Primitive Basis Research Articles

Orbits of normal basis generators

Let L/K be a finite‐dimensional Galois field extension with Galois group G, B the set of normal basis generators for this extension, and C={γ ∈L∣γ B=B}. Then C is a group under multiplication. This group was introduced and characterized in Theorem 1.15 of a well‐known paper ‘Primitive normal bases for finite fields’ by H. W. Lenstra Jr. and R. J. Schoof. This result was stated in that paper without a proof. The purpose of this paper is to give a proof of the theorem.

Numerical atomic basis orbitals from H to Kr

We present a systematic study for numerical atomic basis orbitals ranging from H to Kr, which could be used in large scale $\mathrm{O}(N)$ electronic structure calculations based on density-functional theories (DFT). The comprehensive investigation of convergence properties with respect to our primitive basis orbitals provides a practical guideline in an optimum choice of basis sets for each element, which well balances the computational efficiency and accuracy. Moreover, starting from the primitive basis orbitals, a simple and practical method for variationally optimizing basis orbitals is presented based on the force theorem, which enables us to maximize both the computational efficiency and accuracy. The optimized orbitals well reproduce convergent results calculated by a larger number of primitive orbitals. As illustrations of the orbital optimization, we demonstrate two examples: the geometry optimization coupled with the orbital optimization of a ${\mathrm{C}}_{60}$ molecule and the preorbital optimization for a specific group such as proteins. They clearly show that the optimized orbitals significantly reduce the computational efforts, while keeping a high degree of accuracy, thus indicating that the optimized orbitals are quite suitable for large scale DFT calculations.

Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon

The finite Fourier representation of a function f( x) exhibits oscillations where the function or its derivatives are nonsmooth. This is known as the Gibbs phenomenon. A robust and accurate reconstruction method that resolves the Gibbs oscillations was proposed in a previous paper (J. Comput. Appl. Math. 161 (2003) 41) based on the inversion of the transformation matrix which represents the projection of a set of basis functions onto the Fourier space. If the function is a polynomial, this inverse polynomial reconstruction method (IPRM) is exact. In this paper, we develop the IPRM by requiring that the proper error be orthogonal to the Fourier or polynomial space. The IPRM is generalized to any set of basis functions. The primitive basis polynomials, nonclassical orthogonal polynomials and the Gegenbauer polynomials are used to illustrate the wide validity of the IPRM. It is shown that the IPRM yields a unique reconstruction irrespective of the basis set for any analytic function and yields spectral convergence. The ill-posedness of the transformation matrix due to the exponential growth of the condition number of the matrix is also discussed.

Characterizing Normal Bases via the Trace Map#

In their recent article Chang et al. [Chang, Y., Troung, T. K., Reed, I. S. (2001). Normal bases over GF(q). J. Algebra 241:89–101] have determined all those extensions of Galois fields for which the normal basis generators are characterized by the (obviously necessary) property of having nonzero trace. In the present article, we present a simpler proof of a generalization of that result and discuss an application concerning the existence of trace-compatible sequences of primitive normal bases for certain primary closures of Galois fields.

Iterative solutions with energy selected bases for highly excited vibrations of tetra-atomic molecules.

The use of energy selected bases (ESB) with iterative diagonalization of the Hamiltonian matrix is described for vibrations of tetra-atomic systems. The performance of the method is tested by computing vibrational states of HOOH below 10,000 cm(-1) (1296 A+ symmetry states) and H(2)CO below 13,500 cm(-1) (729 A(1) symmetry states). For iterative solutions, we tested both the implicitly restarted Lanczos method (IRLM) and the standard (nonreorthogonalizing) Lanczos approach. Comparison with other contracted basis approach as well as direct product grid representation shows superior performance of the ESB/IRLM approach. Of the two systems, H(2)CO is found to be more challenging than HOOH since it has much stronger couplings among vibrational modes, which leads to a drastically larger primitive basis set. For H(2)CO we also discuss some interesting behavior of the molecule in the high internal energy regime.

Rapid computation of all sets of electron-repulsion integrals for large-scale molecules

Abstract A concept of nonredundant sets of primitive gaussian basis functions is described, which proved highly efficient for rapid computing all sets of electron-repulsion integrals. Combination of the RT parallel algorithm with the nonredundant method demonstrated performance faster than the G amess program by a margin greater than fivefold even with a single processor calculation, making it ideal for large-scale SCF calculations. Further performance gains require accelerating calculations of initial auxiliary integrals.

A contracted basis-Lanczos calculation of vibrational levels of methane: Solving the Schrödinger equation in nine dimensions

We present a contracted basis-iterative method for calculating numerically exact vibrational energy levels of methane (a 9D calculation). The basis functions we use are products of eigenfunctions of bend and stretch Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wavefunctions well, yet are simple enough that matrix-vector products may be evaluated efficiently. We use Radau polyspherical coordinates. The bend functions are computed in a nondirect product finite basis representation [J. Chem. Phys. 118, 6956 (2003)] and the stretch functions are computed in a product potential optimized discrete variable (PODVR) basis. The memory required to store the bend basis is reduced by a factor of ten by storing it on a compacted grid. The stretch basis is optimized by discarding PODVR functions with high potential energies. The size of the primitive basis is 33 billion. The size of the product contracted basis is six orders of magnitude smaller. Parity symmetry and exchange symmetry between two of the H atoms are employed in the final product contracted basis. A large number of vibrational levels are well converged. These include almost all states up to 8000 cm−1 and some higher local mode stretch bands.

THE PRIMITIVE NORMAL BASIS THEOREM – WITHOUT A COMPUTER

Given q, a power of a prime p, denote by F the finite field GF(q) of order q, and, for a given positive integer n, by E its extension GF(qn) of degree n. A primitive element of E is a generator of the cyclic group E*. Additively too, the extension E is cyclic when viewed as an FG-module, G being the Galois group of E over F. The classical form of this result – the normal basis theorem – is that there exists an element α ∈ E (an additive generator) whose conjugates { α , α q , … , α q n − 1 } form a basis of E over F; α is a free element of E over F, and a basis like this is a normal basis over F. The core result linking additive and multiplicative structure is that there exists α ∈ E, simultaneously primitive and free over F. This yields a primitive normal basis over F, all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).

Generators for primary closures of Galois fields

We continue to study the existence of (norm- and) trace-compatible sequences of primitive normal bases for prime power extensions of finite fields, introduced by the author in Hachenberger (Finite Fields Appl. 5 (1999) 378–385; in: D. Jungnickel, H. Niederreiter (Eds.), Proceedings of the Fifth International Conference on Finite Fields and Applications, Augsburg, August 1999, Springer, Heidelberg, 2001, pp. 208–223), and improve on some aspects of these papers.

Shallow Linear Action Graphs and their Embeddings

Abstract.Action calculi, which generalise process calculi such as Petri nets, π-calculusand ambient calculus, have been presented in terms ofaction graphs. We here offerlinearaction graphs as a primitive basis for action calculi. This paper presents the category of embeddings of undirected linear action graphs without nesting, using a novel form of graphical reasoning which simplifies some otherwise complex manipulations in regular algebra. The results are adapted in a few lines to directed graphs. This work is part of a long-term search for a uniform behavioural theory for process calculi.

The geometry and forcefield of acetylene

The variational method has been used to determine the geometry and ground state potential surface of acetylene. All the parameters were refined through a least-squares fit to J = 0, 1 levels for C2H2 and C2D2. A new program was written to evaluate the rovibrational energy levels; in particular, primitive basis sets were developed for all values of J taking into account the singularity for linear geometries. Thus Σ, II, δ states can be refined. The full theory for tetraatomic linear molecules is presented. In this refinement 150 observed levels were used as data, below 10 000cm−1. The geometry was refined and gives Re(CC) = 1.2028 Å, Re(CH) = 1.0618 Å, to be compared with the best experimentally derived values of 1.2027 ± 0.0005 Å, 1.062 ± 0.001 Å, respectively. The zero point energies are 5771.1cm−1 for C2H2 and 4571.1cm−1 for C2D2.

On the inherent space complexity of fast parallel multipliers for GF(2/sup m/)

A lower bound to the number of AND gates used in parallel multipliers for GF(2/sup m/), under the condition that time complexity be minimum, is determined. In particular, the exact minimum number of AND gates for primitive normal bases and optimal normal bases of Type II multipliers is evaluated. This result indirectly suggests that space complexity is essentially a quadratic function of m when time complexity is kept minimum.

Polarization consistent basis sets: Principles

The basis set convergence of Hartree–Fock energies for the H2, H3+, C2, N2, N4, O2, O3, F2, HF, and CH4 molecules is analyzed using optimized basis functions. Based on these analysis a sequence of polarization consistent basis sets are proposed which should be suitable for systematically improving Hartree–Fock and density functional energies. Analogous to the correlation consistent basis sets designed for correlation energies, higher angular momentum functions are included based on their energetical importance. In contrast to the correlation consistent basis sets, however, the importance of higher angular momentum functions decreases approximately geometric, rather than arithmetic. It is shown that it is possible to design a systematic sequence of basis sets for which results converge monotonic to the Hartree–Fock limit. The primitive basis sets can be contracted by a general contraction scheme. It is found that polarization consistent basis sets provide a faster convergence than the correlation consistent basis sets. Results obtained with polarization consistent basis sets can be further improved by extrapolation.

Primitive complete normal bases for regular extensions

The extension E of degree n over the Galois field F={\text GF}(q)} is called regular over F, if {\text ord}_r(q) and n have greatest common divisor 1 for all prime divisors r of n which are different from the characteristic p of F (here, \order{r}{q} denotes the multiplicative order of q modulo r). Under the assumption that E is regular over F and that q-1 is divisible by 4 if q is odd and n is even, we prove the existence of a primitive element w \in E which is also completely normal over F (the latter means that w simultaneously generates a normal basis for E over every intermediate field K of E/F). Our result achieves, for the class of extensions under consideration, a common generalization of the theorem of Lenstra and Schoof on the existence of primitive normal bases [12] and the theorem of Blessenohl and Johnsen on the existence of complete normal bases [1].

Model of crystal lattice strained along the preferential direction by anisotropic stress for GaAs heteroepitaxial films grown on vicinal Si(001) and Si(110) substrates by molecular-beam epitaxy

I estimated anisotropy of stress and strain for GaAs heteroepitaxial films grown on vicinal Si(001) and Si(110) substrates by molecular-beam epitaxy from measurements of in-plane substrate curvatures and components of lattice parameters by a Bond method using x-ray diffraction. It is shown that the GaAs heteroepitaxial film is anisotropically strained through different relaxation processes of stress due to anisotropic dislocations along the [110] and [11̄0] directions. The crystal lattice of the GaAs heteroepitaxial film on Si substrate is qualitatively represented considering the crystal structure elastically strained with the [110]-, [11̄0]-, and [001]-primitive axes as a primitive basis in the orthogonal rhombic system.

Calculation of molecular integrals for systems confined by hard spherical walls: Use of the single‐center expansion of floating spherical gaussians

AbstractAssuming a gaussian basis set representation of atomic and molecular wave functions, the single‐center expansion of off‐centered spherical gaussian orbitals is exploited to calculate the one and two‐electron integrals for multielectronic atoms and molecules confined within hard spherical walls. As a validating test, the ground‐state energy of a helium atom positioned off‐center in a spherical box is calculated by applying the simplest form of the floating spherical gaussian orbital (FSGO) scheme, i.e., the use of a primitive basis set consisting of a single FSGO per electron pair. Comparison with corresponding recent accurate calculations gives supporting evidence of the adequacy of the method for its application to more elaborate gaussian‐type basis set representations for confined atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 271–278, 2001

A time-dependent discrete variable representation method

We have developed a novel discrete variable representation (DVR) method where not only the amplitudes of the wave function at the DVR grid points can change but also the positions of these grid points can move as a function of time. Since the Gauss–Hermite basis set is used as the primitive basis functions (PBF) to construct the DVR basis set, the method appears as a semiclassical one with a small number of PBF but converges very fast to the quantum with an increasing PBF. We have investigated the dynamics of a reaction coordinate with or without coupling to a heat bath of harmonic oscillators to demonstrate the validity of the proposed method. The excellent agreement of the calculated tunneling probabilities with numbers obtained by traditional quantum grid method (FFT) and the fast computability of the present method compared to the latter are remarkable.

Primitive Normal Bases for Towers of Field Extensions

Given an extension E/F of Galois fields and an intermediate field K, we consider the problem whether the (E, K)-trace of a primitive F-normal element of E can be a prescribed F-normal element of K. An interesting application is the existence of trace-compatible sequences of primitive F-normal elements for certain towers of Galois fields. In this respect, particular emphasis is laid on extensions having prime power degree.

Fitted electronic density functions from H to Rn for use in quantum similarity measures: cis-diamminedichloroplatinum(II) complex as an application example

A consistent set of fitted electronic density functions was generated for the elements from hydrogen to radon using an algorithm based on the elementary Jacobi rotations (EJR) technique. The main distinguishing attribute of this fitting procedure is the production of approximated electronic density functions with positive definite expansion coefficients; in this way, the statistical meaning of the probability distribution is preserved. The methodology, which was fully described previously, was modified in this work to improve and accelerate the fitting procedure. This variation concerns the optimization method employed to obtain the optimal angle of the EJR, implementing an algorithm based on a Taylor series expansion. Additionally, a new 1S-Type Gaussian basis set for atoms H to Rn is presented, that was fitted from a primitive basis set of Huzinaga. Fitted density functions facilitate theoretical calculations over large molecules and may be employed in many areas of computational chemistry, for example, in quantum similarity measures (QSM). To verify the basis set, a sound example related to QSM applications is given. This corresponds to the comparison of experimental structures obtained from X-ray determination for cis-diamminedichloroplatinum(II) complex with optimized molecular geometries using several theoretical methods to quantify the differences between the analyzed levels of theory. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 911–920, 1999

Primitive Normal Bases with Prescribed Trace

Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a∈F be nonzero. We prove the existence of an element w in E satisfying the following conditions: - w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers). - the set {w g ∣g∈G} of conjugates of w under G forms a normal basis of E over F. - the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.