Self-consistent charge and dipole density functional tight binding method and application to carbon-based systems
Graphical abstract
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 in the SCC-DFTB scheme [2] is expressed as a superposition of a reference density and small charge fluctuation . The total energy iswhere is the Fermi-Dirac occupation function of the state k, and 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 can be expressed as a superposition of a density associated with charge difference , and a density associated with dipole difference :
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 have been generated separately based on the ‘pbc-0-1’ parametrization, and calculation of /, Mulliken dipole difference and inclusion of an external field 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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