氦和氢负离子波函数的赝谱计算
Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion
摘要：
本文用赝谱（PS）方法研究了双电子原子基态和激发态S态非相对论性薛定谔方程的数值解。计算实现收敛速度的能量，柯西误差的波函数，和方差的本地能量是指数快速的所有实际用途。该方法需要三个独立的子域来处理波函数的尖点附近的两粒子合并的行为。使用三个子域是必不可少的，以保持指数收敛。几种不同的治疗方法的尖瓣和半无限域的比较表明，最简单的处方是足够的。在许多情况下，不需要用特殊的方法处理三粒子聚并附近的对数行为。PS方法具有许多优点：不需要对波函数在尖点附近或大距离处的渐近行为作任何明确的假设，在所有配点处的局部能量与计算的整体能量完全相等，局部误差随分辨率的增加处处减小，使用Chebyshev多项式的有效基是完整和简单的，并且该方法易于推广到其他束缚态。这项研究作为更一般的两个和可能的三个电子的应用程序的方法的原理证明。
英文摘要：
We study the numerical solution of the non-relativistic Schrödinger equation for two-electron atoms in ground and excited S-states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wavefunction, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wavefunction's cusp-like behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence. A comparison of several different treatments of the cusps and the semi-infinite domain suggest that the simplest prescription is sufficient. For many purposes it proves unnecessary to handle the logarithmic behavior near the three-particle coalescence in a special way. The PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wavefunction near cusps or at large distances, the local energy is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. This study serves as a proof-of-principle of the method for more general two- and possibly three-electron applications.
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