Departament de Química Física, Universitat d'Alacant, E-03080 Alacant, Spain.
J Chem Phys. 2010 Jan 14;132(2):024110. doi: 10.1063/1.3291345.
A new method for solving the Schrödinger equation is proposed, based on the following details. First, a map u=u(r) from Cartesian coordinates r to a new coordinate system u is chosen. Second, the solution (orbital) psi(r) is written in terms of a function U depending on u so that psi(r)=/J(u)/(-1/2)U(u), where /J(u)/ is the Jacobian determinant of the map. Third, U is expressed as a linear combination of plane waves in the u coordinate, U(u)= sum (k)c(k)e(ik x u). Finally, the coefficients c(k) are variationally optimized to obtain the best energy, using a generalization of an algorithm originally developed for the Coulomb potential [J. M. Perez-Jorda, Phys. Rev. B 58, 1230 (1998)]. The method is tested for the radial Schrödinger equation in the hydrogen atom, resulting in micro-Hartree accuracy or better for the energy of ns and np orbitals (with n up to 5) using expansions of moderate length.
提出了一种求解薛定谔方程的新方法，其基本思想如下。首先，选择一个从笛卡尔坐标系 r 到新坐标系 u 的映射 u=u(r)。其次，将解（轨道）psi(r)表示为依赖于 u 的函数 U，使得 psi(r)=/J(u)/(-1/2)U(u)，其中 /J(u)/是映射的雅可比行列式。第三，将 U 表示为 u 坐标中的平面波的线性组合，U(u)= sum (k)c(k)e(ik x u)。最后，使用最初为库仑势开发的算法的推广，通过变分优化系数 c(k)以获得最佳能量[J. M. Perez-Jorda, Phys. Rev. B 58, 1230 (1998)]。该方法已针对氢原子的径向薛定谔方程进行了测试，对于 ns 和 np 轨道（n 高达 5），使用中等长度的展开，其能量的精度达到微哈特里或更高。
