不需要网格：含时薛定谔方程的自适应全柔性高斯
No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schrödinger equation
摘要：
复杂的高斯函数的线性组合，其中的线性和非线性参数是允许变化的，被证明是一个非常灵活和有效的方法来解决时间依赖的薛定谔方程在一个空间维度。众所周知，在狄拉克-弗伦克尔变分原理的系统学中，灵活基组的使用是非常困难的。在这项工作中，我们提出了一种替代的时间传播计划，不强调最佳参数的演变，但直接通过Rothe的方法，也被称为垂直时间层的方法的残差最小化的目标。我们使用一个简单的模型系统模拟原子受到极端的激光脉冲来测试该方案。这样的脉冲产生系统的复杂电离动力学。该计划被证明在这个模型上表现得非常好，特别是不依赖于计算网格。只需要少量的高斯函数就可以达到与高分辨率、基于网格的求解器相当的精度。这为在玻恩-奥本海默近似下和玻恩-奥本海默近似之外的原子和分子的含时薛定谔方程的精确和可负担的解铺平了道路。
英文摘要：
Linear combinations of complex gaussian functions, where the linear and nonlinear parameters are allowed to vary, are shown to provide an extremely flexible and effective approach for solving the time-dependent Schrödinger equation in one spatial dimension. The use of flexible basis sets has been proven notoriously hard within the systematics of the Dirac--Frenkel variational principle. In this work we present an alternative time-propagation scheme that de-emphasizes optimal parameter evolution but directly targets residual minimization via the method of Rothe's method, also called the method of vertical time layers. We test the scheme using a simple model system mimicking an atom subjected to an extreme laser pulse. Such a pulse produces complicated ionization dynamics of the system. The scheme is shown to perform very well on this model and notably does not rely on a computational grid. Only a handful of gaussian functions are needed to achieve an accuracy on par with a high-resolution, grid-based solver. This paves the way for accurate and affordable solution of the time-dependent Schrödinger equation for atoms and molecules within and beyond the Born--Oppenheimer approximation.
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