The equidistant H chain: a 1D metal on a cyclic ring¶

This is the equal-spacing companion to the dimerized H-chain tutorial. That one strings hydrogen into H₂ pairs (a short bond, then a long gap) and gets a gapped, band-insulating Peierls chain. Here every H-H distance is the same, and that one change turns the chain into a one-dimensional metal, the Peierls-unstable parent the pairs chain dimerizes away from. The metal is the harder system, and it is exactly the case the cyclic cluster model handles most gracefully.

Two H-chains, kept separate on purpose

The pairs chain (dimerized, gapped) and this equidistant chain (metallic) are different physics and stay separate tutorials. The pairs chain demonstrates Born-von-Kármán equivalence across supercell and k-mesh representations on vibe-qc’s shipped periodic SCF. This one demonstrates why the finite cyclic ring is the elegant route when the infinite chain is metallic, using the published CCM-HF benchmark from the project author’s thesis.

Equal spacing makes a metal¶

Put one hydrogen atom per cell, spaced a uniform \(a\) apart, one electron each. The single \(1s\) band is then half filled, and a half-filled band with no gap at the Fermi level is a metal. Peierls’ theorem says such a 1D chain is unstable: it lowers its energy by dimerizing into H₂ pairs, which opens a gap at the zone boundary and turns the metal into the insulator of the pairs tutorial. The equidistant chain is the undistorted, metallic reference that instability starts from.

Metals are the awkward case for reciprocal-space sampling. The Brillouin-zone integrand is discontinuous at the Fermi surface, so an integer-occupation calculation on a coarse k-mesh will not settle, and the cure is Fermi-Dirac smearing on a fine mesh. vibe-qc is explicit about this: ask its Γ-only periodic route for smearing and it refuses rather than silently running an integer-Aufbau metal, pointing you at the multi-k driver instead.

The cyclic cluster makes a ring¶

The cyclic cluster model takes the other road. Wrap \(N\) equidistant atoms into a ring by imposing the cyclic Born-von-Kármán boundary conditions (the derivation here), and the infinite metallic chain becomes a finite system: a particle on a ring. A ring of \(N\) atoms holds exactly \(N\) crystal orbitals, the Bloch states \(e^{ikx}\) at the discrete wavevectors

\[ k = \frac{n}{N}\cdot\frac{2\pi}{a}, \qquad n = 0,\ \pm1,\ \pm2,\ \dots \]

folded onto the single Γ point of the cluster. For the \(N=6\) ring the wavevectors are \(k = 0\ (\Gamma),\ \pm\tfrac16,\ \pm\tfrac13,\ \tfrac12\) (in units of \(2\pi/a\)). Qualitatively the energies follow the tight-binding band \(E(k) \approx \alpha + 2\beta\cos(ka)\), so they order from the fully bonding \(\Gamma\) state (\(\cos 0 = +1\)) up to the fully antibonding zone-boundary state (\(\cos\pi = -1\)), and the \(+k\) and \(-k\) partners are degenerate because \(\cos(ka) = \cos(-ka)\). That degeneracy pattern, a single bonding level, then degenerate pairs, then a single antibonding level, is the same one Hückel theory gives for the \(\pi\) system of a ring such as benzene.

The decisive point: at \(N = 6\) the six electrons fill the lowest three orbitals, \(\Gamma\) plus the \(k=\pm\tfrac16\) pair, and stop at a real gap below the empty \(k=\pm\tfrac13\) pair. The finite ring is closed-shell and gapped even though the infinite chain it samples is metallic. The cyclic cluster has converted the awkward metallic integral into a clean molecular diagonalization, no smearing, no k-mesh convergence, one Γ-point solve.

The CCM-HF benchmark¶

The orbital energies of the equidistant H₆ ring, computed at Hartree-Fock / STO-3G with the ab-initio cyclic cluster model, are tabulated in the project author’s dissertation (M. F. Peintinger, Elektronenstruktur von Molekülkristallen, Univ. Bonn, 2013, §8.6.4, Table 8.6; published as Peintinger and Bredow, J. Comput. Chem. 35, 839, 2014):

Crystal orbital	Occupation	Energy (Ha)	\(k\) (\(2\pi/a\))
1	2.00	\(-0.74295030\)	\(0\) (\(\Gamma\))
2	2.00	\(-0.45266305\)	\(+\tfrac16\)
3	2.00	\(-0.45266305\)	\(-\tfrac16\)
4	0.00	\(+0.44076948\)	\(+\tfrac13\)
5	0.00	\(+0.44076948\)	\(-\tfrac13\)
6	0.00	\(+1.30010143\)	\(+\tfrac12\)

The degeneracies are exact (\(k\) and \(-k\) agree to every printed digit), the filled set ends at the \(k=\pm\tfrac16\) pair, and the HOMO-LUMO gap is a clean \(0.893\) Ha. The crystal-orbital coefficients (Table 8.7) make the band endpoints concrete: the lowest orbital is uniform, every atom \(+0.272185\), the fully in-phase \(\Gamma\) combination, while the highest is strictly alternating, \(\pm0.856484\) around the ring, the fully out-of-phase zone-boundary combination. The middle four spread phase smoothly between those limits.

What this benchmark is, and is not

These numbers are the ab-initio CCM-HF result computed in the thesis (with the AICCM program of that work), not a vibe-qc run. vibe-qc’s own CCM-HF is the v2.0 roadmap target; when it lands, this Table 8.6 is its regression fixture. The ab-initio CCM theory section works through the Wigner-Seitz weighting that produces it.

Why the ring is the right tool here¶

The equidistant chain draws the contrast between the two periodic pictures sharply. The supercell-and-k-mesh route has to integrate a discontinuous metallic band and needs smearing to do it; the cyclic-cluster route folds the same physics onto a finite ring that simply has a gap. For a perfect crystal the symmetry of the k-mesh still makes reciprocal-space sampling cheaper, but the moment the symmetry is broken, by a defect, a surface, or an adsorbate, the real-space ring keeps its clean molecular character while the k-mesh does not. That is the argument the cyclic cluster model is built on, and the equidistant H₆ ring is its smallest honest illustration.