tisdag 21 juni 2016

New Quantum Mechanics 2: Computational Results

I have now tested the atomic model for an atom with $N$ electrons of the previous post formulated as a classical free boundary problem in $N$ single-electron charge densities with non-overlapping supports filling 3d space with joint charge density as a sum of electron densities being continuously differentiable across inter-electron boundaries.

I have computed in spherical symmetry on an increasing sequence of radii dividing 3d space into a sequence of shells filled by collections of electrons smeared into spherically symmetric shell charge distribution. The electron-electron repulsive energy is computed with a reduction factor of $\frac{n-1}{n}$ for the electrons in a shell with $n$ electrons to account for lack of self repulsion.

Below is a typical result for Xenon with 54 electrons organised in shells with 2, 8, 18, 18 and 8 electrons with ground state energy -7413 to be compared with -7232 measured and with the energy distribution in the 5 shells displayed in the order of total energy, kinetic energy, kernel potential energy and inter-electron energy. Here the blue curve represents electron charge density, green is kernel potential and red is inter-electron potential. The inter-shell boundaries are adaptively computed  so as to represent a preset 2-8-18-18-8 configuration in iterative relaxation towards a ground state of minimal energy.

In general computed ground state energies agree with measured energies within a few percent for all atoms up to Radon with 86 electrons.

The computations indicate that it may well be possible to build an atomic model based on non-overlapping electronic charge densities as a classical continuum mechanical model with electrons keeping individuality by occupying different regions of space, which agrees reasonably well with observations. The model is an $N$-species free boundary problem in three space dimensions and as such is readily computable for any number of $N$ for both ground states, excited states and dynamic transitions between states.

We recall the the standard model in the form of Schrödinger's equation for a wave function depending on $3N$ space dimensions, is computationally demanding already for $N=2$ and completely beyond reach for larger $N$. As a result the full $3N$-dimensional Schrödinger equation is always replaced by some radically reduced model such as Hartree-Fock with optimization over a "clever choice" of a few "atomic orbitals", or Thomas-Fermi and Density Functional Theory with different forms of electron densities.

The present model is an electron density model, which as a free boundary problem with electric individuality is different from Thomas-Fermi and DFT.

We further recall that the standard Schrödinger equation is an ad hoc model with only formal justification as a physical model, in particular concerning the kinetic energy and the time dependence, and as such should perhaps better not be taken as a given ready-made model which is perfect and as such canonical (as is the standard view).

Since this standard model is uncomputable, it is impossible to show that the results from the model agree with observations, and thus claims of perfection made in books on quantum mechanics rather represent an ad hoc preconceived idea of unquestionable ultimate perfection than true experience.

2 kommentarer:

  1. Atomic energies are not that interesting, do you have any result for chemical quantities. Like bond lengths and angles for molecules or lattice sizes for compounds?

  2. I agree, but it is a necessary starting point.