Stability of the Peierls instability for ring-shaped molecules

Abstract

We investigate the conventional tight-binding model of L π-electrons on a ring-shaped molecule of L atoms with nearest neighbor hopping. The hopping amplitudes, t(w), depend on the atomic spacings, w, with an associated distortion energy V (w). A Hubbard type on-site interaction as well as nearest-neighbor repulsive potentials can also be included. We prove that when L = 4k +2 the minimum energy E occurs either for equal spacing or for alternating spacings (dimerization); nothing more chaotic can occur. In particular this statement is true for the Peierls-Hubbard Hamiltonian which is the case of linear t(w) and quadratic V (w), i.e., t(w) = t 0 − αw and V (w) = k(w − a) 2 , but our results hold for any choice of couplings or functions t(w) and V (w). When L = 4k we prove that more chaotic minima can occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit L → ∞. Our analysis suggests three interesting conjectures about how dimerization stabilizes for large systems. We also treat the spin-Peierls problem and prove that nothing more chaotic than dimerization occurs for L = 4k + 2 and L = 4k. INTRODUCTIONTo derive the shape (and other properties) of molecules from first principles has been an actively pursued goal since the early days of quantum mechanics. Many of the insights into the structure of molecules like benzene and its relatives have, however, been obtained using drastically simplified models. The Schrödinger equation for all the nuclei and electrons in such a molecule involves many dozens or even hundreds of degrees of freedom and, therefore, simpler models with a reduced number of degrees of freedom are a necessity. In the case of benzene the introduction of the Hückel model [1] played an important role. This model is standard textbook material for organic chemistry students (see e.g. [2]). London [3] used the Hückel model to explain the large diamagnetic anisotropy of aromatic compounds and certain other materials quite successfully. A similar approach was used earlier by Jones in his work on bismuth and bismuth alloys [4].In this paper we are interested in ring-shaped molecules of the type (CH) L , for even L, the so-called annulenes (sometimes called cyclic polyenes). The Hückel model for [L]-annulene describes the L π−electrons (one for each carbon atom) as hopping from one carbon atom to the next (tight-binding approximation). The carbon atoms are located at the L sites of a ring-shaped geometry, so L + 1 ≡ 1. The Coulomb interaction between the electrons is ignored in the Hückel model but we will include the Hubbard [5] on-site interaction as in the work of Pariser-Parr [6] and Pople [7], in our study (a nearest neighbour repulsion can also be included).

Ferromagnetic Ordering of Energy Levels

“…The analogous result for d = 2 and s x ≥ 1 as well as d ≥ 3 and s x ≥ 1/2 was subsequently proved by Kennedy, Lieb and Shastry [8]. Many interesting results on a variety of topics later followed using reflection positivity [9,13,15,17,16]. However this technique never succeeded to prove a phase transition, at positive temperatures, for the ferromagnetic Heisenberg model, despite the fact that it is completely trivial to prove a phase transition for the ground states.…”

Variational states for the spin-Peierls system

“…For ␦ 0 the variational ansatz reproduces the explicit dimerization of the model. 10 For vanishing bond alternation ␦ϭ0 the ground state of the finite system has a well-defined lattice momentum k s ͕0, ͖ ͑Ref. 26͒ while the lowest excitations ͑triplet, singlet͒ have k t ,k s * ϭ Ϫk s (mod 2 ).…”

“…9. Extending the discussion to general couplings in the ␣-␦ plane, the system has gap above a dimerized ground state for any nonzero ␦, 10 with the valence-bond ground states ͑1.2͒ on the line 2␣ϩ␦ϭ1. 11 Going to larger ␦ the Hamiltonian ͑1.1͒ corresponds to a ladder system of two coupled Heisenberg chains.…”

Ground states and low-temperature phases of itinerant electrons interacting with classical fields: A review of rigorous results

“…Hamiltonian ͑3.11͒ is reflection positive so that a direct application of the Dyson-Lieb-Simon inequality 16 or its ground state version 18 implies that at least one of the two Hamiltonians,…”