Polynomial and piecewise polynomial fitting in tight-binding model of carbon molecule

Abstract

The tight-binding (TB) method is a semi-empirical method that is primarily used to calculate the energy band structure and single-particle Bloch states of material. Semi-empirical method is the method used in the electron system quantum mechanical involving the Schrödinger equation where the Hamiltonian is replaced with a parameterized model. The parameters of the model are fitted to reproduce the reference data which is obtained from the experimental data. The semi-empirical tight-binding method is one of the main approaches to compute the total energy of a system and it is computationally very fast. Therefore it tends to be used in the calculation of every large system, with more than a few thousand atoms in a unit cell. The main contribution of this research is to develop two new tight-binding energy models for carbon which associate with parameters fitting in the Hamiltonian system by using a minimization approach. This research aims to implement the polynomial and piecewise polynomial interpolation as new scaling function to the old TB model which denoted by Model 1 and Model 2, respectively. These new models are developed based on polynomial and piecewise polynomial approach. Mathematical techniques such as eigenvalues problem, minimization method, Newton iteration method and basic principles of the semi-empirical method are also applied in this research. The new methods obey the concept of semi-empirical approach which assumes that only one electron is free to move around the whole system. The elements of the overlapping Hamiltonian matrix in Model 1 and Model 2 are approximated by the polynomial function and piecewise polynomial function respectively. The models are then applied into 2-carbon, 3-carbon, and 4-carbon bond system with fixed geometry coordinates. Each parameter is calculated by using Newton iteration method which also involves differentiation of eigenvalues in the eigensystem. The energy of the bonding is then compared with the reference data from the well-established method of previous research. Both new models have been successfully reduced the computational time execution in the calculations by reducing the floating-point operations per second (FLOPS) in the algorithms. The results of Model 1 and Model 2 are compared whereby for 2-carbon simulation, the absolute error of these two models remains unchanged. For 3-carbon simulation, the absolute error has been reduced by 0.000006. Meanwhile, for 4-carbon simulation, the absolute error has been reduced by 0.116213. It is found that the efficiency of the new models has been significantly improved from Model 1 to Model 2 when the number of atom increases. The results suggest that Model 2 is more suitable for bigger molecule calculation due to its nature of piecewise polynomial advantages. Most of the results obtained gave positive feedback except some calculations which produced a trivial solution. Good agreement with data collection indicates that proposed models can be used as an alternative solution to the existing models and significant for the advancement of new knowledge.