Comparison study of finite element and basis set methods for finite size scaling

Abstract

We compare two methods of obtaining critical parameters for a quantum Hamiltonian using a finite size scaling approach. A finite element and basis set method were used in conjunction with the finite size scaling to obtain the critical parameters for the Hulthen potential. The critical parameters obtained analytically were the coupling constant λc=12, the critical exponents for the energy α=2 and for the “correlation length” ν=1. The extrapolated results for finite size scaling with the basis set method are λc=0.49999, α=1.9960, and ν=0.99910. The results for the finite element solutions are λc=0.50184, α=1.99993, and ν=1.00079 for the linear interpolation and λc=0.50000, α=2.00011, and ν=1.00032 for the Hermite interpolation. The results for each method compare very well with the analytical results obtained for the Hulthen potential. However, the finite element method is easier to implement and may be combined with ab initio and density functional theory to obtain quantum critical parameters for more complex systems.