Dimension-adaptive sparse grid product methods for the Schrödinger equation

由于电子薛定谔方程的高维性，任何直接的数值解都是不可能的。因此，使用不同的近似，如HF，CI/CC和DFT。然而，这些方法更类似于简化模型，而不是离散化程序。在这个项目中，我们建议使用稀疏网格方法直接离散薛定谔方程。传统的稀疏网格技术允许减少的d维问题的复杂性，从指数缩放的d几乎线性缩放的d，提供了一定的平滑假设得到满足。它采用多能级基表示单粒子态，采用一定的行列式积方法表示多粒子态，并考虑了反对称性（泡利原理）。相应的多级级数展开的某种截断直接导致总电子空间的成本最优离散化。在这里，维度自适应过程允许检测单粒子状态之间的相关性。这种新的方法给出的观点，以减少计算复杂性的N-电子问题的一个电子的问题。对于单粒子态的多级基（真实的空间，傅立叶空间）的不同选择，我们将实现由此产生的维数自适应稀疏网格方法，并比较它们对薛定谔方程的性质。此外，这些代码稍后将被并行化并在分布式内存处理器上实现。

Any direct numerical solution of the electronic Schrödinger equation is impossible due to its high dimensionality. Therefore, different approximations like HF, CI/CC, and DFT are used. However, these approaches more resemble simplified models than discretization procedures. In this project, we propose to use a sparse grid method for the direct discretization of Schrödinger `s equation. The conventional sparse grid technique allows to reduce the complexity of a d-dimensional problem from exponential scaling in d to almost linear scaling in d, provided that certain smoothness assumptions are fulfilled. It uses a multi-level basis to represent one-particle states and employs a certain determinant-product approach to represent many-particle states, which takes anti-symmetry (Pauli priciple) into account. A certain truncation of the corresponding multi-level series expansion directly results in a cost-optimal discretization of the total electronic space. Here, a dimension-adaptive procedure allows to detect correlations between one-particle states. This new approach gives the perspective to reduce the computational complexity of a N-electron problem to that of a one-electron problem. For different choices of multi-level bases (real space, Fourier space) for the one-particle state, we will implement the resulting dimension-adaptive sparse grid approaches and compare their properties for Schrödinger ´s equation. Furthermore, this code will later be parallelized and implemented on distributed memory processors.