The density matrix renormalization group method is used to investigate the Peierls transition for the extended Hubbard model coupled to quantized phonons. Following our earlier work on spin-Peierls systems, we use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit to a gapless, dispersive (Debye) limit to investigate the entire frequency range.A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite size scaling analysis, and bipartite quantum entanglement. All these probes indicate that a transition of Berezinskii-Kosterlitz-Thouless-type is observed at a non-zero electron-phonon coupling, g c , for a non-vanishing electron-electron interaction. An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in g c for a fixed optical (q = π) phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons, with the introduction of phonon dispersion renormalizing the effective electron-lattice coupling for the Peierls-active mode.By varying the Coulomb interaction, U , we observe a generalized Peierls transition, intermediate to the uncorrelated (U = 0) and spin-Peierls (U → ∞) limits, where U is the Hubbard Coulomb parameter.Using the extended Hubbard model with Debye phonons, we investigate the Peierls transition in trans-polyacetylene and show that the transition is close to the critical regime.
The density matrix renormalization group method is used to investigate the spin-Peierls transition for Heisenberg spins coupled to quantized phonons. We use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit to a gapless, dispersive (Debye) limit. A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite size scaling analysis, bond order auto-correlation functions, and bipartite quantum entanglement. All these probes indicate that in the antiadiabatic phonon limit a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type is observed at a non-zero spin-phonon coupling, g c . An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in g c for a fixed optical (q = π) phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons, with the introduction of phonon dispersion renormalizing the effective spin-lattice coupling for the Peierls-active mode. We also show that the staggered spin-spin and phonon displacement order parameters are unreliable means of determining the transition.
Irradiation of 4-diazopyrazolidine-3,5-diones (I 5) in the presence of alcohols or water gave mixtures of the isomeric 1,2-diazetidinones (I 6) and (1 7), formed b y competing photochemical Wolff rearrangement of the two nitrogen groups, followed by reaction of the resulting ketenes with the nucleophile. Some regioselectivity is observed in the ring contraction process, and the relative order of migration of nitrogen groups is NPh > NCHPh, -NCH,Ph -N M e > NCH,CO,Et. The structures of the 1,2-diazetidinones (1 7c) and (24) were confirmed by X-ray crystallography, and a crystal structure of the diazo compound (15g) was also obtained. Possible reasons for the regioselectivity in the ring contraction are discussed.In the preceding paper we reported the details of a new route to 1,2-diazetidinones, aza analogues of p-lactams, based on the photochemical ring contraction of symmetrically substituted 4-diazopyrazolidine-3,5-diones in the presence of a nucleophile (Scheme l). ' The success of this reaction demonstrated that despite the participation of the nitrogen lone pair in amide resonance, the nitrogen atom would migrate to the electron deficient centre to effect the desired ring contraction. In general N-C bonds show a marked reluctance to migrate in the photochemical Wolff rearrangement,, and only a few examples are known.3 Scheme 1. R = PhCH,, Pr, or RR = (CH,),; Nu = EtO, Bu'O, Et,N or H OWe have now investigated this reaction in more detail with a view to determining the electronic and steric requirements for nitrogen atoms to migrate in the Wolff rearrangement, and to extending the range of aza-p-lactams available by this route. Our results on the photochemical ring contraction of unsymmetrically substituted 4-diazopyrazolidine-3,5-diones are reported in detail herein4 Results and DiscussionPreparation of 4-Diazopyrazolidine-3,5-diones.-Eigh t monoand bi-cyclic unsymmetrical 4-diazopyrazolidine-3,5-diones (15) were prepared, containing a range of substituents. The choice of substituents bearing carboxylate groups is influenced by the fact that any potential p-lactam analogue would almost certainly require such a substituent. The 2-methyl-and 2-phenyl-1 -benzylpyrazolidine-3,5-diones (4a,c) were prepared by a similar route to that already used for the symmetrical 1,2-dialkyl derivatives. Thus the ethyl benzylidenecarbazates (la), prepared by methylation of ethyl benzylidenecarbazate itself, and (lc), prepared by acylation of benzaldehyde phenylhydrazone, were hydrogenated to give the corresponding benzyl derivatives (2). Acylation of (2) with ethoxycarbonylacetyl chloride followed by base mediated cyclisation, hydrolysis and decarboxylation gave the pyrazolidine-3,5-diones (4a,c) (Scheme 2). EIOZCNR
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.