Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada.
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada.
J Chem Phys. 2014 Feb 28;140(8):084104. doi: 10.1063/1.4866366.
We analyze two approaches to the quantum-classical Liouville (QCL) formalism that differ in the order of two operations: Wigner transformation and projection onto adiabatic electronic states. The analysis is carried out on a two-dimensional linear vibronic model where geometric phase (GP) effects arising from a conical intersection profoundly affect nuclear dynamics. We find that the Wigner-then-Adiabatic (WA) QCL approach captures GP effects, whereas the Adiabatic-then-Wigner (AW) QCL approach does not. Moreover, the Wigner transform in AW-QCL leads to an ill-defined Fourier transform of double-valued functions. The double-valued character of these functions stems from the nontrivial GP of adiabatic electronic states in the presence of a conical intersection. In contrast, WA-QCL avoids this issue by starting with the Wigner transform of single-valued quantities of the full problem. As a consequence, GP effects in WA-QCL can be associated with a dynamical term in the corresponding equation of motion. Since the WA-QCL approach uses solely the adiabatic potentials and non-adiabatic derivative couplings as an input, our results indicate that WA-QCL can capture GP effects in two-state crossing problems using first-principles electronic structure calculations without prior diabatization or introduction of explicit phase factors.
我们分析了量子经典刘维尔（QCL）形式主义的两种方法，它们在两个操作的顺序上有所不同：Wigner 变换和绝热电子态投影。分析是在二维线性振子模型上进行的，其中来自圆锥交叉的几何相位（GP）效应深刻地影响了核动力学。我们发现，Wigner 先绝热（WA）QCL 方法可以捕获 GP 效应，而绝热先 Wigner（AW）QCL 方法则不能。此外，AW-QCL 中的 Wigner 变换导致双值函数的傅里叶变换不明确。这些函数的双值性质源于在圆锥交叉存在的情况下绝热电子态的非平凡 GP。相比之下，WA-QCL 通过从全问题的单值量的 Wigner 变换开始来避免这个问题。因此，WA-QCL 中的 GP 效应可以与相应运动方程中的动力学项相关联。由于 WA-QCL 方法仅将绝热势和非绝热导数耦合作为输入，因此我们的结果表明，WA-QCL 可以使用第一性原理电子结构计算在两态交叉问题中捕获 GP 效应，而无需先进行 diabatic 化或引入显式相位因子。
