Efficient Calculation of Derivatives of Integrals in a Basis of Non-Separable Gaussians Through Exploitation of Sparsity

A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the \textsc{Crystal} code is presented and the considerably improved computational efficiency over the previous implementation is illustrated. To this purpose, three different tasks involving the use of analytical forces are considered: i) geometry optimization; ii) harmonic frequency calculation; iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: i) a metallic 2D model of the Cu (111) surface; ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and iii) a porous metal-organic crystal, namely the ZIF-8 Zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched

Algebraic tools for dealing with the atomic shell model. I. Wavefunctions and integrals for hydrogen--like ions

Today, the 'hydrogen atom model' is known to play its role not only in teaching the basic elements of quantum mechanics but also for building up effective theories in atomic and molecular physics, quantum optics, plasma physics, or even in the design of semiconductor devices. Therefore, the analytical as well as numerical solutions of the hydrogen--like ions are frequently required both, for analyzing experimental data and for carrying out quite advanced theoretical studies. In order to support a fast and consistent access to these (Coulomb--field) solutions, here we present the Dirac program which has been developed originally for studying the properties and dynamical behaviour of the (hydrogen--like) ions. In the present version, a set of Maple procedures is provided for the Coulomb wave and Green's functions by applying the (wave) equations from both, the nonrelativistic and relativistic theory. Apart from the interactive access to these functions, moreover, a number of radial integrals are also implemented in the Dirac program which may help the user to construct transition amplitudes and cross sections as they occur frequently in the theory of ion--atom and ion--photon collisions.Comment: 23 pages, 1 figur

One-loop kink mass shifts: a computational approach

In this paper we develop a procedure to compute the one-loop quantum correction to the kink masses in generic (1+1)-dimensional one-component scalar field theoretical models. The procedure uses the generalized zeta function regularization method helped by the Gilkey-de Witt asymptotic expansion of the heat function via Mellin's transform. We find a formula for the one-loop kink mass shift that depends only on the part of the energy density with no field derivatives, evaluated by means of a symbolic software algorithm that automates the computation. The improved algorithm with respect to earlier work in this subject has been tested in the sine-Gordon and $\lambda(\phi)_2^4$ models. The quantum corrections of the sG-soliton and $\lambda(\phi^4)_2$-kink masses have been estimated with a relative error of 0.00006% and 0.00007% respectively. Thereafter, the algorithm is applied to other models. In particular, an interesting one-parametric family of double sine-Gordon models interpolating between the ordinary sine-Gordon and a re-scaled sine-Gordon model is addressed. Another one-parametric family, in this case of $\phi^6$ models, is analyzed. The main virtue of our procedure is its versatility: it can be applied to practically any type of relativistic scalar field models supporting kinks.Comment: 35 pages, 6 figures, to be published in Nuclear Physics

ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers

Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory. This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to access different strategies that address the Kohn-Sham eigenvalue problem. Currently supported algorithms include the dense generalized eigensolver library ELPA, the orbital minimization method implemented in libOMM, and the pole expansion and selected inversion (PEXSI) approach with lower computational complexity for semilocal density functionals. The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by offering (a) a unified software framework designed for the electronic structure solvers in Kohn-Sham density-functional theory; (b) reasonable default parameters for a chosen solver; (c) automatic conversion between input and internal working matrix formats, and in the future (d) recommendation of the optimal solver depending on the specific problem. Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800 basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table

Probabilistic Model Checking for Energy Analysis in Software Product Lines

In a software product line (SPL), a collection of software products is defined by their commonalities in terms of features rather than explicitly specifying all products one-by-one. Several verification techniques were adapted to establish temporal properties of SPLs. Symbolic and family-based model checking have been proven to be successful for tackling the combinatorial blow-up arising when reasoning about several feature combinations. However, most formal verification approaches for SPLs presented in the literature focus on the static SPLs, where the features of a product are fixed and cannot be changed during runtime. This is in contrast to dynamic SPLs, allowing to adapt feature combinations of a product dynamically after deployment. The main contribution of the paper is a compositional modeling framework for dynamic SPLs, which supports probabilistic and nondeterministic choices and allows for quantitative analysis. We specify the feature changes during runtime within an automata-based coordination component, enabling to reason over strategies how to trigger dynamic feature changes for optimizing various quantitative objectives, e.g., energy or monetary costs and reliability. For our framework there is a natural and conceptually simple translation into the input language of the prominent probabilistic model checker PRISM. This facilitates the application of PRISM's powerful symbolic engine to the operational behavior of dynamic SPLs and their family-based analysis against various quantitative queries. We demonstrate feasibility of our approach by a case study issuing an energy-aware bonding network device.Comment: 14 pages, 11 figure

Symbolic-Numeric Algorithms for Computer Analysis of Spheroidal Quantum Dot Models

A computation scheme for solving elliptic boundary value problems with axially symmetric confining potentials using different sets of one-parameter basis functions is presented. The efficiency of the proposed symbolic-numerical algorithms implemented in Maple is shown by examples of spheroidal quantum dot models, for which energy spectra and eigenfunctions versus the spheroid aspect ratio were calculated within the conventional effective mass approximation. Critical values of the aspect ratio, at which the discrete spectrum of models with finite-wall potentials is transformed into a continuous one in strong dimensional quantization regime, were revealed using the exact and adiabatic classifications.Comment: 6 figures, Submitted to Proc. of The 12th International Workshop on Computer Algebra in Scientific Computing (CASC 2010) Tsakhkadzor, Armenia, September 5 - 12, 201