.] |

A little thought will show you, that the tangent spherical bananas as charge clouds do not work in a quantitative Kimball model:^{+4} cores.^{+4}!In order to attack these problems beyond mincing words we are actually doing the calculations, beginning with a literal reconstruction of J.H. van't Hoff's and G.N.Lewis' approach. This is an assignment for the reader (to be submitted at next class!). You may use the info given in the main introduction and links, and the four program templates below. Here are the exercises:- We take two methane molecules, remove two H-atoms on both and then fuse the dangling bonds together on a common edge. For a start, the clouds' radii and all the distances are those of methane. We then compute the models energy and then let it relax under the conditions of thermodynamic and mechanical equilibrium. a) What is the outcome? b) Compare to ethene, computed with Perego's kimball.exe, see below, and c) computed with Gamess.
- If you are at it, why don't you solve the same type of problem for acetylene's triple bond? (Tip: the two tetrahedra share a common face - as you certainly have proposed by now - template3).
- next we do the case of a common vertex - ethane, where rotation of the two CH3 moieties by 60° leads to eclipsed and staggered conformations (and, hence, the internal rotation barrier as their energy difference),
- and extend the same type of calculations to polyethenes of, say, 140 C atoms, i.e. C
_{140}H_{282}(see at page end) - finally, we do diborane, where the bananas are protonated (template4).
template2 is a C2H4 computation with van't Hoff's/Lewis'/Bent's approach based on the methane results of template1. It is a Mathematica-4.0 program run that was saved as HTML. This is the runnable program. template3 is a Mathematica program for acetylene with triple bond as common face. template4 is a B2H6 computation with relaxed requirements for tangency, but still in the spirit of common edge tetrahedra. If you are not familiar with Mathematica, it is very easy to translate these notebooks into any programming language from BASIC, FORTRAN to C++. I can provide a Borland Pascal variety for the same templates (and others). Mail me, click below, if you want any of them. I'd like to motivate any reader of this Kimball site - and even more: any teacher, who exposes his students to a qualitative Kimball model and has never validated it - to try this adventure. If you get stuck and would really like to continue, I may be able to help.Here is the result of the second to last assignment. You may download the program alk194.zip, unzip and run it in a 32-bit Windows by clicking on alk194.bat. It computes staggered, saturated C194H390 in just 12 sec on a simple computer with one CPU and produces a table similar to this one (Comment).This stereo is from Perego's Kimball.exe. The input contained two C(p _{z}) orbitals sitting at the C atoms. During the optimization the spheres fused to a π-bond in the center of the molecule, overlaps exchange corrected. Perego's energy is A comparison with variations on Kimball ethene and diborane is here. 1) The π-bond consists of C(p _{z}) orbitals located above the cores. 2) Same as 1) but p_{z} orbitals shifted towards the molecule center. 3) p_{z} orbitals fused as in Perego's ethene model left. It is nearest to QC results: Etot=-78.0375 Eh.W.A.Goddard III and his coworkers have recently started a renaissance of the FSGO approach with their "Electron Force Field" method. For ethene a very interesting comparison to Kimball with practically identical dimensions results: Julius T. Su, Candidacy Report, California Institute of Technology, Feb. 28, 2003, left picture, and J.T.Su & W.A.Goddard III, JCP 131,244501,2009, Fig7, right picture. They also showed that the double banana bond is considerably less stable then the σ-π double bond. |

QC results: Ethene computed with Gaussian03™, model chemistry RHF/6-311++G(2d,p) opt followed by a Natural Bond Orbital analysis (NBO Version 3.1, E.D.Glendening, A.E.Reed, J.E.Carpenter, and F.Weinhold, QCPE Bull. 10(1990)58), and visualized with WebMO, 13.0.011p.- No bananas! Top: C-C π-bond, bottom: C-C σ-bond. Energy -78.059212 Eh. The Kimball model is similar, considering its simplifications. |