Density Matrix Renormalisation Group Calculations for Conjugated Polymers and Quantum Chemistry

Significant Achievements, Anticipated Outcomes and Future Work

In this project we are developing and applying state-of-the-art large scale numerical codes to solve quantum lattice models of conjuagted molecules and polymers. The method we use is ``density matrix renormalisation group'' (DMRG). This method has proven very powerful in its ability to attack molecules of hitherto unreachable size. For example, our codes have allowed us to solve polyenes (linear chains of conjugated carbon atoms) with many hundreds of atoms witin the fully correlated Parier-Parr-Pople (P-P-P) theory without recourse to uncontrolled approximations, whereas previous efforts were restricted to around a dozen atoms. In 2005 we published a study of relaxation effects and excited state geometries of the complex but technologically important polyphenylene system. We also published a study of a quantum spin-Peierls model that dynamically models the ionic degrees of freedom (phonon modes) in polymers such as polyacetylene. An interesting new phase was discovered in the model as a consequence of our accurate numerical method.
Dr Bursill and Dr Barford are in the process of producing a further general study of exciton properties, including the scaling of the exciton size and energy with conjugation length. We have also been working on extending the DMRG calculational method from the relatively simple P-P-P theory to the ZINDO semi-empirical theory, which dynamically models sigma electrons in addition to pi electrons. In our first demonstration application we have been calculating excited states of the benzene molecule. We are also completing a comprehensive study of the excited state properties of poly(phenylenevinylene).

Computational Techniques Used

DMRG techniques represent a powerful alternative to ``exact diagonalisation'' of lattice models of quantum systems on finite lattices. In quantum lattice models, the size of the Hilbert space (on which acts the energy operator that must be diagonalised to obtain the excitation spectrum and correlation functions that are measured in experiments) grows exponentially with the size of the lattice, which must be as close as possible to the bulk limit for comparison with experiment. By employing a systematic Hilbert space truncation, the DMRG allows computational physicists to attack very large lattices with very little loss in accuracy. DMRG involves intesive linear algebra operations - disgonalisation of extremely large, sparse operators, and large, dense matrices, and the accuracy and scope of the method is ultimately limited by the available memory. We have been developing Open MP versions of our codes that can exploit the large momery spaces offered by the large compute nodes on the APAC Altix cluster. We hope to perform some of the world's largest scale DMRG simulations of conjugated systems and we plan to investigate the computationally demanding oligophenylenevinylenes, and 2D systems such as carbon nanotubes, graphite sheets and fullurenes.

Publications, Awards and External Funding

External Funding and Awards

In 2005 this project was supported by an ARC QEII Research Fellowship (Dr Bursill) and by an ARC Discovery grant. Dr Bursill works closely with Dr William Barford from the University of Oxford, who is supported by a number of EPSERC and Royal Society grants in the UK.

Publications

E. E. Moore, W. Barford and R. J. Bursill, Relaxation energies and excited state geometries of poly(para-phenylene), Physical Review B 71, 115107 (2005).
W. Barford and R. J. Bursill, Phase transition in the quantum spin-Peierls model, Physical Review Letters 95, 137207 (2005).