Material modeling for large scale and complex nanostructures: A semi-empirical Hamiltonian method

The advent of the era of nano-structures has also brought about critical issues regarding the determination of stable structures and the associated properties of such systems. From the theoretical perspective, it requires to consider systems of sizes of up to tens of thousands atoms to obtain a realistic picture of thermodynamically stable nano-structure. This is certainly beyond the scope of DFT-based methods. On the other hand, conventional semi-empirical Hamiltonians, which are capable of treating systems of those sizes, do not possess the rigor and accuracy that can lead to a reliable determination of stable structures in nano-systems. During the last dozen years, extensive effort has been devoted to developing methods that can handle systems of nano-sizes on the one hand, while possess first principles-level accuracy on the other. In this review, we present just such a recently developed and well-tested semi-empirical Hamiltonian, referred in the literature as the SCED-LCAO Hamiltonian. Here SCED is the acronym for self-consistent/environment-dependent while LCAO stands for linear combination of atomic orbitals. Compared to existing conventional two-center semiempirical Hamiltonians, the SCED-LCAO Hamiltonian distinguishes itself by remedying the deficiencies of conventional two-center semi-empirical Hamiltonians on two important fronts: the lack of means to determine charge redistribution and the lack of involvement of multi-center interactions. Its framework provides a scheme to self-consistently determine the charge redistribution and includes multi-center interactions. In this way, bond-breaking and bond-forming processes associated with complex structural reconstructions can be described appropriately. With respect to first principles methods, the SCED-LCAO Hamiltonian replaces the time-consuming energy integrations of the self-consistent loop in first principles methods by simple parameterized functions, allowing a speed-up of the self-consistent determination of charge redistribution by two orders of magnitudes. Thus the method based on the SCED-LCAO is no more cumbersome than the conventional semi-empirical methods on the one hand and can achieve the first principle-level accuracy on the other. The parameters and parametric functions for SCED-LCAO Hamiltonian are carefully optimized to model electron-electron correlations and multi-center interactions in an efficient fitting process including a global optimization scheme. To ensure the transferability of the Hamiltonian, the data base chosen in the fitting process contains large amount of physical properties, including (i) the binding energies, the bond lengths, and the symmetries of various clusters covering not only the ground state but also the excited phases, (ii) the binding energies as a function of atomic volume for various crystal phases including also the high pressure phases, and (iii) the electronic band structures of the crystalline systems. In particular, the data bases for excited phases of clusters and high pressure phases in bulk systems are more important when performing molecular dynamics simulations where correct transferable phases are required, such as the excited phases. The validity and the robustness of the SCED-LCAO Hamiltonian have been tested for more complicated Si-, C-, and B-based systems. The success of the SCED-LCAO Hamiltonian will be elucidated through the following applications: (i) the phase transformations of carbon bucky-diamond clusters upon annealing, (ii) the initial stage of growth of single-wall carbon nanotubes (SWCNTs), (iii) the discovery of bulky-diamond SiC clusters, (iv) the morphology and energetics of SiC nanowires (NWs), and (v) the self-assembly of stable SiC based caged nano-structures. A recent upgrade of the SCED-LCAO Hamiltonian, by taking into account the effect on the atomic orbitals due to the atomic aggregation, will also be discussed in this review. This upgrade Hamiltonian has successfully characterized the electron-deficiency in trivalent boron element captured complex chemical bonding in various boron allotropes, which is a big challenge for semi-empirical Hamiltonians.