Charge Overlap Effects Research Articles

Charge-Overlap Effects on Electronic Transitions in Moderate-Energy K+–Ar and Cs+–Ar Collisions

Excitation mechanisms in K+–Ar and Cs+–Ar collisions at moderate laboratory energies Elab = 0.05–2 keV are investigated by calculating repulsive potentials V(R) = 0.01–1 keV at distances R = 0.8–10...

Localized overlap algorithm for unexpanded dispersion energies

First-principles-based, linearly scaling algorithm has been developed for calculations of dispersion energies from frequency-dependent density susceptibility (FDDS) functions with account of charge-overlap effects. The transition densities in FDDSs are fitted by a set of auxiliary atom-centered functions. The terms in the dispersion energy expression involving products of such functions are computed using either the unexpanded (exact) formula or from inexpensive asymptotic expansions, depending on the location of these functions relative to the dimer configuration. This approach leads to significant savings of computational resources. In particular, for a dimer consisting of two elongated monomers with 81 atoms each in a head-to-head configuration, the most favorable case for our algorithm, a 43-fold speedup has been achieved while the approximate dispersion energy differs by less than 1% from that computed using the standard unexpanded approach. In contrast, the dispersion energy computed from the distributed asymptotic expansion differs by dozens of percent in the van der Waals minimum region. A further increase of the size of each monomer would result in only small increased costs since all the additional terms would be computed from the asymptotic expansion.

Distributed molecular polarisabilities and asymptotic intermolecular interaction energies†

Strategies for obtaining distributed molecular polarisabilities and using them in calculations of interaction energies have been examined. A mathematically rigorous strategy applying constrained density fitting that practically removes the so-called charge-flow terms has been proposed. The resulting polarisabilities give asymptotic dispersion energies without the artefacts plaguing the previously used methods. In particular, since the charge-flow polarisabilities are extremely small in our approach, the terms in the distributed expansion of the dispersion energy which decay slower than the sixth inverse power of the intermonomer separation are negligible. Furthermore, we show that the usual practice of approximately locating or neglecting two-centre (nonlocal) distributed polarisabilities in calculations of these energies can now be abandoned since in the algorithm developed by us the inclusion of the nonlocal polarisabilities increases the computational requirements only by one power of the number of atoms and the summations can be restricted to within a small cutoff radius. Our method gives dispersion energies that are practically identical to the values computed from exact (unexpanded) formulas for all separations where charge-overlap effects are small, i.e. this approach gives the best possible asymptotic representation of dispersion energies. Thus, it should be possible to replace the current empirical dispersion functions by ab initiocomputed dispersion energies in a range of applications.

Comprehensive Benchmarking of a Density-Dependent Dispersion Correction.

Standard density functional approximations cannot accurately describe interactions between nonoverlapping densities. A simple remedy consists in correcting for the missing interactions a posteriori, adding an attractive energy term summed over all atom pairs. The density-dependent energy correction, dDsC, presented herein, is constructed from dispersion coefficients computed on the basis of a generalized gradient approximation to Becke and Johnson's exchange-hole dipole moment formalism. dDsC also relies on an extended Tang and Toennies damping function accounting for charge-overlap effects. The comprehensive benchmarking on 341 diverse reaction energies divided into 18 illustrative test sets validates the robust performance and general accuracy of dDsC for describing various intra- and intermolecular interactions. With a total MAD of 1.3 kcal mol(-1), B97-dDsC slightly improves the results of M06-2X and B2PLYP-D3 (MAD = 1.4 kcal mol(-1) for both) at a lower computational cost. The density dependence of both the dispersion coefficients and the damping function makes the approach especially valuable for modeling redox reactions and charged species in general.

A new representation of the dispersion interaction

The conventional formulation of the dispersion energy as a power series in R −1 with damping at small R is re-examined. Symmetry-adapted and intermolecular perturbation theory calculations have been performed and compared for the argon dimer. We show that conventional Tang-Toennies damping functions fit the data better when exchange as well as charge-overlap effects are included. We also construct a new representation of the dispersion energy which provides a more accurate, but still quite compact, description of the dispersion energy, and suggest several simpler forms which would be suitable for describing dispersion more accurately in larger systems. Some preliminary results for the HF dimer are also described.

Coulomb interaction energy including overlap effects for the ground state of RbCs beyond 7 Å

We use a perturbative model based on a partition of the configuration space into four regions to calculate the Coulomb energy including charge-overlap effects (induction and dispersion terms) for the interaction between two atoms Rb(5s2S) and Cs(6s2S). Results for first-order electrostatic as well as second-order induction and dispersion energies will be presented in the range 13 < R < 25 a.u. The damping functions for the various dispersion terms varying as R−6, R−8, and R−10 in the range of large R have also been determined, and fitted analytical forms will be displayed for these terms. From comparison with accurate experimental results, the present method is seen to be valid for values of R beyond 7 Å (1 Å = 10−10 m).

Perturbation model for the X1 Σ+ and a3 Σ+ states of NaK at intermediate internuclear distances beyond 6 Å

The energy of the X1Σ+ and a3Σ+ states of NaK was estimated from a perturbative model for intermediate- and long-range internuclear distances. Both the charge overlap effects calculated from a model previously published and the exchange contribution estimated from the asymptotic formula were taken into account. Fitted analytical expressions are proposed for electrostatic and induction energy terms as well as for various dispersion damping functions. From comparison with accurate experimental results, the present model is seen to be valid for values of R greater than 6 Å (1 Å = 10−10 m) for the X1Σ+ state and R greater than 8 Å for the a3Z+ state.

Gaussian multipole functions for describing molecular charge distributions

Differentiating spherical Gaussians produces a set of Gaussian multipole functions that behave asymptotically in the same way as point multipoles but are more useful for describing molecular charge densities as they also include charge overlap effects at short range. Coulomb matrix elements between Gaussian multipoles are readily evaluated, including the anisotropy. A simple general equation for interactions between point multipoles is also derived.

Coulomb interaction energy including overlap effects for the ground states of LiNa and Na 2

A recently proposed method to calculate first-order electrostatic as well as second-order induction and dispersion energies including charge-overlap effects for the interaction between two atoms each with one active electron is applied to the systems Li(2s) + Na(3s) and Na(3s) + Na(3s), giving the induction energy for both systems. The variation with R of the relative contribution of the overlapping and non-overlapping configuration space regions is discussed for the largest dispersion and induction terms.

Effect of three-body interactions on third virial coefficients of rare gases

Third virial coefficients of neon, argon, krypton, and xenon have been calculated using the reliable XC and HFB pair potentials and three-body dispersion energies. Comparison between calculated and selected experimental values shows that the agreement is within the experimental error. The influence of three-body charge-overlap effects and exchange energies on the third virial coefficients is discussed.

Charge overlap effects for first-order molecule-molecule interactions, through high partial wave order, using the N2-N2interaction as a model

Results are presented for the individual partial wave components of the first-order non-expanded Coulomb energy for the N2-N2 interaction as a function of intermolecular separation R, for 3·5 ˇ- R ˇ- 9·5a 0 and for partial wave indices l a and l b corresponding to (l a + l b) ˇ- 8. These results permit the evaluation of the first-order Coulomb energy, its R -1 multipole expansion, and the charge overlap corrections to the multipole expansion, as a function of R, relative molecular configuration, and (l a + l b). Calculations of this type are used in a discussion of the relative importance of first-order charge overlap effects, and of the convergence of the partial wave calculations, as a function of R, configuration, and (l a + l b). The difficulties in obtaining reliable results for charge overlap, and related effects, are also discussed briefly.

Hartree–Fock approximate correlation energy (HFACE) potential for diatomic interactions. Molecules and van der Waals molecules

A model potential function recently developed for van der Waals molecules has been extended to bound-state diatomics. It is written as the sum of the (extended) Hartree–Fock energy and the (remaining) correlation energy, each of these being elaborated separately from available data pertaining to the regions of the potential where they stand as the dominant energy contributions. The correlation energy is approximated semiempirically from the asymptotic expansion for the dispersion energy, the terms of which are individually damped to account for charge overlap and exchange effects. To determine the values of the C8 and C10 dispersion coefficients, which are missing in the literature, it is proposed the semiempirical rule Cn/C6=knR[a(n–6)/2]0, which is believed to be accurate within 15% for C8 and 30% for C10. Results are presented for thirteen chemically stable bound-state diatomics, in addition to three van der Waals molecules which are typical members of as many families of non-bonding interactions: the ground state of the rare-gas dimers and the alkali earth dimers, and the lowest triplet state of the alkali-metal dimers. The results are all good.

Explicit three-body non-additive triple-dipole dispersion energy term including charge-overlap effects

The Axilrod-Teller-Muto non-additive triple-dipole dispersion energy term is modified to model the available non-expanded triple-dipole ab initio perturbation results of O'Shea and Meath for the lowest quartet state of H3 and hence accounts for the charge-overlap effects at small interatomic separations. It is suggested that the new energy term can be used for trimers of other spherically symmetric systems (for example inert gas trimers) by means of a reduced coordinate approach.

Induction and dispersion energies including charge overlap effects: Green's‐function‐type methods

Induction and dispersion energies including charge overlap effects: Green's‐function‐type methods

Charge overlap effects and the validity of the multipole results for first-order molecule-molecule intermolecular forces using pair interactions involving the ground state H2, N2, CO2, HF and LiH molecules as models

All the pair interactions arising from the ground state H2, N2, CO2, HF and LiH molecules are used as models for investigating the validity of the multipolar results (corresponding to the classical interaction of the permanent multipole moments of the interacting molecules) as a representation of the first-order Coulomb interaction energy. Charge overlap effects, the neglect of which are responsible for the misbehaviour of the multipolar results as the intermolecular distance R becomes small, are discussed as a function of the nature of the interacting molecules. Useful analytical formulae are presented which represent the numerical results. Some of the difficulties in using the multipolar results to represent the anisotropic part of a dimer potential are discussed in general and more specifically with respect to the evaluation of the mean square torque of nitrogen molecules.

Charge overlap effects in dispersion energies

Charge overlap effects in dispersion energies are calculated using frequency dependent atomic polarizabilities. The dispersion energy is expanded in the usual multipole interaction series but the dispersion coefficients are found to be given as a product, C (LA,LB) χ (LA,LB;R) where C (LA,LB) is the usual coefficient for nonoverlapping fragments and χ is called a damping function which equals one asymptotically. Damping functions have been calculated for He(11S)–He(11S), Be(11S)–Be(11S), and combination of He(23S), He(21S), He(11S), and H(12S). The results are used to analyze qualitatively the extent of the dispersion contribution to the energy of interactions at either the equilibrium separation or the critical distance for orbiting collisions.

Validity of the multipole results for first-order molecule-molecule interaction energies.

The HF-HF, LiH-LiH and N2-N2 interactions, together with previous work on the H2-H2 interaction, are used as models for discussing how charge overlap effects in the first-order Coulomb energy vary as a function of orientation and intermolecular separation as the nature of the interacting species changes. For interactions such as H2-H2 and HF-HF, where the interacting molecules have positive even permanent multipole moments, charge overlap effects are relatively small and the usual R -1 multipole results for the interaction energy between the permanent moments of the interacting molecules give a remarkably good representation of the first-order Coulomb energy until R becomes quite small. For interactions such as LiH-LiH and N2-N2, where the interacting molecules have negative even permanent moments, charge overlap effects are large as a relative function of R and the multipolar results yield a poor representation of the interaction energy even for moderately large values of R. Some of the problems and erro...

Charge overlap effects and the validity of the multipole results for first-order molecule-molecule interaction energies. Formalism and an application to H2-H2

For the interaction of two molecules, the first-order coulomb energy, E e (1), is expressed as a sum of non-expanded partial wave components which exhibit the orientation dependence of the energy explicitly. By comparing these non-expanded results with the analogous multipolar results for the first-order energy, the effects of charge overlap on the multipole representation of the energy can be investigated as a function of intermolecular separation and orientation, and the validity of the multipole results for E e (1) assessed. The interaction of two ground-state hydrogen molecules is considered as a specific example of this approach for studying the validity of the multipole expansion as a representation of molecular first-order Coulomb energies. All calculations are presented as a function of four molecular wave functions of varying degrees of sophistication in order to assess the sensitivity of (1) the representation of charge overlap effects and (2) the calculated quadrupole and hexadecapole moments o...

Charge-overlap effects in the non-additive triple-dipole interaction

A formal partial wave analysis of the third-order energy for the interaction of three non-degenerate S-state atoms is used to identify those non-additive terms which, in the limit of negligible charge overlap between the interacting atoms, reduce to the usual long-range expanded form of the triple-dipole energy. Pseudo-state techniques are used to evaluate these terms, both with and without the inclusion of charge-overlap effects, over a range of interatomic separations and angles for the interaction of three ground-state hydrogen atoms. These results are used as a model to discuss the way in which charge-overlap effects modify the usual long-range result for the triple-dipole energy. In addition to the familiar modification of the radial dependence of the interaction energy, charge-overlap effects are found to give rise to a modification of the angular dependence; the latter effect has no analogue in two-atom interactions. The significance of these results is discussed briefly.

Gaussian representations of charge overlap effects in intermolecular forces

AbstractThe ground state HH+ and HH interactions are used as model interactions for investigating the feasibility of using Gaussian basis sets for representing charge overlap effects in intermolecular forces. The non‐expanded charge‐induced dipole energy and the non‐expanded dipole‐dipole dispersion energy, respectively, for these interactions are calculated using two types of Gaussian basis functions to represent the first order wave function, Ψ(1). Very good results for these interaction energies, which include charge overlap effects, are obtained for all interatomic separations by using small Gaussian basis sets to represent the interaction, that is Ψ(1), and/or the isolated atoms (the zeroth order wave function).