高激发矩阵乘积态的优化及其在振动光谱中的应用
Optimization of highly excited matrix product states with an application to vibrational spectroscopy
摘要：
电子和振动结构的双作用型计算通常是可靠地近似多粒子波函数和能量的方法。然而，指数缩放限制了它们的应用范围。利用密度矩阵重整化群（DMRG）算法可以得到一个有效的近似全组态相互作用解，而不受预先设定的激发能级的限制。然而，在标准DMRG实现中，利用与所有低位波函数解正交的空间中的基态优化来计算激发态。因此，不可能进行简单的并行化，并且高激发态的计算变得极其昂贵，特别是在具有高态密度的区域中。在这里，我们介绍了两个变种的密度矩阵重整化群算法，使我们能够直接针对特定的能量区域，因此高度激发态。第一个，基于移位和反转技术，是特别有效的低洼状态，但不稳定的区域具有高密度的状态。第二种基于折叠辅助算子，效率较低，但在靶向高能态方面更准确。我们将该算法应用于核薛定谔方程的解，但强调它也可以应用于一般哈密顿量的对角化，例如电子库仑哈密顿量，以解决X射线光谱。结合几种根归巢算法和行列式空间的随机采样，可以在优化过程中充分跟踪和分析感兴趣的激发态。我们证明，我们可以准确地计算突出的光谱特征的大分子，如肌氨酸-甘氨酸二肽。
英文摘要：
Configuration-interaction-type calculations on electronic and vibrational structure are often the method of choice for the reliable approximation of many-particle wave functions and energies. The exponential scaling, however, limits their application range. An efficient approximation to the full configuration interaction solution can be obtained with the density matrix renormalization group (DMRG) algorithm without a restriction to a predefined excitation level. In a standard DMRG implementation, however, excited states are calculated with a ground-state optimization in the space orthogonal to all lower lying wave function solutions. A trivial parallelization is therefore not possible and the calculation of highly excited states becomes prohibitively expensive, especially in regions with a high density of states. Here, we introduce two variants of the density matrix renormalization group algorithm that allow us to target directly specific energy regions and therefore highly excited states. The first one, based on shift-and-invert techniques, is particularly efficient for low-lying states, but is not stable in regions with a high density of states. The second one, based on the folded auxiliary operator, is less efficient, but more accurate in targeting high-energy states. We apply the algorithm to the solution of the nuclear Schroedinger equation, but emphasize that it can be applied to the diagonalization of general Hamiltonians as well, such as the electronic Coulomb Hamiltonian to address X-ray spectra. In combination with several root-homing algorithms and a stochastic sampling of the determinant space, excited states of interest can be adequately tracked and analyzed during the optimization. We demonstrate that we can accurately calculate prominent spectral features of large molecules such as the sarcosyn-glycine dipeptide.
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