Infinite Basis Research Articles

Nuclear dynamics near conical intersections in the adiabatic representation: I. The effects of local topography on interstate transitions

The local topography of a conical intersection can be represented by four parameters, readily determined from multireference configuration interaction wave functions, describing the pitch and tilt of the double cone. The time-dependent Schrödinger equation is solved in the vicinity of a conical intersection in the adiabatic basis using an approach tailored to this representation. It is shown that an adiabatic state treatment, which offers conceptual advantages is, in the appropriate set of internal coordinates, not qualitatively more difficult than the equivalent calculation in a diabatic basis. The present treatment is fully hermitian and takes full account of the geometric phase effect being, for example, gauge invariant (in the infinite basis limit) and could be used to develop a fully adiabatic description of nonadiabatic dynamics. The gauge invariant formulation provides interesting insights into the consequences of neglecting the geometric phase. The algorithm is used to study the effects of the double cone’s topography on the outcome of a nonadiabatic transition. Transitions from both the upper state to the lower state and from the lower state to upper state are considered for representative sets of conical parameters. The effects of the local topography on the outcome of nonadiabatic transitions can be dramatic.

On ‘infinite basis set limits’ in molecular electronic structure calculations

Sequences of basis sets can be constructed so as to approach the ‘infinite basis set limit’ in molecular electronic structure calculations. Extrapolation procedures can also be devized to provide accurate values of this limit. However, some care is required since the ‘infinite basis set limit’ may not correspond to the limit supported by a complete basis set. Extrapolation procedures can then lead to limiting values, which are not estimates of the exact solutions.

Examples of T-spaces with an infinite basis

This work is devoted to the construction of T-spaces with an infinite basis over a field of finite characteristic and over some other rings. Examples of T-spaces are given that are generated by polynomials in two variables or by polynomials of bounded degree in each variable.

Structure constants for new infinite-dimensional Lie algebras ofU(N+,N-) tensor operators and applications

The structure constants for Moyal brackets of an infinite basis of functions on the algebraic manifolds Mof pseudo-unitary groups U (N+ ,N- ) are provided. They generalize the Virasoro and algebras to higher dimensions. The connection with volume-preserving diffeomorphisms on M , higher generalized-spin and tensor operator algebras of U (N+ ,N- ) is discussed. These centrally extended, infinite-dimensional Lie algebras also provide the arena for nonlinear integrable field theories in higher dimensions, residual gauge symmetries of higher-extended objects in the light-cone gauge and C* -algebras for tractable non-commutative versions of symmetric curved spaces.

Infinite basis limits in electronic structure theory

We have developed a database of 29 molecules for which we have estimated the complete-one-electron-basis-set limit of the zero-point-exclusive atomization energy for five levels of electronic structure theory: Hartree–Fock (HF) theory, Mo/ller–Plesset second- and fourth-order perturbation theory, coupled cluster theory based on single and double excitations (CCSD), and CCSD plus a quasiperturbative treatment of triple excitations [CCSD(T)], all at a single set of standard geometries. Convergence checks indicate that the estimates are within a few tenths of a kcal/mol of the n=infinity limit of the cc-pVnZ basis set sequence. This data is then used to obtain optimized power-law exponents for extrapolating to the basis-set-limit from correlation-consistent polarized valence double and triple zeta (cc-pVDZ and cc-pVTZ) basis sets. This allows one to get thermochemical accuracy comparable to polarized quadruple or quintuple zeta (cc-pVQZ or cc-pV5Z) basis sets with a cost very comparable to polarized triple zeta, which is one order of magnitude less expensive than polarized quadruple zeta and two orders of magnitude less expensive than polarized quintuple zeta.

The Gaussian-2 method with proper dissociation, improved accuracy, and less cost

The Gaussian-2 method (G2) is modified by deleting the empirical high-level correction and instead using empirical coefficients to extrapolate to full configuration interaction and an infinite basis set. The resulting method, called multicoefficient Gaussian-2 (MCG2) is less expensive than G2 but a factor of 1.7 more accurate for molecules composed of H and first-row atoms.

Accurate structures and binding energies for small water clusters: The water trimer

The global minimum on the water trimer potential energy surface has been investigated by means of second-order Mo/ller-Plesset (MP2) perturbation theory employing the series of correlation-consistent basis sets aug-cc-pVXZ (X = D, T, Q, 5, 6), the largest of which contains 1329 basis functions. Definitive predictions are made for the binding energy and equilibrium structure, and improved values are presented for the harmonic vibrational frequencies. A value of 15.82±0.05 kcal mol−1 is advanced for the infinite basis set frozen core MP2 binding energy, obtained by extrapolation of MP2 correlation energies computed at the aug-cc-pVQZ MP2 geometry. Inclusion of core correlation, using the aug-cc-pCV5Z basis set, has been found to increase the binding energy by 0.08 kcal mol−1, and after consideration of core correlation and higher-order correlation effects, the classical binding energy for the water trimer is estimated to be 15.9±0.2 kcal mol−1. A zero-point vibrational correction of −5.43 kcal mol−1 has been computed from aug-cc-pVTZ MP2 harmonic vibrational frequencies. The accuracy of different computational schemes for obtaining the binding energies of the water dimer and trimer has been investigated, and computationally feasible methods are suggested for obtaining accurate structures and binding energies for larger water clusters.

Geometry Optimization with an Infinite Basis Set

We show that geometries can be optimized directly at a level corresponding to extrapolation to an infinite basis set. Numerical examples demonstrate that geometries obtained with gradients extrapolated to the infinite-basis limit agree well with geometries calculated with much bigger basis sets than those used for the calculation. The method may also be used to calculate reaction paths or classical trajectories directly at the extrapolated infinite-basis limit.

Improved coefficients for the scaling all correlation and multi-coefficient correlation methods

We have re-optimized the coefficients for ten scaling all correlation (SAC) methods, five empirical infinite basis (EIB) methods, and 18 multi-coefficient (MC) correlation methods, including the special cases of multi-coefficient SAC and multi-coefficient Gaussian-2 and Gaussian-3. The new parameterization is based on a training set of 82 atomization energies except for multi-coefficient Gaussian-2, which is restricted to H and the first period (nuclear charge ( 9) and is based on a training set of 52 atomization energies. Each method may be employed with or without including core-correlation effects, which are based on a new set of parameters optimized on a 125-molecule training set. The mean unsigned error in the atomization energies of the 82-molecule set is reduced on average by 20% when the new parameters used here are adopted.

Dangers of counterpoise corrected hypersurfaces. Advantages of basis set superposition improvement

The convergence properties of counterpoise corrected hypersurfaces towards the limit of infinite basis set size within a specific method are discussed. In the case of concerted hydrogen exchange of the cyclic hydrogen fluoride trimer counterpoise correction is shown to be highly counterproductive for the convergence speed towards the basis set limit. A best estimate for the MP2 basis set limit of −15.2 kcal/mol for the stabilization energy and of 18.7 kcal/mol for the concerted hydrogen exchange barrier of the cyclic hydrogen fluoride trimer is deduced by calculations using basis sets up to aug-cc-pV5Z. Both results are very well predicted in the uncorrected case already with the smaller basis sets in the MP2/aug-cc-pVxZ series, whereas counterpoise correction completely fails to improve predictions mainly due to a wrong correction of the dynamic electron correlation contribution. Only with very large basis sets it is possible to obtain counterpoise corrected results of comparable quality.

An assessment of theoretical procedures for the calculation of reliable free radical thermochemistry: A recommended new procedure

The ability to predict reliable thermochemical properties of molecules and ions has led to an ever increasing application of ab initio molecular orbital theory. Methods such as G2 theory have been shown to generally give accurate heats of formation (ΔfH) for closed-shell molecules and ions. Open-shell systems have been less thoroughly examined to date and the present paper attempts to redress this situation through a detailed assessment of the performance of a variety of levels of theory in calculating ΔfH values for free radicals. Representatives of three families of theoretical procedures have been studied: the infinite basis set extrapolation techniques of Martin, the CBS procedures of Petersson et al., and the G2 methods of Pople et al. Among the specific influences investigated are choice of geometry, zero-point vibrational energy, high level electron correlation treatment and basis set size. We recommend a new procedure called CBS-RAD for the treatment of free radicals. CBS-RAD is a modification of the CBS-Q method in which the geometry and zero-point energies are obtained at the QCISD/6-31G(d) level, and coupled-cluster theory is used in place of quadratic configuration interaction in single-point energy calculations. We find that for free radicals with low spin contamination G2 theory also performs adequately, but as 〈S2〉 increases the results of G2 calculations can become increasingly unsatisfactory. The recommended CBS-RAD procedure should yield more reliable results over a broader range of free radicals.

A Reassessment of the Groundwater Inverse Problem

This paper presents a functional formulation of the groundwater flow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a “forward operator” which describes the measurement process. In the particular case considered here, the parameter of interest is the large‐scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwater flow equation. The inverse algorithm seeks the “most probable” or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness‐of‐fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well‐posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss‐Newton or gradient‐based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer‐Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.

Density Functional Theory of Molecular Solids: Local versus Periodic Effects in the Two-Dimensional Infinite Hydrogen-Bonded Sheet of Formamide

The performance of an ab initio computational scheme for molecular crystals based on density functional theory (DFT) was investigated by computing several structural and energetic properties of hydrogen bonded infinite chains and of two-dimensional infinite periodic networks of formamide. The applied DFT potentials covered a wide range in quality, starting with a simple local exchange (X) without correlation (C), and then gradually introducing C and gradient corrections for both X and C. At the same time, five atomic basis sets of systematically increasing size, in the range of DZ to TZ(2df,2pd) were used to construct the Bloch-type crystal orbitals, to optimize the structures, and to extrapolate different physical quantities to the limit of a hypothetical infinite basis set. Infinite lattice sums were computed by the multipole expansion technique, and basis set superposition errors were (partly) eliminated by the counterpoise method. To be able to assess the accuracy of the theoretical models, the formam...

Electron correlation and dimerization in trans-polyacetylene: Many-body perturbation theory versus density-functional methods.

Structural and energetic aspects of the Peierls-type lattice dimerization were investigated in infinite, one-dimensional, periodic trans-polyacetylene (t-PA) using many-body perturbation theory (MBPT) and density-functional theory (DFT). Cohesive properties and dimerization parameters were obtained first for the classical Coulomb potential in the Hartree approximation and then by gradually turning on exchange and correlation potentials. Besides the nonlocal Hartree-Fock exchange, several other exchange functionals were used incorporating gradient corrections as well. For MBPT, electron correlation was included up to the fourth order of the Mo/ller-Plesset scheme and the behavior of lattice sums for different PT terms was analyzed in detail. The electrostatic part of the infinite lattice sums was computed by the multipole expansion technique. In solving the polymer Kohn-Sham equations, the performance of several different correlation potentials was studied again including different gradient corrections. Atomic basis sets of systematically increasing size, in the range of double-zeta to triple-zeta (TZ) up to TZ (3df,3p2d), were used in all calculations to construct the symmetry-adapted (Bloch-type) polymer wave functions, to fully optimize the structures, and to extrapolate different physical quantities to the limit of a hypothetical infinite basis set. Comparison of the different DFT results with MBPT and with experiments demonstrated the importance of gradient terms both for exchange and correlation. On the other hand, the best DFT functional, using a medium-size atomic basis set, excellently reproduced the cohesive and dimerization energies obtained for infinite t-PA at the MP4/TZ(3d2f,3p2d) level and provided dimerization parameters close to experiment. The experimentally observed lattice spacing of 2.46\ifmmode\pm\else\textpm\fi{}0.01 \AA{} will be correctly predicted both at the MBPT and DFT levels with 2.48 and 2.44 \AA{}, respectively.

A basis for the type 〈n〉 hyperidentities of the medial variety of semigroups

The medical varietyMV of semigroups is the variety defined by the medial identityxyzw = xzyw. This variety is known to satisfy the medial hyperidentitiesF(G(x 11 ,⋯, x 1n ),⋯, G(x n1 ,⋯, x nn )) = G(F(x 11 ,⋯, x n1 ),⋯, F(x 1n ,⋯, x nn )), forn ≥ 1. Taylor has observed in [2] thatMV also satisfies some other hyperidentities, which are not consequences of the medial ones. In [4] the author introduced a countably infinite family of binary hyperidentities called transposition hyperidentities, which are natural generalizations of then = 2 medial hyperidentity. It was shown that this family is irredundant, and that no finite basis is possible for theMV hyperidentities with one binary operation symbol. In this paper, we generalize the concept of a transposition hyperidentity, and extend it to cover arbitrary arityn ≥ 2. We show that theMV hyperidentities with onen-ary operation symbol have no finite basis, but do have a countably infinite basis consisting of these transposition hyperidentities.

On the image of derivations

Introduction. Let L be a field. An additive mapping d : L −→ L is said to be a derivation of L if d(ab) = d(a)b+ ad(b) for any a, b ∈ L. If K ⊂ L is a field extension, then a derivation d of L is said to be a K-derivation if d(αa) = αd(a) for any α ∈ K and a ∈ L. If d is a derivation of L such that d(L) = L, then we say that d is epimorphic. (a) Let K ⊂ L be a finite dimensional field extension and let d be a Kderivation of L. Then L has a direct sum decomposition L = Ker d ⊕ L′, as a K-module, where L′ is K-isomorphic to d(L). Thus dimK L > dimK d(L) yields d(L) 6= L for any K-derivation d. (b) Assume now that L is a purely transcendental extension with infinite transcendental basis {xλ; λ ∈ Λ} over Q, the field of rational numbers. Since |L| = |Λ| (the cardinalities of L and Λ), we may put L = {aλ; λ ∈ Λ}. Then the Q-derivation d of L defined by d(xλ) = aλ is epimorphic. Thus it is natural to ask whether a field extension K ⊂ L has (or has not) an epimorphic K-derivation. The purpose of this paper is to study on this problem.

Cooperative effects in hydrogen bonding: Fourth-order many-body perturbation theory studies of water oligomers and of an infinite water chain as a model for ice

As a step toward the first principles quantum mechanical modeling of the structural and electronic properties of ice, hydrogen-bonded periodic infinite chains of water molecules have been investigated by the ab initio crystal orbital method at the Hartree–Fock (HF) level and by including electron correlation up to the complete fourth order of Mo/ller–Plesset perturbation theory (MP4). The Bloch functions of the crystal have been expanded in a series of high quality atomic orbital basis sets complemented by extended sets of polarization functions, up to TZ(3d2f,3p2d). Basis set superposition errors have been (partly) eliminated by the counterpoise method and the infinite lattice sums have been computed using the multipole expansion technique. The systematically increasing size of the basis sets has allowed the extrapolation of structural and electronic indices of this ice model to the limit of an infinite atomic basis at both the HF and various correlated levels, respectively. For each theoretical model, detailed comparisons have been made with the corresponding physical properties of water monomers, dimers, and some larger linear oligomers. The results convincingly prove that hydrogen bonding in ice is a highly cooperative phenomenon, both from the structural and energetic points of view. The cohesive energy per hydrogen bond of the crystal is −5.30 kcal/mol at the HF level (with RHFO,O=2.88 Å) as compared with the dimer value of −3.60 kcal/mol (at the optimized distance of 3.03 Å). At the MP2 level of theory, the crystalline binding energy decreases to −6.60 kcal/mol and the lattice contracts to RMP2O,O=2.73 Å (compared with −4.50 kcal/mol at 2.88 Å for the dimer). The correlation corrections at third and fourth order slightly expand the crystal lattice (to RMP4O,O=2.75 Å) and reduce the cohesion by 0.15 kcal/mol. A decomposition of the intermolecular interactions according to different terms of MP4 theory suggests that the cohesive energy of ice results from a delicate balance between different repulsive and attractive terms in third and fourth order, which exhibit different long-range behaviors. The detailed study of the role of high-energy virtual energy bands in computing electron correlation effects in ice provides further insight into the important role that basis set flexibility plays in such investigations. The resulting cohesive energy of −6.83 kcal/mol at the MP4 level is in reasonable agreement with the experimental energy per hydrogen bond in ice I, −6.7 kcal/mol.

An introduction to commutative and noncommutative Gröbner bases

In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger algorithm) for their computation; since the end of the seventies, Gröbner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in Algebraic Geometry and Commutative Algebra; moreover, Gröbner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras, of interest in Differential Algebra (e.g. Weyl algebras, enveloping algebras of Lie algebras). The aim of this paper is to give an introduction, as elementary as I was able to make it, to both commutative and noncommutative algebras: Gröbner bases are in a sense a finite model of an infinite linear Gauss-reduced basis of an ideal viewed as a vector space and Buchberger algorithm is the corresponding generalization of the Gaussian elimination algorithm. Moreover the paper contains a survey of some applications of Buchberger theory to noncommutative algebras; together with these results surveyed, this paper contains some minor new points: e.g. the “useless pair criteria” in the noncommutative case and the final result on the existence and “computability” of Gröbner bases for two-sided ideals in any finitely presented algebra.

Theory of the magnetic properties of an infinite classical spin-chain showing axial anisotropic couplings: Low-temperature behavior.

We consider a classical spin chain characterized by axial anisotropic couplings between nearest neighbors. In particular the antisymmetrical part of exchange is taken into account by means of a Dzialoshinski coupling characterized by a regular or an alternate direction from pair to pair. We expand each operator exp(-\ensuremath{\beta}${\mathit{H}}_{\mathit{i}}$) on the infinite basis of spheroidal functions. We show that the zero-field partition function ${\mathit{Z}}_{\mathit{N}}$(0) has a closed-form expression. The spin-spin correlations can be derived as well as the susceptibilities ${\mathrm{\ensuremath{\chi}}}_{\mathit{u}\mathit{u}}$ (with u=x or y) and ${\mathrm{\ensuremath{\chi}}}_{\mathit{z}\mathit{z}}$. We achieve a thorough investigation of the chain low-temperature behaviors. It appears that there are predictable and original behaviors. In particular, when the z axis is favored, the lowest-energy configuration corresponds to a ferromagnetic or an antiferromagnetic arrangement, with the spins lying along this axis. When the X-Y plane is favored, the spins lie within this plane and exhibit a ferromagnetic, an antiferromagnetic, a helical, or a canted structure. Finally, we point out interesting crossover phenomena, but they are not discussed in detail.

On rigidity of an end under conformal mappings preserving the infinite homology basis

Let f be a quasiconformal self-mapping of an open Riemann surface R. Assume that every cycle γ in R is homologous to f(γ) and f is conformal on an end D of R. Then we show that f is the identity mapping on D if the first homology group of D is infinitely generated.