Wodraszka Robert, Carrington Tucker
Chemistry Department, Queen's University, Kingston, Ontario K7L 3N6, Canada.
J Chem Phys. 2020 Apr 30;152(16):164117. doi: 10.1063/5.0006081.
Although very useful, the original multi-configuration time-dependent Hartree (MCTDH) method has two weaknesses: (1) its cost scales exponentially with the number of atoms in the system; (2) the standard MCTDH implementation requires that the potential energy surface (PES) be in the sum-of-product (SOP) form in order to reduce the cost of computing integrals in the MCTDH basis. One way to deal with (1) is to lump coordinates into groups. This is mode combination (MC). One way to deal with (2) is to reformulate MCTDH using collocation so that there are no integrals. In this paper, we combine MC and collocation to formulate a MC collocation multi-configuration time-dependent Hartree (MC-C-MCTDH) method. In practice, its cost does not scale exponentially with the number of atoms, and it can be used with any general PES; the PES need not be an SOP and need not have a special form. No integrals and, hence, no quadratures are necessary. We demonstrate the accuracy and efficiency of the new method by computing vibrational energy eigenstates of methyl radical, methane, and acetonitrile. To do this, we use MC-C-MCTDH with a variant of improved relaxation, derived by evaluating a residual at points. Because the MC basis functions are multivariate, collocation points in multi-dimensional spaces are required. We use two types of collocation points: (1) discrete variable representation-like points obtained from (approximate) simultaneous diagonalization of matrices and (2) Leja points, which are known to be good interpolation points, determined from a generalized recipe suitable for any basis.
尽管原始的多组态含时 Hartree（MCTDH）方法非常有用，但它有两个缺点：（1）其计算成本随系统中原子数量呈指数增长；（2）标准的 MCTDH 实现要求势能面（PES）为乘积和（SOP）形式，以便降低在 MCTDH 基中计算积分的成本。处理（1）的一种方法是将坐标归为组。这就是模式组合（MC）。处理（2）的一种方法是使用配置法重新构建 MCTDH，这样就无需积分。在本文中，我们将 MC 和配置法相结合，构建了一种 MC 配置多组态含时 Hartree（MC-C-MCTDH）方法。在实际应用中，其成本不会随原子数量呈指数增长，并且可以与任何一般的 PES 一起使用；PES 不必是 SOP 形式，也不必具有特殊形式。无需积分，因此也无需求积。我们通过计算甲基自由基、甲烷和乙腈的振动能量本征态来证明新方法的准确性和效率。为此，我们使用带有改进弛豫变体的 MC-C-MCTDH，该变体通过在点处评估残差得出。由于 MC 基函数是多元的，所以需要多维空间中的配置点。我们使用两种类型的配置点：（1）从矩阵的（近似）同时对角化得到的类似离散变量表示的点，以及（2）Leja 点，已知它是很好的插值点，由适用于任何基的通用方法确定。
