Expansion of a wave function in a Gaussian basis. I. Local versus global approximation
Expansion of a wave function in a Gaussian basis. I. Local versus global approximation
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高斯基波函数的展开。
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
W. Kutzelnigg
中科院分区：
文献类型：
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作者：
W. Kutzelnigg
类氢离子的基态波函数ψ(R)在高斯基下的展开式是由高斯积分变换的离散化得到的。这种推导涉及到积分变量的上下限，以及在变量变换为指数衰减的钟形被积之后，通过梯形近似来计算截断的积分。这会自动产生一个优化的均匀调和基础。研究了两个判据：(A)当r=0时ψ(R)的最佳逼近；(B)哈密顿量期望值的最佳逼近。在任何一种情况下，如果n是基数大小，则收敛速度服从带有误差估计的平方根指数规律。常数a的值取决于所考虑的性质和基优化的标准。给出了均匀回火高斯基的基参数α和β的简单解析表达式。本地和全球标准的比较提供了一些新的、甚至是意想不到的见解。©2012 Wiley期刊，Inc.
The expansion of the ground state wave function ψ(r) of hydrogen-like ions in a Gaussian basis is derived from a discretization of a Gaussian integral transformation. This derivation involves an upper and a lower cut-off of the integration variable and the evaluation of the truncated integral by means of the trapezoid approximation, after a variable transformation to an exponentially decaying bell-shaped integrand. This automatically leads to an optimized even-tempered basis. Two criteria are studied, (a) the best approximation of ψ(r) for r = 0, (b) the best approximation for the expectation value of the Hamiltonian. In either case, the rate of convergence obeys a square-root exponential law with an error estimate , if n is the basis size. The value of the constant a depends on the considered property and on the criterion for the optimization of the basis. Simple analytic expressions for the basis parameters α and β of an even-tempered Gaussian basis are derived. The comparison of the local and global criteria gives some new and even unexpected insight. © 2012 Wiley Periodicals, Inc.