论文标题
在量子化学中，汇总序列中的奇异值和新的轨道订购方法的反转对称性
Inversion symmetry of singular values and a new orbital ordering method in tensor train approximations for quantum chemistry
论文作者
论文摘要
电子波函数的张量序列近似位于QC-DMRG（量子化学密度矩阵重归化组）的核心，这是一种用于数值求解$ n $ electronSchrödinger方程的最新方法。众所周知，TT近似值的准确性受相关奇异值的尾部的控制，这反过来又大大取决于单体基础的排序。 Here we find that the singular values $s_1\ge s_2\ge ... \ge s_d$ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, $s_1s_d=s_2s_{d-1}=s_3s_{d-2}=...$, thus reducing the tail behaviour to a single hidden invariant, which moreover depends explicitly on the订购基础。对于相关的波形，我们发现尾巴是由不变的合适叠加的上限。因此，优化不变性或其叠加，为QC-DMRG提供了新的订购方案。对简单示例的数值测试，即一些Slater决定因素的线性组合，表明新方案将单数值的尾巴降低了现有方法的几个级数，包括广泛使用的Fiedler订单。
The tensor train approximation of electronic wave functions lies at the core of the QC-DMRG (Quantum Chemistry Density Matrix Renormalization Group) method, a recent state-of-the-art method for numerically solving the $N$-electron Schrödinger equation. It is well known that the accuracy of TT approximations is governed by the tail of the associated singular values, which in turn strongly depends on the ordering of the one-body basis. Here we find that the singular values $s_1\ge s_2\ge ... \ge s_d$ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, $s_1s_d=s_2s_{d-1}=s_3s_{d-2}=...$, thus reducing the tail behaviour to a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wavefunctions, we find that the tail is upper bounded by a suitable superposition of the invariants. Optimizing the invariants or their superposition thus provides a new ordering scheme for QC-DMRG. Numerical tests on simple examples, i.e. linear combinations of a few Slater determinants, show that the new scheme reduces the tail of the singular values by several orders of magnitudes over existing methods, including the widely used Fiedler order.
