"Bent" bonds, "Baeyer strain"


The notion of strain energy is one of the concepts which evade the exact definition just like, for example, aromaticity and antiaromaticity, and yet are of immense importance in understanding chemical reactivity. It is not an observable property and cannot be defined as an expectation value of a quantum mechanical operator. In other words, it cannot be determined in a unique way. Rather, it can be stated that a strained compound is less stable than a reference “strain– free” system due to some unfavorable intramolecular interactions. It straightforwardly follows that strained species are more reactive as educts and less abundant as products. As early as 1885 it was realized by Adolf von Baeyer, professor of organic chemistry at the University of München, that small three– and four–membered ring compounds should be angularly strained, because their bond angles are smaller than ideal tetrahedral ones [1]. Since then, the angular or Baeyer strain was a subject matter of numerous studies [2] being a topic of all textbooks on chemistry. One way in estimating the strain energy in cycloalkanes, for example, would be to take cyclohexane as a strainless gauge molecule and to measure its heats of formation ΔHf. It turns out that ΔHf per CH2 group is -4.92 kcal/mol. Hence the expected ΔHf in cyclopropane is -14.75 kcal/mol. Since the measured ΔHf value is 12.73 kcal/mol, one derives the angular strain of 27.5 kcal/mol. Analogously, the obtained strain energy in cyclobutane is 26.5 kcal/mol. This approach has two drawbacks: (1) cyclohexane is not a completely strain–free molecule and (2) the strain energy in cyclo-propane and cyclo-butane cannot be reduced to angular bending only, as one might wish (vide infra).
quoted from the Introduction of
D. Barić & Z.B.Maksić, Theor.Chim.Acc (2005) 114:222-228:
"On the origin of Baeyer strain in molecules"


What is a Bond ?


"Bonds" are drawn as dashes, "valence strokes", or "sticks" between symbols of nuclei (or "atoms"). They are a useful and timeproven abstract concept! What are they?
Here we go:
In molecules and extended structures there is an electromagnetic field which generates attracting and repelling electrostatic forces. These keep positively charged nuclei near a fixed position within a negative electronic charge distribution. All electrons attract each nucleus, all the other nuclei of the molecule repel it. The equilibrium between these forces creates the structure which we describe by "bonds". The electrons are in continuous motion owing to their zeropoint kinetic energy (Tutorials) which prevents a collapse of the opposite charges. We cannot enumerate electrons, hence we never know "which one" is where. However, we are able to measure where they are most often, rarely, or almost never, in a statistical sense. This is shown in a contour plot of a spatial electron density distribution with "imbedded" nuclei as in the experimentally determined graph below. The negative charge distribution extends to infinity but has a rapidly diminishing density if one looks a bit over the "rim" of a molecule. W. Pauli's exclusion principle imposes constraints on the electron distribution and thus on the molecular structure. In order to understand it we try to approximately compute its complex shape by the tools of Quantum Chemistry or less detailed e.g. by Kimball's free clouds, representing an average measure for it. It satisfies the electrostatic equilibrium as well as W. Pauli's exclusion principle and the virial theorem:  In a stationary molecule the kinetic energy Ekin of its electrons and nuclei is numerically exactly half of the negative potential energy Epot. This is summed over the Coulomb attraction between electrons and nuclei, and over the repulsions between nuclei and between electrons. Hence, the total energy Etot = Ekin + Epot is < 0 and has the same numerical value but opposite sign as Ekin. The energy minimum Etot must at least be deep enough to carry the small zeropoint kinetic energy of the nuclei to render a molecule stable, i.e. "bonded" and, hence, da capo.

[Read the first chapter of Chr. Klixbüll Jørgensen, Orbitals in Atoms and Molecules, London 1962)!]

This is about all we can say in general about chemical "bonds" on the microscopic scale. We assume "bond", if the measured distance between two nuclei in the structure discussed above is within a margin defined by experience! Hence, the answer to "What is a chemical bond?" is simply: It is a convention!
fig4b1.gifThis is an electron density map computed from a measured X-ray diffraction pattern of a small molecular crystal. In the plane shown, there are 8 peaks with a truncated high density, probably the signatures of known nuclei. Connecting those with least distances suggests a 5-ring with three substituents. Two nuclei of the ring, bottom and middle right, apparently do not have substituents. They probably are not exactly in the plane visualized, perhaps >CH2 ring members (small density cusp around protons). Of course, the X-ray scientists know the constitution of the molecule! This allows to assign nuclei to the peaks and suggests "bonds".

Before anyone understood the content of the summary above, the idea of a "bond" had been formed from many macroscopic observations which are discussed in any chemistry text, or even in Wikipedia. We do not repeat it here but want to address some aspects of it.

Many organic molecules exhibit structures which have been interpreted as carbon rings. The smallest of them contains 3 C nuclei, cyclo-propane, with angle CCC of 60°. In the pedagogical literature there are weird stories about the "construction" of C-C bonds in such a system, so much smaller than the "correct" CCC angle of 109.47° with normal propane H3C-CH2-CH3 (early summary of detailed bonding ideas in c-propane: A. de Meijere, Angew.Chem. 18,809-826(1979)). Let us look at this problem going from c-propane up to c-hexane, where the tetrahedral angle is fully developed.

Small carbon rings: C3 to C6

HTML version
  • Cyclo-propane
  • Cyclo-butane
  • 1,3-dimethylidene-c-butane
  • Cyclo-pentane
  • Cyclo-hexane
  • Wolfram Notebook
  • Cyclo-propane
  • Cyclo-butane
  • 1,3-dimethylidene-c-butane
  • Cyclo-pentane
  • Cyclo-hexane


  • In recent times, many carbon 3D structures have been synthesized, from tetrahedrane to vertex hydrides of other Platonic solids and more. We test tetrahedrane and derivatives. Most of these need "bent bonds" to reduce "strain", as above for 2D rings. We use the variable bf again which equals 1 for C-C bonding clouds on the bond direction. bf>1 moves the clouds out - bent bonds, bf<1 shifts them towards the center of the 3D structure.

    3D carbon cages

    HTML version
  • Tetrahedrane
  • Tetramethyl-tetrahedrane
  • Tetra-t-butyltetrahedrane


  • Wolfram Notebook
  • Tetrahedrane
  • Tetramethyl-tetrahedrane
  • Tetra-t-butyltetrahedrane
  • Cubane
  • Dodecahedrane