Slater-type Functions Research Articles

The evaluation of some four-electron correlated integrals with a Slater basis arising in linear and nonlinear r12 theories

This study considers the analytic evaluation of correlated integrals involving explicit inter-electron separation distances using a basis of standard Slater-type functions. These basis functions are employed in Hylleraas-CI calculations, Hylleraas calculations on atomic four-electron systems, and in non-Born–Oppenheimer calculations on small molecular systems. A Fourier transform approach allows a principal auxiliary integral to be evaluated analytically, permitting a number of four-electron integrals to be reduced to various one-dimensional Cauchy–Frullani integrals, which can be evaluated in closed form. The stability of some of the formulas with respect to numerical evaluation, particularly for the high-order derivative cases, is discussed in detail. Numerical values for a selection of test integrals are provided.

Method for Slater-Type Density Fitting for Intermolecular Electrostatic Interactions with Charge Overlap. I. The Model.

The effects of charge overlap, or charge penetration, are neglected in most force fields and interaction terms in QM/MM methods. The effects are however significant at intermolecular distances near the van der Waals minimum. In the present study, we propose a method to evaluate the intermolecular Coloumb interaction using Slater-type functions, thus explicitly modeling the charge overlap. The computational cost of the method is low, which allows it to be used in large systems with most force fields as well as in QM/MM schemes. The charge distribution is modeled as a distributed multipole expansion up to quadrupole and Slater-type functions of angular momentum up to L = 1. The exponents of the Slater-type functions are obtained using a divide-and-conquer method to avoid the curse of dimensionality that otherwise is present for large nonlinear optimizations. A Levenberg-Marquardt algorithm is applied in the fitting process. A set of parameters is obtained for each molecule, and the process is fully automated. Calculations have been performed in the carbon monoxide and the water dimers to illustrate the model. Results show a very good accuracy of the model with relative errors in the electrostatic potential lower than 3% over all reasonable separations. At very short distances where the charge overlaps is the most significant, errors are lower than 8% and lower than 3.5% at distances near the van der Waals minimum.

Formation of non-excited and excited hydrogen in proton–lithium inelastic scattering

The collisions of a proton with a lithium atom are treated for the first time as a three-channel problem under the assumption that the elastic and hydrogen formation in non-excited, H(1s), and excited, H(2s), channels are open. The effect of polarization potentials of the target and hydrogen formation is considered. A one-valence-electron model for the target, based on the Clementi–Roetti Slater-type basis functions, as well as a modified coupled-static approximation are used to calculate the partial and total cross sections of seven partial waves (0 ≤ ℓ ≤ 6, where ℓ is the total angular momentum) at incident energies between 50 and 500 keV. Our values of the total cross section are in good agreement with previous results.

Metallization of Atomic Solid Hydrogen within the Extended Hubbard Model with Renormalized Wannier Wave Functions

We refer to our recent calculations (Eur. Phys. J. B, \textbf{86}, 252 (2013)) of metallization pressure of the three-dimensional simple-cubic crystal of atomic hydrogen and study the effect on the crucial results concocting from approximating the $1s$ Slater-type orbital function with a series of $p$ Gaussians. As a result, we find the critical metallization pressure $p_C = 102\ GPa$. The latter part is a discussion of the influence of zero-point motion on the stabilizing pressure. We show that in our model the estimate magnitude of zero-point motion carries a little effect on the critical metallization pressure at zero temperature.

Calculation of screened Coulomb potential matrices and its application to He bound and resonant states

We present two analytical methods, Taylor expansion and Gegenbauer expansion, to efficiently and accurately calculate the two-electron screened Coulomb potential matrix elements with Slater-type configuration-interaction basis functions. The former permits great advantages in fast computation of the potential matrices at small screening parameters and the latter allows accurate calculation of the matrices at all screening parameters. The bound and resonant states of a He atom embedded in the screening environment are calculated by employing the variational and complex-scaling methods, respectively, and the results are compared with other theoretical predictions. The expectation values of some physical quantities for He ground state are compared with the recent calculation of Ancarani and Rodriguez [Phys. Rev. A 89, 012507 (2014)] and extended to stronger screening environment. The energies and widths for the doubly excited resonant states are in good agreement with previous calculations, while the interelectronic angle $arccos\ensuremath{\langle}cos({\ensuremath{\theta}}_{12})\ensuremath{\rangle}$ show significant discrepancies with the Feshbach projection calculation of Ord\'o\~nez-Lasso et al. [Phys. Rev. A 88, 012702 (2013)]. The expectation values of $\ensuremath{\langle}{\mathbf{p}}_{\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{p}}_{\mathbf{2}}\ensuremath{\rangle}$ are also calculated for the resonant states investigated here. We conclude that the present methods in the framework of complex scaling enable us to get reliable energy, width, and other physical quantities of the resonant states in a variety of screening conditions.

Excitation Cross Section within Schwinger Variational Principal and Pseudo State Method in a Set of Slater Type Functions

We use Schwinger variational principle to evaluate excitation cross section of atomic hydrogen by proton impact at energy range between 2keV and 200keV. Contribution of the continuum near ionization threshold was introduced in a model using L2 method developed in a pseudo-space spanned by a set of Slater type functions. Our results were compared with those of Bouamoud (1988-1989) using 1s, 2s and 2p state of hydrogen atom.

Calculation of the ground state energy of neutral atoms (Z ≤ 54) by the algebraic variant of the Hartree-Fock method

High-precision calculations of the ground-state energy of atoms He through Xe are performed in the algebraic approximation of the Hartree-Fock method. The orbital exponents of Slater-type basis functions are optimized using the second-order minimization methods, which allows the virial theorem to be fulfilled to within 10−15–10−17 for the first time. The energies of atoms calculated with rather limited basis sets are, in terms of accuracy, as good as the results obtained by using the numerical procedure for solving the Hartree-Fock equations.

Repulsion integrals involving Slater-type functions and Yukawa potential

In range-separated functionals using basis sets of exponential functions, the Yukawa potential appears as the natural choice for the attenuated electron–electron repulsion. The development of these methods requires one- and two-center integrals involving two exponential functions and the Yukawa potential. In this work, general expressions are derived for these integrals, and stable algorithms for their efficient calculation, covering a wide range of screening parameters, are developed and tested.

Bound states for the quantum dipole moment in two dimensions

We calculate accurate eigenvalues and eigenfunctions of the Schrödinger equation for a two-dimensional quantum dipole. This model proved useful for the study of elastic effects of a single edge dislocation. We show that the Rayleigh–Ritz variational method with a basis set of Slater-type functions is considerably more efficient than the same approach with the basis set of point-spectrum eigenfunctions of the two-dimensional hydrogen atom used in earlier calculations.

Using the Screened Coulomb Potential To Illustrate the Variational Method

The screened Coulomb potential, or Yukawa potential, is used to illustrate the application of the single and linear variational methods. The trial variational functions are expressed in terms of Slater-type functions, for which the integrals needed to carry out the variational calculations are easily evaluated in closed form. The variational energies of the first two states of the screened Coulomb potential are calculated by minimizing them with respect to the exponential parameters included in the Slater functions using elementary minimization algorithms. The project allows students to concentrate on the fundamentals and the performance of the variational method at its increasingly accurate levels of application to realistic systems.

An Improvement on Noninteger n-Generalized Exponential Type Orbitals with Hyperbolic Cosine in Atomic Calculations

Abstract An improvement on the total energy results reported in Ref. 9 in atomic calculations based on a modified type hyperbolic cosine function cosh(βrμ + γ) is presented. It is shown that the noninteger n-generalized exponential type orbitals rn* - 1e- ζ rμ with modified type hyperbolic cosine as radial basis functions are a much better approximation to the Hartree–Fock orbitals than a double-zeta basis set of Slater type functions. The efficiency of the new basis function is tested by application to some closed and open shell neutral atoms and their ions. A substantial improvement in both the total and orbital energies is obtained within the minimal basis sets framework. The total energy values obtained in this work are significantly close to the numerical Hartree–Fock results. These results supersede all previous minimal basis function total energies achieved in the literature.

Range-Separated Exchange Functionals with Slater-Type Functions.

An implementation of range-separated density functionals utilizing the Yukawa potential and Slater-type functions is described. The density-functional part of the range-separated regime is straightforward. The exact exchange part makes use of established methods for evaluating exchange integrals over Slater-type functions but still requires new one- and two-center integrals. Equations for the one-center integrals are derived. The two-center integrals are evaluated through a combination of new equations and techniques taken from procedures for evaluating two-center Coulomb integrals over Slater-type functions. In a first application, the performance of range-separated functionals in the prediction of transition metal thermochemistry is evaluated using a database of average ligand removal energies. The range-separated functionals perform better than a GGA parent and similarly to commonly used hybrid and meta-hybrid functionals. The results were relatively insensitive to the chosen value of the attenuation parameter.

Exchange-only optimized-effective-potential calculations using Slater-type basis functions: Atoms and diatomic molecules

The exchange-only optimized-effective-potential method is implemented with the use of Slater-type basis functions, seeking an alternative to the standard methods of solution with some computational advantages. This procedure has been tested in a small group of closed-shell atoms and diatomic molecules, for which numerical solutions are available. The results obtained with this implementation have been compared to the exact numerical solutions and to the results obtained when the optimized effective equations are solved using the Gaussian-type basis sets. This Slater-type basis approach leads to a more compact expansion space for representing the potential of the optimized-effective-potential method and to considerable computational savings when compared to both the numerical solution and the more traditional one in terms of the Gaussian basis sets.

On the mathematical nature of Guseinov’s rearranged one-range addition theorems for Slater-type functions

Starting from one-range addition theorems for Slater-type functions, which are expansion in terms of complete and orthonormal functions based on the generalized Laguerre polynomials, Guseinov constructed addition theorems that are expansions in terms of Slater-type functions with a common scaling parameter and integral principal quantum numbers. This was accomplished by expressing the complete and orthonormal Laguerre-type functions as finite linear combinations of Slater-type functions and by rearranging the order of the nested summations. Essentially, this corresponds to the transformation of a Laguerre expansion, which in general only converges in the mean, to a power series, which converges pointwise. Such a transformation is not necessarily legitimate, and this contribution discusses in detail the difference between truncated expansions and the infinite series that result in the absence of truncation.

Compact wave functions for the beryllium isoelectronic series, Li− to Ne6+: A standard Hylleraas approach

Variational calculations have been carried out for the ground states of several members of the beryllium isoelectronic series using a standard Hylleraas approach involving Slater-type basis functions. The species examined are Li(-), Be, B(+), C(2+), N(3+), O(4+), F(5+), and Ne(6+). For each species, the nonrelativistic energy, the electronic density at the nucleus, the expectation value <∇(i)·∇(j)>, the moments <r(i)(n)> for n = -1, 1, 2, and 3, and <r(ij)(n)> for n = -1, 1, and 2, are reported. With relatively compact basis sets, the ground state energies are obtained with uncertainties ranging from 50 parts per million to just under 4 parts per million.

Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.

Energy and density analysis on the H2 molecule from the united atom to dissociation: The Σ, Π, Δ, ϕ, and Γ manifolds

We present 140 accurate potential energy curves, PECs, for the Σ, Π, Δ, ϕ, and Γ manifolds for the H2 molecule, mapping all the states with energy below the H ground state. The full configuration interaction, nonrelativistic Born–Oppenheimer computations are performed with large and optimized basis sets of Slater-type and spherical Gaussian functions; these new basis sets are somewhat larger than those used in recent published studies on the 60 Σ state PECs. The full CI computations are performed twice, with Hartree–Fock and with Heitler–London-type functions, allowing the identification of the ionic component in the total energy. The computed energies are within 10−5 hartree from the most accurate PECs in literature. We aim (a) at the evaluation of the PECs starting at very short and unexplored internuclear distances (0.01 bohrs) and ending at full dissociation, (b) at the systematic prediction of high excited state PECs dissociating as 1s + 4l and 1s + 5l, and (c) at the characterization of the evolution of the 140 PEC electronic densities from united atom to dissociation. With this work we fill a gap in today literature, which has dealt mainly with low excited states, generally excluding short internuclear distances. The electronic configuration at the united atom persists as dominant configuration well beyond the equilibrium separation, and it switches to that at dissociation often with energy patterns seemingly irregular, in particular when the values of the principal quantum number at dissociation and at the united atom differ by one or more unit. The Hund's singlet-triplet splitting, which propagates from the united atom to the molecule, is discussed. The singlet and triplet states are rather close in energy in the Π manifolds, and approach degeneracy in the Δ and ϕ manifolds, to become fully degenerate in the Γ manifolds. Discussions on the correlation energy correction, adiabatic correction, spectroscopic constants and on general features of the H2 excited states are presented. The H2 molecule is a system, which—to be understood—needs consideration of both the very short internuclear distances in approaching the united atom and of the very high excited states below H. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Review of theories on ionization in fast ion-atom collisions with prospects for applications to hadron therapy

This study emphasizes the need for a systematic and in-depth connection between the progress in quantum theory of energetic ion collisions and applications to hadron therapy. Scattering theory for fast ion beams has reached its stage of development where accurate and robustly applicable methodologies can advantageously be exported to applied fields such as space research, fusion energy program, medicine, etc. In particular, distorted wave collision theories at high energies readily provide total, partial and fully differential cross sections for inelastic collisions of ionic projectiles with any target system. By numerous and thorough testings, such theoretical cross sections were found to exhibit excellent agreement with experimental data on atomic targets. Adequate extensions of these methods to molecular targets were also accomplished with computational efforts that are approximately comparable to that for multi-electron atomic targets. This was done by using the standard Slater-type atomic basis functions for any molecular targets, including tissue-equivalent materials (e.g. water) of relevance to hadron therapy. This expertize needs to be brought to medicine through ion transport physics, which most frequently employs the crude Bragg sum rule for obtaining molecular cross sections as linear combination of atomic cross sections. Relativistic distorted wave theories are also available, but not currently in use for modeling the passage of relativistic ions through tissue, as needed in hadron therapy of deep-seated tumors. It is high time for extensive and thorough applications of the well-established distorted wave scattering theories to fast collisions of bare and partially clothed multiple charged ions with water molecule. This type of application would provide the most accurate data bases for various cross sections (on electron capture, excitation, ionization, etc) that can be used as reliable entry data for subsequent Monte Carlo simulations of energy losses of ions during their passage through tissue. In order to gain in overall efficiency, these theoretical cross sections could be precomputed at sufficiently dense multi-variable grids, thus yielding modules for advantageous direct sampling during stochastic simulations. Such a comprehensive strategy could provide both accurate and efficient algorithms that would incorporate the state-of-the-art methodologies from high-energy atomic scattering theory involving ion beams. This is currently missing in the physics part of hadron therapy, since all the major Monte Carlo codes customarily employ atomic cross section data bases that rely almost exclusively upon the Bethe–Bloch formula and some phenomenological expressions with fitting parameters adjusted to the limited sets of experimental data. Crucially, the need is emphasized for the introduction of a still missing Monte Carlo code which could simulate transport of ions together with secondary electrons in tissue. The current main Monte Carlo codes simulate transport of either ions or electrons, but not both simultaneously. However, energetic ions produce a large number of electrons by densely ionizing the traversed tissue and many of them are δ-electrons i.e. capable on their own of ionizing various targets. Due to their light mass and considerable energy, δ-electrons undergo multiple scatterings. Because of this cumulative effect, among all the double strand breaks of DNA molecules of tissue treated by ion therapy, some 70% are produced by δ-electrons. Hence the necessity to simulate transport of δ-electrons produced by primary ion beams. Such types of computations are presently missing from the major ion transport codes. Overall, this work thoroughly analyzes conceptual and computational advances of the leading quantum-mechanical distorted wave theories for energetic ion collisions aimed at applications to medicine. Additionally, the main strategic directions are also indicated to further cross-disciplinary fertilization between medicine and basic research on collision theory of fast heavy ions of relevance to hadron therapy.

Sub-meV accuracy in first-principles computations of the ionization potentials and electron affinities of the atoms H to Ne

Recent advances in coupled-cluster methods for the treatment of electron correlation in atoms (and molecules) have now made possible the computation of the ionization potentials and electron affinities of the atoms H through Ne with a mean absolute deviation of 0.35 meV with respect to experimental data. In this article, coupled-cluster wave functions are parametrized using orbital and Slater-type geminal basis functions. Many-body electron interactions are accounted for through connected quintuple excitations, and relativistic effects (spin-orbit as well as scalar) and diagonal Born-Oppenheimer corrections are included.

Energy and density analysis of the H2 molecule from the united atom to dissociation: The Σ3g+ and Σ3u+ states

The first 14 (3)Sigma(g)(+) and the first 15 (3)Sigma(u)(+) states of the H(2) molecule are computed with full configuration interaction both from Hartree-Fock molecular orbitals and Heitler-London atomic orbitals within the Born-Oppenheimer approximation, following recent studies for the (1)Sigma(g)(+) and (1)Sigma(u)(+) manifolds [Corongiu and Clementi, J. Chem. Phys. 131, 034301 (2009) and J. Phys. Chem. (in press)]. The basis sets utilized are extended and optimized Slater-type functions and spherical Gaussian functions. The states considered correspond to the configurations (1s(1)nl(1)) with n from 1 to 5; the internuclear separations sample the distances from 0.01 to 10,000 bohrs. For the first three (3)Sigma(g)(+) and (3)Sigma(u)(+) states and for the fourth and fifth (3)Sigma(g)(+) states, our computed energies at the equilibrium internuclear separation, when compared to the accurate values by Staszewska and Wolniewicz and by Kołos and Rychlewski, show deviations of about 0.006 kcal/mol, a test on the quality of our computations. Motivation for this work comes not only from obtaining potential energy curves for the high excited states of H(2) but also from characterizing the electronic density evolution from the united atom to dissociation to provide a detailed analysis of the energy contributions from selected basis subsets and to quantitatively decompose the state energies into covalent and ionic components. Furthermore, we discuss the origin of the seemingly irregular patterns in potential energy curves in the two manifolds, between 4 and 6-9 bohrs--there are two systems of states: the first, from the united atom to about 4 bohrs, is represented by functions with principal quantum number higher than the one needed at dissociation; this system interacts at around 4 bohrs with the second system, which is represented by functions with principal quantum number corresponding to one of the dissociation products.