The speed of calculating the electron densities is enhanced by the finite element method and large scale problems are accessed to replace the ab-initio method with finite element method. A three dimensional Thomas-Fermi-Amaldi equation for electron densities is formulated with newly-derived conditio...
The speed of calculating the electron densities is enhanced by the finite element method and large scale problems are accessed to replace the ab-initio method with finite element method. A three dimensional Thomas-Fermi-Amaldi equation for electron densities is formulated with newly-derived condition to remove the singularities at the nuclei. The suggested equation is discretized by the finite element method and verified by comparing with those from Hartree-Fock method. The modified Thomas-Fermi euqation introdued by Parr who removes the singularity on the position of an atom is extended to three dimensional modified Thomas-Fermi equation. Instead of Dirac exchange form, Amaldi exchange energy is adopted. The molecular cusp condition describes the slope of the electron density at the nucleus of molecule. Therefore, an extra equation to satisfy the molecular cusp condition is derived to complete the set of three dimensional TFA equation. Three dimensional TFA equation is discretized to determine the discrete electron densities directly by the finite element method and the electron density is chosen as the fundamental variable. The weak form of the TFA equation is obtained from vari-ational form of total energy functional and then collocation finite element method is applied to the weak form. Electron-electron repulsion term contributes to nonzero off-diagonal term in the Jacobian matrix making it a full matrix. Therefore, the electron-electron repulsion term should be replaced with the electrostatic potential, so the Jacobian matrix becomes sparse matrix. Poisson equation and three dimen-sional TFA equation are solved alternatively until self-consistency is reached and electron densities and elestrostatic potentials are updated at each step of iteration. Poisson equation is linear partial differential equation while three dimensional equation is nonlinear algebraic equation. Therefore the numerical solution for the three dimension TFA equation needs iterative scheme. On the other hand, the electron density should always satisfy the constraint of the non-negative electron density, thus direction of updating the electron densities also should not be violated this constraint. Rosen's gradient projection method is used to insure the non-negativity of the electron density. However, since the convergence rate is too slow, the arc-length method for nonlinear equation is exploited to enhance the convergence rate. To verify the present method, the energy and electron densities for atoms such as He, Be, Ne, Mg, Ar, and Ca atoms and molecules, such as H_(2), CH_(4), fullerene, carbon nanotube, and metal cluster of Zn are compared with the results from Hartree-Fock method. The electron densities matches reasonably well with those determined by the HF/6-31G method. Next, kinetic energy, electron-electron repulsion energy, electron-nucleus attraction energy, nucleus-nucleus repulsion energy, and total energy for the molecules mentioned above were also computed using the present method. Chemical potential for some atoms are close to the experimental data. The computational expense of the present method used to determine the electron density and energy is compared to that of the HF/6-31G. In conculsion, the computational expense of determining the electron density and its corresponding energy for a large scale structure, such as a carbon nanotube, is shown that the present method is much more efficient compared to that of conventional Hartree-Fock method using the 6-31G Pople basis set.
The speed of calculating the electron densities is enhanced by the finite element method and large scale problems are accessed to replace the ab-initio method with finite element method. A three dimensional Thomas-Fermi-Amaldi equation for electron densities is formulated with newly-derived condition to remove the singularities at the nuclei. The suggested equation is discretized by the finite element method and verified by comparing with those from Hartree-Fock method. The modified Thomas-Fermi euqation introdued by Parr who removes the singularity on the position of an atom is extended to three dimensional modified Thomas-Fermi equation. Instead of Dirac exchange form, Amaldi exchange energy is adopted. The molecular cusp condition describes the slope of the electron density at the nucleus of molecule. Therefore, an extra equation to satisfy the molecular cusp condition is derived to complete the set of three dimensional TFA equation. Three dimensional TFA equation is discretized to determine the discrete electron densities directly by the finite element method and the electron density is chosen as the fundamental variable. The weak form of the TFA equation is obtained from vari-ational form of total energy functional and then collocation finite element method is applied to the weak form. Electron-electron repulsion term contributes to nonzero off-diagonal term in the Jacobian matrix making it a full matrix. Therefore, the electron-electron repulsion term should be replaced with the electrostatic potential, so the Jacobian matrix becomes sparse matrix. Poisson equation and three dimen-sional TFA equation are solved alternatively until self-consistency is reached and electron densities and elestrostatic potentials are updated at each step of iteration. Poisson equation is linear partial differential equation while three dimensional equation is nonlinear algebraic equation. Therefore the numerical solution for the three dimension TFA equation needs iterative scheme. On the other hand, the electron density should always satisfy the constraint of the non-negative electron density, thus direction of updating the electron densities also should not be violated this constraint. Rosen's gradient projection method is used to insure the non-negativity of the electron density. However, since the convergence rate is too slow, the arc-length method for nonlinear equation is exploited to enhance the convergence rate. To verify the present method, the energy and electron densities for atoms such as He, Be, Ne, Mg, Ar, and Ca atoms and molecules, such as H_(2), CH_(4), fullerene, carbon nanotube, and metal cluster of Zn are compared with the results from Hartree-Fock method. The electron densities matches reasonably well with those determined by the HF/6-31G method. Next, kinetic energy, electron-electron repulsion energy, electron-nucleus attraction energy, nucleus-nucleus repulsion energy, and total energy for the molecules mentioned above were also computed using the present method. Chemical potential for some atoms are close to the experimental data. The computational expense of the present method used to determine the electron density and energy is compared to that of the HF/6-31G. In conculsion, the computational expense of determining the electron density and its corresponding energy for a large scale structure, such as a carbon nanotube, is shown that the present method is much more efficient compared to that of conventional Hartree-Fock method using the 6-31G Pople basis set.
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