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Register a new userOff-center impurity in alkali halides: reorientation, electric polarization and pairing to F center. III. Numerical calculations
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We carried out numerical calculations by an extended-Hueckel program in order to check the analytical results reported in the preceding Part I and Part II. We typically consider alkali halide clusters composed of some tens of constituent atoms to calculate electronic energies under static conditions or versus the displacements of particular atoms. Among other things, the off-center displacement of substitutional Li+ impurity in most alkali halides is evidenced. The trigonometric profile of the rotational barriers is also confirmed for KCl.
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We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, $textrm{erfc}(omega r_{12})/r_{12}$. These estimators are much tighter than one based on the Schwarz inequality and can be viewed as a complement to the distance-dependent estimators for four-center short-range Coulomb integrals and for two-center and three-center full Coulomb integrals previously reported. Because the short-range Coulomb potential is commonly used in solid-state calculations, including those with the HSE functional and with our recently introduced range-separated periodic Gaussian density fitting, we test our estimators on a diverse set of periodic systems using a wide range of the range-separation parameter $omega$. These tests demonstrate the robust tightness of our estimators, which are then used with integral screening to calculate periodic three-center short-range Coulomb integrals with linear scaling in system size.
We revisit the well-known Mollwo-Ivey relation that describes the universal dependence of the absorption energies of F-type color centers on the lattice constant $a$ of the alkali-halide crystals, $E_{mbox{abs}}propto a^{-n}.$ We perform both state-of-the-art ab-initio Quantum Chemistry and post-DFT calculations of F-center absorption spectra. By tuning independently the lattice constant and the atomic species we show that the scaling of the lattice constant alone (keeping the elements fixed) would yield $n=2$ in agreement with the particle-in-the-box model. Keeping the lattice constant fixed and changing the atomic species enables us to quantify the ion-size effects which are shown to be responsible for the exponent $n approx 1.8$.
We investigate the Josephson transport through a thin semiconductor barrier containing impurity centers with the on-site Hubbard interaction $u$ of an arbitrary sign and strength. We find that in the case of the repulsive interaction the Josephson current changes sign with the temperature increase if the energy of the impurity level $epsilon$ (measured from the Fermi energy of superconductors) falls in the interval $(-u,0)$. We predict strong temporal fluctuations of the current if only a few centers present within the junction. In the case of the attractive impurity potential ($u<0$) and at low temperatures, the model is reduced to the effective two level Hamiltonian allowing thus a simple description of the nonstationary Josephson effect in terms of pair tunneling processes.
A common way to evaluate electronic integrals for polyatomic molecules is to use Beckes partitioning scheme [J. Chem. Phys.88, 2547 (1988)] in conjunction with overlapping grids centered at each atomic site. The Becke scheme was designed for integrands that fall off rapidly at large distances, such as those approximating bound electronic states. When applied to states in the electronic continuum, however, Becke scheme exhibits slow convergence and it is highly redundant. Here, we present a modified version of Becke scheme that is applicable to functions of the electronic continuum, such as those involved in molecular photoionization and electron-molecule scattering, and which ensures convergence and efficiency comparable to those realized in the calculation of bound states. In this modified scheme, the atomic weights already present in Beckes partition are smoothly switched off within a range of few bond lengths from their respective nuclei, and complemented by an asymptotically unitary weight. The atomic integrals are evaluated on small spherical grids, centered on each atom, with size commensurate to the support of the corresponding atomic weight. The residual integral of the interstitial and long-range region is evaluated with a central master grid. The accuracy of the method is demonstrated by evaluating integrals involving integrands containing Gaussian Type Orbitals and Yukawa potentials, on the atomic sites, as well as spherical Bessel functions centered on the master grid. These functions are representative of those encountered in realistic electron-scattering and photoionization calculations in polyatomic molecules.
The numerical computation of chemical potential in dense, non-homogeneous fluids is a key problem in the study of confined fluids thermodynamics. To this day several methods have been proposed, however there is still need for a robust technique, capable of obtaining accurate estimates at large average densities. A widely established technique is the Widom insertion method, that computes the chemical potential by sampling the energy of insertion of a test particle. Non-homogeneity is accounted for by assigning a density dependent weight to the insertion points. However, in dense systems, the poor sampling of the insertion energy is a source of inefficiency, hampering a reliable convergence. We have recently presented a new technique for the chemical potential calculation in homogeneous fluids. This novel method enhances the sampling of the insertion energy via Well-Tempered Metadynamics, reaching accurate estimates at very large densities. In this paper we extend the technique to the case of non-homogeneous fluids. The method is successfully tested on a confined Lennard-Jones fluid. In particular we show that, thanks to the improved sampling, our technique does not suffer from a systematic error that affects the classic Widom method for non-homogeneous fluids, providing a precise and accurate result.
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