Time development of many-electron wave functions based on the time-dependent variational principle toward separation of collective coordinates

含时变分原理(TDVP)是分子中电子转移过程的多电子波函数形式。TDVP中的定常作用原理导致了变分参数的伪经典力学。所有为研究经典力学而开发的丰富仪器都可以应用于TDVP。对TDVP的灵敏度分析也进行了研究。灵敏度分析是一种评估非线性系统的解对外部参数的依赖程度的技术。所得到的灵敏度方程可以看作是Jacobi方程对由外部参数引起的测地线偏差的推广。例如，在分子的Born-Oppenheimer近似中，灵敏度分析使我们能够计算由于核坐标而引起的电子态的动态变化。从TDVP的伪经典力学观点出发，灵敏度分析评估了附近…之间的偏差更多的轨迹，即由外部参数引起的测地线偏差。此外，如果我们考虑TDVP中的约束，伪经典力学的结构使我们能够利用狄拉克的约束经典力学。根据狄拉克的术语，TDVP中的约束被分为第一类和第二类。约束的拉格朗日乘子由相容条件决定。灵敏度分析继承了TDVP中的约束。然而，在灵敏度分析中，约束的一致性条件被自动满足。我们只需要对灵敏度函数的初值施加约束。文中给出了灵敏度分析的两个实例。一种是对在绝热基上线性展开的波函数的简单分析。另一个例子是一般波函数在定态邻域内的测地线偏差，动力学稳定性和静态稳定性是相互联系的。较少

Time-dependent variational principle (TDVP) is formulated in many-electron wave functions for electron-transfer processes in molecules. The stationary action principle in the TDVP leads to the pseudo classical-mechanics of variational parameters. All the rich apparatus developed for the study of classical mechanics can be applied to the TDVP. The sensitivity analysis for the TDVP is also investigated. The sensitivity analysis is a technique to assess how the solutions in nonlinear systems depend on external parameters. The obtained sensitivity equations can be considered as an extension of the Jacobi equation to the geodesic deviation which is caused by external parameters. For example, in the Born-Oppenheimer approximation for molecules, the sensitivity analysis enables us to calculate the dynamical variation of electronic states due to nuclear coordinates. From the viewpoint of the pseudo classical-mechanics of the TDVP, the sensitivity analysis assesses the deviation between nearby … More trajectories, i.e., the geodesic deviation caused by external parameters. Moreover, if we consider constraints in the TDVP, the structure of the pseudo classical-mechanics enables us to utilize Dirac's constrained classical-mechanics. The constraints in the TDVP are classified into the first-and second-classes according to Dirac's terminology. The Lagrange multipliers for the constraints are determined by the consistency conditions. The sensitivity analysis inherits the constraints in the TDVP. The consistency conditions for the constraints are, however, satisfied automatically in the sensitivity analysis. We need to impose the constraints only on initial values of the sensitivity functions. Two illustrative examples of the sensitivity analysis are investigated. One is a simple analysis for wave functions which are linearly expanded in adiabatic bases. The other example is the geodesic deviation in the neighborhood of stationary states for general wave functions and the dynamical and the statical stabilities are related with each other. Less