An Efficient Chebyshev–Lanczos Method for Obtaining Eigensolutions of the Schrödinger Equation on a Grid

✍ Scribed by M. Braun; S.A. Sofianos; D.G. Papageorgiou; I.E. Lagaris

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 432 KB
• Volume: 126
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.0140

✦ Synopsis

been employed in bound state calculations, where the spectrum of the Hamiltonian can be obtained from the solution A grid method for obtaining eigensolutions of bound systems is presented. In this, the block-Lanczos method is applied to a Chebyof the TDSE. The problem usually encountered is the shev approximation of exp(ϪH/⌬), where ⌬ is the range of eigenvaldiagonalization of the Hamiltonian H in (1). Since this in ues we are interested in. With this choice a preferential convergence practice cannot be done, it is impossible to compute the of the eigenvectors corresponding to low-lying eigenvalues of H is action of the propagator e ϪitH on an arbitrary wave function achieved. The method is used to solve a variety of one-, two-, and three-dimensional problems. To apply the kinetic energy operator [2] and, thus, the time-evolution is accomplished using we use the fast sine transform instead of the fast Fourier transform, various approximations to the exponential. We mention thus fullfilling, a priori, the box boundary conditions. We further here the Crank-Nicolson scheme [3-5] and the Kosloff extend the Chebyshev approximation to treat general functions of and Tal-Ezer method [6]. For a detailed discussion we refer matrices, thus allowing its application to cases for which no analytithe reader to the review article of de Raedt [2]. The existing cal expressions of the expansion coefficients are available. ᮊ 1996 Academic Press, Inc.

methods, although simple to use, can be quite CPU-time intensive, especially in calculating a large number of bound states in more than one dimension.