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

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 λ_c = 1/2, the critical exponents for the energy α=2 and for the "correlation length" =1. The extrapolated results for finite size scaling with the basis set method are λ_c =0.499 99, α=1.9960, and ν =0.999 10. The results for the finite element solutions are λ_c =0.501 84, α=1.999 93, and ν =1.000 79 for the linear interpolation and λ_c =0.500 00, α=2.000 11, and ν =1.000 32 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.