Large-scale numerical investigation of excited states in poly(para-phenylene)

Abstract

A density matrix renormalisation group scheme is developed, allowing for the first time essentially exact numerical solutions for the important excited states of a realistic semi-empirical model for oligo-phenylenes. By monitoring the evolution of the energies with chain length and comparing them to the experimental absorption peaks of oligomers and thin films, we assign the four characteristic absorption peaks of phenyl-based polymers. We also determine the position and nature of the nonlinear optical states in this model.The phenyl-based conjugated polymers have attracted immense interest from physicists and chemists as a result of the discovery of electroluminescence in poly(phenylenevinylene) (PPV) [1]. In particular, a great deal of theoretical and experimental effort has been devoted to understanding the important neutral excitations driving nonlinear optical processes, the positions of triplet states and the nature of exciton binding and decay [2][3][4][5][6]. Correlation between π-electrons has proved to be important in obtaining an accurate description of excited states in conjugated systems. It is thus desirable to have accurate numerical solutions of correlated electron models for these systems (without recourse to uncontrolled approximation schemes such as configuration interaction approximation schemes, or perturbative schemes starting from a noninteracting molecular orbital or k-space band description). In this paper we provide, for the first time, accurate results for a realistic semi-empirical model for poly(phenylene) and its oligomers, a model system for phenyl-based conjugated polymers.As our starting point for describing poly(phenylene), we adopt the Pariser-Parr-Pople (P-P-P) model [7],where <> represents nearest neighbors, c iσ destroys a π-electron on conjugated carbon atom i, n iσ = c † iσ c iσ and n i = n i↑ + n i↓ . We use the Ohno parameterisation for the Coulomb interaction, V ij = U/ 1 + (U r ij /14.397) 2 , where r ij is the inter-atomic distance inÅ and U = 10.06 eV [8]. The transfer integrals (t ij ) and bond lengths are 2.539 eV and 1.4Å respectively for phenyl bonds, and 2.22 eV and 1.51Å respectively for single bonds [8]. In the following we consider oligophenyls of N repeat units (phenyl rings), as depicted in Fig. 3.When solved accurately, the P-P-P theory has proved remarkably accurate in describing many of the low-lying excitations in a wide range of conjugated molecules [8,9]. Given such debates as to the nature of exciton binding in PPV and the related issues of interpreting various spectral features [2][3][4][5][6], it is intriguing as to what the P-P-P theory predicts for large phenyl systems. Exact diagonalisation is currently restricted to systems with two phenyl rings. The advent of the density matrix renormalisation group (DMRG) method [10] has been fortuitous in that it has the potential to provide effectively exact numerical solutions of the P-P-P theory for systems with hundreds of conjugated carbon atoms. The DMRG has obstensively been restricted to systems with...

Symmetry-adapted density matrix renormalization group calculations of the primary excited states of poly(para-phenylene vinylene)

“…͑The efficiency of the renormalized phenyl basis within the context of biphenyl itself is discussed in Ref. 26.͒ We see, for example, that in the second calculation we recover the ground state energy to around six to seven significant figures using around 0.3% of the Hilbert space.…”

“…5 of Ref. 26.͔ Notice that this assignment places a lower bound on the binding energy of the n = 1 exciton as E͑mA g ͒ − E͑1B u ͒.…”

“…The particulars of the threeblock method for the present class of problem, particularly the procedure of deriving an optimized reduced basis for the phenyl ring are described in some detail in Ref. 26. In this section we describe the three-block DMRG construction we have used for the PPV geometry, together with convergence data that demonstrate that we can obtain essentially exact results for the low-lying excited states of Eq.…”

“…26. The key idea is to derive efficient local bases for the phenyl units so that DMRG system blocks can be expanded by a whole phenyl unit in one step, and an effectively one-dimensional ͑1D͒, infinite lattice DMRG algorithm can be employed, avoiding the need for finite lattice approaches and maintaining the spatial symmetries of the molecule at every step.…”

“…DMRG calculations on the full PPP model were presented for poly͑para-phenylene͒ by Bursill and Barford 26 using a DMRG approach that we call the "three-block" DMRG. In this approach we employ local optimized Hilbert spaces for the phenyl units so that the DMRG blocks can be expanded by whole phenyl rings in one step.…”