Symmetry beneath the table

Introduction

All elementary rationalizations of the structure of the periodic table begin with an account of the electronic structure of hydrogenic atoms where an electron is distributed in the presence of a nucleus of atomic number Z and experiences a Coulomb field. The resulting array of states has a major peculiarity: all orbitals of the same shell are degenerate. In other words, there is what seems to be an accidental degeneracy for states with the same n but different l. This degeneracy is exact (within the Schrödinger formalism) and there appears to be no operation that rotates orbitals of different l into each other (the normal underlying reason for exact degeneracy). How, for instance, can a 2p orbital be rotated into a 2s orbital? The issue is resolved when it is realized that the Coulomb potential is not merely spherically symmetrical (that is, is a basis for the special orthogonal group in three dimensions, SO(3)) but is hyperspherically symmetrical (that is, is a basis of the special orthogonal group in four dimensions, SO(4)) [1].

Atoms in higher dimensions

To illustrate how stepping up a dimension to display a hidden symmetry the illustratively more attainable example is the process of stepping from two to three dimensions. Thus Fig. 1 shows the analogues of two p orbitals and one s orbital, and it is clear that although the p orbitals are related by a 90° rotation about a perpendicular axis, the s orbital cannot be generated by a rotation. However, if the two p orbitals are the projection of a sphere with its hemispheres different colours, then it is easy to see that a rotation of the sphere results in projections that include the s orbital. The same is true for all the orbitals of hydrogenic atoms: if they are regarded as projections from a hypersphere, then all orbitals of a given n can be generated from rotations in the higher dimension. Hydrogenic accidental degeneracy is therefore non-accidental: it is a hypersymmetry [2].

Fig. 1:

Although the disk in the central plane cannot be generated by rotation of either of the shapes in the outer two planes, if it is regarded as a projection of the coloring of a sphere, then all three shapes can be interrelated by rotation. That is, a higher dimensionality might reveal a hidden symmetry.

Were the array of states of a hydrogenic atom to be used directly to formulate a periodic table, then successive periods would be of length 2n², with n=1, 2, … (viz. 2, 8, 18, 32, 50, 72, …). That the actual periodic table has periods of lengths 2, 8, 8, 18, 18, 32, 32, …) shows the naivety of this approach. As is well known, the observed order is achieved by allowing for electron–electron repulsions. In the language underlying this article, although these repulsions are Coulombic (and therefore in a certain sense SO(4)), they are not based on a single centrosymmetric point and thus represent symmetry breaking. That is, the periodic table can be regarded as a symmetry-broken version of a pure Coulombic field. How that symmetry breaking results in the transformation {2, 8, 18, 32, 50, 72, …} → {2, 8, 8, 18, 18, 32, 32, …} is treated in all elementary accounts of the structure of the table and will not be treated here.

Gauge symmetry

The Coulomb interaction itself can be traced to another symmetry, namely gauge symmetry. Once again, it is possible to give a pictorial explanation of what is involved in this otherwise challenging part of theoretical physics. As everyone knows, a particle travelling with a certain linear momentum is represented by a monochromatic wave. In elementary accounts, this wave is commonly represented as cos kx. However, it is best to represent it as the complex wave e^ikx=cos kx+i sin kx, for only complex wavefunctions correspond to net motion. Such a wave is therefore best represented by a uniform helix (Fig. 2). A gauge transformation (specifically in this case an Abelian gauge transformation) is then simply a shift in the phase of the helix. As can readily be verified by noting that the amplitude of the helix is a measure of probability density, a global shift in phase, a global gauge transformation, does not change the probability of finding the particle in a region, so is a symmetry of the system. The immediate consequence of global gauge invariance (the invariance of observables under a gauge transformation) is that whatever flows into an arbitrary region of space is matched by what flows out, as suggested by Fig. 2. That is, global gauge invariance implies the conservation of charge, one of the fundamental features of electromagnetism and through that, atomic structure. A point to note, for it will be important in a moment, is that the wavefunction that has been modified by a gauge transformation continues to satisfy the same Schrödinger equation as it did before its phase was shifted [3].

Fig. 2:

(a) A particle with a definite linear momentum is represented by a helix of constant amplitude and therefore constant probability density. That probability density is unchanged by a global gauge transformation (a uniform change of phase), and as the probability density within the region marked remains the same, global gauge invariance implies charge conservation. (b) The phase of the helix can also be changed by different amounts in different regions, and still correspond to the same observable (the probability density). However, the new wave no longer satisfies the original Schrödinger equation; the latter needs to be augmented by additional terms that imply Maxwell’s equations for electromagnetism. Thus, local gauge invariance implies the existence of electromagnetic forces.

A very important point is that the probability density represented (through its amplitude) by the helix is also invariant under local gauge transformations, the modification of its phase by different amounts at different locations, for these modifications leave its amplitude unchanged. However, the locally gauge transformed helix no longer satisfies the same Schrödinger equation as the original helix. For it to satisfy a Schrödinger equation, an additional term must be included in the equation. Remarkably, that additional term is what is entailed by combining Maxwell’s equations for electromagnetism with the Schrödinger equation in elementary treatments. In other words, local gauge invariance implies electromagnetism, and specifically the Coulomb field. Thus, deep below the discussion of the structure of the periodic table lies yet another symmetry, namely local (abelian) gauge invariance.

I have slipped in ‘Abelian’ in that last sentence, to open yet another window on the symmetries underlying the periodic table, for the strong and weak forces that govern nuclear structure, and which are manifest to us chemists as atomic number can also be traced to local gauge invariance of a more complicated, non-Abelian kind. A discussion of that symmetry would take us too far from our subject, but it should be noted that the existence of the ordinal number Z that underpins the sequence of elements portrayed in the table is yet another consequence of symmetry.

Atoms in lower dimensions

I have explored the consequences of going up in dimensionality; what happens on going down in symmetry? In two dimensions (2D), Flatland, the Coulomb potential is proportional to ln r. (That form ensures that the total force over a shell surrounding the point charge is independent of the radius, just as 1/r, corresponding to a force proportional to 1/r², ensures the same constancy in three dimensions.) Although replacing ln r by 1/r is somewhat different, in part because at large r the latter approaches zero so permits the existence of unbound states whereas ln r approaches infinity, albeit very slowly, and therefore has only bound states, the 1/r form captures the general form of the lower energy states and has the advantage that the Schrödinger equation can be solved analytically; it also preserves the symmetry of the system. Elementary methods then show that the energies of the bound states are proportional to –1/(n+½)² with n=0, 1, … and m_l=0,±1, …,±n. Note that the degeneracies are 1, 3, 5, … and in general 2n+1. Then, assuming that the normal Aufbau procedure is relevant, the initial part of the resulting periodic table is as shown in Fig. 3.

Fig. 3:

The periodic table implied by a simple interpretation of the solutions of a two-dimensional hydrogenic atom.

One further descent in dimensionality takes us into Lineland, where the atoms are one-dimensional (1D) entities. The Coulomb potential in Lineland is proportional to r itself, giving a V-shaped potential energy well, for then the corresponding force is independent of r, like the size of the surrounding shell (which consists simply of two points). The solutions of the Schrödinger equation can be constructed by matching fragments of Airy functions, Ai(r), on either side of r=0, and although numerical values for the energy levels can be given, there is no simple analytical expression for them (the first few are at 1.02, 2.34, 3.25, … in appropriate units). Once again, the essence of the structure is obtained, and the symmetry of the system preserved, by replacing the true Coulomb potential by the usual 1/r form, and intriguingly the energy levels are exactly like those of a 3D hydrogenic atom, being proportional to –1/n² with n=1, 2, …, though all of them necessarily nondegenerate in this 1D system. The resulting periodic table is therefore extremely simple, Fig. 4.

Fig. 4:

A very primitive periodic table is implied by the structure of a one-dimensional hydrogenic atom.

Inverse atoms

Another aspect of symmetry is to turn an atom inside-out. That means removing the nucleus and ensuring that the electrons are confined in an infinitely deep well. In Lineland, the system is the elementary problem of a particle in an infinitely deep square well, and the nondegenerate energy levels are proportional to n², with n=1, 2, …. The resulting periodic table is therefore the same as for the 1D atom. In Flatland, an electron is confined to a disk-like region in a plane. The solutions of the Schrödinger equation for such a system are well-known and the wavefunctions are essentially Bessel functions in the radius, J(r), multiplied by the usual angular functions for a cylindrically symmetrical system (that is, eimlϕ ). There are no simple expressions for the energy levels, which are obtained by ensuring that the relevant Bessel function has a node at the edge of the disk, and no clear pattern of degeneracies, other than for the lowest six levels, which are 1, 2, 2, 1, 2, 2. Note that these degeneracies stem from |m_l|=0, 1, 2, 0, 3, 1, respectively, so they are more subtle than they might seem in the sense that the ‘blocks’ of the periodic table are in the order s, p, d, s, f, p. As there is no central nucleus, the elementary Aufbau rules based on penetration and shielding arguments do not apply and without carrying out a numerical SCF calculation (which I leave to others), there is no obvious reason why the pattern of energy levels should change from the one-electron version. Provided that is so, the resulting periodic table is shown in Fig. 5 (as far as Ni, Z=28).

Fig. 5:

The periodic table of the elements now interpreted as atoms confined to two-dimensional cavities.

Finally, we come back to 3D and the realistic issue of electrons trapped in deep spherical wells. The Schrödinger equation has well-established solutions for this model, the wavefunctions being spherical Bessel functions, j_l(r), multiplied by spherical harmonics [2]. Quantization arises from the requirement that the j_l(r) have a node at the edge of the well. The rather complex spectrum of energy levels is shown in Fig. 6, with the degeneracies of the s, p, d, f, g, h wavefunctions being 1, 3, 5, 7, 9, 11, respectively (that is, 2l+1). As the order of energy levels is s, p, d, s, f, p, g, … and ignoring electron–electron repulsion effects, the resulting periodic table will have the complicated block structure shown in Fig. 7.

Fig. 6:

The spectrum of energy levels and their degeneracies for a particle trapped in a spherical cavity of radius R with impenetrable walls.

Fig. 7:

The periodic table of the elements now interpreted as atoms confined to three-dimensional cavities.

Spacetime symmetry

The final symmetry I shall consider is that between space and time. As Z increases, so (classically) do the speeds of the electrons, especially but not only those in inner shells, and it is necessary to consider relativistic effects. These are covered by others in this collection or articles, but it is possible to deploy simple arguments to predict the value of Z at which space has been sufficiently rotated into time, and vice versa, for relativity to play a role, the Dirac equation has to replace the Schrödinger equation, and familiar periodicity is no longer a feature of the sequence of elements.

To estimate the value of Z at which relativistic effects become important, consider the ground state energy of a hydrogenic atom written in terms of the fine-structure constant, α, which is E=–½α²m_ec²Z². According to the virial theorem, the contribution of the kinetic energy to this total is E_kinetic=½α²m_ec²Z², which can be identified for the present purposes with ½m_e⟨v²⟩, where v is the speed of the electron, which suggests that v≈αcZ. The condition for v=kc, where k is the appropriate fraction, is therefore Z≈k/α≈137k. With k≈½, when space has been rotated into time by around 45°, we can expect relativistic effects to become significant at around Z=70 (Yb), in Period 6. Thus, in this period, the familiar periodicity can be expected progressively to fail.

What the periodic table never does, though, is fail to provide a stimulus for reflection and understanding.