Terms Of Gaussian Basis Functions Research Articles

Semi-analytical model of an acoustic black hole piezoelectric bimorph cantilever for energy harvesting

An acoustic black hole (ABH) beam termination can be achieved by decreasing its thickness according to a power-law profile. Waves entering the ABH slow down and vibrational energy strongly concentrates at the tip of the beam. This can be exploited for energy harvesting, as suggested in some recent works. The finite element method (FEM) is commonly used to carry out the simulations, which hampers long parametric analyzes. In this paper, we develop a semi-analytical approach to characterize the performance of a piezolectric bimorph cantilever with an ABH termination. The method can be easily extended to further configurations and allows one to determine ABH harvesting capabilities when varying system parameters, in a fast and efficient way. The Lagrangian of the ABH beam plus piezoelectric layers is constructed and the coupled equations for the flexural vibrations and voltage are derived from it. The flexural displacement field is expanded in terms of Gaussian basis functions. Vibration shapes and harvested power are computed with the proposed method and validated against FEM simulations. The ABH piezolectric bimorph cantilever is shown to substantially enhance the harvesting capabilities of a cantilever with uniform cross-section. The semi-analytical approach is then used to examine the influence of several ABH and piezoelectric layer parameters on energy harvesting efficiency. As regards the former, the effects of the tip truncation thickness and ABH order are explored. In what concerns the piezoelectric layer, we investigate the effects of its location, thickness, splitting it into several patches and varying the load resistors to enhance its performance in a broad frequency range. The proposed method constitutes a valuable tool for the design of ABH energy harvesting devices.

First Ionization Energy as the Asymptotic Limit of the Average Local Electron Energy.

The first vertical ionization energy of an atom or molecule is encoded in the rate of exponential decay of the exact natural orbitals. For natural orbitals represented in terms of Gaussian basis functions, this property does not hold even approximately. We show that it is nevertheless possible to deduce the first ionization energy from the long-range behavior of Gaussian-basis-set wave functions by evaluating the asymptotic limit of a quantity called the average local electron energy (ALEE), provided that the most diffuse functions of the basis set have a suitable shape and location. The ALEE method exposes subtle qualitative differences between seemingly analogous Gaussian basis sets and complements the extended Koopmans theorem by being robust in situations where the one-electron reduced density matrix is ill-conditioned.

Toward First-Principles-Level Polarization Energies in Force Fields: A Gaussian Basis for the Atom-Condensed Kohn-Sham Method.

The last 20 years of force field development have shown that even well parametrized classical models need to at least approximate the dielectric response of molecular systems-based, e.g., on atomic polarizabilities-in order to correctly render their structural and dynamic properties. Yet, despite great advances most approaches tend to be based on ad hoc assumptions and often insufficiently capture the dielectric response of the system to external perturbations, such as, e.g., charge carriers in semiconducting materials. A possible remedy was recently introduced with the atom-condensed Kohn-Sham density-functional theory approximated to second order (ACKS2), which is fully derived from first principles. Unfortunately, specifically its reliance on first-principles derived parameters so far precluded the widespread adoption of ACKS2. Opening up ACKS2 for general use, we here present a reformulation of the method in terms of Gaussian basis functions, which allows us to determine many of the ACKS2 parameters analytically. Two sets of parameters depending on exchange-correlation interactions are still calculated numerically, but we show that they could be straightforwardly parametrized owing to the smoothness of the new basis. Our approach exhibits three crucial benefits for future applications in force fields: i) efficiency, ii) accuracy, and iii) transferability. We numerically validate our Gaussian augmented ACKS2 model for a set of small hydrocarbons which shows a very good agreement with density-functional theory reference calculations. To further demonstrate the method's accuracy and transferability for realistic systems, we calculate polarization responses and energies of anthracene and tetracene, two major building blocks in organic semiconductors.

Two-dimensional dielectric profile reconstruction based on spectral-domain moment method and nonlinear optimization

A novel method for two-dimensional (2-D) profile reconstruction of dielectric objects, based on nonlinear optimization, is presented in this paper. The unknown dielectric profile is expressed in terms of Gaussian basis functions. The scattering integral equation (SIE) is discretized using a spectral-domain moment technique, where the unknown internal field is described as a superposition of a limited number of plane waves, resulting in a significant reduction of the associated computational cost. The inverse-scattering problem is solved by minimizing a cost function consisting of two terms: the first term represents the error between measured and predicted values of the scattered field, while the second term corresponds to the error in satisfying the SIE for the field in the interior of the scatterer. Accurate and efficient dielectric profile reconstructions of 2-D lossy scatterers of circular and square cross sections using synthetic scattered field data are presented, while the effect of various discretization parameters on the convergence of the method is studied in detail.

Kohn-Sham potentials corresponding to Slater and Gaussian basis set densities

A new method based on linear response theory is proposed for the determination of the Kohn-Sham potential corresponding to a given electron density. The method is very precise and affords a comparison between Kohn-Sham potentials calculated from correlated reference densities expressed in Slater-(STO) and Gaussian-type orbitals (GTO). In the latter case the KS potential exhibits large oscillations that are not present in the exact potential. These oscillations are related to similar oscillations in the local error function δ i (r)=(−ɛ i )ϕ i (r) when SCF orbitals (either Kohn-Sham or Hartree-Fock) are expressed in terms of Gaussian basis functions. Even when using very large Gaussian basis sets, the oscillations are such that extreme care has to be exercised in order to distinguish genuine characteristics of the KS potential, such as intershell peaks in atoms, from the spurious oscillations. For a density expressed in GTOs, the Laplacian of the density will exhibit similar spurious oscillations. A previously proposed iterative local updating method for generating the Kohn-Sham potential is evaluated by comparison with the present accurate scheme. For a density expressed in GTOs, it is found to yield a smooth “average” potential after a limited number of cycles. The oscillations that are peculiar to the GTO density are constructed in a slow process requiring very many cycles.

Dual-space approach for density-functional calculations of two- and three-dimensional crystals using Gaussian basis functions.

We reformulate theories for electronic structure calculations of periodic systems in a way suitable for large scale calculations using Gaussian basis functions. An accurate grid is introduced for efficient calculation of matrix elements. A dual-space approach is used to calculate the Coulomb potential with computational cost that scales linearly with the size of basis set. A preconditioned generalized conjugate gradients approach is introduced for rapidly converging wave functions expressed in terms of Gaussian basis functions. This method is applied to a variety of crystals (including diamond, GaN, AlN, CdTe, and ${\mathrm{C}}_{60}$) and surfaces [including GaAs (110) and BN (110)] with excellent results.

The evaluation of electronic extracule and intracule densities and related probability functions in terms of Gaussian basis functions

The evaluation of the basic two-electron integrals involved in the calculation of extracule and intracule densities is described. Expressions are given for the evaluation of the related spherically averaged, longitudinal, and transverse probability functions from wave functions constructed from Gaussian basis sets. All results are expressed in closed analytical forms which are suited to efficient coding. Given that certain pair densities can be related to experimental scattering cross sections, the formulae reported herein will facilitate further comparison between experiment and theory and lead to a more comprehensive understanding of the electronic structures of molecules.