Linear Scaling Techniques Research Articles

QM/AMOEBA description of properties and dynamics of embedded molecules

AbstractWe describe the development, implementation, and application of a polarizable QM/MM strategy, based on the AMOEBA polarizable force field, for calculating molecular properties and performing dynamics of molecular systems embedded in complex matrices. We show that polarizable QM/MM is a well‐understood, mature technology that can be deployed using a state‐of‐the‐art implementation that combines efficient numerical methods and linear scaling techniques. Thanks to these numerical advances and to the availability of parameters for a wide manifold of systems in the AMOEBA force field, polarizable QM/AMOEBA can be used for advanced production applications, that range from the prediction of spectroscopies to ground‐ and excited‐state multiscale ab initio molecular dynamics simulations.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Electronic Structure Theory > Combined QM/MM Methods

Structure optimization with stochastic density functional theory.

Linear-scaling techniques for Kohn-Sham density functional theory are essential to describe the ground state properties of extended systems. Still, these techniques often rely on the localization of the density matrix or accurate embedding approaches, limiting their applicability. In contrast, stochastic density functional theory (sDFT) achieves linear- and sub-linear scaling by statistically sampling the ground state density without relying on embedding or imposing localization. In return, ground state observables, such as the forces on the nuclei, fluctuate in sDFT, making optimizing the nuclear structure a highly non-trivial problem. In this work, we combine the most recent noise-reduction schemes for sDFT with stochastic optimization algorithms to perform structure optimization within sDFT. We compare the performance of the stochastic gradient descent approach and its variations (stochastic gradient descent with momentum) with stochastic optimization techniques that rely on the Hessian, such as the stochastic Broyden-Fletcher-Goldfarb-Shanno algorithm. We further provide a detailed assessment of the computational efficiency and its dependence on the optimization parameters of each method for determining the ground state structure of bulk silicon with varying supercell dimensions.

Statistical bias correction on the climate model for el nino index prediction

El Nino can harm many sectors in Indonesia by reducing precipitation levels in some areas. The occurrence of El Nino can be estimated by observing the sea surface temperature in Nino 3.4 region. Therefore, an accurate model on sea surface temperature prediction in Nino 3.4 region is needed to optimize the estimation of the occurrence of El Nino, such as ECMWF. However, the prediction model released by ECMWF still consists of some systematic errors or biases. This research aims to correct these biases using statistical bias correction techniques and evaluate the prediction model before and after correction. The statistical bias correction uses linear scaling, variance scaling, and distribution mapping techniques. The results show that statistical bias correction can reduce the prediction model bias. Also, the distribution mapping and variance scaling are more accurate than the linear scaling technique. Distribution mapping has better RMSE in December-March, and variance scaling has better RMSE in April-June also in October and November. However, in July-September, prediction from ECMWF has better RMSE. The application of statistical bias correction techniques has the highest refinement in January-March at the first lead time and in April at the fifth until the seventh lead time.

Evaluation of adaptation options for reducing soil erosion due to climate change in the Swat River Basin of Pakistan

Changes in climate have a direct impact on basin hydrology, soil erosion, and sediment yield and this could adversely affect the water availability and efficiency of hydraulic structures. Hence, there is a need to assess the relationship between climate, hydrology, and related processes. In this study, future climate data on three regional climate models: REMO, CNRM, and MPI under RCP 4.5 and 8.5 scenarios are downscaled using linear scaling techniques to create climate change scenarios in the Swat River Basin of Pakistan. In this study, the relationships between climate, discharge, and sediment yield are analysed using the Soil and Water Assessment Tool (SWAT), while also evaluating the effectiveness of two adaptation options: sediment management, i.e., check dams, silt fences, terraces, and sediment basins; and two erosion reduction options, i.e., filter strips and grassed waterways. Based on the results, an overall increase in temperature of 0.6 to 2.3 °C in the 2020s, 1.1 to 3.3 °C in the 2050s, and 2 to 5.6 °C in the 2080s is projected relative to the baseline period (1979–2005). The precipitation change varies from −24 to 34%. The model R2 values are 0.67 and 0.75, while NSE values are 0.63 and 0.74 during calibration (1984–1993) and validation periods (1994–2001), respectively. The overall change in future streamflow is predicted to vary from −28 to 28%. The projected annual average sediment yield at the outlet of the basin follows a pattern similar to that exhibited by precipitation and discharge. The results indicate that the provision of terraces, silt fences, a sediment basin, check dam, filter strips, and grassed waterway reduce the sediment yield in the basin by 73, 66, 65, 58, 54, and 48%, respectively. We conclude that a sediment basin is the most economical option for the management of soil erosion under climate change in the case of the Swat River.

Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks.

The fast and reliable determination of wave functions and electron densities of macromolecules has been one of the goals of theoretical chemistry for a long time, and in this context, several linear scaling techniques have been successfully devised over the years. Different approaches have been adopted to tackle this problem, and one of them exploits the fact that, according to the traditional chemical perception, molecules can be seen as constituted of recurring units (e.g., functional groups) with well-defined chemical features. This has led to the development of methods in which the global wave functions or electron densities of macromolecules are obtained by simply transferring density matrices or fuzzy electron densities associated with molecular fragments. In this context, we propose an alternative strategy that aims at quickly reconstructing wave functions and electron densities of proteins through the transfer of extremely localized molecular orbitals (ELMOs), which are orbitals strictly localized on small molecular units and, for this reason, easily transferable from molecule to molecule. To accomplish this task we have constructed original libraries of ELMOs that cover all the possible elementary fragments of the 20 natural amino acids in all their possible protonation states and forms. Our preliminary test calculations have shown that, compared to more traditional methods of quantum chemistry, the transfers from the novel ELMO databanks allow to obtain wave function and electron densities of large polypeptides and proteins at a significantly reduced computational cost. Furthermore, notwithstanding expected discrepancies, the obtained electron distributions and electrostatic potentials are in very good agreement with those obtained at Hartree-Fock and density functional theory (DFT) levels. Therefore, the results encourage to use the new libraries as alternatives to the popular pseudoatom-databases of crystallography in the refinement of crystallographic structures of macromolecules. In particular, in this context, we have already envisaged the coupling of the ELMO databanks with the promising Hirshfeld atom refinement technique to extend the applicability of the latter to very large systems.

EVALUATING THE SCALE FACTORS OF PEDO-TRANSFER MODELING THE SOIL WATER CHARACTERISTIC CURVE OF ZAKHO SOILS

The soil-water characteristic curve is the basis for estimating unsaturated soil properties. The negative pore pressure measurement is costly, time consuming, and needs accurate techniques to be measured. However, the pedo-transfer modeling of Arya and Paris introduces solution for the problem. The constant, linear, and Vaz scaling techniques in association with Arya and Paris model was used to predict SWCC from particle-size distribution. A total of 48 soil suction vs. volumetric water content experimental tests representing the SWCC for five different regions of Zakho district was processed for this purpose. The scaling parameters were obtained and the results show that linear and constant techniques have interpret the experimental results of the SWCC with appreciable confidence, however Vas scaling method is not recommended to be used for Zahko soil for the relatively weak correlations obtained.

Libraries of Extremely Localized Molecular Orbitals. 1. Model Molecules Approximation and Molecular Orbitals Transferability.

Despite more and more remarkable computational ab initio results are nowadays continuously obtained for large macromolecular systems, the development of new linear-scaling techniques is still an open and stimulating field of research in theoretical chemistry. In this family of methods, an important role is occupied by those strategies based on the observation that molecules are generally constituted by recurrent functional units with well-defined intrinsic features. In this context, we propose to exploit the notion of extremely localized molecular orbitals (ELMOs) that, due to their strict localization on small molecular fragments (e.g., atoms, bonds, or functional groups), are in principle transferable from one molecule to another. Accordingly, the construction of orbital libraries to almost instantaneously build up approximate wave functions and electron densities of very large systems becomes conceivable. In this work, the ELMOs transferability is further investigated in detail and, furthermore, suitable rules to construct model molecules for the computation of ELMOs to be stored in future databanks are also defined. The obtained results confirm the reliable transferability of the ELMOs and show that electron densities obtained from the transfer of extremely localized molecular orbitals are very close to the corresponding Hartree-Fock ones. These observations prompt us to construct new ELMOs databases that could represent an alternative/complement to the already popular pseudoatoms databanks both for determining electron densities and for refining crystallographic structures of very large molecules.

EMPIRE: a highly parallel semiempirical molecular orbital program: 1: self-consistent field calculations.

EMPIRE is a massively parallel semiempirical (NDDO) molecular-orbital program designed to scale well both on single multi-core nodes (using open MP) and on large clusters (using a hybrid open MP/MPI model). The program design and performance are discussed for single self-consistent-field calculations on up to 55,000 (the adamantane crystal shown in the graphic) atoms and on both single- and multi-node machines using either Windows 7 or Linux. EMPIRE currently carries out the full SCF calculation with no local approximations or other linear-scaling techniques. The single-node version is available free of charge to bona fide academic groups.

Real‐time quantum chemistry

AbstractSignificant progress in the development of efficient and fast algorithms for quantum chemical calculations has been made in the past two decades. The main focus has always been the desire to be able to treat ever larger molecules or molecular assemblies—especially linear and sublinear scaling techniques are devoted to the accomplishment of this goal. However, as many chemical reactions are rather local, they usually involve only a limited number of atoms so that models of about 200 (or even less) atoms embedded in a suitable environment are sufficient to study their mechanisms. Thus, the system size does not need to be enlarged, but remains constant for reactions of this type that can be described by less than 200 atoms. The question then arises how fast one can obtain the quantum chemical results. This question is not directly answered by linear‐scaling techniques. In fact, ideas such as haptic quantum chemistry (HQC) or interactive quantum chemistry require an immediate provision of quantum chemical information which demands the calculation of data in “real time.” In this perspective, we aim at a definition of real‐time quantum chemistry, explore its realm and eventually discuss applications in the field of HQC. For the latter, we elaborate whether a direct approach is possible by virtue of real‐time quantum chemistry. © 2012 Wiley Periodicals, Inc.

Pore‐Scale Simulations to Determine the Applied Hydrodynamic Torque and Colloid Immobilization

Values of the applied hydrodynamic torque (Tapplied) and the resisting adhesive torque (Tadhesion) will determine whether a colloid will be immobilized (Tapplied ≤ Tadhesion) or roll (Tapplied > Tadhesion) on a solid water interface. Previous literature has demonstrated in 1–2 collector (grain) systems that the influence of Tapplied on colloid retention can be significant under unfavorable attachment conditions and that only a fraction of the solid surface may contribute to retention. However, many questions remain on how to obtain, analyze, and upscale information on the forces and torques that act on colloids near solid surfaces in porous media. To address some of these gaps in knowledge, high resolution pore‐scale water flow simulations were conducted for sphere packs (25 spheres) over a range of Darcy velocities, grain sizes and distributions, and porosities. The spatial variability of Tapplied was calculated from this information, and successfully described using a lognormal cumulative density function (CDF). Linear interpolation and scaling techniques were subsequently used to predict the lognormal CDF of Tapplied for various colloid sizes, grain sizes and distributions, and water velocities. The lognormal CDF of Tapplied was then evaluated at select values of Tadhesion (i.e, interaction energy) to quantify the fraction and locations on the solid surface that contributes to colloid retention (Sf), and the theoretical maximum solid phase concentration of retained colloids (Smax).

Optimal Power Flow Using an Extended Conic Quadratic Formulation

Recent research has shown that the load flow equations describing the steady-state conditions in a meshed network can be placed in extended conic quadratic (ECQ) format. This paper presents a study of the implementation of the new load flow equations format in an optimal power flow (OPF) program which accounts for control devices such as tap-changing transformers, phase-shifting transformers, and unified power flow controllers. The proposed OPF representation retains the advantages of the ECQ format: 1) it can be easily integrated within optimization routines that require the evaluation of second-order derivatives, 2) it can be efficiently solved for using primal-dual interior-point methods, and 3) it can make use of linear programming scaling techniques for improving numerical conditioning. The ECQ-OPF program is employed to solve the economic dispatch and active power loss minimization problems. Numerical testing is used to validate the proposed approach by comparing against solution methods and results of standard test systems.

Linear scaling density fitting

Two modifications of the resolution of the identity (RI)/density fitting (DF) approximations are presented. First, we apply linear scaling and J-engine techniques to speed up traditional DF. Second, we develop an algorithm that produces local, accurate fits with effort that scales linearly with system size. The fits produced are continuous, differentiable, well-defined, and do not require preset fitting domains. This metric-independent technique for producing a priori local fits is shown to be accurate and robust even for large systems. Timings are presented for linear scaling RI/DF calculations on large one-, two-, and three-dimensional carbon systems.

RM1: A reparameterization of AM1 for H, C, N, O, P, S, F, Cl, Br, and I

Twenty years ago, the landmark AM1 was introduced, and has since had an increasingly wide following among chemists due to its consistently good results and time-tested reliability--being presently available in countless computational quantum chemistry programs. However, semiempirical molecular orbital models still are of limited accuracy and need to be improved if the full potential of new linear scaling techniques, such as MOZYME and LocalSCF, is to be realized. Accordingly, in this article we present RM1 (Recife Model 1): a reparameterization of AM1. As before, the properties used in the parameterization procedure were: heats of formation, dipole moments, ionization potentials and geometric variables (bond lengths and angles). Considering that the vast majority of molecules of importance to life can be assembled by using only six elements: C, H, N, O, P, and S, and that by adding the halogens we can now build most molecules of importance to pharmaceutical research, our training set consisted of 1736 molecules, representative of organic and biochemistry, containing C, H, N, O, P, S, F, Cl, Br, and I atoms. Unlike AM1, and similar to PM3, all RM1 parameters have been optimized. For enthalpies of formation, dipole moments, ionization potentials, and interatomic distances, the average errors in RM1, for the 1736 molecules, are less than those for AM1, PM3, and PM5. Indeed, the average errors in kcal x mol(-1) of the enthalpies of formation for AM1, PM3, and PM5 are 11.15, 7.98, and 6.03, whereas for RM1 this value is 5.77. The errors, in Debye, of the dipole moments for AM1, PM3, PM5, and RM1 are, respectively, 0.37, 0.38, 0.50, and 0.34. Likewise, the respective errors for the ionization potentials, in eV, are 0.60, 0.55, 0.48, and 0.45, and the respective errors, in angstroms, for the interatomic distances are 0.036, 0.029, 0.037, and 0.027. The RM1 average error in bond angles of 6.82 degrees is only slightly higher than the AM1 figure of 5.88 degrees, and both are much smaller than the PM3 and PM5 figures of 6.98 degrees and 9.83 degrees, respectively. Moreover, a known error in PM3 nitrogen charges is corrected in RM1. Therefore, RM1 represents an improvement over AM1 and its similar successor PM3, and is probably very competitive with PM5, which is a somewhat different model, and not fully disclosed. RM1 possesses the same analytical construct and the same number of parameters for each atom as AM1, and, therefore, can be easily implemented in any software that already has AM1, not requiring any change in any line of code, with the sole exception of the values of the parameters themselves.

Special Issue Current Trends in Electronic Structure: Real‐space, Embedding and Linear Scaling Techniques

AbstractThis Special Issue of physica status solidi (b) is devoted to the topic of first‐principles simulation of solids, liquids and complex matter and presents articles on the progress in linear scaling methods, electronic transport, simulation of STM images, and time‐dependent DFT methods.The cover picture is taken from the paper by Torsti et al. and shows an isosurface of the positron density localized at a Ga vacancy in GaN [1]. The positron state is calculated by the real‐space code Mika/doppler based on the Rayleigh‐quotient multigrid method. The code predicts positron lifetimes and momentum densities of annihilating electron–positron pairs measured in positron annihilation spectroscopy for defects in solids.

The calculation of indirect nuclear spin-spin coupling constants in large molecules.

We present calculations of indirect nuclear spin-spin coupling constants in large molecular systems, performed using density functional theory. Such calculations, which have become possible because of the use of linear-scaling techniques in the evaluation of the Coulomb and exchange-correlation contributions to the electronic energy, allow us to study indirect spin-spin couplings in molecules of biological interest, without having to construct artificial model systems. In addition to presenting a statistical analysis of the large number of short-range coupling constants in large molecular systems, we analyse the asymptotic dependence of the indirect nuclear spin-spin coupling constants on the internuclear separation. In particular, we demonstrate that, in a sufficiently large one-electron basis set, the indirect spin-spin coupling constants become proportional to the inverse cube of the internuclear separation, even though the diamagnetic and paramagnetic spin-orbit contributions to the spin-spin coupling constants separately decay as the inverse square of this separation. By contrast, the triplet Fermi contact and spin-dipole contributions to the indirect spin-spin coupling constants decay exponentially and as the inverse cube of the internuclear separation, respectively. Thus, whereas short-range indirect spin-spin coupling constants are usually dominated by the Fermi contact contribution, long-range coupling constants are always dominated by the negative diamagnetic spin-orbit contribution and by the positive paramagnetic spin-orbit contribution, with small spin-dipole and negligible Fermi contact contributions.

An embedding scheme based on quantum linear-scaling methods

We point out a close connection between linear-scaling electronic structure techniques and the embedding problem (treatment of a defect in an infinite perfect crystal), the key to the connection being the locality of the density matrix. We show how an existing linear-scaling technique can be turned into an embedding technique, which appears to have advantages over earlier methods. The effectiveness of the new technique is illustrated by practical tight-binding calculations on three systems: a substitutional impurity and the vacancy in silicon, and the Si (0 0 1) surface. We indicate the developments needed to perform embedding calculations by the new approach with first-principles density-functional theory (DFT).

Parallel sparse matrix multiplication for linear scaling electronic structure calculations

Linear-scaling electronic-structure techniques, also called O(N) techniques, rely heavily on the multiplication of sparse matrices, where the sparsity arises from spatial cut-offs. In order to treat very large systems, the calculations must be run on parallel computers. We analyze the problem of parallelizing the multiplication of sparse matrices with the sparsity pattern required by linear-scaling techniques. We show that the management of inter-node communications and the effective use of on-node cache are helped by organizing the atoms into compact groups. We also discuss how to identify a key part of the code called the ‘multiplication kernel’, which is repeatedly invoked to build up the matrix product, and explicit code is presented for this kernel. Numerical tests of the resulting multiplication code are reported for cases of practical interest, and it is shown that their scaling properties for systems containing up to 20,000 atoms on machines having up to 512 processors are excellent. The tests also show that the CPU efficiency of our code is satisfactory.

Towards excitation energies and (hyper)polarizability calculations of large molecules. Application of parallelization and linear scaling techniques to time‐dependent density functional response theory

Towards excitation energies and (hyper)polarizability calculations of large molecules. Application of parallelization and linear scaling techniques to time‐dependent density functional response theory

Linear-scaling density-functional theory with Gaussian orbitals and periodic boundary conditions: Efficient evaluation of energy and forces via the fast multipole method

We report methodological and computational details of our Kohn-Sham density-functional method with Gaussian orbitals for systems with periodic boundary conditions. Our approach for the Coulomb problem is based on the direct space fast multipole method, which achieves not only linear scaling of computational time with system size but also very high accuracy in all infinite summations. The latter is pivotal for avoiding numerical instabilities that have previously plagued calculations with large bases, especially those containing diffuse functions. Our program also makes extensive use of other linear-scaling techniques recently developed for large clusters. Using these theoretical tools, we have implemented computational programs for energy and analytic energy gradients (forces) that make it possible to optimize geometries of periodic systems with great efficiency and accuracy. Vibrational frequencies are then accurately obtained from finite differences of forces. We demonstrate the capabilities of our methods with benchmark calculations on polyacetylene, polyphenylenevinylene, and a (5,0) carbon nanotube, employing basis sets of double zeta plus polarization quality, in conjunction with the generalized gradient approximation and kinetic-energy density-dependent functionals. The largest calculation reported in this paper contains 244 atoms and 1344 contracted Gaussians in the unit cell.

Quantum chemistry in the 21st century (Special topic article)

Abstract Quantum chemistry is the field in which solutions to the Schrödinger equation are used to predict the properties of molecules and solve chemical problems. This paper considers possible future research directions in light of the discipline's past successes. After decades of incremental development—accompanied by a healthy dose of skepticism from the experimental community—the ready availability of fast computers has ushered in a "golden age" of quantum chemistry. In this new era of acceptance, theoretical predictions often precede experiment in small molecule chemistry, and quantum chemical methods play an ever greater role in biochemical and other larger systems. Quantum chemists increasingly divide their efforts along three fronts: high-level (spectroscopic) accuracy for small molecules, characterized by such techniques as Brueckner methods, r12 formalisms, and multireference calculations; parameterization- or extrapolation-based intermediate-level schemes (such as Gaussian-N theory) for medium molecules; and lower-level (chemical) accuracy for large molecules, characterized by density functional theory and linear scaling techniques. These tools, and quantum chemistry as a whole, are examined here from a historical perspective and with a view toward their future applications.