The Periodic Law?

[exchemist]

As one develops more adequate Nilsson mappings of energies versus deformations, the finding of conserved individual total shell energies can guide creation of new models. We'll know where to expect to see the appearances of particular single particle levels. These models depend on experimental measurements for confirmation. So far we don't have one based on first principles. Rather the correction terms have numerical coefficient 'constants' which aren't constant at all, but change from shell to shell, apparently without any overarching mathematical motif that can be recognized. The constant goes up, then it comes down, as we move from shell to shell progressively. I'm aiming at a first-principles model.

 

Jess Tauber

Are you talking about nuclear energy states or electronic states in the atom?

[tetrahedron]

Throughout most of my arguments I've been talking about the nuclear system. People just claim, without really thinking about it, that there isn't any connection between the electronic and nuclear systems. But there are a number of such connections. Just to start they all use same-sized orbital sets. Both systems are amenable to expression as left-step depictions. Both see effects from spin-orbit coupling. And both have mathematical underpinnings from Pascal Triangle mathematics, just because the quantum harmonic oscillator equation always delivers numbers of stable states that themselves are terms in Pascal Triangle diagonals. The dimensionality of the oscillator system under consideration determines which diagonal is chosen.

 

My analyses of electronic shells are much less developed than for the nuclear system, where things were far more obvious. On the other hand, with the electronic periodic table I was able to reconfigure it as a tetrahedron of close-packed spheres, one sphere per element. I had thought I was the first to do this, but a decade after my initial effort I found that the same configuration had been arrived at by the Russian physician Dmitry Weise. See Fig.s 4 and 8 at http://weise.symmetry-us.com/pythagorean-approach-in-chemistry-tetrahedron/. 

 

More recently I came up with somewhat different mappings, based on the same skew-rhombic (rhombi bent up to a tetrahedral dihedral angle) motif, where each individual period (left-step) is a kind of tile (think Penrose tiles) that partially wraps around a tetrahedral edge on the face of the tetrahedron. Tetrahedra of close-packed spheres can be created by co-centered layers. The first 'core' tet is 20 spheres (so the first 20 elements), surrounded by 100 spheres (so 21 to 120). Within each tile each period starts as the longest edge, representing the largest orbital, leftmost in the period. Then the next smaller orbital begins where the last ends, so there is no numerical break in the atomic number sequence. And so on, until finally the s-orbital is reached. Organized in this fashion, the terminals of each period contacts the initials of the next in turn. And the same is also true for the end of the inner tetrahedral core and the beginning of the outer jacket. Thus there are NO numerical breaks in the sequence of atomic numbers anywhere in these models (there are 4 different tetrahedral configurations which will accomplish this). See https://www.facebook.com/photo.php?fbid=10155616726521346&set=gm.762705150603772&type=3&theater&ifg=1.

 

Jess Tauber

[exchemist]

Throughout most of my arguments I've been talking about the nuclear system. People just claim, without really thinking about it, that there isn't any connection between the electronic and nuclear systems. But there are a number of such connections. Just to start they all use same-sized orbital sets. Both systems are amenable to expression as left-step depictions. Both see effects from spin-orbit coupling. And both have mathematical underpinnings from Pascal Triangle mathematics, just because the quantum harmonic oscillator equation always delivers numbers of stable states that themselves are terms in Pascal Triangle diagonals. The dimensionality of the oscillator system under consideration determines which diagonal is chosen.

 

My analyses of electronic shells are much less developed than for the nuclear system, where things were far more obvious. On the other hand, with the electronic periodic table I was able to reconfigure it as a tetrahedron of close-packed spheres, one sphere per element. I had thought I was the first to do this, but a decade after my initial effort I found that the same configuration had been arrived at by the Russian physician Dmitry Weise. See Fig.s 4 and 8 at http://weise.symmetry-us.com/pythagorean-approach-in-chemistry-tetrahedron/. 

 

More recently I came up with somewhat different mappings, based on the same skew-rhombic (rhombi bent up to a tetrahedral dihedral angle) motif, where each individual period (left-step) is a kind of tile (think Penrose tiles) that partially wraps around a tetrahedral edge on the face of the tetrahedron. Tetrahedra of close-packed spheres can be created by co-centered layers. The first 'core' tet is 20 spheres (so the first 20 elements), surrounded by 100 spheres (so 21 to 120). Within each tile each period starts as the longest edge, representing the largest orbital, leftmost in the period. Then the next smaller orbital begins where the last ends, so there is no numerical break in the atomic number sequence. And so on, until finally the s-orbital is reached. Organized in this fashion, the terminals of each period contacts the initials of the next in turn. And the same is also true for the end of the inner tetrahedral core and the beginning of the outer jacket. Thus there are NO numerical breaks in the sequence of atomic numbers anywhere in these models (there are 4 different tetrahedral configurations which will accomplish this). See https://www.facebook.com/photo.php?fbid=10155616726521346&set=gm.762705150603772&type=3&theater&ifg=1.

 

Jess Tauber

There is no physical connection between the nuclear states and the electronic states. That is what people mean when they say they are  not connected. It is however quite obvious that there will be some analogous quantum behaviour, given that both involve systems confined by a spherical potential. As you will presumably be aware, the stationary state solutions of Schrödinger's equation are determined in part by the shape of the confining potential. So you might expect in both cases a series of spherical harmonic-related solutions, identified by similar sorts of quantum numbers. 

 

I'll leave you to it where nuclear structure is concerned as I am not a nuclear physicist.

 

Regarding electronic structure in the atom, this is already well described and understood. I do not as yet see what your numerical analysis will contribute by way of further understanding. 

[tetrahedron]

Well since you like to call numerical analysis 'numerology' I'll leave you with this little tidbit.  If you take those atomic numbers which are also Fibonacci numbers, then their patterning, out to atomic number 89 (the last one in the real periodic system, since the next would be 144, far beyond synthetic capacities) is neither random nor arbitrary. In every single case they fall upon first (leftmost) lobes in either the first half-orbital row (where only singlet electrons appear in lobes) or the second half-orbital row (where opposite-spin partners are accumulating). And all the ODD Fib numbers are on the singlet side, while the EVEN Fib numbers are on the doublet side. There are no exceptions, out to and including 89. The mapping doesn't care about the identity of the actual orbitals, so when we have electronic configurational anomalies the mappings still go to first singlets and doublets.  Interestingly, though less perfectly, the related Lucas numbers, taken as atomic numbers, map to last (RIGHTMOST) lobes, again with the odd=singlet and even=doublet bias. This only works out to atomic number 18. However the PT shows behavioral adjustments allowing the trend to continue despite positional mismatches. 29 and 47 (copper and silver, coincidentally in the same group) both have anomalous electronic configurations where the d orbital has been filled by internal shift of one s electron to the underlyingly (by position in the orbital row) d9 configuration. And 76 (osmium), with a d6 configuration (which would otherwise match the Fibonacci trend) behaves in the monatomic gaseous state as if it were akin to a noble gas (with a p6 filled configuration).

 

THAT'S numerology, though without any of the metaphysics or supernatural stuff.

 

Jess Tauber