Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain.
Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, E-46022 Valencia, Spain.
J Chem Phys. 2018 Jun 28;148(24):244109. doi: 10.1063/1.5036838.
We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of the said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similar to the well-known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples.
我们考虑用含时哈密顿量的薛定谔方程的数值积分，其中含时哈密顿量被表示为动能和时变位势之和。无对易（CF）传播子是指数传播子，已被证明对一般的时变哈密顿量非常有效。我们提出了针对上述结构哈密顿量的新的 CF 传播子，表现出相当大的性能提升。我们得到了新的四阶和六阶 CF 传播子，以及一种新的六阶传播子，其中包含一个仅依赖坐标的双对易项，因此这个项可以被认为是免费的。该算法需要计算指数在向量上的作用，类似于著名的指数中点传播子，这是通过 Lanczos 方法完成的。我们在几个数值例子上展示了新方法的性能。
