Isomorphism in electron-pair densities of atoms

The spherically averaged electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are a couple of densities which characterize the motion of electron pairs in atomic systems. We study a generalized electron-pair density (q; a, b) that represents the probability density function for the magnitude of two-electron vector a r _j +b r _k of any pair of electrons j and k to be q, where a and b are nonzero real numbers. In particular, h(u)=g(u;1, −1) and d(R) = . It is shown that the scaling property of the Dirac delta function and the inversion symmetry of orbitals in atoms due to the central force field generate several isomorphic relations in the electron-pair density (q; a, b) with respect to the two parameters a and b. The approximate isomorphism d(R)≅8h(2R) known in the literature between the intracule and extracule densities is a special case of the present results. Received: 24 May 2000 / Accepted: 18 July 2000 / Published online: 27 September 2000     

The calculation of electron localization and delocalization indices at the Hartree–Fock, density functional and post-Hartree–Fock levels of theory

 Localization, λ(A), and delocalization indices, δ(A,B), as defined in the atoms in molecules theory, are a convenient tool for the analysis of molecular electronic structure from an electron-pair perspective. These indices can be calculated at any level of theory, provided that first- and second-order electron densities are available. In particular, calculations at the Hartree–Fock (HF) and configuration interaction (CI) levels have been previously reported for many molecules. However, λ(A) and δ(A,B) cannot be calculated exactly in the framework of Kohn–Sham (KS) density functional theory (DFT), where the electron-pair density is not defined. As a practical workaround, one can derive a HF-like electron-pair density from the KS orbitals and calculate approximate localization and delocalization indices at the DFT level. Recently, several calculations using this approach have been reported. Here we present HF, CI and approximate DFT calculations of λ(A) and δ(A,B) values for a number of molecules. Furthermore, we also perform approximate CI calculations using the HF formalism to obtain the electron-pair density. In general, the approximate DFT and CI results are closer to the HF results than to the CI ones. Indeed, the approximate calculations take into account Coulomb electron correlation effects on the first-order electron density but not on the electron-pair density. In summary, approximate DFT and CI localization and delocalization indices are easy to calculate and can be useful in the analysis of molecular electronic structure; however, one should take into account that this approximation increases systematically the delocalization between covalently bonded atoms, with respect to the exact CI results. Received: 13 February 2002 / Accepted: 24 April 2002 / Published online: 18 June 2002     

Correlated electron-pair properties of the Li atom in momentum space

On the basis of multiconfiguration Hartree–Fock calculations, correlated electron-pair intracule (relative motion) and extracule (center-of-mass motion) properties are reported for the Li atom in momentum space. The present results are more accurate and consistent than those in the literature. Received: 10 September 2001 / Accepted: 11 December 2001 / Published online: 22 March 2002     

Second-order quantum similarity measures from intracule and extracule densities

The calculation of quantum similarity measures from second-order density functions contracted to intracule and extracule densities obtained at the Hartree-Fock level is presented and applied to a series of atoms, (He, Li, Be, and Ne), isoelectronic molecules (C₂H₂, HCN, CNH, CO, and N₂), and model hydrogen-transfer processes (H₂/H⁺, H₂/Hot, H₂/H⁻). Second-order quantum similarity measures and indices are found to be suitable measures for quantitatively analyzing electron-pair density reorganizations in atoms, molecules, and chemical processes. For the molecular series, a comparative analysis between the topology of pairwise similarity functions as computed from one-electron, intracule, and extracule densities is carried out and the assignment of each particular local similarity maximum to a molecular alignment discussed. In the comparative study of the three hydrogen-transfer reactions considered, second-order quantum similarity indices are found to be more sensitive than first-order indices for analyzing the electron-density reorganization between the reactant complex and the transition state, thus providing additional insights for a better understanding of the mechanistic aspects of each process. Received: 7 July 1997 / Accepted: 29 October 1997     

Electron-pair densities of singly charged atoms

For the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground states, spherically averaged electron-pair intracule (relative-motion) density h(u), extracule (center-of-mass-motion) density d(R), and their moments uⁿ and Rⁿ are examined, where u and R are the interelectronic distance and the center-of-mass radius of a pair of electrons, respectively. The intracule and extracule densities of all the 96 ions are found to be monotonically decreasing functions, as for neutral atoms. Approximate relations d(R)8h(2R) and uⁿ/Rⁿ2ⁿ are confirmed to be valid for the charged atoms as well.     

Average inner product sums of electron linear momenta for atoms and ions

From the second moments of the electron-pair densities in momentum space, accurate Hartree–Fock values of the average inner product sum 〈∑ _i<j p _i ·p _j 〉 of electron linear momenta are evaluated for the 102 neutral atoms from He to Lr, the 53 singly charged cations from Li⁺ to Cs⁺, and the 43 stable anions from H⁻ to I⁻ in their experimental ground states. The present results are new for 38 species and improve the literature values for 68 species. Received: 18 July 2002 / Accepted: 4 September 2002 / Published online: 8 November 2002 Acknowledgement. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. Correspondence to: H. Matsuyama e-mail: hisashi@mmm.muroran-it.ac.jp     

Two-electron distribution functions and intracules

 Two-electron distribution functions and intracules are functions of electronic coordinates and occupy an important, and frequently overlooked, middle ground between the beguiling simplicity of electron densities and the bewildering complexity of wavefunctions. We survey the functions that have been considered by earlier workers and introduce two new ones, the Wigner intracule and the action intracule, that have not previously been discussed. To illustrate their usefulness, we consider the intracules of jellium, a few small atoms and the dissociating hydrogen molecule. Received: 26 July 2002 / Accepted: 20 October 2002 / Published online: 30 January 2003 Correspondence to: P.M.W. Gill e-mail: peter.gill@nott.ac.uk     

Structure and stability of hypervalent NLi n Na2 (n = 1–4) and related species at density functional theory level

A number of configurations of NLi _n Na₂ (n = 1–4) species were optimized using the B3LYP–density functional theory method; the 6-31G^* basis set was used in this calculation. In order to study all possible dissociation energies, some related species such as NLi₂Na, NLi _n (n = 1–4), Li _n (n = 1, 2) and Na _n (n = 1, 2) were also considered. Optimizations of these species were followed by fundamental frequency calculations at the same level. Global minima of these species were shown to adopt C _{2 v} (NLi₄Na₂, NLi₂Na₂), D _{3 h} (NLi₃Na₂) and C _s (NLiNa₂ and NLi₂Na) configurations. All possible dissociation energies were obtained. Received: 30 November 1998 / Accepted: 15 October 1999 / Published online: 14 March 2000     

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.     

Electron-pair distributions and chemical bonding

The electron-pair (intracule and extracule) densities of the first-row hydrides may be understood on the basis of the different chemical-bonding schemes in this series. Thus, the LiH pair densities show a strongly ionic nature and may be fairly well described by Li⁺H^?. The CH-pair densities, on the other hand, may be approximated by the promolecular superposition of a carbon and a hydrogen atom with an accumulation of pair density in the internuclear region signifying covalency. In the case of FH, its pair densities show a predominately ionic structure and are closer to those of F^? than to those of the promolecular superposition of a fluorine and a hydrogen atom. The slight deformation of longitudinal pair densities observed in FH is largely due to the presence of the H⁺. © 1994 John Wiley & Sons, Inc.     

Synthesis and characterization of poly(styrene/α-t-butoxy-ω-vinylbenzyl-polyglycidol) microspheres

Polystyrene microspheres with polyglycidol (polyGL) in a surface layer were synthesized in batch radical emulsifier-free emulsion copolymerizations of styrene and surfmers, α-t-butoxy-ω-vinylbenzyl-polyGL macromonomers (VB-polyGL). Macromonomers with number-average molecular weight Mˉ _n=950 (VB-polyGL950) and Mˉ _n=2700 (VB-polyGL2700) were used for these polymerizations. In all syntheses the initial concentrations of styrene and initiator (K₂S₂O₈) were constant. The initial macromonomer-to-styrene ratios were varied from 1.10 × 10⁻³ to 1.64 × 10⁻² mol/mol and from 3.46 × 10⁻⁴ to 3.47 × 10⁻³ mol/mol for VB-polyGL950 and VB-polyGL2700, respectively. The diameters of microspheres obtained were smaller for the syntheses with higher concentrations of macromonomers. Syntheses with VB-polyGL950 yielded microspheres with number-average diameters (Dˉ _n) from 216 to 900 nm and with a bimodal diameter distribution. The number-average diameters of microspheres obtained with VB-polyGL2700 varied from 220 to 650 nm, depending on the initial concentration of macromonomer. Their diameter distributions were monomodal, with a diameter polydispersity parameter (ratio of weight-average and number-average diameters) in the range 1.007≤Dˉ _w/Dˉ _n≤1.022. For each type of microsphere the fraction of polyGL in a surface layer and the surface concentration of sulfate anions were determined. The fraction of polyGL in the surface layer was related to the initial monomer composition in the polymerizing mixture. Adsorption of human serum albumin onto surfaces of some poly(styrene/VB-polyGL) microspheres was up to 10 times lower than for the polystyrene microspheres obtained in a similar emulsifier-free emulsion polymerization of styrene. Received: 26 September 2000/Accepted: 19 February 2001     

Pseudopotential study of lanthanum and lutetium dimers

Relativistic energy-consistent small-core lanthanide pseudopotentials of the Stuttgart–Bonn variety and extended valence basis sets have been used for the investigation of the dimers La₂ and Lu₂. It was found that the ground states for La₂ and Lu₂ are most likely ¹∑ _g ⁺ (σ _g ²π _u ⁴) and ³∑ _g ⁻ (4f ¹⁴4f ¹⁴σ _g ²σ _u ²π_u ²), respectively. The molecular constants including error bars were derived from multireference configuration interaction as well as coupled-cluster calculations, taking into account corrections for atomic spin–orbit splitting as well as possible basis set superposition errors. The theoretical values for La₂ (R _e=2.70±0.03 ?, D _e=2.31±0.13 eV, ω_e=186±13 cm⁻¹) show good agreement with the experimental binding energy (D _e=2.52±0.22 eV), but the experimental vibrational constant in an Ar matrix (ω_e=236±0.8 cm⁻¹) is significantly higher. For Lu₂ the theoretical values (R _e=3.07±0.03 ?, D _e=1.40±0.12 eV, ω_e=123±1 cm⁻¹) are in overall excellent agreement with experimental data (D _e=1.43±0.34 eV, ω_e=122± 1 cm⁻¹). The electronic structures of La₂ and Lu₂ are compared to those other lanthanide dimers and trends in the series are discussed. Received: 25 March 2002 / Accepted: 2 June 2002 / Published online: 21 August 2002     

Density functional computation of 55Mn NMR parameters

Gradient-corrected (GGA) and hybrid variants of density functional theory are used to compute geometries and ⁵⁵Mn chemical shifts of MnO₄ ⁻, Mn(CO)₆ ⁺, Mn₂(CO)₁₀, Mn(CO)₅ X [X=H, Cl, C(O)Me], Mn(CO)₅ ⁻, Mn(NO)₃(CO), and Mn(C₅H₅)L _x [L _x =(CO)₃, C₆H₆, C₇H₈]. For this set of compounds, substituent effects on δ(⁵⁵Mn) are significantly underestimated with the pure GGA functional BPW91 and are well described with hybrid functionals such as mPW1PW91 and, in particular, B3LYP. The computed data provide evidence for solvent and counterion effects on δ(⁵⁵Mn) of MnO₄ ⁻ and Mn(CO)₆ ⁺, respectively. The latter, in the presence of Cl⁻, may be described as highly fluxional Mn(CO)₅C(O)Cl. Electric field gradients computed with the B3LYP functional can be used for a qualitative rationalization of observed trends in ⁵⁵Mn NMR line widths. Electronic supplementary material to this paper can be obtained by using the Springer LINK server located at http://dx.doi.org/10.1007/s002140020338x Received: 17 January 2002 / Accepted: 13 March 2002 / Published online: 3 June 2002     

Signatures of reactive resonance: three case studies

 Recent advances in the study of short-lived reactive resonances are reviewed. Special emphasis is given to addressing the question of how reactive resonances might be observed in molecular beam scattering experiments. Three case studies are presented for simple triatomic systems that are believed to exhibit resonance phenomena: F + HD → D + HF, F + H₂ → H + HF, and H + HD → D + H₂. It is seen that reactive resonances do strongly influence collision observables, but in a different way for each case. At this stage, there does not appear to be a unique resonance signature that can be applied to all reactions. Received: 2 March 2002 / Accepted: 2 June 2002 / Published online: 17 October 2002 Acknowledgements. For their valuable input into the research discussed here, we are grateful to our collaborators, K. Liu, S.H. nk;Lee, F. Dong, D.E. Manolopoulos, D, Skouteris, X. Yang, S. Harich, D. Dai, and C.C. Wang. This work was supported by the National Science Foundation. Correspondence to: R. T. Skodje e-mail: skokje@spot.colorado.edu     

Structure and stability of gold-substituted diborane, boranes, and borohydride ions

Pseudopotential ab initio calculations were performed for species of the type BH _n (AuPH₃) _m ^k , where n+m=3 or 4, and the charge k is −2,…,+1. Some derivatives of these and diaurated diboranes were also studied. The structural data agree well with the available experimental evidence. Factors affecting the stability of these systems, including the role of aurophilic attraction, are discussed. The singly charged anions and the diaurated diboranes are predicted to be the most stable members of these series. Received: 22 January 1999 / Accepted: 2 June 1999 / Published online: 4 October 1999     

An atom-in-molecules and electron-localization-function study of the interaction between O2 and VxOy+/VxOy (x = 1, 2, y = 1–5) clusters

 The most stable structures of V _x O _y ⁺/V _x O _y (x=1, 2, y=1–5) clusters and their interaction with O₂ are determined by density functional calculations, the B3LYP functional with the 6-31G* basis set. The nature of the bonding of these clusters and the interaction with O₂ have been studied by topological analysis in the framework of both the atoms-in-molecules theory of Bader and the Becke–Edgecombe electron localization function. Bond critical points are localized by means of the analysis of the electron density gradient field, ∇ρ(r), and the electron localization function gradient field, ∇η(r). The values of the electron density properties, i.e., electron density, ρ(r), Laplacian of the electron density, ∇²ρ(r), and electron localization function, η(r), allow the nature of the bonds to be characterized, and linear correlation is found for the results obtained in both gradient fields. Vanadium-oxygen interactions are characterized as unshared-electron interactions, and linear correlation is observed between the electron density properties and the V–O bond length. In contrast, O₂ units involve typical shared-electron interactions, as for the dioxygen molecule. Four different vanadium–oxygen interactions are found and characterized: a molecular O₂ interaction, a peroxo O₂ ²⁻ interaction, a superoxo O₂ ⁻ interaction and a side-on O₂ ⁻ interaction. Received: 15 October 2001 / Accepted: 30 January 2002 / Published online: 24 June 2002     

Reaction of Tetracopper(I)-Phosphonitocavitand with PhSeSiMe3: Synthesis and Structure of a Polyanionic Cluster [pyH]6[(CuCl)9(μ3-SePh)5(μ4-SePh)]

Treatment of tetracopper(I)-phosphonitocavitand [1·Cu₄(μ-Cl)₄(μ₄-Cl)] (2) (1 = tetraphosphonitocavitand [rccc-2,8,14,20-tetrakis-(iso-butyl)-phosphonitocavitand (C₄₄H₄₈O₈P₄Ph₄)]) with PhSeSiMe₃ in THF at low temperature afforded a novel polyanionic cluster [pyH]₆[(CuCl)₉(μ₃-SePh)₅(μ₄-SePh)] (4) as a major product along with a new tetracopper(I)-phosphonitocavitand (3) with a centered μ₃-Cl. Molecular structure of anionic cluster in 4 consists of six PhSe⁻ bridging ligands containing five μ₃-SePh and one exceptional μ₄-SePh bridging nine copper atoms, of which eight copper atoms have trigonal coordination geometry and the other has distorted tetrahedral geometry. Dedicated to Professor Han-Qin Liu on the occasion of his 70th birthday.     

Self-consistent reaction field calculation of solvent reorganization energy in electron transfer: a dipole-reaction field interaction model

Based on the continuum dielectric model, this work has established the relationship between the solvent reorganization energy of electron transfer (ET) and the equilibrium solvation free energy. The dipole-reaction field interaction model has been proposed to describe the electrostatic solute-solvent interaction. The self-consistent reaction field (SCRF) approach has been applied to the calculation of the solvent reorganization energy in self-exchange reactions. A series of redox couples, O₂/O⁻ ₂, NO/NO⁺, O₃/O⁻ ₃, N₃/N⁻ ₃, NO₂/NO⁺ ₂, CO₂/CO⁻ ₂, SO₂/SO⁻ ₂, and ClO₂/ClO⁻ ₂, as well as (CH₂)₂C-(-CH₂-) _n -C(CH₂)₂ (n=1 ∼ 3) model systems have been investigated using ab initio calculation. For these ET systems, solvent reorganization energies have been estimated. Comparisons between our single-sphere approximation and the Marcus two-sphere model have also been made. For the inner reorganization energies of inorganic redox couples, errors are found not larger than 15% when comparing our SCRF results with those obtained from the experimental estimation. While for the (CH₂)₂C–(–CH₂–) _n –C(CH₂)₂ (n=1 ∼ 3) systems, the results reveal that the solvent reorganization energy strongly depends on the bridge length due to the variation of the dipole moment of the ionic solute, and that solvent reorganization energies for different systems lead to slightly different two-sphere radii. Received: 19 April 2000 / Accepted: 6 July 2000 / Published online: 27 September 2000     

Structures and stability of N7+ and N7− clusters

Ab initio molecular orbital theory and density functional theory have been used to study nine isomers of N₇ ionic clusters with low spin at the HF/6-31G*, MP2/6-31G*, B3LYP/6-31G*, and B3LYP/6-311(+)G* levels of theory. All stationary points are examined with harmonic vibrational frequency analyses. Four N₇ ⁺ isomers and five N₇ ⁻ isomers are determined to be local minima or very close to the minima on their potential-energy hypersurfaces, respectively. For N₇ ⁺ and N₇ ⁻, the energetically low lying isomers are open-chain structures (C _{2 v} and C _{2 v} or C₂). The results are very similar to those of other known odd-number nitrogen ions, such as N₅ ⁺, N₉ ⁺, and N₉ ⁻, for which the open-chain structures are also the global minima. This research suggests that the N₇ ionic clusters are likely to be stable and to be potential high-energy-density materials if they could be synthesized. Received: 16 July 2001 / Accepted: 8 October 2001 / Published online: 21 January 2002