### 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 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.

Original language | English |
---|---|

Article number | 104105 |

Journal | Journal of Chemical Physics |

Volume | 131 |

Issue number | 10 |

DOIs | |

Publication status | Published - 25 Sep 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Physical and Theoretical Chemistry

### Cite this

*Journal of Chemical Physics*,

*131*(10), [104105]. https://doi.org/10.1063/1.3207909

**Comparison study of finite element and basis set methods for finite size scaling.** / Antillon, Edwin; Moy, Winton; Wei, Qi; Kais, Sabre.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 131, no. 10, 104105. https://doi.org/10.1063/1.3207909

}

TY - JOUR

T1 - Comparison study of finite element and basis set methods for finite size scaling

AU - Antillon, Edwin

AU - Moy, Winton

AU - Wei, Qi

AU - Kais, Sabre

PY - 2009/9/25

Y1 - 2009/9/25

N2 - 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 = 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 λc =0.499 99, α=1.9960, and ν =0.999 10. The results for the finite element solutions are λc =0.501 84, α=1.999 93, and ν =1.000 79 for the linear interpolation and λc =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.

AB - 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 = 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 λc =0.499 99, α=1.9960, and ν =0.999 10. The results for the finite element solutions are λc =0.501 84, α=1.999 93, and ν =1.000 79 for the linear interpolation and λc =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.

UR - http://www.scopus.com/inward/record.url?scp=70349283276&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349283276&partnerID=8YFLogxK

U2 - 10.1063/1.3207909

DO - 10.1063/1.3207909

M3 - Article

AN - SCOPUS:70349283276

VL - 131

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 10

M1 - 104105

ER -