Primitive Basis Research Articles

Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis.

We develop and implement an exact conical intersection nonadiabatic wave packet dynamics method that combines the local diabatic representation, Strang splitting for the total molecular propagator, and discrete variable representation with uniform grids. By employing the local diabatic representation, this method captures all nonadiabatic effects, including nonadiabatic transitions, electronic coherences, and geometric phase. Moreover, it is free of singularities in the first and second derivative couplings and does not require the electronic wave function to be continuous with respect to the nuclear coordinates. We further show that in contrast to the adiabatic representation, the split-operator method can be directly applied to the full molecular propagator with the locally diabatic ansatz. The Fourier series, employed as the primitive nuclear basis functions, is universal and can be applied to all types of reactive coordinates. The combination of local diabatic representation, Strang splitting, and Fourier basis allows numerically exact modeling of conical intersection quantum dynamics directly with adiabatic electronic states that can be obtained from standard electronic structure computations.

Two-component transformation inclusive contraction scheme in the relativistic molecular orbital theory

The incorporation of a two-component transformation into the contraction coefficients of basis functions is suggested. Such a contraction, referred to as “two-component transformation inclusive contraction (TIC)”, effectively eliminates the computational cost associated with the two-component transformation and can be readily implemented into conventional quantum chemistry programs. TIC was verified through numerical calculations using the second- and third-order unitarized Douglas–Kroll method, which is newly derived for the validation of TIC, and the infinite-order two-component method. The numerical validation results suggested that TIC could sufficiently reproduce the results of primitive basis sets for both atoms and molecules.

Matrix-decomposed two-electron integrals in the infinite-order two-component Hamiltonian

The Cholesky decomposition (CD) and lower–upper decomposition (LUD) of two-electron integral (TEI) matrices were applied to a spin-free infinite-order two-component (IOTC) relativistic Hamiltonian. In the Hamiltonian, the evaluation of three types of TEIs of primitive basis functions and the unitary transformation of TEIs are required. CD was used to calculate the symmetric TEI matrices: Coulomb-like and specific spin-free interaction terms. LUD was used for the asymmetric TEI matrix: Darwin-like term. Unitary transformation was performed on the decomposed matrices. Numerical assessments suggest that the proposed method can reduce the computational cost without loss of accuracy.

The structure of abelian number fields with Dirichlet characters

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree.

Discontinuous Galerkin method with Voronoi partitioning for quantum simulation of chemistry

To circumvent a potentially dense two-body interaction tensor and obtain lower asymptotic costs for quantum simulations of chemistry, the discontinuous Galerkin (DG) basis set with a rectangular partitioning strategy was recently introduced [McClean et al, New J. Phys. 22, 093015, 2020]. We propose and numerically scrutinize a more general DG basis set construction based on a Voronoi decomposition with respect to the nuclear coordinates. This allows the construction of DG basis sets for arbitrary molecular and crystalline configurations. We here employ the planewave dual basis set as primitive basis set in the supercell model; as a set of grid-based nascent delta functions, the planewave dual functions provide sufficient flexibility for the Voronoi partitioning. The presented implementation of this DG-Voronoi approach is in Python and solely based on PySCF. We numerically investigate the performance, at the mean-field and correlated level of theory for quasi-1D, quasi-2D and fully 3D systems, and exemplify the application to crystalline systems.

Ground state of asymmetric tops with DMRG: Water in one dimension.

We propose an approach to compute the ground state properties of collections of interacting asymmetric top molecules based on the density matrix renormalization group method. Linear chains of rigid water molecules of varying sizes and density are used to illustrate the method. A primitive computational basis of asymmetric top eigenstates with nuclear spin symmetry is used, and the many-body wave function is represented as a matrix product state. We introduce a singular value decomposition approach in order to represent general interaction potentials as matrix product operators. The method can be used to describe linear chains containing up to 50 water molecules. Properties such as the ground state energy, the von-Neumann entanglement entropy, and orientational correlation functions are computed. The effect of basis set truncation on the convergence of ground state properties is assessed. It is shown that specific intermolecular distance regions can be grouped by their von-Neumann entanglement entropy, which in turn can be associated with electric dipole-dipole alignment and hydrogen bond formation. Additionally, by assuming conservation of local spin states, we present our approach to be capable of calculating chains with different arrangements of the para and ortho spin isomers of water and demonstrate that for the water dimer.

A brute-force code searching for cell of non-identical displacement for CSL grain boundaries and interfaces

Atomic simulation of coincidence-site lattice (CSL) grain boundaries (GBs) and interfaces is of importance for understanding GBs in polycrystalline materials and for making functional films. A common process of high-throughput simulation for CSL GBs and interfaces is to explore rigid body translation (RBT) of one crystal respect to the other. Cell of non-identical displacement (CNID) is the minimum cell including all non-identical RBTs in the GB or interface plane and is important for effective sampling. This work proposes an algorithm to compute the CNID of any two-dimensional CSL GB or interface based on a reciprocity relation between the displacement shift complete (DSC) and CSL. Program summaryProgram title: cnidcalCPC Library link to program files:https://doi.org/10.17632/wzy67jd56d.1Code Ocean capsule:https://codeocean.com/capsule/1648641Licensing provisions: MIT licenseProgramming language: Python Nature of problemDetermination of atomic structure of crystalline interfaces and grain boundaries is a key issue for understanding the properties of polycrystalline materials, where atomic simulation has made notable contribution in this field. Simulation of crystalline interfaces often applies a bicrystal model of a coincidence-site lattice (CSL) [1, 2] interface comprising two slabs of crystals which coincides in the interface plane. To determine the most stable structure of this interface model, a main task is often to tailor the initial bicrystal model to obtain many non-identical and non-relaxed structures for subsequent relaxation to explore the energy landscape of the system to find the ‘global minimum’ at 0 K, where one of the most important operations to tailor the model is applying rigid body translation (RBT) [1] of one crystal respect to the other by a vector confined in the interface plane. Because of symmetry of the two crystals, there exist periodicity of this translation vector which is reflected by a special cell called ‘cell of non-identical displacement’ (CNID) [1] which is the minimum cell including all non-identical choices of this vector and is key for effective sampling. Although CNID has a straightforward definition that it is a lattice including both the two overlapping two-dimensional lattices, finding a method capable to determine a primitive basis of CNID in general cases can be non-trivial and there has not been a reported programme capable to do so. Solution methodCNID in nature is a displacement shift complete (DSC) lattice [2]. While there has not been effective reported programme to compute DSC, several programmes have been made to compute CSL [3, 4, 5]. Besides, Grimmer [6] has proposed a mathematical relationship between the DSC and the CSL. In this way, based on a previously reported programme computing the CSL by brute-force method [5] and the relationship between DSC and CSL proposed by Grimmer, we generated a python code to compute CNID. This code is available to compute the CNID of a CSL interface comprising any two lattices if only they coincide even including complex cases involving heterogeneous lattices with difference in both shape and size.

Hyperfine Coupling Constants in Local Exact Two-Component Theory

We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete derivative of the X2C Hamiltonian is formed, that is, the derivatives of the unitary decoupling transformation are considered. This requires the solution of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the density functional approximation is carried out for a set of 12-17 transition-metal compounds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amount of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure density functional r2SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C6Cl5)4]- with more than 4700 primitive basis functions and four rare-earth single-molecule magnets: [La(OAr*)3]-, [Lu(NR2)3]-, [Lu(OAr*)3]-, and [TbPc2]-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the experimental findings of all Pt, La, Lu, and Tb molecules.

Implementation of a time-dependent multiconfiguration self-consistent-field method for coupled electron-nuclear dynamics in diatomic molecules driven by intense laser pulses

We present an implementation of a time-dependent multiconfiguration self-consistent-field (TD-MCSCF) method [R. Anzaki et al., Phys. Chem. Chem. Phys. 19, 22008 (2017)] with the full configuration interaction expansion for coupled electron-nuclear dynamics in diatomic molecules subject to a strong laser field. In this method, the total wave function is expressed as a superposition of different configurations constructed from time-dependent electronic Slater determinants and time-dependent orthonormal nuclear basis functions. The primitive basis functions of nuclei and electrons are strictly independent of each other without invoking the Born-Oppenheimer approximation. Our implementation treats the electronic motion in its full dimensionality on curvilinear coordinates, while the nuclear wave function is propagated on a one-dimensional stretching coordinate with rotational nuclear motion neglected. We apply the present implementation to high-harmonic generation and dissociative ionization of a hydrogen molecule and discuss the role of electron-nuclear correlation.

Optimal radial basis for density-based atomic representations.

The input of almost every machine learning algorithm targeting the properties of matter at the atomic scale involves a transformation of the list of Cartesian atomic coordinates into a more symmetric representation. Many of the most popular representations can be seen as an expansion of the symmetrized correlations of the atom density and differ mainly by the choice of basis. Considerable effort has been dedicated to the optimization of the basis set, typically driven by heuristic considerations on the behavior of the regression target. Here, we take a different, unsupervised viewpoint, aiming to determine the basis that encodes in the most compact way possible the structural information that is relevant for the dataset at hand. For each training dataset and number of basis functions, one can build a unique basis that is optimal in this sense and can be computed at no additional cost with respect to the primitive basis by approximating it with splines. We demonstrate that this construction yields representations that are accurate and computationally efficient, particularly when working with representations that correspond to high-body order correlations. We present examples that involve both molecular and condensed-phase machine-learning models.

Existence of primitive 2-normal elements in finite fields

An element α∈Fqn is normal over Fq if B={α,αq,αq2,⋯,αqn−1} forms a basis of Fqn as a vector space over Fq. It is well known that α∈Fqn is normal over Fq if and only if gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1 and xn−1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al. (2013): an element α∈Fqn is k-normal over Fq if the greatest common divisor of the polynomials gα[x] and xn−1 in Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal.Huczynska et al. made the question about the pairs (n,k) for which there exist primitive k-normal elements in Fqn over Fq and they got a partial result for the case k=1, and later Reis and Thomson (2018) completed this case. The Primitive Normal Basis Theorem solves the case k=0. In this paper, we solve completely the case k=2 using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of k-normal elements in Fqn.

The computational origin of representation.

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations-whatever they are-must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an "assembly language" for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems.

Discontinuous Galerkin discretization for quantum simulation of chemistry

All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much of the foundation of modern correlated quantum chemistry calculations—on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block-diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N4.5) in molecular orbital representations to O(N2.6) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1–2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.

Revisiting IRKA: Connections with Pole Placement and Backward Stability

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order ${\mathscr{H}}_{2}$ -approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid—but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.

Фразеологизмдердегі түркі әлемінің тілдік бейнесі

The article considers the semantic features of phraseological units in the language picture of the Turkic world. The phraseological foundation in any Turkic language is a mirror of the nation’s life. The whole life of the tradition, the system of values of the people is gathered here. The fund of cognitive knowledge of Turkic-speaking peoples 64ISSN 2664-5157. Turkic Studies Journal, Number 1, Volume 2, 2020Guldarhan Smagulova replenishes each other. Since the world of shared values, stereotypical skills and cultural lifestyles and their similarities in religion are based on a holistic mental space. The image of this integrity is collected and preserved through the cumulative quality of the language, and proves the cultural identity in the past way of life of the Turkic peoples. At the same time, the cultural interpretation of Turkic phraseology and paremiology has of particular importance. Today the phraseology of the Turkic languages is a comprehensive object of research and its scientific results are based on anthropocentric approaches. One nation- one language. Only ethnolinguistic and linguo-cultural studies can analyze the ethnic content of images preserved in the language and the differences between the Turkic world and other nations. Ethnic culture is the most ancient layer, the primitive basis of national culture. The traditions and customs of this layer reflect the commonness and similarity of the Turkic culture. Therefore, the general vocabulary in the fund of the Turkic languages, identical phraseological units, linguistic traces of cultural rhemes in the history of the ethnic group, and common subjects in our literature require scientific analysis.

The ANO-R Basis Set.

In this work, the new ANO-R basis set for all elements of the first six periods is introduced. The ANO-R basis set is an all-electron basis set that was constructed including scalar-relativistic effects of the exact-two component (X2C) Hamiltonian and modeling the atomic nucleus by a Gaussian charge distribution, which makes the basis set suitable for calculations of both light and heavy elements. For high accuracy, it takes advantage of the general contraction scheme and was developed at the CASSCF/CASPT2 level of theory. The distinguishing feature of the ANO-R basis set is its compactness in terms of both primitive and contracted basis functions, thus containing no superfluous functions for a given quality. An optimum number of primitive basis functions was selected based on studying the convergence toward the complete basis set limit for each element individually. The primitive basis sets were then contracted using the density-averaged atomic-natural-orbital (ANO) scheme, and suitable contraction levels were determined solely based on the natural orbital occupation numbers that describe the contribution of each natural orbital to the one-particle density matrix. Rather than following the common "split-valence n-tuple zeta plus polarization functions" structure, the resulting basis sets ANO-R0 to ANO-R3 possess a unique composition for each element, ensuring that no unnecessary functions are included while the basis sets are still balanced across the first six periods (H-Rn).

Ovoids and primitive normal bases for quartic extensions of Galois fields

We determine lower bounds for the number of primitive normal elements in a four-dimensional extension E over a Galois field \(F={\mathrm{GF}}(q)\). Our approach is based on viewing E as the three-dimensional projective space \(\Gamma =\mathrm{PG}(3,q)\). In any of the three cases, whether q is even, or \(q\equiv 3 { \text{ mod } }4\), or \(q\equiv 1 { \text{ mod } }4\), we use a decomposition of the multiplicative group of E in order to determine a (canonical) partition of the point set of \(\Gamma \) into \(q+1\) ovoids. The points of \(\Gamma \) are distinguished into primitive and non-primitive ones, and an ovoid is called primitive if it contains at least one primitive point. The bounds are derived by studying the intersections of the primitive ovoids with the configuration of those points of \(\Gamma \) which do not give rise to normal elements of E / F. Given that \(q^2+1\) is a prime number when q is even, or that \(\frac{1}{2}(q^2+1)\) is a prime number when q is odd, we actually achieve the exact number of all primitive normal elements for the quartic extension over F.

Quasi-Atomic Bond Analyses in the Sixth Period: I. Relativistic Accurate Atomic Minimal Basis Sets for the Elements Cesium to Radon

Full-valence relativistic accurate atomic minimal basis set (AAMBS) orbitals are developed for the sixth-row elements from cesium to radon, including the lanthanides. Saturated primitive atomic basis sets are developed and subsequently used to form the AAMBS orbitals. By virtue of the use of a saturated basis, properties computed based on the AAMBS orbitals are basis set independent. In molecules, the AAMBS orbitals can be used to construct valence virtual orbitals (VVOs) that provide chemically meaningful ab initio lowest unoccupied molecular orbitals (LUMOs) with basis set independent orbital energies. The optimized occupied molecular orbitals complemented with the VVOs form a set of full-valence molecular orbitals. They can be transformed into a set of oriented quasi-atomic orbitals (QUAOs) that provide information on intramolecular bonding via an intrinsic density analysis. In the present work, the development of the AAMBS for the sixth row is presented.

Molscat: A program for non-reactive quantum scattering calculations on atomic and molecular collisions

molscat is a general-purpose program for quantum-mechanical calculations on nonreactive atom–atom, atom–molecule and molecule–molecule collisions. It constructs the coupled-channel equations of atomic and molecular scattering theory, and solves them by propagating the wavefunction or log-derivative matrix outwards from short range to the asymptotic region at long range. It then applies scattering boundary conditions to extract the scattering matrix (S matrix). Built-in coupling cases include atom + rigid linear molecule, atom + vibrating diatom, atom + rigid symmetric top, atom + asymmetric or spherical top, rigid diatom + rigid diatom, rigid diatom + asymmetric top, and diffractive scattering of an atom from a crystal surface. Interaction potentials may be specified either in program input (for simple cases) or with user-supplied routines. For the built-in coupling cases, molscat can loop over total angular momentum (partial wave) and total parity to calculate elastic and inelastic integral cross sections and spectroscopic line-shape cross sections. Post-processors are available to calculate differential cross sections, transport, relaxation and Senftleben–Beenakker cross sections, and to fit the parameters of scattering resonances. molscat also provides an interface for plug-in routines to specify coupling cases (Hamiltonians and basis sets) that are not built in; plug-in routines are supplied to handle collisions of a pair of alkali-metal atoms with hyperfine structure in an applied magnetic field. For low-energy scattering, molscat can calculate scattering lengths and effective ranges and can locate and characterise scattering resonances as a function of an external variable such as the magnetic field. Program summaryProgram Title:molscatProgram Files doi:http://dx.doi.org/10.17632/rtzgf5mwpn.1Licensing provisions: GPLv3Programming language: Fortran 90External routines/libraries: LAPACK, BLASNature of problem: Quantum-mechanical calculations of scattering properties for non-reactive collisions between atoms and molecules.Solution method: The Schrödinger equation is expressed in terms of coupled equations in the interparticle distance, R. Solutions of the coupled-channel equations are propagated outwards from the classically forbidden region at short range to the asymptotic region. The program calculates scattering S matrices and uses them to calculate scattering properties including elastic, inelastic and line-shape cross sections.Unusual features:1. molscat contains numerous features to handle quantities important in low-energy collisions. It can propagate very efficiently to very long range, and it can calculate low-energy properties such as the scattering length (complex and energy-dependent if required) and the effective range.2. molscat can construct and solve sets of coupled equations using basis sets that are not eigenfunctions of the Hamiltonians of the colliding particles at long range. The propagation is carried out in the primitive basis set, and the solutions are transformed to the asymptotic basis set before applying long-range boundary conditions to extract the S matrix.3. molscat provides an interface that allows users to specify the coupled equations that arise from additional Hamiltonians and basis sets.4. molscat provides an interface that allows users to include multiple external fields in the Hamiltonian. This same interface also allows users to include a factor which scales the interaction potential, and to investigate how properties vary with this factor.5. molscat can locate and characterise low-energy Feshbach resonances either as a function of collision energy or as a function of external fields or of the potential scaling factor.

Error-consistent segmented contracted all-electron relativistic basis sets of double- and triple-zeta quality for NMR shielding constants.

We present property-tailored all-electron relativistic Karlsruhe basis sets for the elements hydrogen to radon. The modifications described herein use at most four additional primitive basis functions and re-optimized contraction coefficients of the inner-most segment. Thus, the shielding constants are improved while maintaining the compactness of the basis set. A large set of 255 closed-shell molecules was used to assess the quality of the developed bases throughout the periodic table of elements.