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At least 505 records · Page 28
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Clean and Convenient Tessellations for Number Counting Jastrow Factors

We introduce a basis of counting functions that, by cleanly tessellating three-dimensional space, allows real space number counting Jastrow factors to be straightforwardly applied to general molecular situations. Here, by exerting direct control over electron populations in local regions of space and encoding pairwise correlations between these populations, these Jastrow factors allow even very simple reference wave functions to adopt nodal surfaces well suited to many strongly correlated settings. Being trivially compatible with traditional Jastrow factors and diffusion Monte Carlo and having the same cubic per-sample cost scaling as a single determinant trial function, these Jastrow factors thus offer a powerful new route to the simultaneous capture of weak and strong electron correlation effects in a wide variety of molecular and materials settings. In multiple strongly correlated molecular examples, we show that even when paired with the simplest possible single determinant reference, these Jastrow factors allow quantum Monte Carlo to out-perform coupled cluster theory and approach the accuracy of traditional multireference methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Liquid chloroform structure from computer simulation with a full ab initio intermolecular interaction potential

We have calculated the intermolecular interaction energies of the chloroform dimer in 12 orientations using the second-order Møller-Plesset perturbation theory. Single point energies of important geometries were calibrated by the coupled cluster with single and double and perturbative triple excitation method. Dunning's correlation consistent basis sets up to aug-cc-pVQZ have been employed in extrapolating the interaction energies to the complete basis set limit values. With the ab initio potential data we constructed a 5-site force field model for molecular dynamics simulations. We compared the simulation results with recent experiments and obtained quantitative agreements for the detailed atomwise radial distribution functions. Our results were also consistent with previous results using empirical force fields with polarization effects. Moreover, the calculated diffusion coefficients reproduced the experimental data over a wide range of thermodynamic conditions. To the best of our knowledge, this is the first ab initio force field which is capable of competing with existing empirical force fields for liquid chloroform.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Appointing silver and bronze standards for noncovalent interactions: A comparison of spin-component-scaled (SCS), explicitly correlated (F12), and specialized wavefunction approaches

A systematic examination of noncovalent interactions as modeled by wavefunction theory is presented in comparison to gold-standard quality benchmarks available for 345 interaction energies of 49 bimolecular complexes. Quantum chemical techniques examined include spin-component-scaling (SCS) variations on second-order perturbation theory (MP2) [SCS, SCS(N), SCS(MI)] and coupled cluster singles and doubles (CCSD) [SCS, SCS(MI)]; also, method combinations designed to improve dispersion contacts [DW-MP2, MP2C, MP2.5, DW-CCSD(T)-F12]; where available, explicitly correlated (F12) counterparts are also considered. Dunning basis sets augmented by diffuse functions are employed for all accessible ζ-levels; truncations of the diffuse space are also considered. After examination of both accuracy and performance for 394 model chemistries, SCS(MI)-MP2/cc-pVQZ can be recommended for general use, having good accuracy at low cost and no ill-effects such as imbalance between hydrogen-bonding and dispersion-dominated systems or non-parallelity across dissociation curves. Moreover, when benchmarking accuracy is desirable but gold-standard computations are unaffordable, this work recommends silver-standard [DW-CCSD(T**)-F12/aug-cc-pVDZ] and bronze-standard [MP2C-F12/aug-cc-pVDZ] model chemistries, which support accuracies of 0.05 and 0.16 kcal/mol and efficiencies of 97.3 and 5.5 h for adenine·thymine, respectively. Choice comparisons of wavefunction results with the best symmetry-adapted perturbation theory [T. M. Parker, L. A. Burns, R. M. Parrish, A. G. Ryno, and C. D. Sherrill, J. Chem. Phys. 140, 094106 (2014)] and density functional theory [L. A. Burns, Á. Vázquez-Mayagoitia, B. G. Sumpter, and C. D. Sherrill, J. Chem. Phys. 134, 084107 (2011)] methods previously studied for these databases are provided for readers' guidance.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Independent amplitude approximations in coupled cluster valence bond theory: Incorporation of 3-electron-pair correlation and application to spin frustration in the low-lying excited states of a ferredoxin-type tetrametallic iron-sulfur cluster

We report that coupled cluster valence bond (CCVB) is a simple electronic structure method based on a perfect pairing (PP) reference with 2-pair recouplings for strong electron correlation problems. CCVB is spin-pure, size-consistent, and can exactly (in its active space) separate any molecule into atoms for which unrestricted Hartree-Fock (UHF) at dissociation is the sum of the ground state UHF energies of the atoms. However CCVB is far from a complete description of strong correlations. Its first failure to exactly describe spin-recouplings arises at the level of 3 electron pairs, such as the recoupling of 3 triplet oxygen atoms in the dissociation of singlet ozone. Such situations are often associated with spin frustration. To address this limitation, an extension of CCVB, termed CCVB+i3, is reported here that includes an independent (i) amplitude approximation to the 3-pair recouplings. CCVB+i3 thereby has the same basic computational requirements as those of CCVB, which has previously been shown to be an efficient method. CCVB+i3 correctly separates molecules that CCVB cannot. As a by-product, an independent 2-pair amplitude approximation to CCVB, called PP+i2, is also defined. Remarkably, PP+i2 can also correctly separate systems that CCVB cannot. CCVB+i3 is validated on the symmetric dissociation of D 3h ozone. CCVB+i3 is then used to explore the role of 3-pair recouplings in an [Fe 4 S 4 (SCH 3 ) 4 ] 2- cluster that has been used to model the iron-sulfur core of [Fe 4 S 4 ] ferredoxins. Using localized PP orbitals, such recouplings are demonstrated to be large in some low-lying singlet excited states of the cluster. Lastly, significant 3 pair recoupling amplitudes include the usual triangular motif associated with spin frustration and other geometric arrangements of the 3 entangled pairs across the 4 iron centers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

The unusual symmetry of hexafluoro-o-xylene—A microwave spectroscopy and computational study

The rotational constants and quartic centrifugal distortion coefficients of hexafluoro-o-xylene were precisely derived from the 8 GHz to 18 GHz gas phase microwave spectrum. In addition, the rotational constants of all singly substituted 13 C isotopologues were determined. Instead of the intuitively expected symmetry of C 2v , as in o-xylene, calculations with a variety of methods (B3LYP, CAM-B3LYP, ωB97XD, MP2, and coupled-cluster singles, doubles, and perturbative triples) predict a C 2 symmetry structure in which the two CF 3 groups rotate in opposite directions by about 16°. While the experimental results in this study are not capable of proving this unusual symmetry, they can support future microwave, gas phase electron diffraction or nuclear magnetic resonance studies. From the presented data, a preliminary r 0 structure was determined, reproducing the experimental rotational constants with deviations of no more than 15 kHz. Analysis of the interactions between the two CF 3 groups using an effective fragment potential approach identifies two major contributions to their interaction, due to exchange-repulsion and electrostatic repulsion, with electrostatic repulsion responsible for the barrier at the C 2v geometry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Diffusion quantum Monte Carlo approach to the polaritonic ground state

Making and using polaritonic states (i.e., hybrid electron-photon states) for chemical applications has recently become one of the most prominent and active fields that connects the communities of chemistry and quantum optics. Modeling of such polaritonic phenomena using ab initio approaches calls for new methodologies, leading to the reinvention of many commonly used electronic structure methods, such as Hartree-Fock, density functional, and coupled cluster theories. Here in this work, we explore the formally exact diffusion quantum Monte Carlo approach to obtain numerical solutions to the polaritonic ground state during the dissociation of the H 2 molecular system. Here, we examine various electron-nuclear-photon properties throughout the dissociation, such as changes to the minimum of the cavity Born-Oppenheimer surface, the localization of the electronic wave function, and the average mode occupation. Finally, we directly compare our results to that obtained with state-of-the-art, yet approximate, polaritonic coupled cluster approaches.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2023

Trends of Neutron Skins and Radii of Mirror Nuclei from First Principles

The neutron skin of atomic nuclei impacts the structure of neutron-rich nuclei, the equation of state of nucleonic matter, and the size of neutron stars. Here we predict the neutron skin of selected light- and medium-mass nuclei using coupled-cluster theory and the auxiliary field diffusion Monte Carlo method with two- and three-nucleon forces from chiral effective field theory. We find a linear correlation between the neutron skin and the isospin asymmetry in agreement with the liquid-drop model and compare with data. We also extract the linear relationship that describes the difference between neutron and proton radii of mirror nuclei and quantify the effect of charge symmetry breaking terms in the nuclear Hamiltonian. In conclusion, our results for the mirror-difference charge radii and binding energies per nucleon agree with existing data.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

16 O Electroweak Response Functions from First Principles

We present calculations of various electroweak response functions for the 16 O nucleus obtained using coupled-cluster theory in conjunction with the Lorentz integral transform method. We employ nuclear forces derived at next-to-leading order and next-to-next-to-leading order in chiral effective field theory and perform a Bayesian analysis to assess uncertainties. Our results are in good agreement with available electron-scattering data at |𝐪|≈326 MeV/c. Additionally, we provide several predictions for the weak response functions in the quasielastic peak region at |𝐪| =300 and 400 MeV/c, which are critical for long-baseline neutrino experiments.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1996

On the Reaction of FNO2 with CH3, t-butyl, and C13H21

Theoretical studies are reported for the reaction of FN02 with the radicals CH3, t-butyl, and C13H21, which are templates for the radical site of a hydrogenated diamond surface. All structures axe fully optimized using density functional theory (DFT) based on the B3LYP functional. Calibration calculations axe performed for CH3 + FNO2 using the coupled cluster approach, the the internally contracted multireference configuration. interaction method, and second order perturbation theory based upon complete active space SCF (CASSCF) reference wave function. These calibration calculations support the B3LYP approach for the calculation of bond energies, but show the B3LYP barrier is too low. Combining the calibration calculations with the larger clusters yields our best estimate of a barrier of about 10 kcal/mol. for the reaction of FNO2 with a radical site on hydrogenated diamond.

Thuemmel, H. T.↗
DOE OSTI2017

Accurate Isomerization Enthalpy and Investigation of the Errors in Density Functional Theory for Dihydroazulene/Vinylheptafulvene Photochromism Using Diffusion Monte Carlo

In this paper, we investigate the isomerization enthalpy of the dihydroazulene/vinylheptafulvene (DHA/VHF) molecular photoswitch system derivatives using electronic structure calculation methods including density functional theory (DFT), quantum Monte Carlo (QMC), and coupled cluster (CCSD(T)). Recent efforts have focused on tuning the isomerization enthalpy of the photoswitch for solar thermal energy storage applications using substitutional functional groups on its five- and seven-membered carbon rings, predominantly using DFT for the energy predictions. However, using the higher accuracy QMC and CCSD(T) methods, we show that in many cases DFT incorrectly predicts the isomerization enthalpy, and the errors depend on the functional groups substituted and the choice of the DFT functional. Isomerization of the DHA to VHF molecule is an electrocyclic ring-opening reaction on the five-membered ring of the DHA isomer. We find that the DFT errors are correlated to the electrocyclic ring-opening reactions of cyclobutene and cyclo-1,3-hexadiene, such that the DFT error changes monotonically with the size of the carbon ring, although QMC and CCSD(T) results are in a good agreement irrespective of the ring size. Using the QMC and CCSD(T) isomerization enthalpies, we predict gravimetric energy densities of the DHA derivatives for solar thermal storage applications. Our results show that suitable substitutions on DHA can yield gravimetric storage densities as large as 732 kJ/kg.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

Approximate solutions to the ab initio electronic structure problem have been a focus of theoretical and computational chemistry research for much of the past century, with the goal of predicting relevant energy differences to within “chemical accuracy” (1 kcal/mol). For small organic molecules, or in general, for weakly correlated main group chemistry, a hierarchy of single-reference wave function methods has been rigorously established, spanning perturbation theory and the coupled cluster (CC) formalism. For these systems, CC with singles, doubles, and perturbative triples is known to achieve chemical accuracy, albeit at O(N7) computational cost. In addition, a hierarchy of density functional approximations of increasing formal sophistication, known as Jacob’s ladder, has been shown to systematically reduce average errors over large datasets representing weakly correlated chemistry. However, the accuracy of such computational models is less clear in the increasingly important frontiers of chemical space including transition metals and f-block compounds, in which strong correlation can play an important role in reactivity. A stochastic method, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), has been shown to be capable of producing chemically accurate predictions even for challenging molecular systems beyond the main group, with relatively low O(N3 − N4) cost and near-perfect parallel efficiency. Herein, we present our perspectives on the past, present, and future of the ph-AFQMC method. We focus on its potential in transition metal quantum chemistry to be a highly accurate, systematically improvable method that can reliably probe strongly correlated systems in biology and chemical catalysis and provide reference thermochemical values (for future development of density functionals or interatomic potentials) when experiments are either noisy or absent. Finally, we discuss the present limitations of the method and where we expect near-term development to be most fruitful.

Chemistry↗
DOE OSTI2013

Reliable Modeling of the Electronic Spectra of Realistic Uranium Complexes

We present an EOMCCSD (equation of motion coupled cluster with singles and doubles) study of excited states of the small [UO2]2+ and [UO2]+ model systems as well as the larger UV IO2(saldien) complex. In addition, the triples contribution within the EOMCCSDT and CR-EOMCCSD(T) (completely renormalized EOMCCSD with non-iterative triples) approaches for the [UO2]2+ and [UO2]+ systems as well as the active-space variant of the CR-EOMCCSD(T) method | CREOMCCSd(t) | for the UV IO2(saldien) molecule are investigated. The coupled cluster data was employed as benchmark to chose the "best" appropriate exchange--correlation functional for subsequent time-dependent density functional (TD-DFT) studies on the transition energies for closed-shell species. Furthermore, the influence of the saldien ligands on the electronic structure and excitation energies of the [UO2]+ molecule is discussed. The electronic excitations as well as their oscillator dipole strengths modeled with TD-DFT approach using the CAM-B3LYP exchange{correlation functional for the [UV O2(saldien)]- with explicit inclusion of two DMSOs are in good agreement with the experimental data of Takao et al. [Inorg. Chem. 49, 2349-2359, (2010)].

Environmental Molecular Sciences Laboratory↗
DOE OSTI2019

Distinguishing artificial and essential symmetry breaking in a single determinant: approach and application to the C 60 , C 36 , and C 20 fullerenes

We present a thorough analysis of symmetry breaking observed in Hartree–Fock (HF) solutions of the C 60 , C 36 , and C 20 fullerenes in order to characterize the nature of electron correlation in them. Our analysis is based on (1) the critical regularization strength to restore symmetry breaking in the recently developed regularized orbital optimized second-order Møller–Plesset perturbation theory (κ-OOMP2), (2) singlet–triplet gaps from various MP2 methods, and (3) natural orbital occupation numbers from restricted coupled-cluster with singles and doubles (RCCSD) and coupled-cluster valence bond with singles and doubles (CCVB-SD). Based on these three independent probes, we conclude that C 36 (D6h) exhibits genuine strong correlation and symmetry breaking whereas C 60 exhibits artificial HF symmetry breaking and is not strongly correlated. Investigating the critical regularization strength, we discuss strong correlation in C 20 at the Jahn–Teller distorted geometries (C 2h , D 2h , C i , and D 3h ) and the Ih geometry. Only C 20 (I h ) was found to be strongly correlated while others exhibit artificial HF symmetry breaking. This analysis highlights a useful feature of the recommended κ-OOMP2 method. It is an electronic structure method that describes dynamic correlation, and attenuates strong correlation in MP2 towards zero by regularization. Therefore, κ-OOMP2 will exhibit symmetry breaking in its reference determinant only when correlation is strong (i.e., essential symmetry breaking). Furthermore, artificial symmetry breaking (arising in HF due to neglect of dynamic correlation) thus appears to be removed in κ-OOMP2.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Thermofield Theory for Finite-Temperature Electronic Structure

Wave function methods have offered a robust, systematically improvable means to study ground-state properties in quantum many-body systems. Theories like coupled cluster and their derivatives provide highly accurate approximations to the energy landscape at a reasonable computational cost. Analogues of such methods to study thermal properties, though highly desirable, have been lacking because evaluating thermal properties involve a trace over the entire Hilbert space, which is a formidable task. Besides, excited-state theories are generally not as well studied as ground-state ones. In this mini-review, we present an overview of a finite-temperature wave function formalism based on thermofield dynamics to overcome these difficulties. Thermofield dynamics allows us to map the equilibrium thermal density matrix to a pure state, i.e., a single wave function, albeit in an expanded Hilbert space. Ensemble averages become expectation values over this so-called thermal state. Around this thermal state, we have developed a procedure to generalize ground-state wave function theories to finite temperatures. As explicit examples, we highlight formulations of mean-field, configuration interaction, and coupled cluster theories for thermal properties of Fermions in the grand-canonical ensemble. To assess the quality of these approximations, we also show benchmark studies for the one-dimensional Hubbard model, while comparing against exact results. We will see that the thermal methods perform similarly to their ground-state counterparts, while merely adding a prefactor to the asymptotic computational cost. Furthermore, they also inherit all the properties, good or bad, from the ground-state methods, signifying the robustness of our formalism and the scope for future development.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

First principle calculations of effective exchange integrals: Comparison between SR (BS) and MR computational results

First principle calculations of effective exchange integrals (J) in the Heisenberg model for diradical species were performed by both symmetry-adapted (SA) multi-reference (MR) and broken-symmetry (BS) single reference (SR) methods. Mukherjee-type (Mk) state specific (SS) MR coupled-cluster (CC) calculations by the use of natural orbital (NO) references of ROHF, UHF, UDFT and CASSCF solutions were carried out to elucidate J values for di- and poly-radical species. Spin-unrestricted Hartree Fock (UHF) based coupled-cluster (CC) computations were also performed to these species. Comparison between UHF-NO(UNO)-MkMRCC and BS UHF-CC computational results indicated that spin-contamination of UHF-CC solutions still remains at the SD level. In order to eliminate the spin contamination, approximate spin-projection (AP) scheme was applied for UCC, and the AP procedure indeed corrected the error to yield good agreement with MkMRCC in energy. The CC double with spin-unrestricted Brueckner's orbital (UBD) was furthermore employed for these species, showing that spin-contamination involved in UHF solutions is largely suppressed, and therefore AP scheme for UBCCD removed easily the rest of spin-contamination. We also performed spin-unrestricted pure- and hybrid-density functional theory (UDFT) calculations of diradical and polyradical species. Three different computational schemes for total spin angular momentums were examined for the AP correction of the hybrid (H) UDFT. HUDFT calculations followed by AP, HUDFT(AP), yielded the S-T gaps that were qualitatively in good agreement with those of MkMRCCSD, UHF-CC(AP) and UB-CC(AP). Thus a systematic comparison among MkMRCCSD, UCC(AP) UBD(AP) and UDFT(AP) was performed concerning with the first principle calculations of J values in di- and poly-radical species. It was found that BS (AP) methods reproduce MkMRCCSD results, indicating their applicability to large exchange coupled systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Kinetics of the Strongly Correlated CH 3 O + O 2 Reaction: The Importance of Quadruple Excitations in Atmospheric and Combustion Chemistry

Kinetics measurements on radical–radical reactions are often unavailable experimentally, and obtaining quantitative rate constants for such reactions by theoretical methods is challenging because the transition states and the reactants are often strongly correlated. Treating strongly correlated systems by coupled cluster theory limited to single, double, and triple connected excitations is often inadequate. We therefore use a new method, called GMM(P), for extrapolation to the complete configuration interaction limit to go beyond triple excitations and in particular to approximate the CCSDTQ(P)/CBS limit. Here, we present this method and use it to investigate the CH 3 O + O 2 reaction. The contribution of connected quadruple excitations to the barrier height energy is found to be -3.13 kcal/mol, and adding a quasiperturbative calculation of the effect of connected pentuple excitations brings the post-connected-triples contributions to -3.44 kcal/mol, which corresponds to Boltzmann factors that increase calculated rate constants by factors of 1.0 × 10 3 , 3.3 × 10 2 , and 18 at 250, 298, and 600 K, respectively. We present rate constants for temperatures from 250 to 2000 K, and we find that the Arrhenius activation energy increases from 0.58 to 9.68 kcal/mol over this range. We also find reasonably good accuracy for the barrier height with the MN15-L exchange–correlation functional, and we calculate rate constants by a combination of GMM(P) and MN15-L electronic structure calculations and conventional and variational transition state theory, in particular canonical variational theory with small-curvature tunneling. The present findings have broad implications for obtaining quantitative rate constants for complex reaction systems in atmospheric and combustion chemistry.

Chemistry↗
DOE OSTI2022

Geometry optimizations with spinor-based relativistic coupled-cluster theory

Development of analytic gradients for relativistic coupled-cluster singles and doubles augmented with a non-iterative triples [CCSD(T)] method using an all-electron exact two-component Hamiltonian with atomic mean-field spin–orbit integrals (X2CAMF) is reported. This enables efficient CC geometry optimizations with spin–orbit coupling included in orbitals. Here, the applicability of the implementation is demonstrated using benchmark X2CAMF-CCSD(T) calculations of equilibrium structures and harmonic vibrational frequencies for methyl halides, CH 3 X (X = Br, I, and At), as well as calculations of rotational constants and infrared spectrum for RaSH+, a radioactive molecular ion of interest to spectroscopic study.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Unsupervised machine learning based on non-negative tensor factorization for analyzing reactive-mixing

Analysis of reactive-diffusion simulations representing complex mixing processes requires a large number of independent model runs. For each high-fidelity model simulation, the model inputs are varied and the predicted mixing behavior is represented by temporal and spatial changes in species concentration. It is then required to discern how the model inputs (such as diffusivity, dispersion, anisotropy, and velocity field properties) impact the mixing process. This task is challenging and typically involves interpretation of large model outputs representing temporal and spatial changes of species concentration within the model domain. Yet, the task can be automated and substantially simplified by applying Machine Learning (ML) methods. Here, we present an application of an unsupervised ML method (called NTF k ) using Non-negative Tensor Factorization (NTF) coupled with a custom clustering procedure based on k-means to reveal the temporal and spatial features in product concentrations. An attractive and unique aspect of the proposed ML method is that it ensures the extracted features are non-negative, which is important to obtain a meaningful deconstruction of the mixing processes. The ML methodology is applied to a large set of high-resolution finite-element model simulations representing anisotropic reaction-diffusion processes in perturbed vortex-based velocity fields. The applied finite-element method ensures that spatial and temporal species concentration are always non-negative, even in the case of high anisotropic contrasts. The simulated reaction is a fast irreversible bimolecular reaction A+B→C , where species A and B react to form species C. The reactive-diffusion model input parameters that control mixing include properties of the velocity field (such as vortex structures), anisotropic dispersion, and molecular diffusion. We demonstrate the applicability of the ML feature extraction method to produce a meaningful deconstruction of model outputs to discriminate between different physical processes impacting the reactants, their mixing, and the spatial distribution of the product C. The introduced ML analysis allowed us to identify additive temporal and spatial features that characterize mixing behavior. The application of the proposed NTF k approach is not limited to reactive-mixing. NTFk can be readily applied to any observed or simulated datasets that can be represented as tensors (multi-dimensional arrays) and have separable latent signatures or features.

97 MATHEMATICS AND COMPUTING↗