Calculation of electron density of periodic systems using non-orthogonal localised orbitals

Methods for calculating an electron density of a periodic crystal constructed using non-orthogonal localised orbitals are discussed. We demonstrate that an existing method based on the matrix expansion of the inverse of the overlap matrix into a power series can only be used when the orbitals are highly localised (e.g. ionic systems). In other cases including covalent crystals or those with an intermediate type of chemical bonding this method may be either numerically inefficient or fail altogether. Instead, we suggest an exact and numerically efficient method which can be used for orbitals of practically arbitrary localisation. Theory is illustrated by numerical calculations on a model system.Comment: 12 pages, 4 figure

Fr\'echet frames, general definition and expansions

We define an {\it $(X_1,\Theta, X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot|_1 \leq |\cdot|_2$, and a $BK$-space (\Theta, \snorm[\cdot]). Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\infty$ and of sequence spaces ${\Theta_s}_{s=0}^\infty$, we define a general Fr\' echet frame on the Fr\' echet space $X_F=\bigcap_{s=0}^\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\' echet frame to be a general Fr\' echet frame, as well as for the complementedness of the range of the analysis operator $U:X_F\to\Theta_F$.Comment: A new section is added and a minor revision is don

Simulated structure and imaging of NTCDI on Si(1 1 1)-7 × 7 : a combined STM, NC-AFM and DFT study

The adsorption of naphthalene tetracarboxylic diimide (NTCDI) on Si(1 1 1)-7 × 7 is investigated through a combination of scanning tunnelling microscopy (STM), noncontact atomic force microscopy (NC-AFM) and density functional theory (DFT) calculations. We show that NTCDI adopts multiple planar adsorption geometries on the Si(1 1 1)-7 × 7 surface which can be imaged with intramolecular bond resolution using NC-AFM. DFT calculations reveal adsorption is dominated by covalent bond formation between the molecular oxygen atoms and the surface silicon adatoms. The chemisorption of the molecule is found to induce subtle distortions to the molecular structure, which are observed in NC-AFM images

Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion

We show that the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric down-conversion enables the generation of entangled Hermite-Gaussian modes. We derive an analytical expression for the two-photon state in terms of these modes and show that there are restrictions on both the parity and order of the down-converted Hermite-Gaussian fields. Using these results, we show that the two-photon state is indeed entangled in Hermite-Gaussian modes. We propose experimental methods of creating maximally-entangled Bell states and non-maximally entangled pure states of first order Hermite-Gaussian modes.Comment: 9 pages, 4 figures. Corrections made as per referee comments, references updated. Submitted PR

Managing the supercell approximation for charged defects in semiconductors: finite size scaling, charge correction factors, the bandgap problem and the ab initio dielectric constant

The errors arising in ab initio density functional theory studies of semiconductor point defects using the supercell approximation are analyzed. It is demonstrated that a) the leading finite size errors are inverse linear and inverse cubic in the supercell size, and b) finite size scaling over a series of supercells gives reliable isolated charged defect formation energies to around +-0.05 eV. The scaled results are used to test three correction methods. The Makov-Payne method is insufficient, but combined with the scaling parameters yields an ab initio dielectric constant of 11.6+-4.1 for InP. Gamma point corrections for defect level dispersion are completely incorrect, even for shallow levels, but re-aligning the total potential in real-space between defect and bulk cells actually corrects the electrostatic defect-defect interaction errors as well. Isolated defect energies to +-0.1 eV are then obtained using a 64 atom supercell, though this does not improve for larger cells. Finally, finite size scaling of known dopant levels shows how to treat the band gap problem: in less than about 200 atom supercells with no corrections, continuing to consider levels into the theoretical conduction band (extended gap) comes closest to experiment. However, for larger cells or when supercell approximation errors are removed, a scissors scheme stretching the theoretical band gap onto the experimental one is in fact correct.Comment: 11 pages, 3 figures (6 figure files). Accepted for Phys Rev

Temperature control in molecular dynamic simulations of non-equilibrium processes

Thermostats are often used in various condensed matter problems, e.g. when a biological molecule undergoes a transformation in a solution, a crystal surface is irradiated with energetic particles, a crack propagates in a solid upon applied stress, two surfaces slide with respect to each other, an excited local phonon dissipates its energy into a crystal bulk, and so on. In all of these problems, as well as in many others, there is an energy transfer between different local parts of the entire system kept at a constant temperature. Very often, when modelling such processes using molecular dynamics simulations, thermostatting is done using strictly equilibrium approaches serving to describe the NV T ensemble. In this paper we critically discuss the applicability of such approaches to non-equilibrium problems, including those mentioned above, and stress that the correct temperature control can only be achieved if the method is based on the generalized Langevin equation (GLE). Specifically, we emphasize that a meaningful compromise between computational efficiency and a physically appropriate implementation of the NV T thermostat can be achieved, at least for solid state and surface problems, if the so-called stochastic boundary conditions (SBC), recently derived from the GLE (Kantorovich and Rompotis 2008 Phys. Rev. B 78 094305), are used. For SBC, the Langevin thermostat is only applied to the outer part of the simulated fragment of the entire system which borders the surrounding environment (not considered explicitly) serving as a heat bath. This point is illustrated by comparing the performance of the SBC and some of the equilibrium thermostats in two problems: (i) irradiation of the Si(001) surface with an energetic CaF2 molecule using an ab initio density functional theory based method, and (ii) the tribology of two amorphous SiO2 surfaces coated with self-assembled monolayers of methyl-terminated hydrocarbon alkoxylsilane molecules using a classical atomistic force field. We discuss the differences in behaviour of these systems due to applied thermostatting, and show that in some cases a qualitatively different physical behaviour of the simulated system can be obtained if an equilibrium thermostat is used

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 Taylor & Francis.In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.The work was supported by the grant EP/H020497/1 "Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients" of the EPSRC, UK

On homogenization of electromagnetic crystals formed by uniaxial resonant scatterers

Dispersion properties of electromagnetic crystals formed by small uniaxial resonant scatterers (magnetic or electric) are studied using the local field approach. The goal of the study is to determine the conditions under which the homogenization of such crystals can be made. Therefore the consideration is limited by the frequency region where the wavelength in the host medium is larger than the lattice periods. It is demonstrated that together with known restriction for the homogenization related with the large values of the material parameters there is an additional restriction related with their small absolute values. From the other hand, the homogenization becomes allowed in both cases of large and small material parameters for special directions of propagation. Two unusual effects inherent to the crystals under consideration are revealed: flat isofrequency contour which allows subwavelength imaging using canalization regime and birefringence of extraordinary modes which can be used for beam splitting.Comment: 16 pages, 12 figures, submitted to PR

Fast Optimal Transport Averaging of Neuroimaging Data

Knowing how the Human brain is anatomically and functionally organized at the level of a group of healthy individuals or patients is the primary goal of neuroimaging research. Yet computing an average of brain imaging data defined over a voxel grid or a triangulation remains a challenge. Data are large, the geometry of the brain is complex and the between subjects variability leads to spatially or temporally non-overlapping effects of interest. To address the problem of variability, data are commonly smoothed before group linear averaging. In this work we build on ideas originally introduced by Kantorovich to propose a new algorithm that can average efficiently non-normalized data defined over arbitrary discrete domains using transportation metrics. We show how Kantorovich means can be linked to Wasserstein barycenters in order to take advantage of an entropic smoothing approach. It leads to a smooth convex optimization problem and an algorithm with strong convergence guarantees. We illustrate the versatility of this tool and its empirical behavior on functional neuroimaging data, functional MRI and magnetoencephalography (MEG) source estimates, defined on voxel grids and triangulations of the folded cortical surface.Comment: Information Processing in Medical Imaging (IPMI), Jun 2015, Isle of Skye, United Kingdom. Springer, 201