Comparison between the point-charge electrostatic model and the angular overlap model: The general case

The aim of this paper is to explore in detail the mathematical connection between the point-charge electrostatic model and the angular overlap model. More precisely, for any configuration nl^x in any symmetry, the matrix elements, and therefore the energy levels, as given by the angular overlap model restricted to the antibonding effects λ (for any of the possible λ's), are deduced from an effective potential constructed according to the point-charge electrostatic model. This allows us to give a mathematical proof of the equivalence between the point-charge electrostatic model and the angular overlap model taking into account all the antibonding effects λ. Finally, the sum rules for the angular overlap model are examined in the light of the formulae connecting the two models considered. Numerical tables for obtaining the results as given by the angular overlap model restricted to any of the antibonding effects from those as given by the point-charge electrostatic model are listed for the configurations nl^x with l = p, d, f and g.     

Extension of adaptive tree code and fast multipole methods to high angular momentum particle charge densities

The development and implementation of a tree code (TC) and fast multipole method (FMM) for the efficient, linear-scaling calculation of long-range electrostatic interactions of particle distributions with variable shape and multipole character are described. The target application of these methods are stochastic boundary molecular simulations with polarizable force fields and/or combined quantum mechanical/molecular mechanical potentials. Linear-scaling is accomplished through the adaptive decomposition of the system into a hierarchy of interacting particle sets. Two methods for effecting this decomposition are evaluated: fluc-splitting and box-splitting, for which the latter is demonstrated to be generally more accurate. In addition, a generalized termination criterion is developed that delivers optimal performance at fixed error tolerance that, in the case of quadrupole-represented Drude water, effects a speed-up by a factor of 2-3 relative to a multipole-independent termination criteria. The FMM is shown to be approximately 2-3 times faster than the TC, independent of the system size and multipole order of the particles. The TC and FMM are tested for a variety of static and polarizable water systems, and for the the 70S ribosome functional complex containing an assembly of transfer and messenger RNAs.     

Accuracy of distributed multipoles and polarizabilities: comparison between the LoProp and MpProp models

Localized multipole moments up to the fifth moment as well as localized dipole polarizabilities are calculated with the MpProp and the newly developed LoProp methods for a total of 20 molecules, predominantly derived from amino acids. A comparison of electrostatic potentials calculated from the multipole expansion obtained by the two methods with ab initio results shows that both methods reproduce the electrostatic interaction with an elementary charge with a mean absolute error of approximately 1.5 kJ/mol at contact distance and less than 0.1 kJ/mol at distances 2 A further out when terms up to the octupole moments are included. The polarizabilities are tested with homogenous electric fields and are found to have similar accuracy. The MpProp method gives better multipole moments unless diffuse basis sets are used, whereas LoProp gives better polarizabilities.     

The self-interaction correction to the local spin density model: Effect on atomic momentum space properties

Perdew and Zunger showed that the exact energy density functional for the ground state is strictly self-interaction-free. However, the local spin density (LSD) approximation lacks this self-interaction correction (SIC). Perdew and Zunger studied the effect on coordinate-space properties of incorporating the SIC. In the present study we examine the effect of SIC on momentum space properties, viz. the electron momentum distribution, ⊂p″⊃ values, electron momentum densities and the Compton profiles for atoms He to Ar. A remarkable improvement is seen in all the momentum space properties of the SIC LSD model over the LSD model when compared to their near Hartree-Fock counterparts.     

Combinatorics of angular momentum recoupling theory: spin networks,their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and mathematical physics. This paper starts with an account of the basics of such a theory, which represents the most exhaustive framework in dealing with interacting many-angular momenta quantum systems. We then outline the essential features of this algebra, that can be encoded, for each fixed number N = (n + 1) of angular momentum variables, into a combinatorial object, the spin network graph, where vertices are associated with finite-dimensional, binary coupled Hilbert spaces while edges correspond to either phase or Racah transforms (implemented by 6j symbols) acting on states in such a way that the quantum transition amplitude between any pair of vertices is provided by a suitable 3nj symbol. Applications of such a combinatorial setting—both in fully quantum and in semiclassical regimes—are briefly discussed providing evidence of a unifying background structure.     

Gaussian vs. Slater representations of d orbitals: An information theoretic appraisal based on both position and momentum space properties

A detailed appraisal of Gaussian-type orbital (GTO) and Slater-type orbital (STO) expansions of 3d orbitals is carried out for the ²S state of copper—a case that should be maximally unfavorable for STOs. The appraisal is based on a wide variety of both position and momentum space properties and utilizes an information theoretic quality assessment technique. It is found that GTO expansions are not as useful as STO expansions for the prediction of 〈p⁸〉, 〈p⁷〉, and 〈r^?6〉 because these properties probe the functional deficiencies of GTOs at small r and large p. On the other hand, GTO expansions can predict accurate values of large r properties like 〈r⁸〉 despite the fact that their position space asymptotic decay is too fast. Unlike the case of s orbitals in helium, there does not seem to be any consistent ordering between accuracy in position space and accuracy in momentum space. The quality measures are found to be very useful for pinpointing the deficiencies of various expansions. This information enables us to construct easily a new GTO and a new STO expansion that are more accurate than any of the others in the literature. It is suggested that one STO is worth no more than two GTOs in the case of d orbitals.     

Transformed Dirac equation for the hydrogen atom,comparison with previous approaches in momentum space,and the anomalous Zeeman effect in momentum representation

The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v²/c² but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Formaldehyde photodissociation: dependence on total angular momentum and rotational alignment of the CO product

Quasiclassical trajectory calculations are reported to investigate the effects of rotational excitation of formaldehyde on the branching ratios of the fragmentation products, H2+CO and H+HCO. The results of tens of thousands of trajectories show that increased rotational excitation causes suppression of the radical channel and enhancement of the molecular channel. Decomposing the molecular channel into "direct" and "roaming" channels shows that increased rotation switches from suppressing to enhancing the roaming products across our chosen energy range. However, decomposition into these pathways is difficult because the difference between them does not appear to have a distinct boundary. A vector correlation investigation of the CO rotation shows different characteristics in the roaming versus direct channels and this difference is a potentially useful signature of the roaming mechanism, as first speculated by Kable and Houston in their experimental study of photodissociation of acetaldehyde [P. L. Houston and S. H. Kable, Proc. Nat. Acad. Sci. 103, 16079 (2006)].     

Quantum level structures at a Fermi resonance with angular momentum: classical periodic orbits, catastrophe maps and quantum monodromy

The Xiao-Kellman catastrophe map, for the classification of classical periodic orbits of the standard 2:1 Fermi resonance Hamiltonian is extended to species with finite vibrational angular momentum. The influence of the classical periodic orbit structure on different organizations of the quantum mechanical eigenvalues, in the four regions of the map, is strikingly demonstrated. The quantum eigenvalue lattices in angular momentum and energy space show dislocations attributable to a topological effect, termed quantum monodromy. Analogues with quantum monodromy in quasi-linear molecules and LiCN/LiNC isomerisation are demonstrated.     

A multilayer model for the study of space distributed redox modified electrodes: Part I. Description and discussion of the model

A multilayer model for the study of space distributed redox modified electrodes (redox polymer electrodes, or adsorption of an electroactive substance in several layers) is described and discussed in detail. It is shown in particular that it is equivalent to a system in which the electrons diffuse in the coating.     

Multireference coupled-cluster methods using an incomplete model space: Application to ionization potentials and excitation energies of formaldehyde

A recent fully linked multireference coupled-cluster method using an incomplete model space is applied to the direct calculation of the difference energies of formaldehyde. For the calculation of excitation energies (EE) use is made of a reference space composed of particle-hole excited configurations built from a set of active orbitals. Ionization potentials are obtained from a model space of singly ionized configurations. Results are compared with experiment and previous calculations.     

Two-photon state selection and angular momentum polarization probed by velocity map imaging: application to H atom photofragment angular distributions from the photodissociation of two-photon state selected HCl and HBr

A formalism for calculating the angular momentum polarization of an atom or a molecule following two-photon excitation of a J-selected state is presented. This formalism is used to interpret the H atom photofragment angular distributions from single-photon dissociation of two-photon rovibronically state selected HCl and HBr prepared via a Q-branch transition. By comparison of the angular distributions measured using the velocity map imaging technique with the theoretical model it is shown that single-photon dissociation of two-photon prepared states can be used for pathway identification, allowing for the identification of the virtual state symmetry in the two-photon absorption and/or the symmetry of the dissociative state. It is also shown that under conditions of excitation with circularly polarized light, or for excitation via non-Q-branch transitions with linearly polarized light the angular momentum polarization is independent of the dynamics of the two-photon transition and analytically computable.     

Hybrid coupled cluster methods: combining active space coupled cluster methods with coupled cluster singles, doubles, and perturbative triples

Based on the coupled-cluster singles, doubles, and a hybrid treatment of triples (CCSD(T)-h) method developed by us [J. Shen, E. Xu, Z. Kou, and S. Li, J. Chem. Phys. 132, 114115 (2010); and ibid. 133, 234106 (2010); and ibid. 134, 044134 (2011)], we developed and implemented a new hybrid coupled cluster (CC) method, named CCSD(T)q-h, by combining CC singles and doubles, and active triples and quadruples (CCSDtq) with CCSD(T) to deal with the electronic structures of molecules with significant multireference character. These two hybrid CC methods can be solved with non-canonical and canonical MOs. With canonical MOs, the CCSD(T)-like equations in these two methods can be solved directly without iteration so that the storage of all triple excitation amplitudes can be avoided. A practical procedure to divide canonical MOs into active and inactive subsets is proposed. Numerical calculations demonstrated that CCSD(T)-h with canonical MOs can well reproduce the corresponding results obtained with non-canonical MOs. For three atom exchange reactions, we found that CCSD(T)-h can offer a significant improvement over the popular CCSD(T) method in describing the reaction barriers. For the bond-breaking processes in F(2) and H(2)O, our calculations demonstrated that CCSD(T)q-h is a good approximation to CCSDTQ over the entire bond dissociation processes.     

Quadrupole, octopole, and hexadecapole electric moments of Sigma, Pi, Delta, and Phi electronic states: cylindrically asymmetric charge density distributions in linear molecules with nonzero electronic angular momentum

The number of independent components, n, of traceless electric 2(l)-multipole moments is determined for C(infinity v) molecules in Sigma(+/-), Pi, Delta, and Phi electronic states (Lambda=0,1,2,3). Each 2(l) pole is defined by a rank-l irreducible tensor with (2l+1) components P(m)((l)) proportional to the solid spherical harmonic r(l)Y(m)(l)(theta,phi). Here we focus our attention on 2(l) poles with l=2,3,4 (quadrupole Theta, octopole Omega, and hexadecapole Phi). An important conclusion of this study is that n can be 1 or 2 depending on both the multipole rank l and state quantum number Lambda. For Sigma(+/-)(Lambda=0) states, all 2(l) poles have one independent parameter (n=1). For spatially degenerate states--Pi, Delta, and Phi (Lambda=1,2,3)--the general rule reads n=1 for l<2/Lambda/ (when the 2(l)-pole rank lies below 2/Lambda/ but n=2 for higher 2(l) poles with l>or=2/Lambda/. The second nonzero term is the off-diagonal matrix element [formula: see text]. Thus, a Pi(Lambda=1) state has one dipole (mu(z)) but two independent 2(l) poles for l>or=2--starting with the quadrupole [Theta(zz),(Theta(xx)-Theta(yy))]. A Delta(Lambda=2) state has n=1 for 2((1,2,3)) poles (mu(z),Theta(zz),Omega(zzz)) but n=2 for higher 2((l>or=4)) poles--from the hexadecapole Phi up. For Phi(Lambda=3) states, it holds that n=1 for 2(1) to 2(5) poles but n=2 for all 2((l>or=6)) poles. In short, what is usually stated in the literature--that n=1 for all possible 2(l) poles of linear molecules--only applies to Sigma(+/-) states. For degenerate states with n=2, all Cartesian 2(l)-pole components (l>or=2/Lambda/) can be expressed as linear combinations of two irreducible multipoles, P(m=0)((l)) and P/m/=2 Lambda)((l)) [parallel (z axis) and anisotropy (xy plane)]. Our predictions are exemplified by the Theta, Omega, and Phi moments calculated for Lambda=0-3 states of selected diatomics (in parentheses): X (2)Sigma(+)(CN), X (2)Pi(NO), a (3)Pi(u)(C(2)), X (2)Delta(NiH), X (3)Delta(TiO), X (3)Phi(CoF), and X (4)Phi(TiF). States of Pi symmetry are most affected by the deviation from axial symmetry.     

Adhesive transition from noncontacting to contacting elastic spheres: extension of the Maugis-Dugdale model

In this paper, an adhesion model for spherical noncontact is proposed based on the Maugis-Dugdale (MD) adhesive contact model. The proposed noncontact model is combined with the MD contact model, thus providing a full range adhesion model with analytical transition from noncontacting to contacting asperity geometry. The proposed model is favorably compared with the full range improved DMT model for low surface energy values. The transition process from noncontact to contact and the adhesion instability that occurs during this transition are also investigated. It is found that jump-off points are different for displacement control and force control. Moreover, under displacement control, jump-on and jump-off points are different when the adhesion parameter lambda is greater than 0.95, whereas they are identical for lambda<0.95. By curve fitting a relationship between the critical approach under displacement and force control separately and the adhesion parameter lambda, approach prediction equations for jump-on and jump-off under different adhesion levels were obtained.     

Auger parameter shifts in the case of the non-local screening mechanism: Applications of the electrostatic model to molecules,solids and adsorbed species

A simple electrostatic model is applied to predict and rationalize the auger parameter shifts in molecules, solids and adsorbed species. The model is valid in the case of the non-local screening mechanism, i.e. when the screening of the core hole is due to the polarization of the nearest-neighbour ligands (no charge transfer from the ligands to localized electronic levels of the atom with the core hole, as occurs with the main peak of the core-ionized heavier 3d_n transition metal ions or the light lanthanide ions). To good approximation it is shown that: (1) the Auger parameter shift is a function of the number, distance, electronic polarizability and local symmetry of the first-neighbour ligands of the core-ionized atom; (2) the dipole–dipole interactions can play a key role in determining the extent of the extra-atomic relaxation energy and hence of the Auger parameter shift.