Roberto Reyes-García, Salvador A. Cruz, Remigio Cabrera-Trujillo
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On the wavefunction cutoff factors of atomic hydrogen confined by an impenetrable spherical cavity
The Schrödinger equation for the hydrogen atom enclosed by an impenetrable spherical cavity is solved through a Finite-Differences approach to gain an insight on the actual nature and structure of the ansatz wavefunction cutoff factor widely used in an ad hoc manner in corresponding variational calculations to comply with the Dirichlet boundary conditions. The results of this work provide a theoretical foundation for the choice of the appropriate analytical cutoff functions that fulfill the boundary conditions. We find three different regions for the behavior of the cutoff functions. Small cavity radius where the cutoff function has a parabolic behavior, an intermediate region where the cutoff function is quasi-linear, and a large cavity region where the cutoff function is a step-like function. We deduce the traditional linear and quadratic cutoff functions used in the literature as well as its validity region for the confining radius. Finally, we provide a mathematical deduction of the exact cutoff function in terms of the nodal structure of the free hydrogenic wavefunctions and a relation to the Laguerre polynomials for some cavity radii where the free atomic energy level coincides with a confined energy level. We find that the cutoff function transit over several unconfined solutions in terms of its nodal structure as the system is compressed.
期刊介绍:
Since its first formulation quantum chemistry has provided the conceptual and terminological framework necessary to understand atoms, molecules and the condensed matter. Over the past decades synergistic advances in the methodological developments, software and hardware have transformed quantum chemistry in a truly interdisciplinary science that has expanded beyond its traditional core of molecular sciences to fields as diverse as chemistry and catalysis, biophysics, nanotechnology and material science.