含时薛定谔方程的R矩阵本征态基组与有限差分传播方法：单电子情形
A combined R-matrix eigenstate basis set and finite-differences propagation method for the time-dependent Schr\"{od}dinger equation: the one-electron case
摘要：
在这项工作中，我们提出了强电磁场下的原子和分子系统的含时薛定谔方程（TDSE）的解决方案的理论框架与电子的坐标的配置空间分离的两个区域，即区域$I$和$II$。在区域$I$中，TDSE的解由含时波函数的R-矩阵基组表示获得。在区域$II$中，考虑波函数的网格表示，并通过有限差分法获得空间和时间中的传播。这似乎是第一次提出基组和网格方法相结合来处理多区域时间相关问题。在这两个区域中，时间传播采用高阶显式格式。然而，在纯类氢系统中，由于这种分离而不涉及近似，在多电子系统中，本方法的有效性和有用性依赖于R-矩阵理论的基本假设，也就是说，在一定距离之外（包含区域$I$）单个喷射电子可与多电子系统的其他电子区分开，并在那里演化（区域II）有效地作为单电子系统。本文对单激活电子系统的方法作了详细的发展，并应用于强激光场中氢原子的典型情况。
英文摘要：
In this work we present the theoretical framework for the solution of the time-dependent Schrödinger equation (TDSE) of atomic and molecular systems under strong electromagnetic fields with the configuration space of the electron's coordinates separated over two regions, that is regions $I$ and $II$. In region $I$ the solution of the TDSE is obtained by an R-matrix basis set representation of the time-dependent wavefunction. In region $II$ a grid representation of the wavefunction is considered and propagation in space and time is obtained through the finite-differences method. It appears this is the first time a combination of basis set and grid methods has been put forward for tackling multi-region time-dependent problems. In both regions, a high-order explicit scheme is employed for the time propagation. While, in a purely hydrogenic system no approximation is involved due to this separation, in multi-electron systems the validity and the usefulness of the present method relies on the basic assumption of R-matrix theory, namely that beyond a certain distance (encompassing region $I$) a single ejected electron is distinguishable from the other electrons of the multi-electron system and evolves there (region II) effectively as a one-electron system. The method is developed in detail for single active electron systems and applied to the exemplar case of the hydrogen atom in an intense laser field.
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