Composite Basis Set of Plane Wave and Gaussian Function or Spline Function

Abstract

By combining plane waves with Gaussian or spline functions, this paper constructs a new composite basis set. As a non local basis vector, the plane wave basis group needs a large number of plane waves to expand all parts of the physical space, including the intermediate regions that are not important to our problems. Our basis set uses the local characteristics of Gaussian function or spline function at the same time, and controls the energy interval by selecting different plane wave vectors, so as to realize the partition solution of Hamiltonian matrix. Orthogonal normalization of composite basis sets is performed using Gram-Schmidt's orthogonalization method or Löwdin's orthogonalization method. Considering the completeness of plane wave vector, a certain value of positive and negative should be selected at the same time. Here, by changing the absolute value of wave vector, we can select the eigenvalue interval to be solved. The plane wave with a specific wave vector value is equivalent to a trial solution in the region with gentle potential energy. The algorithm automatically combines local Gaussian or spline functions to match the wave vector value difference between the trial solution and the strict solution. By selecting the absolute value of the wave vector in the plane wave function, this paper turns the calculation of large Hamiltonian matrices into the calculation of multiple small matrices, together with reducing the basis numbers in the region where the electron potential changes smoothly, we can significantly reduce the computational time. As an example, we apply this basis set to a one-dimensional finite depth potential well, it can be found that our method significantly reduce the number of basis vectors used to expand the wave function while maintaining a suitable degree of computational accuracy. We also studied the impact of different parameters on calculation accuracy. Finally, the above calculation method can be directly applied to the DFT calculation of plasmons in silver nanoplates or other metal nanostructures. Given a reasonable tentative initial state, the ground state electron density distribution of the system can be solved by self consistent solution using DFT theory, and then the electromagnetic field distribution and optical properties of the system can be solved using time-dependent density functional theory theory (TDDFT).