The primal definition of first-order non-adiabatic couplings among electronic states relies on the knowledge of how electronic wavefunctions vary with nuclear coordinates. However, the non-adiabatic coupling between two electronic states can be obtained in the vicinity of a conical intersection from energies only, as this vector spans the branching plane along which degeneracy is lifted to first order. The gradient difference and derivative coupling are responsible of the two-dimensional cusp of a conical intersection between both potential-energy surfaces and can be identified to the non-trivial eigenvectors of the second derivative of the square energy difference, as first pointed out in Köppel and Schubert [Mol. Phys. 104(5-7), 1069 (2006)]. Such quantities can always be computed in principle for the cost of two numerical Hessians in the worst-case scenario. Analytic-derivative techniques may help in terms of accuracy and efficiency but also raise potential traps due to singularities and ill-defined derivatives at degeneracies. We compare here two approaches, one fully numerical, the other semianalytic, where analytic gradients are available but Hessians are not, and investigate their respective conditions of applicability. Benzene and 3-hydroxychromone are used as illustrative application cases. It is shown that non-adiabatic couplings can thus be estimated with decent accuracy in regions of significant size around conical intersections. This procedure is robust and could be useful in the context of on-the-fly non-adiabatic dynamics or be used for producing model representations of intersecting potential energy surfaces with complete obviation of the electronic wavefunctions.

中文翻译：

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无波函数的基于能量的过程在圆锥形交叉点周围生成一阶非绝热耦合的适用性

电子状态间一阶非绝热耦合的原始定义依赖于电子波函数如何随核坐标变化的知识。但是，仅在能量的圆锥形交叉点附近，才能获得两个电子状态之间的非绝热耦合，因为该矢量跨越了将简并性提升至一阶的分支平面。梯度差和导数耦合是两个势能面之间的圆锥形交点的二维尖峰的原因，并且可以确定为平方能量差的二阶导数的非平凡本征向量，如Köppel首先指出的那样。和舒伯特[Mol。物理 104（5-7），1069（2006）]。在最坏的情况下，这样的数量原则上总是可以用两个数字Hessian的成本来计算。解析导数技术可能在准确性和效率方面有所帮助，但由于奇异性和简并性时不确定的导数，也会增加潜在的陷阱。我们在这里比较两种方法，一种是完全数值方法，另一种是半解析方法，在这种方法中可以使用解析梯度，但没有黑森斯方法，并研究了它们各自的适用条件。苯和3-羟基色酮用作说明性应用案例。结果表明，非绝热耦合可以在圆锥形交叉点附近的较大尺寸区域中以适当的精度进行估计。