In this paper, we solve the Schrödinger equation using the finite difference time domain (FDTD) method to determine energies and eigenfunctions. In order to apply the FDTD method, the Schrödinger equation is first transformed into a diffusion equation by the imaginary time transformation. The resulting timedomain diffusion equation is then solved numerically by the FDTD method. The theory and an algorithm are provided for the procedure. Numerical results are given for illustrative examples in one, two and three dimensions. It is shown that the FDTD method accurately determines eigenfunctions and energies of these systems.
Most of the Mie-scattering calculations have been done for a particle embedded in a nonabsorbing host medium. Generalization to an absorbing host medium can be achieved (a) by modifying the calculation of the spherical Bessel functions to account for a complex argument and (b) by accounting properly for the net rate of incident, scattered, and absorbed energy. We present an extended formalism of Mie scattering for the case of an absorbing host medium. Numerical calculations show that for a large spherical particle embedded in an absorbing host medium the extinction efficiency approaches 1 compared with 2 for a nonabsorbing host medium. We conjecture that this difference is due to the suppression of diffraction when the radius of the sphere is large.
In ab initio calculations a finite graphitic cluster model is often used to approximate the interaction energy of a water molecule with an infinite single-layer graphitic surface (graphene). In previous studies, the graphitic cluster model is a collection of fused benzene rings terminated by hydrogen atoms. In this study, the effect of using fluorine instead of hydrogen atoms for terminating the cluster model is examined to clarify the role of the boundary. The interaction energy of a water molecule with the graphitic cluster was computed using ab initio methods at the MP2 level of theory and with the 6-31G(d = 0.25) basis set. The interaction energy of a water molecule with graphene is estimated by extrapolation of two series of increasing size graphitic cluster models (C(6n2)H(6n) and C(6n2)F(6n), n = 1-3). Two fixed orientations of water molecule are considered: (a) both hydrogen atoms of water pointing toward the cluster (mode A) and (b) both hydrogen atoms of water pointing away from the cluster (mode B). The interaction energies for water mode A are found to be -2.39 and -2.49 kcal/mol for C(6n2)H(6n) and C(6n2)F(6n) cluster models, respectively. For water mode B, the interaction energies are -2.32 and -2.44 kcal/mol for C(6n2)H(6n) and C(6n2)F(6n) cluster models, respectively.
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