Dimension-adaptive sparse grid product methods for the Schrödinger equation

由于电子Schrödinger方程的高维性，它的任何直接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，而不是离散化过程。在这个项目中，我们建议使用稀疏网格方法来直接离散Schrödinger的方程。传统的稀疏网格技术允许将d维问题的复杂性从d的指数缩放降低到d的几乎线性缩放，前提是满足某些平滑假设。它采用多级基表示单粒子状态，采用一定的行列式积方法表示多粒子状态，并考虑了非对称性（泡利原理）。对相应的多阶级数展开进行一定的截断，直接导致整个电子空间的成本最优离散化。在这里，一个维度自适应程序允许检测单粒子状态之间的相关性。这种新方法为将n电子问题的计算复杂度降低到单电子问题提供了前景。对于单粒子状态的多级基（实空间，傅立叶空间）的不同选择，我们将实现所得到的自适应维度稀疏网格方法，并比较它们在Schrödinger方程中的性质。此外，这些代码稍后将在分布式内存处理器上并行化和实现。

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.