Slater's Rules Research Articles

Atomic Orbital Bases of Bunge and Koga Studied by Frobenius Products with Moscow–Aachen–Paris (MAP) Orbitals

The minimization of Frobenius angles between functional subspaces spanned by different sets of atomic functions is employed to determine the values of orbital exponents ξ characterizing minimum atomic parameters/Moscow–Aachen–Paris (MAP) basis sets providing the best representation of two Hartree–Fock based atomic basis sets: that of Bunge et al. available for elements H–Xe and that of Koga and Thakkar spanning H to Lr (Z = 103). So-extracted values of exponents follow piecewise linear laws as functions of the nuclear charge Z resembling the prescriptions set by Slater rules for the orbital exponents. In details, however, the rules proposed by Slater are not precisely followed, neither for effective principal quantum numbers n* nor screening increments σ. Nevertheless, the linear pieces of the ξ vs. Z follow the structure of the Periodic Table being specific for the segments corresponding to p-, d- (transition), and f- (Lanthanides and Actinides) elements, respectively.

Efficient basis sets for core-excited states motivated by Slater's rules.

X-Ray photoemission spectroscopy is a commonly applied characterization technique that probes the local chemistry of atoms in molecules and materials via the photoexcitation of electrons from atomic core orbitals. These measurements can be interpreted by comparison with previous literature or through the calculation of core-electron binding energies (CEBEs) for model systems. However, physically and numerically accurate description of the core-excited electronic structures demands specializations beyond routine ground state setups. Inspired by Slater's rules, we focus on developing computationally efficient and physically motivated contractions to reproduce the core-excited atomic orbitals which led to improved numerical accuracy of calculated CEBEs. When applied to carbon 1s excitations in a wide range of molecules, these core-excited basis sets produce total energy differences (ΔSCF) using a hybrid exact-exchange density functional (B3LYP) that can reproduce core-excitation energies within experimental accuracy (∼0.1 eV). Due to missing relativistic effects, heavier elements (N, O) exhibit slightly larger systematic absolute errors, but still maintain a satisfactory 0.2 eV mean average error for relative CEBEs. We also connect the known variability in the core level binding energy with local atomic charge to demonstrate how the transferability of a given model should be measured against a diverse test set. We conclude by exploring one outlier, CO, and the outlook for extending this approach to other elements.

Preserving the half-metallicity at the full Heusler alloy Sc2CrSi surfaces as well as Sc2CrSi/HgTe (1 1 1) interface

Using the first principle study within the density functional theory, the electronic and magnetic properties of the bulk on (111), (110) and (001) surfaces of the full-Heusler alloy and the Sc2CrSi/HgTe (111) interfaces were verified. In this article, our calculations showed that the compound exhibits half-metallic ferromagnetism with an energy gap of 0.544 eV and a spin-flip gap of 0.1 eV in the majority spin channel at an equilibrium lattice constant of 6.39 Å. The total magnetic moment of the studied compound was calculated according to the Slater rule to be 2 μB. The results confirm that the half-metallicity of bulk is completely conserved at the Sc(2)Si-terminated (001) surface, Sc(1)-, and Si-terminated (111) surfaces, but destroyed at Cr- and Sc(2)-terminated (111) surface, the Sc(1)Cr- terminated (001) surface, and Sc(1)Sc(2)CrSi-terminated (110) Surface. From the above-mentioned characteristics of Sc2CrSi exhibit that this alloy is an adequate promising candidate for spin electronics implementations. Moreover, during the calculation of the interfacial adhesion energies, it is found it that Sc(1)-Te and Si-Te configurations are more stable among the others. Regrettably, interfacial configurations show that the half-metallicity of bulk Sc2CrSi is destroyed for all four possible configurations.

Abnormal Site Preference of Al and Ga in Gd3Al2.3Ga2.7O12:Ce Crystals

Large‐sized Gd3Al2.3Ga2.7O12:Ce single crystals are grown by the Czochralski method. The distribution of Al and Ga at octahedral (oct‐, 6‐oxygen‐coordinated) and tetrahedral (tet‐, 4‐oxygen‐coordinated) sites in the crystal is studied. Rietveld refinement is used to analyze the site occupation of Al and Ga. The results show that Ga ions with a larger radius than Al ions prefer to occupy the more compact tetrahedral sites. Density functional theory calculations show that the energy for a Ga ion at the tetrahedral site is lower than that at the octahedral site in the Gd3Al5O12 crystal, whereas the case for Al occupation in the Gd3Ga5O12 crystal is opposite. Slater rules are used to estimate the effective charge number of different ions, and the distribution of coordination ions around the octahedral and tetrahedral sites is analyzed. Then, the interaction between ions is simplified and the repulsive energy between Al/Ga and the coordination ions is calculated, which could provide reasonable understanding on the abnormal occupation. In addition, the occupation of Sc in Gd/Y garnet systems is also discussed using this method.

The Smoothed Complexity of Computing Kemeny and Slater Rankings

The computational complexity of winner determination under common voting rules is a classical and fundamental topic in the field of computational social choice. Previous work has established the NP-hardness of winner determination under some commonly-studied voting rules, such as the Kemeny rule and the Slater rule. In a recent position paper, Baumeister, Hogrebe, and Rothe (2020) questioned the relevance of the worst-case nature of NP-hardness in social choice and proposed to conduct smoothed complexity analysis (Spielman and Teng 2009) under Blaser and Manthey’s (2015) framework. In this paper, we develop the first smoothed complexity results for winner determination in voting. We prove the smoothed hardness of Kemeny and Slater using the classical smoothed runtime analysis, and prove a parameterized typical-case smoothed easiness result for Kemeny. We also make an attempt of applying Blaser and Manthey’s (2015) smoothed complexity framework in social choice contexts by proving that the framework categorizes an always-exponential-time brute force search algorithm as being smoothed poly-time, under a natural noise model based on the well-studied Mallows model in social choice and statistics. Overall, our results show that smoothed complexity analysis in computational social choice is a challenging and fruitful topic.

Preserving Condorcet Winners under Strategic Manipulation

Condorcet extensions have long held a prominent place in social choice theory. A Condorcet extension will return the Condorcet winner as the unique winner whenever such an alternative exists. However, the definition of a Condorcet extension does not take into account possible manipulation by the voters. A profile where all agents vote truthfully may have a Condorcet winner, but this alternative may not end up in the set of winners if agents are acting strategically. Focusing on the class of tournament solutions, we show that many natural social choice functions in this class, such as the well-known Copeland and Slater rules, cannot guarantee the preservation of Condorcet winners when agents behave strategically. Our main result in this respect is an impossibility theorem that establishes that no tournament solution satisfying a very weak decisiveness requirement can provide such a guarantee. On the bright side, we identify several indecisive but otherwise attractive tournament solutions that do guarantee the preservation of Condorcet winners under strategic manipulation for a large class of preference extensions.

An axiomatic characterization of the Slater rule

Despite being a well-studied decision rule, the Slater rule has not been analyzed axiomatically. In this paper, we show that it is the only rule which is unbiased, monotone, tournamental, tie-breaking, and gradual. Thereby we provide a characterization of it for the first time. We also show these axioms to be logically independent.

Electron capture and single ionization in H+ + Ar collisions: classical calculations

A classical model is used to study electron capture and single ionization (SI) following H+ + Ar collisions at projectile energies varying from 400 to 40 keV. In the present model, the Ar electrons are treated independently from each other, and only the 3s and 3p electrons are supposed to be captured by the projectile. In addition, a Coulombic potential with an effective charge Zeff = 6.75, derived from Slater rules, is used in the calculations to simulate the screening of the Ar nucleus due to the presence of the core and 2l electrons. Total cross sections for single electron capture and SI are calculated and compared with previous experiments and earlier calculations based on a semiclassical approach. The reasonable agreement we observed allows a preliminary study of double electron capture (DC). The total cross section for DC is found to be much larger than the experimental one. Possible reasons for this disagreement are discussed.

Methods of Calculating Ionization Energies of Multielectron (Five or More) Isoelectronic Atomic Ions

We have previously used simple empirical equations to reproduce the literature values of the ionization energies of isoelectronic sequences of up to four electrons which gave very good agreement. We reproduce here a kinetic energy expression with corrections for relativity and Lamb shift effects which give excellent agreement with the literature values. These equations become more complex as the number of electrons in the system increases. Alternative simple quadratic expressions for calculating ionization energies of multielectron ions are discussed. A set of coefficients when substituted into a simple expression produces very good agreement with the literature values. Our work shows that Slater's rules are not appropriate for predicting trends or screening constants. This work provides very strong evidence that ionization energies are not functions of complete squares, and when calculating ionization energies electron transition/relaxation has to be taken into account. We demonstrate clearly that for particular isoelectronic sequences, the ionizing electrons may occupy different orbitals and in such cases more than one set of constants are needed to calculate the ionization energies.

A new monotonicity condition for tournament solutions

We identify a new monotonicity condition (called cover monotonicity) for tournament solutions which allows a discrimination among main tournament solutions: The top-cycle, the iterated uncovered set, the minimal covering set, and the bipartisan set are cover monotonic while the uncovered set, Banks set, the Copeland rule, and the Slater rule fail to be so. As cover monotonic tournament solutions induce social choice rules which are Nash implementable in certain non-standard frameworks (such as those set by Bochet and Maniquet (CORE Discussion Paper No. 2006/84, 2006) or Ozkal-Sanver and Sanver (Social Choice and Welfare, 26(3), 607–623, 2006), the discrimination generated by cover monotonicity becomes particularly notable when implementability is a concern.

A Diabatic Representation Including Both Valence Nonadiabatic Interactions and Spin−Orbit Effects for Reaction Dynamics

A diabatic representation is convenient in the study of electronically nonadiabatic chemical reactions because the diabatic energies and couplings are smooth functions of the nuclear coordinates and the couplings are scalar quantities. A method called the fourfold way was devised in our group to generate diabatic representations for spin-free electronic states. One drawback of diabatic states computed from the spin-free Hamiltonian, called a valence diabatic representation, for systems in which spin-orbit coupling cannot be ignored is that the couplings between the states are not zero in asymptotic regions, leading to difficulties in the calculation of reaction probabilities and other properties by semiclassical dynamics methods. Here we report an extension of the fourfold way to construct diabatic representations suitable for spin-coupled systems. In this article we formulate the method for the case of even-electron systems that yield pairs of fragments with doublet spin multiplicity. For this type of system, we introduce the further simplification of calculating the triplet diabatic energies in terms of the singlet diabatic energies via Slater's rules and assuming constant ratios of Coulomb to exchange integrals. Furthermore, the valence diabatic couplings in the triplet manifold are taken equal to the singlet ones. An important feature of the method is the introduction of scaling functions, as they allow one to deal with multibond reactions without having to include high-energy diabatic states. The global transformation matrix to the new diabatic representation, called the spin-valence diabatic representation, is constructed as the product of channel-specific transformation matrices, each one taken as the product of an asymptotic transformation matrix and a scaling function that depends on ratios of the spin-orbit splitting and the valence splittings. Thus the underlying basis functions are recoupled into suitable diabatic basis functions in a manner that provides a multibond generalization of the switch between Hund's cases in diatomic spectroscopy. The spin-orbit matrix elements in this representation are taken equal to their atomic values times a scaling function that depends on the internuclear distances. The spin-valence diabatic potential energy matrix is suitable for semiclassical dynamics simulations. Diagonalization of this matrix produces the spin-coupled adiabatic energies. For the sake of illustration, diabatic potential energy matrices are constructed along bond-fission coordinates for the HBr and the BrCH(2)Cl molecules. Comparison of the spin-coupled adiabatic energies obtained from the spin-valence diabatics with those obtained by ab initio calculations with geometry-dependent spin-orbit matrix elements shows that the new method is sufficiently accurate for practical purposes. The method formulated here should be most useful for systems with a large number of atoms, especially heavy atoms, and/or a large number of spin-coupled electronic states.

Fast Approaches for Molecular Polarizability Calculations

Molecular polarizability of a molecule characterizes the capability of its electronic system to be distorted by the external field, and it plays an important role in modeling many molecular properties and biological activities. In this paper, a set of fast empirical models have been developed to predict molecular polarizability using two types of approaches. The first type of approaches is based on Slater's rules of calculating the effective atomic nuclear shielding constants. The best model (model 1A) of this category has achieved an average unsigned error (AUE), root-mean square error (RMSE), and average percent error (APE) of 2.23 au, 3.29 au, and 2.77%, respectively. The second type of model is based on an additive hypothesis of molecular polarizability. Five models have been constructed using different schemes of atom types. The best model that applies 14 atom types, model 2e, achieves AUE, RMSE, and APE of 0.99 au, 1.48 au, and 1.24%, respectively. This performance is much better than those of the models purely based upon chemical composition (model 2A and the Bosque and Sales model), for which errors are about 2-fold higher. It is expected that both model 1A and model 2E will have broad applications in QSAR and QSPR studies.

A comparison between the prudent order and the ranking obtained with Borda's, Copeland's, Slater's and Kemeny's rules

Arrow and Raynaud suggested that the result of a ranking rule should be a prudent order. We prove that we can construct profiles of linear orders for which the unique prudent order is the exact opposite of the ranking obtained with Borda's rule or Copeland's rules. Furthermore, we show that we can construct profiles of linear orders such that the unique prudent order winner can be found at any position in the corresponding unique order found by Slater's or Kemeny's rules. Finally, we show that there exist profiles where the unique Slater or Kemeny order is the exact opposite of one prudent order.

Density-functional generalized-gradient and hybrid calculations of electromagnetic properties using Slater basis sets.

In this paper we extend our density-functional theory calculations, with generalized gradient approximation and hybrid functionals, using Slater-type orbitals (STOs), to the determination of second-order molecular properties. The key to the entire methodology involves the fitting of all STO basis function products to an auxiliary STO basis, through the minimization of electron-repulsion integrals. The selected properties are (i) dipole polarizabilities, (ii) nuclear magnetic shielding constants, and (iii) nuclear spin-spin coupling constants. In all cases the one-electron integrals involving STOs were evaluated by quadrature. The implementation for (ii) involved some complexity because we used gauge-including atomic orbitals. The presence of two-electron integrals on the right-hand side of the coupled equations meant that the fitting procedure had to be implemented. For (iii) in the hybrid case, fitting procedures were again required for the exchange contributions. For each property we studied a number of small molecules. We first obtained an estimate of the basis set limit using Gaussian-type orbitals (GTOs). We then showed how it is possible to reproduce these values using a STO basis set. For (ii) a regular TZ2P quality STO basis was adequate; for (i) the addition of one set of diffuse functions (determined by Slater's rules) gave the required accuracy; for (iii) it was necessary to add a set of 1s functions, including one very tight function, to give the desired result. In summary, we show that it is possible to predict second-order molecular properties using STO basis sets with an accuracy comparable with large GTO basis sets. We did not encounter any major difficulties with either the selection of the bases or the implementation of the procedures. Although the energy code (especially in the hybrid case) may not be competitive with a regular GTO code, for properties we find that STOs are more attractive.

Transitive closure, proximity and intransitivities

Assignments of weak orders to complete binary relations are considered. Firstly, it is shown that assigning the transitive closure of a complete binary relation does not always assign the closest weak order according to any reasonable metric on complete binary relations. It is then shown that the assignment of a weak order to a complete binary relation assigns its transitive closure if and only if it assigns the closest weak order according to a particular distance function that is not a metric. This permits more direct comparisons between the Transitive Closure Rule and other rules such as the Slater Rule.

Electron Delocalization in Atomic and Molecular Systems

Electron delocalization in atomic and molecular systems in terms of the single particle delocalization index (Fulton, R. L. J. Phys. Chem. 1993, 97, 7516) is analyzed. The single particle delocalization index is a quantitative measure of the degree of sharing of an electron between two disjoint regions in a many electron system. The overall focus of this paper is the determination of spatial regions in atoms from which electrons are greatly delocalized. The delocalization indices from spherical volumes of various radii centered on the nucleus of interest to the volumes outside those regions show remarkably well-defined shell structure. The delocalization shell structure exhibits spatial regions in atoms, determined by the intershell minima, in which electrons are essentially localized. The delocalization shell structure of the electrons is apparent in all the atoms in the last column of the periodic table of the elements (Ne 1S0, Ar 1S0, Kr 1S0, Xe 1S0, and Rn 1S0), even in heavy atoms such as gold (Au 2S1/2) where by traditional methods the shell structure is not clear. The delocalization shell structure of the electrons in zinc (Zn 1S0) is also displayed because the expected shell structure, corresponding to principal quantum number n = 4, was not firmly established by previous indicators until atomic number 32 was reached. The values of distinct maxima of the delocalization indices, corresponding to regions within inner shells, are quite remarkable and related to the number of the electrons in each shell quantitatively. The physical meaning of these values is that the sharing of the core electrons in each shell does not extend to outer regions. The core shell structures in carbon (C 3P0) and silicon (Si 3P0) are mimicked by means of a set of the simulated natural orbitals (based on Slater's rules) used to construct the delocaliztaion index as a function of the distance from the nucleus. The separation of the contributions of definite angular momenta to the delocalization index for Ne 1S0, Ar 1S0, Zn 1S0, Xe 1S0, and Au 2S1/2 is carried out. Interference effects between these contributions are substantially restricted to the valence region. The values of the maxima of the contributions of definite angular momenta to the delocalization index are in agreement with the number of the s-, p-, d-, and f-electrons in each shell quantitatively. This is in agreement with well-known fact that the valence electrons only tend to be chemically active. The delocalization shell structures around the heavy atoms in the molecules LiH, H2O, CH4, SiH4, NH3, and PH3 are also displayed and compared to those around the single atoms Li 2S1/2, O 3P2, C 3P0, Si 3P0, N 4S3/2, and P 4S3/2. The delocalization of the core electrons remains essentially unchanged by incorporating the atoms into the molecules. The modifications of electron delocalization in the valence region are clear, and the chemically active regions in the molecules are nicely visible.

Screening Percentages Based on Slater Effective Nuclear Charge as a Versatile Tool for Teaching Periodic Trends

Slater's rules produce horizontal trends for effective nuclear charge Z *, which is the atomic number, Z, minus a screening constant, S. Using Slater's system, S is found by totaling the screening by all electrons except the one in question. The rules for calculating S have been modified from those of Slater to account for poorer shielding by d and f electrons. When this new value is divided by the atomic number, the result is defined as the "screening percentage", SP. When screening percentages are calculated across the periodic table, horizontal and vertical trends result. These trends provide an indicator of shielding and distance from the nucleus for valence electrons. The new rules for calculating S create vertical trends that mirror ionization energies (IE) and account for more subtle periodic anomalies such as the scandide and lanthanide contractions. An empirical system for relating IE with the screening percentage is included and the experimental IE data are compared to calculated values based on this new relationship. Finally, the horizontal and vertical trends in screening percentage are presented in a "shaded" periodic table suitable for freshman students. More advanced methods for depicting screening percentages are suggested for the advanced student.

3-D computer simulation of single-electron charging in silicon nanocrystal floating gate flash memory devices

The operation of a silicon nanocrystal quantum-dot based flash memory device is simulated numerically with emphasis on energy and charge quantization in the quantum-dot. The simulation involves the self-consistent solution of three-dimensional (3-D) Poisson and Schrodinger-like equations, with the Slater rule for determining the charging voltage. We also compute the capacitance-voltage characteristics of the device and derive the threshold voltage, V/sub T/, variation with single-electron charging as a function of design parameters.

The Genius of Slater's Rules

More than 60 years ago Slater developed a very simple procedure for determining the one-electron energies for atoms and their ions. With only a few modifications a procedure has been developed that yields the one-electron energies for atoms and ions with a level of detail very well suited for instruction in the structure and properties of atoms. It provides for the computation of very reasonable values for such properties as ionization energies, electron affinities, promotion energies, electronic transitions, and even XPS and ESCA spectra. Moreover, because the procedure is simple and intuitive, exercises can be developed that provide insights for students into how atoms work.

On the rapid evaluation of cofactors in the calculation of nonorthogonal matrix elements

The calculation of matrix elements involving nonorthogonal orbitals is speeded up by recognizing the orthogonalities between orbitals, leading to generalized Slater rules. The block structure present in the overlap matrix makes an efficient evaluation of its cofactors possible. These cofactors are calculated per subblock, each with its own parity sign. An adjustment parity sign has to be evaluated, which is added to the combined local signs, to give the correct total sign for the matrix element. An algorithm for the evaluation of this adjustment sign has been developed, making an easy and correct evaluation possible. The current scheme is shown to be very efficient, but possibilities for further improvement remain. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 77–83, 1998