Are Molecular Orbital Hamiltonians actually black-box neural networks?

Regarding quantum physics ”The Copenhagen interpretation was recently summarized as “Shut up and calculate!” That’s blunt, but not completely unfair. It is, in fact, the right injunction

for most physicists most of the time.” say Rosenblum and Kuttner (2006).

A very simplified discussion on molecular orbitals excerpted from Levine (1977) follows. A Slater type orbital (for example, for a 1s orbital) has the form:

Molecular orbitals can be constructed with linear equations:

Following the Hellmann-Feynman theorem

it is possible then to calculate the molecular orbital energies. Levine(1977) cites Parr (1963) saying “We have at our reach precise descriptions of molecules”. That is, Parr's position is that advanced quantum chemistry provides a view of how molecules behave. However, we can see that the energy of an electron in an orbital is calculated by an operator which calculates a number of exponential functions f , whose addition multiplied by a set of parameters allows to calculate a set of functions φ, from which an integral is then calculated. That seems quite similar to a neural network of  neurons in the first row, n neurons in the second row, and one neuron in the third row. An example of a similar neural network applied to the calculation of chemical activities (figure 1) can be found in Nami and Deyhimi (2011).

This brings the question of whether the calculation of energies for molecular orbitals is really mechanistically based, or just a black box model. We go back to Rosenblum and Kuttner (page 119): "Here’s something to ponder: Suppose the cat was placed in the box and the atom sent into the mirror system eight hours before you looked. The system evolves unobserved during those eight hours. If you find the cat alive, since it has gone eight hours without eating, you find a hungry cat. If you find a dead cat, an examination by a veterinary forensic pathologist would determine the cat to have died eight hours ago. Your observation not only creates a current reality, it also creates the history appropriate to that reality. You might consider all this absurd. Precisely Schrödinger’s point! He concocted his cat story to argue that, taken to its logical conclusion, quantum theory, at least its Copenhagen interpretation, was absurd. Therefore, he claimed, it must not be accepted as a description of what’s really going on."

A perverse hypothesis is proposed. Let us assume that electromagnetic fields, as a consequence of the emissions of subatomic particles, combine in an interference pattern instead of leading to homogenenous potential barriers. Such pattern would create regions of maximum and minimum field intensity, and regions of intermediate field intensity in between. The juxtaposition of the individual fields is represented pictorially in Figure 2(left) by means of cross sections of concentric spheres centered on field generating particles. The illustration does not intend to be rigorous (force lines of a magnetic field have nothing to do with an spherical distribution around the originating particle), but to show the complex patterns resulting of the juxtaposition of individual particle-centered patterns. The arrows try to show how a particle would try to negotiate passage through the field through favorable field intensity regions, which can obviously be regarded as a 3-D maze. Figure 2(right) represents even more schematically the now obvious consequence of the hypothesis: random motion through a potential barrier may result in the particle taking “the wrong turns” and getting out the way it entered. Although not illustrated, a potential well’s effect to reject particle may be the consequence of less interference between particle fields at the boundary of said field making it more “impermeable”. The conclusion would be that a wave function Ψ does not describe a quantum behaviour as we think of it, but is simply a function good at being fitted to give values between 0 and 1 within a certain interval, a probability generator.

A final element of this “heretic” quantum model would be the following: let us assume that considering electrons as basic subatomic particles is flawed because the fact they cannot be broken is due to their high mobility, which prevents the stress due to the impacting energy to break up the assembly of sub-particles constituting the electron, instead accelerating the electron and changing the configuration of the particle. If so, then the Hamiltonian of an electron:

HΨ = EΨ

Would be in fact

H_e-nΨ_e-n + ΣH_se-ecΨ_se-ec = Ψ_e-nE_e-n + ΣΨ_se-ecE_se-ec

The subindex _e-n would mean electron-nucleus interaction, and the subindex se-ec would refer to the interactions between the sub-electronic particles and the electronic virtual center. As a consequence, what we accept as wave function for an electron would be a combination:

Ψ = ( Ψ_e-nE_e-n + ΣΨ_se-ecE_se-ec)/E

No wonder the integral of the square of the wave function can assume very intricate forms! And no wonder trying to make sense of any steps of the calculation makes quantum theory “queerer than we can suppose”, as Haldane puts it or that Einstein found its implications “spooky” (Rosenblum and Kuttner, 2006, page 3). The fact is that it may be a black box tool with no significant meaning whatsoever: neural networks are precisely extremely good calculation tools with no pretense of giving us an explanation of why.

REFERENCES

Levine, I..N., Quantum Chemistry, (Sp. trans of the first edition) Editorial AC (1977)

Nami, F., Deyhimi, F.; Prediction of activity coefficients at infinite dilution for organic solutes in ionic liquids by artificial neural network, J. Chem. Thermodynamics 43 22–27 (2011)

Rosemblum, B.; Kuttner, F.; Quantum Enigma, 112 Oxford University Press (2006)