用广义模跟踪法收敛于鞍点计算电子激发态
Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following
摘要：
激发电子态的变分计算是通过在表面上寻找鞍点来进行的，鞍点描述了系统的能量如何作为电子自由度的函数而变化。这种方法有几个优点，通常使用的方法，特别是在密度泛函计算的上下文中，作为崩溃的基态是避免的，然而，轨道变优化的激发态。这种优化使得有可能描述具有大电荷转移的激发，其中基于基态轨道的计算是有问题的，如在线性响应时间依赖性密度泛函理论中。本文提出了一种广义模式跟踪方法，通过对电子Hessian矩阵的$n$最低本征值的本征向量方向上的梯度分量求逆，找到了$n^{\text{th}}$阶鞍点。这种方法具有以下的一个选定的激发态通过原子配置的单一行列式波函数的对称性被打破，在乙烯和二氢分子中的核运动的势能曲线的计算所证明的明显优势。该方法使用广义Davidson算法和指数变换来实现，用于更新能量泛函的广义梯度近似内的轨道。收敛被发现是更强大的比一个直接的优化方法，以前表现出优于标准的自洽场的方法，如这里所示的电荷转移激发硝基苯和N-苯基吡咯，涉及计算的$4^{\text{th}}$ -和$6^{\text{th}}$ -阶鞍点，分别。最后，通过对一个双铂银配合物的计算，说明了该方法对大分子激发态能量曲线的适用性。
英文摘要：
Variational calculations of excited electronic states are carried out by finding saddle points on the surface that describes how the energy of the system varies as a function of the electronic degrees of freedom. This approach has several advantages over commonly used methods especially in the context of density functional calculations, as collapse to the ground state is avoided and yet, the orbitals are variationally optimized for the excited state. This optimization makes it possible to describe excitations with large charge transfer where calculations based on ground state orbitals are problematic, as in linear response time-dependent density functional theory. A generalized mode following method is presented where an $n^{\text{th}}$-order saddle point is found by inverting the components of the gradient in the direction of the eigenvectors of the $n$ lowest eigenvalues of the electronic Hessian matrix. This approach has the distinct advantage of following a chosen excited state through atomic configurations where the symmetry of the single determinant wave function is broken, as demonstrated in calculations of potential energy curves for nuclear motion in the ethylene and dihydrogen molecules. The method is implemented using a generalized Davidson algorithm and an exponential transformation for updating the orbitals within a generalized gradient approximation of the energy functional. Convergence is found to be more robust than for a direct optimization approach previously shown to outperform standard self-consistent field approaches, as illustrated here for charge transfer excitations in nitrobenzene and N-phenylpyrrole, involving calculations of $4^{\text{th}}$- and $6^{\text{th}}$-order saddle points, respectively. Finally, calculations of a diplatinum and silver complex are presented, illustrating the applicability of the method to excited state energy curves of large molecules.
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