A Different Way of Applying Quantum Mechanics to Chemistry

Time to explain what I presented at the conference, and to start, there are interpretive issues to sort. The first issue is either there is a physical wave or there is not. While there is no definitive answer to this, I argue there is, because something has to go through both slits in the two slit experiment, and the evidence is the particle always goes through only one slit. That means something else should be there. 

I differ from the pilot wave theory in two ways. The first is mathematical. The wave is taken as complex because its phase is. (Here, complex means a value includes the square root of minus 1, and is sometimes called an imginary number for obvious reasons.) However, Euler, the mathematician that really invented complex numbers, showed that if the phase evolves, as waves always do by definition of an oscillation and as action does with time, there are two points that are real, and these are at the antinodes of the wave. That means the amplitudes of the quantal matter wave should have real values there. The second difference is if the particle is governed by a wave pulse, the two must travel at the same velocity. If so, and if the standard quantum equations apply, there is a further energy, equal in magnitude to the kinetic energy of the particle. This could be regarded as equivalent to Bohm’s quantum potential, except there is a clear difference in that there is a clear value of it. It is a hidden variable in that you cannot measure it, but then again, so is potential energy; you have never actually measured that.

This gives a rather unexpected simplification: the wave behaves exactly as a classical wave, in which the square of the amplitude, which is a real number, gives the energy of the particle. This is a big advance over the probabilistic interpetation because while it may be correct,  the standard wave theory would say that A^2 = 1 per particle, that is, the probability of one particle being somewhere is 1, which is not particularly informative. The difference now is that for something like the chemical bond, the probabilistic interpretation requires all values of ψ1 to interact with all values of ψ2; The guidance wave method merely needs the magnitude of the combined amplitude.  Further, the wave refers only to a particle’s motion, and not to a state (although the particle motion may define a state). This gives a remarkable simplification for the stationary state, and in particular, the chemical bond. The usual way of calculating this involves calculating the probabilities of where all the electrons will be, then further calculating the interactions between them, recalculating their positions with the new interactions, recalculating the new interactions, etc, and throughout this procedure, getting the positional probabilities requires double integrations because forces give accelerations, which means constants have to be assigned. This is often done, following Pople, by making the calculations fit similar molecules and transferring the constants, which to me is essentially an empirical assignment, albeit disguised with some fearsome mathematics.

The approach I introduced was to ignore the position of the electrons and concentrate on the waves. The waves add linearly, but you introduce two new interactions that effectively require two new wave components. They are restricted to being between the nuclei, the reason being that the force on nucleus 1, from atom 2 must be zero, otherwise it would accelerate and there would be no bond. The energy of an interaction is the energy added to the antinode. For a hydrogen molecule, the electrons are indistinguishable, so the energy of the two new interactions are equivalent to the energy of the two equivalent ones from the linear addition, therefore the bond energy of H2 is 1/3 the Rydberg energy. This is out by about 0.3%. We get better agreement using potential energy, and introduce electric field renormalisation to comply with Maxwell’s equations. Needless to say this did not get much excitement from the conference audience. You cannot dazzle with obscure mathematics doing this.

The first shock for the audience was when I introduced my Certainty Principle, which, roughly stated, is the action of a stationary state must be quantized, and further, because of the force relationship I introduced above, the action must be that of the sum of the participating atoms. Further, the periodic time of the initial electron interactions must be constant (because their waves add linearly, a basic wave property). The reason you get bonding from electron pairing is that the two electrons halve the periodic time on that zone, which permits twice the energy at constant action, or the addition of the two extras as shown for hydrogen. That also contracts the distance to the antinode, in which case the appropriate energy can only arise because the electron – electron repulsion compensates. This action relationship is also the first time a real cause of there being a bond radius that is constant across a number of bonds has been made. The repulsion energy which is such a problem if you consider it from the point of view of electron position is self-correcting if you consider the wave aspect. The waves cannot add linearly and maintain constant action unless the electrons provide exactly the correct compensation, and the proposition is the guidance waves guide them into doing that.

The next piece of “shock” comes with other atoms. The standard theory says their wave functions correspond to the excited states of hydrogen because they are the only solutions of the Schrödinger equation. However, if you do that, you find that the calculated values are too small. By caesium, the calculated energy is out by an order of magnitude. The standard answer – the electrons penetrate the space occupied by the inner electrons and experience stronger electric field. If you accept the guidance wave principle, that cannot be because the energy is determined at the major antinode. (If you accept Maxwell’s equations I argue it cannot be either, but that is too complicated to be put here.) So my answer is that the wave is actually a sum of component waves that have different nodal structures, and an expression for the so-called screening constant (which is anything but constant, and varies according to circumstances) is actually a function of quantum numbers, and I produced tables that shows the functions give good agreement for the various groups, and for excited states.

Now, the next shock. Standard theory has argued that the properties of heavy elements like gold have unusual properties because these “penetrations” lead to significant relativistic corrections. My guidance wave theory requires such relativistic effects to be very minor, and I provided  evidence that showed the properties of elements like francium, gold, thallium, and lead were as good or better fitting with my functions as any of the other elements.

What I did not tell them was that the tables of data I showed them had been published in a peer reviewed journal over thirty years before, but nobody took any notice. As someone said to me, when I mentioned this as an aside, “If it isn’t published in the Physical Review or J. Chem Phys., nobody reads it.” So much for publishing in academic journals.  As to why I skipped those two, I published that while I was starting my company, and $US 1,000 per page did not strike me as a good investment.So, if you have got this far, I appreciate your persistence. I wish you a very Merry Christmas and the best for 2020. I shall stop posting until a Thursday in mid January, as this is the summer holiday season here.