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# Derrel Walters' 3-Body Conjecture

The principle that I layout below should be understandable to anyone who has taken freshman physics and linear algebra. With that realization, it is strange that no one - to my knowledge - ever discovered it in the past. In any case, it will change physics. Outside of a few of my colleagues, the Daily Paul is only the second audience to view this. So, enjoy if you enjoy science.

Consider the following simple experiment where a researcher is measuring the interaction energy of two like-charged particles.

where the interaction energy will be calculated as:

Take notice that the dot product will involve the angle between the measurement axis and the ij-axis. So, it is always practical to measure along the ij-axis to avoid unnecessary complexity.

So, the researcher performs the experiments and always gets consistent results. With only a few more measurements to go, there is confidence that this project will get wrapped up rather quickly.

However, a prankster pulls a fast one on the researcher by placing a third oppositely charged body under the counter:

The measurements are affected. The force he measures now is:

Being the astute fellow that he is, the researcher quickly discovers the treachery, and as a glass-is-half-full type of guy, he sets up the following apparatus and ventures to calculate the potential energy of the system:

The energy he calculates is:

and after the projections are performed and the resulting expression is simplified:

and in general:

where it has been demonstrated that the familiar potential energy expression, namely:

currently used for the calculation of systems with multiple interacting bodies cannot be measured, except for the special case of where two of the angles of the triangle formed by the interacting bodies can be approximated as right angles - ie at long distances - the conjecture.

Why is this important?

First of all, it can immediately be implemented into the classical physics of macroscopic systems - where for instance, a more accurate potential energy surface could be mapped for our solar system. Perhaps, it will allow more precise detection of extraterrestrial planets. Who knows what else.

However, I'm not an astrophysicist.

(WARNING: Technical details below.)

In quantum mechanics, there are mathematical devices called operators which are the measuring tools, so to speak, of the practitioners - the most important of which, or at least the most famous among them, is the Hamiltonian operator. This is the tool used to measure the total energy of quantum mechanical systems. This is what it looks like, currently (neglecting sources of external potential energy, such as magnetic fields):

where the operator, T, is the kinetic energy operator, and one sees that the Hamiltonian operator just acts to sum the internal energy components of the system.

However, the gedankenexperiment that we performed above demonstrates that, in this form, the Hamiltonian is only exact for treatment of two-body systems - such as a hydrogen atom. For the treatment of multiple bodies, the correction derived above must be implemented if one wishes to accurately calculate the energy that would be measured from experiment.

This should come as no surprise to people familiar with the practical application of Density Functional Theory (DFT). Within the orbital approximation, the mathematical machinery is identical to that used for the application of Hartree-Fock (HF) with the addition of the Exchange-Correlation functional (where the Daily Paul moderating code has taken the last name of a famous scientist as a naughty word).

What does the last sentence say when taken literally?

This is my take on what DFT has to say about HF. The HF approximation is accurate, but one needs to perform some additional operations on the density to get the true energy.

I propose that the above correction is the correlation energy - ie the energy that must be added to the HF energy to make it exact within its approximations (primarily, the mean-field approximation). Further, I propose that, for the sake of pedagogy, the operator be given the symbol - C - where it can be taken to mean Correlation.

With regards,

Derrel Walters

PS - When implemented, I suggest STO-3G will likely be a good basis set to begin with. In addition, when solved with spherical harmonics, I predict the nuclear charge will be invariant in the exponents - that is, the correction accounts for the nuclear shielding.

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### Does this mean

we get free energy