Elsevier

Computational Materials Science

Volume 134, 15 June 2017, Pages 206-213
Computational Materials Science

Self-consistent charge and dipole density functional tight binding method and application to carbon-based systems

https://doi.org/10.1016/j.commatsci.2017.03.032Get rights and content

Abstract

The density functional tight binding (DFTB) method is a fast, semi-empirical, total energy electronic structure method based upon and parameterized to density functional theory (DFT). The standard self-consistent charge (SCC) DFTB approximates the charge fluctuations in a system using a multipole expansion truncated to the monopole term. For systems with asymmetric charge distributions, such as might be induced by an applied external field, higher terms in the multipole expansion are likely to be important. We have extended the formalism to include dipoles (SCCD), have implemented the method computationally, and test it by calculating the response of various carbon nanotubes and fullerenes to an applied electric field. A comparison of polarizabilities with experimental data or more sophisticated DFT calculations indicates a substantial improvement over standard SCC-DFTB. We also discuss the issues surrounding parameterization of the new SCCD-DFTB scheme.

Introduction

To accurately calculate the electronic structure of solid state materials, the density functional theory (DFT) has proven to be a trustworthy method if used appropriately. However, for large systems DFT is increasingly expensive. For these systems, a much faster semi-empirical method based upon the DFT framework, density functional tight binding (DFTB) method [1], [2], can provide insight into the physical properties with a balance of accuracy and efficiency. First generation DFTB [1] approximates the total energy as a sum of the eigenvalues of all occupied states (also known as band structure energy) and a two-body repulsive energy, which is fitted to full DFT results. With careful parametrization, this method yields insightful structural and band structure results of various systems [1] possessing relatively small charge redistribution. Elstner et al. [2] extended the method to accommodate systems with considerable charge redistribution by introducing a charge fluctuation determined self-consistently to minimize the total energy. This method, self-consistent-charge-DFTB (SCC-DFTB), fundamentally enables the treatment of charge redistribution, and exhibits better results and transferability [2], [3]. Further extension of the DFTB framework are possible, e.g. as described in Ref. [4].

Standard SCC-DFTB truncates the charge fluctuation around each atom to the monopole term. For systems with significantly asymmetrical charge distributions it is natural to consider achieving greater accuracy by extending the monopole approximation to higher terms. Bodrog and Aradi [4] have proposed using tabulated multipole interaction matrices and discussed formally some of the consequences for computation of the Hamiltonian and total energy. The specific method yielding the multipole interaction matrix and the parameterization have not been presented, nor implemented or applied. Motivated by a need to model with low cost large-scale graphene/graphitic films under the influence of external fields acting on the nanoscale, we develop the extension of the standard second-order DFTB framework to dipole terms proposed in Ref. [4]. We describe and implement a method to construct and tabulate the multipole interaction matrix, discuss parameterization issues, and validate and assess the dipole extension for carbon-based systems.

Section snippets

Self-consistent charge DFTB

First, we briefly summarize the theoretical background of SCC-DFTB. From DFT theory and the Kohn-Sham ansatz [5], the charge density n(r) in the SCC-DFTB scheme [2] is expressed as a superposition of a reference density n0(r) and small charge fluctuation δn(r). The total energy isE[n]=kfkΨk|-22+V^ext+drn0(r)+δn(r)|r-r|+Vxc[n0+δn]|Ψk-EH[n0+δn]+Exc[n0+δn]-drVxc[n0+δn](n0(r)+δn(r))+EII,where fk is the Fermi-Dirac occupation function of the state k, and Ψk is the corresponding

Dipole approximation

The monopole approximation used in SCC-DFTB has fundamentally improved the accuracy of the DFTB allowing for the incorporation of charge transfer effects. However, for highly polarized systems extending the approximation is desirable. At the next level of approximation the atomic charge density fluctuation δni(r) can be expressed as a superposition of a density ρi(r)[Δqi] associated with charge difference Δqi, and a density ρi(r)[Δpi] associated with dipole difference Δpi: δni(r)=ρi(r)[Δqi]+ρ̃i(

Applications to carbon based systems

An implementation of the above SCCD-DFTB scheme has been made, based upon the existing SCC-DFTB code “DFTB+” [3], [9]. The Slater-Koster integrals for the dipole matrix Pμν have been generated separately based on the ‘pbc-0-1’ parametrization, and calculation of Γ̂ij10/Γ̂ij11, Mulliken dipole difference Δpi and inclusion of an external field Eiext are implemented along with the necessary modifications to the total energy and Hamiltonian shift.

We describe both the charge and dipole distribution

Conclusion

In conclusion, we have extended the standard SCC-DFTB method from monopole to dipole approximation. Implementing the extension within the “DFTB+” code, we have applied it to various carbon systems and discussed the parametrization of the dipole extension. Comparing with ab initio DFT calculations, we find calculated polarizabilities of a set of 12 CNTs, are improved using our SCCD-DFTB scheme over those obtained from charge only SCC-DFTB. We expect more generally that SCCD-DFTB method increases

Acknowledgement

This work was supported by a Science and Innovation Award (EP/G036101/1) from the United Kingdom Engineering and Physical Sciences Research Council. YW gratefully acknowledges a University Research Studentship from the University of Bath.

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