Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA.
J Chem Phys. 2019 Feb 7;150(5):054102. doi: 10.1063/1.5058268.
Solving the vibrational Schrödinger equation in the neighborhood of conical intersections in the adiabatic representation is a challenge. At the intersection point, first- and second-derivative nonadiabatic coupling matrix elements become singular, with the singularity in the second-derivative coupling (diagonal Born-Oppenheimer correction) being non-integrable. These singularities result from discontinuities in the vibronic functions associated with the individual adiabatic states, and our group has recently argued that these divergent matrix elements cancel when discontinuous adiabatic vibronic functions sum to a continuous total nonadiabatic wave function. Here we describe the realization of this concept: a novel scheme for the numerically exact solution of the Schrödinger equation in the adiabatic representation. Our approach is based on a basis containing functions that are discontinuous at the intersection point. We demonstrate that the individual adiabatic nuclear wave functions are themselves discontinuous at the intersection point. This proves that discontinuous basis functions are essential to any tractable method that solves the Schrödinger equation around conical intersections in the adiabatic representation with high numerical precision. We establish that our method provides numerically exact results by comparison to reference calculations performed in the diabatic representation. In addition, we quantify the energetic error associated with constraining the density to be zero at the intersection point, a natural approximation. Prospects for extending the present treatment of a two-dimensional model to systems of higher dimensionality are discussed.
在绝热表象中求解圆锥交叉点附近的振动薛定谔方程是一项挑战。在交叉点处，一阶和二阶非绝热耦合矩阵元变得奇异，二阶耦合（对角 Born-Oppenheimer 修正）的奇异是不可积的。这些奇异来自与各个绝热态相关的振子函数的不连续性，我们的研究小组最近认为，当不连续的绝热振子函数加起来形成连续的总非绝热波函数时，这些发散的矩阵元会相互抵消。在这里，我们描述了这一概念的实现：一种在绝热表象中精确求解薛定谔方程的新方案。我们的方法基于在交叉点处不连续的函数基。我们证明了单个绝热核波函数本身在交叉点处是不连续的。这证明了不连续的基函数对于任何在绝热表象中用高数值精度解决圆锥交叉点处的薛定谔方程的可处理方法都是必不可少的。我们通过与在非绝热表象中进行的参考计算进行比较，证明了我们的方法提供了数值精确的结果。此外，我们还量化了将密度约束在交叉点处为零的能量误差，这是一种自然的近似。讨论了将目前对二维模型的处理扩展到更高维系统的前景。
