Since I found nothing during February relevant to my theory of planetary formation, I thought I should outline why I think we need an alternative. The following is a very condensed look at the giant planets, and my ebook (Planetary Formation and Biogenesis) has much more detail.
 
The standard theory for a giant is that solids come together by some unknown mechanism and form planetesimals, and these, through gravity, form larger bodies, and finally planets. It is usually assumed for giants that this takes less than a million years (My), then over a period of time that depends on the assumptions, these collide to form larger planets, until they reach about 10 times the mass of the Earth, then they start accreting gas as well. (Actually, they will accrete gas by the time they get to the size of Mars, but such early atmospheres contribute little to the mass. They then take about 10 – 15 My to get to a size where runaway gas accretion starts. So, what are the problems. I consider some of these to be:
(1) After 60 years, there is still no firm idea how the planetesimals form, therefore the distribution of them is simply an unverified assumption,
(2) Simulations agree that planetesimal collisions to reach Earth take about 30 My, so how does Neptune get there so fast, when matter density is much lower and velocities are much slower? Collision probability depends on the square of particle density, and initial particle density is proportional to r^-q, where q is usually taken as 1.5, although that too is an assumption, and it could be 2. If the average body contains n initial particles, particle density is now 1/n initial particle density.
(3) If material comes together by collision, to get things to go fast enough, relative velocities have to increase as particle size increases, so why do the bodies simply not smash to pieces, assuming they do form?
(4) The star LkCa 15 is approximately 2 My old, it is slightly smaller than the sun, and it apparently has a planet of nearly 6 times Jupiter's mass at about 16 A.U, or about three times as distant from the star as Jupiter.
 
In my opinion, (4) is critical. Accretion disks last between 1 – 10 My after primary accretion, so the LkCa 15 system is a very young one, so how did its gas giant get so big? Obviously, everything has to happen a lot faster than under standard theory. What are the possibilities? To start with, standard theory ignores chemistry, so what happens if we include it?
 
My concept is that the initial cores grow like snowballs. In the outer disk water and silicates condense to form amorphous particles that adsorb other gases (Icarus 63: 317-332) and retain them to past the melting point. As the particles fall inwards, the temperature rises, and at some point, occluded volatiles that have passed their melting point are emitted. If, however, the melting point is not reached, the volatile is retained, more or less as a solid, and fills the pores. Suppose two particles collide. If they are sufficiently below a melting point, they bounce off each other, but if the volatile can melt, the energy of collision is absorbed in melting it, in other words, kinetic energy is converted to heat and the collision is, for the moment, inelastic. Now, the liquid trapped in pores between the particles cannot escape, but it can merge, then when it cools, it solidifies, thus we have pressure-induced melt-welding of the particles. This is similar to how a snow-ball grows with pressure. If so, then we look at the ices, these are (separated into subsets that have similar melting properties) in order of decreasing melting points (temperatures in degrees K): {water (273)}, {methanol/ammonia/water eutectic and CO₂ (164-195)}, {CH₄ and Ar, (84-90)},  {CO and N₂ (63-68)}, and {neon (25)}.
 
There are, therefore, zones where ice can accrete into larger bodies, which depend on the temperatures in the disk. The surfaces of the disks usually have temperature proportional to r^-0.75, but the interior should retain heat better. If we put the index = -0.825, and assume Jupiter is the optimal place for a water-based core, then we predict the solar system as Saturn (water/ammonia/methanol) at 7.8 – 9.6 A.U. (actual, 9.5); Uranus (methane/argon) at 20-21.7 A.U. (actual, 19.7); Neptune (CO and N2) at 28 – 31 A.U. (actual 30) and possibly a planet based on neon at about 95 A.U. The satellites are based on the same compositions so we predict the Jovian system to be based only on water (the rest having volatilized); the Saturnian system to have ammonia and methanol (which can undergo chemistry to produce nitrogen and methane, which explains why Titan has an atmosphere and Ganymede does not); the Uranian system to be the slowest starter, because methane and argon are relatively minor components, but which will grow faster than Neptune once total accretion gets under way because matter density is greater, while Neptune will initially grow faster than Uranus, because nitrogen and carbon monoxide are common, but slower when gravity becomes the driving force. As far as I know, this is the only theory that requires Neptune to be bigger than Uranus to start with, and always to be denser. It also predicts that planets grow proportional to their cross-sectional area, because they grow initially in a flow of ice particles, all of which are continually renewed by the stream of gas heading starwards. By not involving collisions between equally sized objects, the rate of formation increases dramatically. Note it also predicts no life under-ice at Europa because the Europan sea will be deficient in both nitrogen and carbon. There are thin atmospheres around the major Jovian moons (Thinner than the gases in a light bulb!) but on Europa, there appear to be no nitrogenous species to 7 orders of magnitude less than the major species. What do you think? 

I was feeling remarkably happy when my thesis was written, because I felt I had made an important advance, and then, disaster! Maerker and Roberts published a paper (J.A.C.S. 1966, 88, 1742-1759) that asserted that the cyclopropyl ring also stabilized adjacent negative charge. If this were correct, the cyclopropane ring did conjugate with adjacent charge, and my polarization explanation, and my PhD thesis, were just plain wrong. The reason is, of course, that a polarization field will stabilize one charge, but must destabilize the other, because the force between like charge is repulsive. My first response was deep despair; my second was, perhaps I had better read this paper carefully.
 
There are three major complications. The first is that if the lack of stability is indicated through rearrangement, that only means something else is more stable. Thus a Grignard reagent made from cyclopropylmethyl chloride leads to a ring-opened rearranged "carbanion". As it happens, the cation made from cyclopropylmethyl chloride, or cyclopropylmethyl alcohol, or a tosylate, is also unstable and promptly rearranges. (This is the famous bicyclobutonium "non-classical" carbenium ion.) Rearrangements of the carbenium ion are inhibited by bulky substituents, but the stabilizing effect of the cyclopropyl ring is easily shown by considering the rates of formation of the ion, or the energy of the species by mass spectrometry. However, at the time neither of these techniques were available for the anion. The second is solvation. Carbanions tend to be generated in solvents such as ether or petrol, which almost forces ion pairing, whereas the carbenium ions tend to be made in acids more acidic than concentrated sulphuric acid, and hence have very high dielectric constants and strong solvating properties. Thus the cyclopropycarbinyl carbenium ion made in solution is never stabilized to anywhere near the extent as is found by mass spectrometry. The third is that the polarization field is not a simple field. Four orbitals move towards a substituent, but at the corner of the cyclopropane ring, there is weak positive polarization field, due to the movement of the three orbitals about that atom, which, being very close, over-ride the stronger effect of the more distant movement. Further, while the cyclopropyl anion receives some localized stabilization together with the expected destabilization, the associated cation in the ion pair is strongly stabilized, and overall, the "anion" appears to be slightly stabilized. This effect is most strongly seen in calcium carbide, which, of course cannot be stabilized by conjugation without violating the Exclusion Principle. Further, according to the polarization interpretation, the "bare anion" formed on a carbon atom adjacent to the cyclopropyl ring should be destabilized, but by less than half as much as the cation is stabilized (because the charge is more distant, and by applying the virial theorem). In solution, solvation becomes an issue, as does the location of the cation.
 
So what was the evidence Roberts found relating to the conjugative stabilization of the anion? Some of the evidence, in my opinion, falsified the conjugative explanation because the anion refused to form when it should have. Such failures included: treating with butyl lithium (which meant that the protons were less acidic than those of butane), refluxing for 46 hr with phenylpotassium in heptane, treating with pentylpotassium (which reacts smoothly with ethylbenzene), stirring at 80^o in heptane with potassium and sodium monoxide. My theory might be still alive!
 
However, evidence for conjugation was claimed when the phenylmethylcyclopropylcarbinyl anion formed with potassium as the counterion, but it rearranged to the corresponding allylcarbinyl anion with any tendency towards covalent character. Roberts argued (almost certainly correctly) that when something like lithium was the counterion, the lithium would get close to the anionic centre and partial covalent binding would occur. Potassium was big enough that the bulky substituents forced it away. However, it was here that we differed in our interpretation. Roberts claimed that the extra stability with potassium was due to the fact that the "pure anion" was formed, and the cyclopropyl ring provided conjugative stabilization. My interpretation was, the potassium formed the "pure anion", which permitted delocalization, which in turn permitted the anionic charge to be delocalized by the benzene ring, out of range of the cyclopropane ring. Any attempt at localizing the negative charge on the carbinyl carbon led to repulsive interactions from the cyclopropane ring, and hence rearrangement.
 
There were two problems with that explanation. The first is, is it convincing? You, the reader, can judge. The second was, there was no way to publish it. The problem with a scientific paper was, once something was asserted, that explanation stood. Falsification with independent evidence was required, not a simple assertion. Nevertheless there is another lesson here. Just because somebody asserts that something has happened, that does not make it so. Read the evidence carefully!

I apologize for the font-size in the last post. This was done from my laptop while on vacation, and for some reason, with different software, I got the wrong outcome. Sorry.

Let me now revisit my first paper for the last time. This paper was cursed by the referees: they accepted it without comment! Nevertheless, it is a paper that is an example of how not to publish a new concept, and it might be of help to young chemists to consider these errors, so as not to make them themselves. For those interested in the paper itself, see Tetrahedron 25 : 1349-1360 (1969).

The first error was that I put both the strain formulae and the means of explaining the dipole moments into one paper. I should have submitted two. The reason is, with two points to make that may be of value to different audiences, one of the points will be embedded in the paper, and the audience for that paper will never find it. I did not wish to be accused of not having enough material, or of unnecessarily trying to increase my publication count. That is silly thinking. There were a number of other papers out there relatively short on substance, and in any case, if both papers were submitted at the same time, let the referees/editor suggest merger.

The second error was that I did not state clearly, separate from the rest, what I had found. The basic point was that the changes of dipole moments were caused by enhanced electron density over normal alkanes, and the "compression" of the electron cloud present was proportional to the work done "compressing" the electron cloud. I relied on the fact that someone would follow what I had done by reading the paper, and by drawing the obvious conclusion from the table, but that is not good enough. If you want to persuade an audience, say it in the abstract, start the introduction with why you have to say it, say it in detail in the text, show it in the table, and say it again in the summary.

The title, "Ring strain and the negative pole" was, I thought, clever, because it related ring strain to the polarization field in the shortest space. Advice to young scientists: if you think of a clever title, try it out on some other scientists and see what they think. It may merely make them shake their heads!

Perhaps the biggest mistake was that I failed to explain fully what I was doing. The last thing I wanted to do was irritate "the real chemists" by labouring the obvious. What I did not realize was that what was obvious to me was not necessarily obvious to anyone else! I had taught myself physics, and I had bought the cheapest suitable books, which happened to be produced by Mir publications in Moscow. What I had not realized was that what was considered to be interesting additional information in Moscow was simply not present in the more expensive western textbooks. (I do not care what anybody says; for me, Landau and Lifshitz’ Mechanics is still by far the best book for learning about Lagrangian mechanics.) Accordingly, I assumed any physics in a textbook I could follow was well known, and I did not want to be accused of padding the paper with trivia. I thought that taking a divergence of a polarization field to get a pseudocharge (the pole) would be obvious. Obviously I should have said exactly what I was doing, but I was young and unguided. Worse, I did not want it rejected for being too long, which, of course, was all the more reason to submit two papers. I suppose also I was rather pleased with myself for finding a way to get an analytical solution that avoided insoluble partial differential equations.

So what happened? Apparently some chemists thought I was generating charge, which violates a variety of conservation laws. Of course I was not: I was merely trying to give an easier way of putting a numerical cause to the changes in the electric field that must arise when energy is stored in it. Since my undergraduate training in chemistry had given me no enlightenment to Maxwell’s electromagnetic theory, I should have realized it was wrong to assume everyone else would understand.
 
So, I hope young chemists might find something helpful from this. If nothing else, try to get someone else to read your paper and tell you what you said. That way, you will know whether it is comprehensible.

As outlined in a previous post, my PhD  results meant that I had to find a way to account for the strange properties of the cyclopropane ring without invoking conjugation. What I came up with was to account for them through a polarization field that was generated through the work done moving orbitals towards the centre of the strained ring. Thanks to the dimensional equivalence of a polarization field and a displacement field, and thanks to Maxwell’s electromagnetic theory, I could represent the movement of charge (the real cause of  the polarization field) in terms of the addition of a pseudocharge to orbitals that had not moved (equivalent to a change of displacement field). I could do that similarly with increased charge density, and the reason for doing this was that it permitted a known solution to the partial differential equations, with only one constant required to be set. I used the electric moment of cyclopropyl chloride to set that constant, and hence the pseudocharge. Now, the question was, would the same pseudocharge properly account for the stabilization of adjacent positive charge?

 

The cyclopropylcarbinyl cation was known to be stabilized compared with corresponding alkyl cations through studies of solvolysis of tosylates, etc, but these studies were of lesser value because there was the problem of solvation energies. There was also an issue that the cyclopropylcarbinyl cation is also unstable and promptly rearranges. (This is the famous bicyclobutonium "non-classical" carbenium ion.) Substitution can stabilize it, but that adds complications. Then, luck! Well, sort of. The stability of the cyclopropylcarbinyl cation was published in a PhD thesis, and reported in some books on mass spectrometry, but for some reason it never seemed to have made its way into a paper. Someone else was having trouble getting supervisors to publish! 

 

However, I had a reported value for the gas phase cation and I could use my pseudocharge to calculate the energy of the interaction of such a polarization field with the cationic centre. Obviously, this would be a fairly crude calculation, and there would be a number of minor effects overlooked, but I needed to know whether I was even in the right ball park. To my surprise, the agreement with observation was very good.

 

Of course, there was an easier way of doing this calculation. If the strain energy arose through charge in orbitals moving closer together and thus raising repulsive energies, and if four of the six moved closer to the substituent, then the cationic charge would neutralize the repulsive energy on these, hence the maximum energy of interaction with adjacent charge would be 2/3 the strain energy, plus any additional standard stabilizations that would occur in any carbenium ion. However, as far as I was concerned, the triumph was that the same value of my pseudocharge gave correct values for a completely different type of observation for which there was no obvious relationship other than the proposed theory. Being young and inexperienced in the ways of the world, I  thought I was making headway.

January was mainly a good month for my theory of planetary formation because two papers were published that strongly supported two of the proposals in my ebook that are not generally accepted. There was a third paper that is also highly relevant.
 
The first relates to where the volatiles of the rocky planets came from. There are two propositions that are usually debated: the volatiles were delivered by comets, or by chondrites. A review written by Halliday (Geochim. Cosmochim. Acta, 105, 146-171) showed the ratios of H/C/N are inconsistent with any ratio of these. All my review could do was to say that only minor miracles would permit some combination, but now even minor miracles are ruled out. My argument is that the volatiles were accreted chemically along with everything else, and the different ratios of volatiles on these planets arose because the materials from which they were made originated in different temperature zones.
 
The second paper was even more important, although I am not quite sure the authors themselves appreciated the full impact of what they discovered. One problem for planetary formation is, when the planets start forming, they form in a disk full of gas, so the question is, why do not the planets head towards the star? They are, after all, orbiting in the equivalent of a stellar "atmosphere". There are a number of papers written on this, but they generally lead to all planets ending up close to the star. What I proposed was that because the gas is falling towards the star and the azimuthal velocity is less than the Keplerian velocity at that distance, a pressure wave builds up in front of the body. That would normally cause the orbit to decay, but because there are solids in the gas, and because the gas has definite radial velocity, the planet starts spinning up, as if rolling around its orbit as material with an inwards radial velocity is accreted on the leading face. All the giants do this, although Uranus provides a problem. The body then drags gas down across its leading face, and when it becomes more gravitationally significant, it holds onto some and drags it around to its starwards side, where it strikes gas coming the other way (in its sub-Keplerian orbit). Two gas streams flowing in opposite directions cancel their velocities, in which case the gas will stream towards the star, gradually merging into the more general flow. The original angular momentum lost must be conserved, and only the orbiting body can take it up. By gaining angular momentum, it tends to move away from the star, thus offsetting the expected orbit decay.
 
The paper that excited me was due to Casassus et al. (Nature 493: 191-194). What they did was to observe the star HD 142457, which is still in the late accretionary stage, and they found two filaments of gas streaming inwards with a radial velocity greater that the azimuthal velocity by approximately 10% or greater, and further, the flows in these filaments were approximately equal to the rate of stellar accretion. It gives a surprisingly pleasant feeling to find what you predicted, as much out of desperation as anything, actually turns up.
 
There have been a variety of estimates of how much gas is present in the late stage of stellar accretion. This is important because our planetary system requires about 1% of stellar mass, and previous estimates varied by about two orders of magnitude, with only the higher end having sufficient material. In my ebook, I assumed an average, hence I required initial planetary accretion to commence earlier.  The third paper (Bergin et al. Nature 493: 644-646) provides a new means of estimating the mass in the disk by using the emissions from hydrogen deuteride, and from observations on TW Hydrae, which has a disk between 3-10 My old, the disk contains 5% of the stellar mass, which the authors state is easily enough to form planets. As it happens, this system has been reported to have a planet of about 10 Jupiter masses rather close to the star, the star continues to accrete disk gas (For TW Hydrae, approximately 1% of gas per My), and after a period, the great bulk of this disk is blown away by a stellar outburst. Further, as Casassus showed, planetary accretion must be highly inefficient if planets throw gas inwards instead of accreting it. Accordingly, I think that the odds favour my concept that planetary formation must have started earlier. Either way, though, this evidence strongly suggests that planets are more likely to form around stars than not, so the probability that we are alone in the Universe has probably just got a lot lower.

From the material in the two previous posts, I now had the concepts necessary to address the issue of cyclopropane interactions with substituents. I now had to select something to apply them. What I chose was the problem of why the dipole moment of cyclopropyl chloride was about 0.4 D less than an alkyl chloride. This was important for two reasons. The first was that the generally accepted viewpoint was that this was evidence for back-conjugation, indeed it was the only evidence for it. The second was that following classical electrostatic theory, increasing the charge density on the cyclopropane ring should repel electrons, and if my concept was correct, at first sight the dipole moment should increase. On the other hand, there was a serious reason why that classical thinking must be wrong: the dipole moments of methyl acetylene and propene, where the former was approximately twice the latter. How could that be? Conjugation did not seem to be correct here.
 
My answer invoked quantum theory, albeit a version a little at odds with the standard version. I proposed that the wave function can be factorized. (According to the State Vector formalism, it cannot!) The key is that the stationary state is determined by quantized action, which requires that the frequencies in the bond zone cannot lead to destructive interference with components outside the overlap zone, and recall, p orbitals have two lobes, only one of which overlaps. If so, when the electron density increases in the cyclopropane ring, the electron density in bonds to substituents must correspondingly increase close to the cyclopropyl carbon atom. The "back donation" was not from lone pairs, but was simply movement of charge distribution in the bond from the substituent. Interestingly, nuclear quadrapole coupling parameters indicate that there is a small but axially symmetric movement of charge towards the cyclopropane ring. Such was the power of the current paradigm, this was interpreted as indicating conjugation equally from both p orbitals. That, of course, violates all other theory of delocalization of wave functions. (Incidentally, in all reviews, textbooks, etc, this difficulty is avoided by omitting all reference to these nuclear quadrapole parameter data. We cannot have observation getting in the road of a good theory!) Anyway, as far as I was concerned, I had worked out an answer to the key problem of the dipole moment of cyclopropyl chloride.
 
The problem now was to put numbers to it. Returning to my argument that the increase in charge density due to strain is mathematically equivalent (at least in terms of the equations I intended to use) as adding a pseudocharge to the original framework, I could use cyclopropyl chloride to fix the value of that pseudocharge, (via a value for the minor radius of a torus on which the pseudocharge was placed) then apply that to a number of "strained" systems. The change in dipole moment should be equal (so I thought) to the change of dipole moment generated by adding the pseudocharge to the neutral ring. I got almost exact agreement for methyl acetylene (0.75 D) and propene (0.36 D) and close agreement for methyl cyclobutane (calc. 0.07 D, measured 0.05 D) although I overestimated the dipole moment of methyl cyclopropane. Nevertheless, I felt I had achieved something. I had explained why the lower dipole moment of cyclopropyl chloride did not necessarily indicate conjugation, which should have been self-evident from the dipole moment of methyl cyclopropane, and I had an estimate for the source of this polarization field.
 
More interesting, in my opinion, this requirement that action be quantized (a general requirement for quantum theory) and the requirement that all parts of the wave have a common frequency is one of the best ways of considering electronegativity. There is what I feel is a very important point here: in classical physics, increasing electron density in part of a molecule would tend to repel additional electrons. Because of the quantization conditions that fix the wavelengths of stationary waves, it attracts them, which is why fluorine is so electronegative. Thus in my interpretation, electronegativity is determined by the electron density about the atom, and in the bond, the dipole moment gives a measurement of the electronegativity. As you can see, that is not exactly a standard interpretation! You, the reader, understandably, will not be convinced, nor should you be. All I ask is, bear with me. In future posts you will see that this goes somewhat further than you might at first think.

My next problem was that I needed a means of estimating strain energy for molecules. For cyclopropane, I could have used observed values from heats of combustion, but I wanted something for general strained molecules. It may be of some interest to see how I arrived at what I did. Assume a standard carbon-carbon single bond. Now, put the rest of the molecule in place, and consider the bent bond model of Coulson and Moffitt, Phil. Mag. 1949, 40, 1-35.) As the extra parts of the molecule are put in place, the electrons in the chosen bond move outwards, approximately to some fraction of where the orbitals would intersect if all bonds are sp3. Now, as a first guess, I put the strain energy as being proportional to the displacement from the C – C bond axis, which is proportional to sine theta/2, theta being the total deformation of the bond angle from the tetrahedral angle. (With two bonds required to make an angle, the total deformation is divided evenly between the two orbitals. The energy is force times distance, so I started by assuming a constant force as deformation progressed.) This was really more a first guess, but I was hoping the difference DELTA between observed and calculated would help me guess the manner in which the force varied. What surprised me was that this almost worked, and it worked even better if I divided by [square root (bond distance)]. Also, if I used the bond energy scheme of Cox and Pilcher (Thermochemistry of Organic and Organometallic Compounds.  Academic Press: London, 1970) it also correctly calculated the "strain energy" of ethylene and acetylene! Two membered strained rings, and fused two membered rings! Since DELTA was < 10 kJ/mol for every molecule for which I had data, and usually significantly better, I was then happy enough to use this as an empirical relationship for estimating the strain energy of a number of molecules for which no determination had been made. As an aside, this relationship gives a very large strain energy for tetrahedrane, greater than that of the strength of a carbon-carbon bond. Of course that does not mean that tetrahedrane cannot be made, because simply breaking a bond leaves the great majority of the strain still there.
 
There is clear evidence this had little effect on the scientific community. In 1984, Dewar (JACS 106, 669-82) produced an argument that, since bond bending was simple harmonic, the strain energy would be proportional to the square of theta/2, or maybe theta, which gave an enormous value of DELTA. However, molecular orbital theory showed that this energy was greatly reduced by something called sigma conjugation, and sigma conjugation exactly offset DELTA. Then, in 1985, Cremer and Cracka (JACS 107, 3800-3810, 3811-2819) announced that Dewar had the wrong force constant, and his enormous strain energy should be reduced by approximately 100 kJ/mol, leaving only a huge DELTA. But not to worry! Revised molecular orbital calculations showed that there was sigma conjugation that exactly offset this new DELTA. Two computations, using what purported to be the same methodology, got exact agreement with observation, despite the key term differing by 100 kJ/mol. How could that be? Of course, there was no mention of my work, which argued that the whole argument was spurious because there is a very big difference between the square of an angle and its sine. If I were correct, there is no huge discrepancy to explain, and no sigma conjugation.
 
Of course, when I wrote my paper, there was no thought of sigma conjugation. But the question I now have to ask myself is, should I have put this strain formula in a separate paper? On the plus side is the argument that a paper should really make only one point, and ideally the whole point of the paper can be summarized by a single statement. This makes it easier to find, particularly then when "finding" was done by reading journal contents pages, and later through Chemical Abstracts. On the negative side, and what swayed me at the time, was the thought that a complete argument should be in one place. There was also the worry that the strain relationship alone may not have been sufficient to get into a reasonable journal.  Whatever the validity of either argument, the fact of the matter is, I put the strain relationship into the middle of my first paper, and I doubt many people even know about it. 

I received my PhD, but I never heard from supervisor again. He wrote up a paper (published 1969) on the amine dissociation constants and rate constants, which also reported the synthesis and properties of some new compounds, but with no amines with mesomeric withdrawing ability, he ignored the issue of conjugation. As far as I am aware, he ignored the acidities in toluene, which was really his only contribution, and which gave a critical answer but one that conflicted with the emerging consensus. Make of that what you will.
 
In the meantime, I was determined to write up my theory, which had to show how certain effects could occur without cyclopropane having delocalized electrons. The two main observations to explain were the reduced dipole moment of cyclopropyl chloride, and the stabilization of adjacent positive charge. How to go about it? The first objective was to show qualitatively how these effects could be generated.
 
If I take an electron at a distance x from a proton and move it to y, where is the energy stored? In my interpretation of Maxwell's electromagnetic theory, it is stored in the electric field. Accordingly, the stabilization of positive charge adjacent to a cyclopropane ring compared with charge adjacent to a standard aliphatic hydrocarbon fragment can arise simply from the charged site receiving a stronger negative electric field. My qualitative argument to get such a field involved the strain forcing the charge in four of the orbitals around the distal atoms to move closer to the source of charge, while there was little effect from the two geminal orbitals, because their motion was more rotatory. (That may not have been the easiest way of looking at it.  If strain in the cyclopropane ring arises because of the greater electron repulsion through the orbitals being moved closer together, then adjacent positive charge would overturn that repulsion for four of the orbital lobes.)
 
The problem now was to put numbers to the cause, and this is where I had what I thought was an inspiration. Suppose you were beside a wall, and could measure electric fields, and such a field corresponding to D_o was coming from the other side of the wall. Since cyclopropyl is electrically neutral overall and has no electric moment, D_o = 0. Now, suppose you experience an increase in field. This can be explained two ways. The first is that charge q has been added, in which case the displacement field increases from D_o to D₁. In the second case, the original charge has moved, and there are now two fields: the original displacement field D_o and a polarization field P, which are dimensionally equivalent. If charge is added to the original charge, at its point location div D₁ = q. But for the case of the charge having moved, since we measure an electric field we can also write,  div P = q’, and the situation is numerically equivalent to having added a pseudocharge q’. Of course, since their fields are equivalent, q' = q. 
 
Why do that? Because if we wish to calculate how far the charge moved, we must solve the Schrodinger equation, which cannot be done, but if we think in terms of adding a pseudocharge, there is a mathematical simplification. If the cyclopropane ring bonds are represented by a torus, we have an analytical solution to the otherwise impossible differential equations. The work done assembling the pseudocharge on that torus is proportional to the strain energy. There remains one unknown: the minor radius of the torus, but before addressing that there was also the issue of determining the strain energy. More next post.