Configuration interaction matrix elements for atoms using permutation group algebra

A procedure is described for the efficient evaluation of the energy matrix elements necessary for atomic configuration-interaction calculations. With the orbital configurations of an N electron system in spin state S written as the irreducible representations [2^1/2N?S, 1^2S] of the permutation group S( N ), it is possible to evaluate readily the energy matrix elements of a spin-free Hamiltonian expressed in terms of the generators of the unitary group. We show how the use of angular momentum ladder operators permits the effective generation of a basis of eigenstates of ??², ??_z as well as ??² and ??_z, for which the energy matrix elements may be evaluated with ease.     

A method for the rapid calculation of matrix elements with highly oscillatory JWKB radial wavefunctions

A method for the rapid computation of matrix elements with JWKB radial wavefunctions is discussed. The method consists of dividing the range of integration into segments determined by the nodes of the semiclassical wavefunction. The desired matrix elements are calculated by summing the contributions from each segment which are evaluated by integrating between nodes with a Gauss-Mehler quadrature formula. The results are compared with exact quantum mechanical calculations and were found to agree within 1–2%. The calculations with the present method were generally five to ten times faster than the quantum mechanical calculations.     

A semi-empirical model for the charging and discharging of electric-paint displays

Electrochromic displays based on porous nanostructured films, “electric-paint displays”, are promising for electronic-paper applications due to their prominent paint-on-paper qualities. We have investigated the charge–discharge kinetics of electric-paint displays by chronocoulometry experiments. Whereas the charging curves are exponential, A(1−e^−t/τ), up to about 1 s, the discharging (when bleaching the display) cannot be described properly by a single exponential expression in the 0–1 s time regime. A semi-empirical model, in which the display is treated like a serial RC-circuit with a capacity step function, was developed and fitted to the experimental chronocoulometry curves with successful results. The general features of both the charging and discharging curves, as well as their dependences on the applied voltage, are reproduced in the 0–1 s regime by the suggested model.     

A semi-empirical approach for predicting the performance of mixed matrix membranes containing selective flakes

A successful model for mixed matrix membrane performance must address the complex geometry of the problem and accurately treat the diffusion behavior of the host–guest systems being considered. Detailed calculations based on the Maxwell–Stefan equations provide a widely accepted means of treating the diffusion of gases within zeolites. However, a full numerical solution of these equations for a complex mixed matrix membrane geometry does not offer the convenience and transparency that comes with an analytical treatment. At the same time, existing analytical equations which were formulated specifically to address mixed matrix geometry do so under the assumption of very simplistic models for diffusion. Here, an approach is presented for predicting the permeability and selectivity of mixed matrix membranes containing zeolite flakes that combines well-known analytical expressions for mixed matrix membrane performance with Maxwell–Stefan modeling for zeolite diffusion. The constant permeabilities required by the analytical models are calculated by the Maxwell–Stefan equations as a function of operating conditions, and these calculated effective permeabilities are used to predict mixed matrix membrane performance at corresponding operating conditions. The method is illustrated through two case studies: normal- and iso-butane separation by a membrane containing silicalite-1 flakes and carbon dioxide/methane separation by membranes containing CHA-type zeolites. Predictions are compared to experimental results found in the literature for both cases. Also, the applicability of the Maxwell and Cussler analytical models for mixed matrix membrane performance is explored as a function of flake loading and aspect ratio.     

Interaction of low-energy ions and atoms of light elements with a fluorinated carbon molecular lattice

The mechanism of interaction of low-energy atoms and ions of light elements (H, H+, He, Li, the kinetic energy of the particles 2-40 eV) with C6H6, C6F12, C60, and C60F48 molecules was studied by ab initio MD simulations and quantum-chemical calculations. It was shown that starting from 6 A from the carbon skeleton for the "C6H6 + proton" and "C60 + proton" systems, the electronic charge transfer from the aromatic molecule to H+ occurs with a probability close to 1. The process transforms the H+ to a hydrogen atom and the neutral C6H6 and C60 molecules to cation radicals. The mechanism of interaction of low-energy protons with C6F12 and C60F48 molecules has a substantially different character and can be considered qualitatively as the interaction between a neutral molecule and a point charge. The Coulomb perturbation of the system arising from the interaction of the uncompensated proton charge with the Mulliken charges of fluorine atoms results in an inversion of the energies of the electronic states localized on the proton and on the C6F12 and C60F48 molecules and makes the electronic charge transfer energetically unfavorable. On the different levels of theory, the barriers of the proton penetration for the C6F12 and C60F48 molecules are from two to four times lower than those for the corresponding parent systems (C6H6 and C60). The penetration barriers of the He atom and Li+ ion depend mainly on the effective radii of the bombarding particles. The theoretical penetration and escaped barriers for the "Li+ + C60" process qualitatively explain the experimental conditions of synthesis of the Li@C60 complex.     

Charge and radial correlations in helium and helium-like atoms

Summary For two-electron atoms, the method of a variable exponent, which treats the orbital exponent (or effective nuclear charge) of an electron as an explicit function of the radial coordinate of the other electron, is studied. The method is shown to improve the energy and other electronic properties remarkably. An incorporation of the variable exponent into the Kellner approximation for He, for example, gives the energy –2.872 606 1 a.u., which is lower than the original Kellner energy by 0.024 949 8 a.u. and exceeds the Hartree-Fock limit energy by 0.010 926 1 a.u. The improvement due to the variable exponent originates from the inclusion of the charge and radial correlations. Applications of the method to the Eckart and Hylleraas approximations are also presented.     

Distortion of interacting atoms and ions

The distortion of atoms in HeNe and ions in LiF and NaCl is studied by use of non-orthogonal, least distorted, localized molecular orbitals in the restricted Hartree-Fock approximation. We find that the least distortion criterion is effective in producing localized orbitals. In addition we find that point-by-point the localized orbitals differ little from the Hartree-Fock orbitals of the corresponding atoms or ions, although the differences are energetically important. We infer from our calculations that the effective potential which distorts atomic orbitals into localized molecular orbitals must be quite weak.     

Chemistry and the near-sighted nature of the one-electron density matrix

Two different macrospopic pieces of copper have different external potentials and, because of the unique functional relationship between the electron density and the external potential as demanded by density functional theory, should possess different electron density distributions. Experimentally, however, an atom in the bulk exhibits the same electron density in both samples and they possess identical sets of intensive properties. Density functional theory does not account for the fundamental observation underlying the theory of atoms in molecules: that what are apparently identical distributions of charge can be observed for an atom or a grouping of atoms in systems with different external potentials and that these atoms contribute essentially identical amounts to the energies and all other properties of the systems in which they occur. It is shown that, unlike the external potential, the kinetic energy density and the potential energy density, defined by the virial of the Ehrenfest force acting on electron density, are short-range functions. As recorded in the first article on atoms in molecules, they exhibit a local dependence on the electron density that causes them to faithfully mimic the transferability of the atomic charge distributions from one system to another. The electron, the kinetic energy, and the virial densities are all determined directly by the one-electron density matrix, a function termed near-sighted by Professor Kohn. It is this near-sighted property of the one-matrix that underlies the working hypothesis of chemistry—that of a functional group exhibiting a characteristic set of properties. The observations obtained from the theory of atoms in molecules and the atomic theorems it determines demonstrate the existence of a local relationship between the electron density and all properties of a system. © 1995 John Wiley & Sons, Inc.     

Coupling of radial and angular correlations in two-electron atoms

The nature of interelectronic correlation in the ground and low lying excited states of the helium sequence is investigated. It is demonstrated that as a consequence of the coupling between radial and angular correlation the electrons are actually pulled together rather than pushed apart, in the low nuclear charge end of the isoelectronic sequence. A uniform treatment of correlation effects in both the ground and the excited states is thus shown to be feasible.     

Inner and outer radial density functions in many-electron atoms

When any two electrons are considered simultaneously, the radial density function D(r) in many-electron atoms is shown to be rigorously separated into inner D _<(r) and outer D _>(r) radial densities. Accordingly, radial properties such as the electron–nucleus attraction energy V _en and the diamagnetic susceptibility χ _d are the sum of the inner and outer contributions. The electron–electron repulsion energy V _ee has an approximate relation with the minus first moment of the outer density D _>(r). For the 102 atoms He through Lr in their ground states, different characteristics of local maxima in the radial densities D _<(r), D _>(r), and D(r) are reported based on the numerical Hartree-Fock wave functions. Relative contributions of the inner and outer components to V _en and are also discussed for these atoms.     

A semi-empirical method for photofraction determination

A semi-empirical method for photofraction determination is proposed: the photopeak efficiency is plotted as a function of the calculated total number of photons which interact at least once with the crystal for different sample to crystal distances. The tangent to this curve gives the photofraction at the point considered. No background subtraction or shielding is necessary.     

Comparative study of perturbative methods for computing electron transfer tunneling matrix elements with a nonorthogonal basis set

The authors consider the problem of computing tunneling matrix elements for bridge-mediated electron transfer reactions using the Lowdin [J. Math. Phys. 3, 969 (1962); J. Mol. Spectrosc. 13, 326 (1964)] projection-iteration technique with a nonorthogonal basis set. They compare the convergence properties of two different Lowdin projections, one containing the overlap matrix S and the other containing the inverse S-1 in the projected Hamiltonian. It was suggested in the literature that the projected Hamiltonian with S-1 has better convergence properties compared to the projected Hamiltonian with S. The authors test this proposal using a simple analytical model, and ab initio Hartree-Fock calculations on different molecules with several types of basis sets. Their calculations show that, for Gaussian-type basis sets, the projected Hamiltonian containing S has the best convergence properties, especially for diffuse basis sets and in the strong coupling limit. The limit of diffuse basis sets is relevant to tunneling matrix element calculations involving excited states and anionic electron transfer.     

A new one-electron model for extended Hückel type molecular orbital calculations

A new form of Coulomb and resonance integrals for extended Hückel type molecular orbital calculations was derived. In this model an electron moving in a molecule, when it stays in the neighborhood of the center of an atom in the molecule, sees the potential of the corresponding isolated atom. Compared with the well-known forms of Coulomb and resonance integrals, each integral derived from this model considers the influences from all atoms in the molecule, and includes the concept of atomic cell. The similarities of this model to other extended Hückel type formulations are also examined.     

Frequency-dependent response properties and excitation energies from one-electron density matrix functionals

The recent formulation of the time-dependent density matrix functional theory (TD-DMFT) has opened an avenue to calculations of frequency-dependent response properties and excitation energies of atoms and molecules. In practice, the accuracy of the computed data is limited by both the errors inherent to the adiabatic approximation or its modifications and the quality of the energy functionals. The relative importance of these two factors is carefully assessed with test calculations on diatomic molecules with few electrons. The test results clearly demonstrate the superiority of an ad hoc approach that corrects the improper behavior of the adiabatic approximation at the low-frequency limit. Even more importantly, TD-DMFT convincingly removes the ambiguity in the choice of the two-electron integrals that is present in the stationary-state case. On the other hand, paralleling the previously reached conclusions pertinent to ionization potentials, the presently available BBC-type functionals are found to be insufficiently accurate to provide reliable quantitative predictions of excitation energies.     

Ground-state multiplicities of atoms and positive ions

A procedure is developed to establish the ground-state multiplicities for atoms with any number of electrons. The procedure is applied to two- and three-electron systems. The results are that all neutral and positive two- and three-electron atoms have singlet (S=0) and doublet (S=1/2) ground states, respectively. Received: 10 August 1999 / Accepted: 6 October 1999 / Published online: 15 December 1999