Minimum-Error Method for Scattering Problems in Quantum Mechanics: Two Stable and Efficient Implementations

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• 作者：Burcin Temel ; Greg Mills ; Horia Metiu
• 刊名：Journal of Physical Chemistry A
• 出版年：2006
• 出版时间：September 7, 2006
• 年：2006
• 卷：110
• 期：35
• 页码：10513 - 10520
• 全文大小：94K
• 年卷期：v.110,no.35(September 7, 2006)
• ISSN：1520-5215

文摘

We examine here, by using a simple example, two implementations of the minimum error method (MEM),a least-squares minimization for scattering problems in quantum mechanics, and show that they provide anefficient, numerically stable alternative to Kohn variational principle. MEM defines an error-functionalconsisting of the sum of the values of (H - E)² at a set of grid points. The wave function , is forcedto satisfy the scattering boundary conditions and is determined by minimizing the least-squares error. Westudy two implementations of this idea. In one, we represent the wave function as a linear combination ofChebyshev polynomials and minimize the error by varying the coefficients of the expansion and the R-matrix(present in the asymptotic form of ). This leads to a linear equation for the coefficients and the R-matrix,which we solve by matrix inversion. In the other implementation, we use a conjugate-gradient procedure tominimize the error with respect to the values of at the grid points and the R-matrix. The use of the Chebyshevpolynomials allows an efficient and accurate calculation of the derivative of the wave function, by using FastChebyshev Transforms. We find that, unlike KVP, MEM is numerically stable when we use the R-matrixasymptotic condition and gives accurate wave functions in the interaction region.