Quantitative electronic Lewis structure derived from nuclear coordinates of a molecule:
c-Propane


All computations are transparent. The run lasts < 1 sec on a i7-4790 CPU. (ES 16 June 2016/30 Oct 2016).

c-propane is a famous case because the CCC angle in the ring is only 60°, far from the usual tetrahedral angle of 109.47°.

It seems that Th. Förster in 1939 has proposed, for the first time, that the bonding electrons are not necessarily forced to concentrate on the straight line - the not existing bonding stick ! - connecting two nuclei. They might relax the “strain” by yielding to the outside of the edges of the C3 ring. By this token the electron-electron repulsion and the high requirement for exchange corrections (because of Pauli forbidden overlaps within the ring) are lessened. We try to do this with Kimball’s free clouds.

You can manipulate the locations of the C-C bonding clouds by changing the factor bf in the first paragraph: bf=1 means straight bonds, bf > 1 makes the clouds shift their centers away from the symmetry center while maintaining touching bonding and core clouds. bf < 1 pushes the clouds towards the center of the molecule with bf ~0 a total collapse of three clouds onto one heap. Follow the changes in kinetic and total energy of the molecule while increasing or decreasing bf. Try to understand what you are doing. It is not difficult and gives you insight into important interactions. Perhaps, the most important being, that the chemists bonding stick is even more esoteric than spherical electron clouds. “Bent bonds”, “banana bonds” and similar concepts reflect the macroscopic image of atoms holding together by sticks. You can feel (!)  the strain if you try to form c-propane with a Dreiding model. You might even break it! Try to translate these archetypal notions into the quantitative, microscopic, nuclear-electron picture of modern chemistry. But, what would happen to the way organic chemists discuss their science if they would discontinue to draw their molecular stick formulae? Hence, ‘bent bonds’ and friends as well as ‘strained bonds’ and ‘tension in rings’, will be with us and make heated scientific exchange for decades to come ... and some of this is fruitful for scientific progress even if the underlying concepts are ill defined and obsolete.

Strained ring is a beautiful shortcut for expressing the difference in reactivity of c-propane and c-hexane but should not be given in energy units! The derivative of such an energy suggests the existence of forces to move nuclei for relaxing strain. This is nonsense! Every stable, stationary molecule is free of any forces to move nuclei. Hence, there is no strain nor an associated energy! Look at the Hellmann-Feynman force analysis of c-propane at the end of this notebook. It shows vanishing forces on all C3H6 nuclei.

Here I take the coordinates from my optimized, converged, coupled cluster computation CCSD(T)/cc-pVTZ with nearly complete correlation energy using ORCA (Neese, F. (2012) The ORCA program system, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2, 73-78):



It turns out that this theoretically determined structure is almost identical to the experimental one. You can test this by changing the name of the structure file cproporca.txt to the experimental set of coordinates c-propane2.txt in the first paragraph of the Wolfram Notebook c-Propane_orca1.nb and run the calculation with it.

Note: There is presently no exchange correction for cloud overlap in this notebook. The predefined bf = 1.68 presents a molecule with only tiny overlap of the C-C clouds.
Hellmann-Feynman electrostatic analysis at end.

Input and Definitions


The structure is read in Å. We use atomic units, the universally applied system of theoretical chemistry and (micro) physics, see NIST. Length data are in Bohr : 1 a0 = 0.52917721 Å = 52.9177 pm; electric charges in ± electron charges, and energies in Hartrees : 1 Eh = 2 Rydberg = 627.5095 kcal/mol = 2625.50 kJ/mol.

Normal Input for a structure given as : Symbol, x, y, z

c-Propane_orca1_1.gif

c-Propane_orca1_2.png

c-Propane_orca1_3.png

c-Propane_orca1_4.png

c-Propane_orca1_5.png

c-Propane_orca1_6.png

Normal Input for a structure downloaded from Wolfram ChemData repository

c-Propane_orca1_7.png

Analyze the atomic constituents

c-Propane_orca1_8.png

c-Propane_orca1_9.png

c-Propane_orca1_10.png

c-Propane_orca1_11.png

c-Propane_orca1_12.png

c-Propane_orca1_13.png

Analyze Lewis structure

Compute Kimball radii from distance matrix, show core radii derived from CH4, NH3, H2O gauge molecules (cnofhydb.pas), (cnofhydb.ex_ to be renamed into runnable cnofhydb.exe after download), H excentricities, and number of σ bonds.

Distance Matrix :

c-Propane_orca1_14.gif

c-Propane_orca1_15.png

Nuclear repulsion

c-Propane_orca1_16.gif

c-Propane_orca1_17.png

c-Propane_orca1_18.gif

c-Propane_orca1_19.png

Select bonded pairs and their distances

c-Propane_orca1_20.png

c-Propane_orca1_21.png

Subtract proton eccentricities

c-Propane_orca1_22.png

c-Propane_orca1_23.png

c-Propane_orca1_24.gif

c-Propane_orca1_25.png

Subtract core radius

c-Propane_orca1_26.gif

c-Propane_orca1_27.png

Show radii determined

c-Propane_orca1_28.gif

c-Propane_orca1_29.png

Summary of Lewis properties

c-Propane_orca1_30.gif

c-Propane_orca1_31.png

c-Propane_orca1_32.png

c-Propane_orca1_33.png

Compute kinetic energy terms, bonding clouds, core clouds:

c-Propane_orca1_34.gif

c-Propane_orca1_35.png

c-Propane_orca1_36.gif

Partial kinetic energy for cores and σ clouds

c-Propane_orca1_37.png

c-Propane_orca1_38.png

c-Propane_orca1_39.gif

c-Propane_orca1_40.png

c-Propane_orca1_41.png

Determine connectivity matrix:

c-Propane_orca1_42.gif

c-Propane_orca1_43.png

σ Bonding clouds: Connected atom pair, radius of cloud

c-Propane_orca1_44.gif

c-Propane_orca1_45.png

c-Propane_orca1_46.png

c-Propane_orca1_47.png

c-Propane_orca1_48.png

Graphics:Cloud radii (bohr)

Plot molecule and its partial constituents

c-Propane_orca1_50.gif

c-Propane_orca1_51.png

Graphics:c-Propane: straight lines connecting cores,  bonding clouds yield

Graphics:Core skeleton, 'bent bonds'

Graphics:&sigma; skeleton, 'no sticks'

Graphics:C-C clouds on skeleton, 'no sticks!', tiny overlap

Graphics:H atoms, 'bent bonds'

For comparison we look at a quantum mechanical overlap density plot of c-Propanone (optimized HF/6-31G(d) computation) with superimposed Kimball free clouds (computed by Kimball.exe), forced on the bond directions, the black lines. We comment three points: 1) The density plot shows C-C bond density maxima outside the straight lines connecting the nuclei without any sort of “hybridization”; 2) The Kimball clouds, as pink circle projections, on the bond directions produce large, cyan colored, triple overlap lenses, violating Pauli’s principle; 3) correcting for that leaves restvolumes of Kimball clouds which contain the “bent” bond density maxima.-
In this notebook we have moved the bonding spheres away from the center, thus avoiding the Pauli forbidden region, with the maxima of the density plot ~ in the center of the Kimball cloud. In the cyan region, picture below, the density goes to zero towards the center, the last isodense has the same value as isodense Nr.1 on the outside. That falsifies many popular qualitative descriptions of c-Propan(on)e with a construction of hybridized orbitals producing a high density in the middle of the triangle.- [We don’t discuss the O=C bond of c-Propanone here]

c-Propane_orca1_57.gif

Prepare interaction matrices:

c-Propane_orca1_58.png

c-Propane_orca1_59.png

c-Propane_orca1_60.png

Compute energy components

Interactions for i not j

c-Propane_orca1_61.gif

Interactions for i equals j

c-Propane_orca1_62.png

Add components of Ne[10] cores; Politzerratio

c-Propane_orca1_63.png

c-Propane_orca1_64.png

Results (energies in [Eh] Hartree)

c-Propane_orca1_65.png

c-Propane_orca1_66.png

c-Propane_orca1_67.png

c-Propane_orca1_68.png

c-Propane_orca1_69.png

c-Propane_orca1_70.png

c-Propane_orca1_71.png

c-Propane_orca1_72.png

c-Propane_orca1_73.png

c-Propane_orca1_74.png

c-Propane_orca1_75.png

c-Propane_orca1_76.png

c-Propane_orca1_77.png

c-Propane_orca1_78.png

Hellmann-Feynman electrostatic analysis

c-Propane_orca1_79.gif

c-Propane_orca1_80.png

Graphics:Residual Hellmann-Feynman force ellipsoids (5 times magnified), core polarization neglected

Now include polarization of non centrosymmetric C-cores and compute electrostatic forces again:

c-Propane_orca1_84.png

Graphics:Residual Hellmann-Feynman force ellipsoids, core polarization included

Created with the Wolfram Language