Littlejohn Robert, Rawlinson Jonathan, Subotnik Joseph
Department of Physics, University of California, Berkeley, California 94720, USA.
School of Mathematics, University of Manchester, Manchester, United Kingdom.
J Chem Phys. 2024 Mar 21;160(11). doi: 10.1063/5.0192465.
This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.
本文描述了一种在玻恩-奥本海默理论中计算高阶或非绝热修正及其与平动自由度相互作用的方法。该方法利用维格纳-魏尔对应将核算符映射到经典相空间上的函数，并使用莫雅星积来表示这些函数上的算符乘法。本文主体部分对这些内容进行了解释。结果是一个关于κ2的幂级数，其中κ = (m/M)1/4是通常的玻恩-奥本海默参数。最低阶项是通常的玻恩-奥本海默近似，而高阶项是非绝热修正。在计算电子电流、动量和密度时需要这些修正。核自由度和电子自由度的分离是在平动和转动的精确对称性（对于孤立分子）的背景下进行的，并且这些对称性，特别是平动对称性，被明确纳入我们的讨论。本文给出了分子玻恩-奥本海默理论中莫雅展开的独立推导。我们展示了如何在莫雅微扰理论框架内计算电子电流和动量；我们推导了电子哈密顿量、电子本征态以及平动下导数耦合的变换规律；我们详细讨论了玻恩-奥本海默表示中分子质心的直线运动；并且我们展示了消除导数耦合的平动分量如何导致一个幺正变换，该变换具有精确分离平动自由度的效果。
