Exchange energy for two closed shells generated by a bare Coulomb potential energy −<Emphasis Type="Italic">Ze</Emphasis><Superscript>2</Superscript>/<Emphasis Type="Italic">r</Emphasis> in the limit of large <Emphasis Type="Italic">Z</Emphasis>, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze ²/r and (ii) the exchange energy density e_x(r){\varepsilon_x({\bf r})} . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential 2e_x(r)/n(r){2\varepsilon_x(r)/n(r)} , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.     

Reactivity of aromatic compounds toward diphenylcarbonyl oxide

The reactivity of organic compounds (PhH, PhMe, PhF, PhCl, PhOH, PhOEt, PhCHO, Ph₂CO, PhCN, Ph₂S, Ph₂SO, Ph₂SO₂, andp-Me₂C₆H₄) toward diphenylcarbonyl oxide Ph₂COO was characterized by thek ₃₃/k ₃₁ ratio, wherek ₃₃ andk ₃₁ are the rate constants for the reactions of Ph₂COO with the arene and diphenyldiazomethane Ph₂CN₂, respectively. The values ofk ₃₃/k ₃₁ vary from 2.6·10⁻³ (PhCN) to 0.65 (Ph₂S) (70°C, MeCN). The reaction is preceded by formation of a complex with charge transfer from a substrate to Ph₂COO. In the reactions with aromatic substances (except for Ph₂SO, PhCHO, and Ph₂CO), carbonyl oxide behaves as an electrophile. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2197–2201, November, 1998.     

Analytical Hartree-Fock electron densities for singly charged cations and anions

The Hartree-Fock (HF) electron density has an important property that it is identical to the unknown exact density to the first order in the perturbation theory. We generate the spherically averaged HF electron density ρ(r) by using the numerical HF method for the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground state. The resultant density is then accurately fitted into an analytical function F(r), which is expressed by a linear combination of basis functions r ⁿⁱ exp(−ζ _i r). The present analytical approximation F(r) has the following properties: (1) F(r) is nonnegative, (2) F(r) is normalized, (3) F(r) reproduces the HF moments <r ^k > (k=−2 to +6), (4) F(0) is equal to ρ(0), (5) F ^′(0) satisfies the cusp condition and (6) F(r) has the correct exponential decay in the long-range asymptotic region. The present results together with our previous ones for neutral atoms provide a compilation of accurate analytical approximations of the HF electron densities for all the neutral and singly charged atoms with the number of electrons N≤54. Received: 11 July 1997 / Accepted: 27 August 1997     

The influence of substituents on the kinetics of the reaction of carbonyl oxides with benzaldehyde

The reactivity of carbonyl oxides toward benzaldehyde was characterized by thek ₃₃/k ₃₃ ratio, wherek ₃₃ andk ₃₁ are the rate constants of the reactions of RCOO with PhCHO and diphenyldiazomethane Ph₂CN₂, respectively. Thek ₃₃/k ₃₁ ratios obtained at 60°C in acetonitrile range from 0.61·10⁻² (m-BrPh₂CN₂) to 20·10⁻² (Ph₂MeCHO). The reactions are probably preceded by the formation of a charge-transfer complex (CTC) with charge transfer from aldehyde to RCOO. The carbonyl oxide reacts with aldehydes by both the nucleophilic pathway (at the C atom of the—CHO group to form 1,2,4-trioxolane) and electrophilic pathway (by the attack at the aromatic ring with the intermediate formation of CTC). In the latter case, either 1,2,4-trioxolane or oxidation products of the phenyl ring are formed. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 650–654, April, 2000.     

Electronic effects in the reaction of diphenylcarbonyl oxide with aldehydes

The reactivity of 14 aldehydes with diphenylcarbonyl oxide Ph₂COO was characterized by thek ₃₃/k ₃₁ ratio. The values ofk ₃₃/k ₃₁ vary from 1.3·10⁻² (C₆F₅CHO) to 1.0 (p-Me₂N-PhCHO), 70 °C, acetonitrile as the solvent. A charge transfer complex (CTC) was suggested to be primarily formed during the reaction. The electronic effects of substituents in the reaction were analyzed using the published data. Carbonyl oxide reacts with aldehydes as a nucleophile (at the carbon atom of the −CHO fragment to form 1,2,4-trioxolane) and also as an electrophile (at the aromatic ring with the intermediate formation of CTC). The latter is transformed into either 1,2,4-trioxolane or the products of oxidation of the phenyl ring. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1090–1096, June, 1999.     

Equation of motion of the correlated first-order density matrix for the ground-state of the Hookean atom with two electrons

The ground-state wave function Ψ for a given force constant k = 1/4 a.u. of the two-electron Hookean atom is known in exact analytical form. Here the corresponding first-order density matrix γ(r, r′) is studied, particular attention being focussed on its equation of motion. The exact form which results from the known Ψ is displayed, and given a physical interpretation. Harmonic confined model two-electron atoms with arbitrary interaction u(r ₁₂) are also briefly referred to in the present context.     

Analytical Hartree–Fock electron densities for atoms He through Lr

The Hartree-Fock electron density has an important property that it is identical to the exact density to first order in the perturbation theory. For the neutral atoms from He (Z = 2) to Lr (Z = 103) in their ground state, we report an accurate analytical approximation F(r) to the spherically averaged electron densityρ(r) obtained by the numerical Hartree-Fock method. The present density functionF(r) is expressed by a linear combination of reasonable number (not more than 30) of basis functionsr ⁿⁱ exp(- ζ _i r), and has the following properties: (i)F(r) is nonnegative, (ii)F(tr) is normalized, (iii)F(r) reproduces the Hartree-Fock moments <r ^k > (k = −2 to +6), (iv)F(0) is equal toρ(0), (v)F′(0) satisfies the cusp condition, and (vi)F(r) has the correct exponential decay in the long-range asymptotic region.     

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     

Electron-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇ V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.     

Bound states of a more general exponential screened Coulomb potential

An alternative approximation scheme has been used in solving the Schr?dinger equation to the more general case of exponential screened Coulomb potential, V(r) = −(a/r)[1 + (1 + br)e ^−2br ]. The bound state energies of the 1s, 2s and 3s-states, together with the ground state wave function are obtained analytically upto the second perturbation term.      

A theoretical study of the intramolecular interaction between proximal atoms in planar conformations of biphenyl and related systems

The nature of the interaction between proximal hydrogens in planar biphenyl has been recently a matter of debate as arguments in favor of and against the existence of “H–H” bonding have been recently put forward. This issue is addressed here through the study of both the electron density ρ(r) and the electron localization function (ELF) η(r) obtained in quantum calculations on molecular systems with F atoms replacing hydrogens in the moiety that presents this interaction. The analysis of geometries and properties of ρ(r) and η(r) at both planar and twisted equilibrium conformations of those systems along with biphenyl, permits to get information on this intramolecular interaction that is compared with the use of the traditional chemical concepts (steric hindrance and π-resonance effects) involved. It is shown that although the ELF gives information compatible with these classical terms, this does not preclude the existence of bonds between proximal atoms with features rather similar to those of well-established intramolecular hydrogen bonds.     

Effect of counterions on photoprocesses of thiacarbocyanine in a binary solvent blend

A model describing the effect of counterion X⁻ (X = Cl, I) on the deactivation kinetics of the S ₁ state of thiacarbocyanine Cy⁺X⁻ is presented. According to the model, the ion pair Cy⁺X⁻ in a binary solution is characterized by a distribution function f(r) over interatomic distances r, which depends on the composition of the mixture. The assumption of kinetically independent local states of the ion pair, which decay with the rate constants k _i(r)(i = 1–4 is the index of the decay channel), is made. The statistic analysis of the experimental data in terms of the model permitted us to find the functions f(r) and to estimate the parameters of the constants k _i(r).     

Statistical–mechanical models with separable many-body interactions: especially partition functions and thermodynamic consequences

We start from a classical statistical–mechanical theory for the internal energy in terms of three- and four-body correlation functions g ₃ and g ₄ for homogeneous atomic liquids like argon, with assumed central pair interactions f(r_ij){\phi(r_{ij})} . The importance of constructing the partition function (pf) as spatial integrals over g ₃, g ₄ and f{\phi} is stressed, together with some basic thermodynamic consequences of such a pf. A second classical example taken for two-body interactions is the so-called one-component plasma in two dimensions, for a particular coupling strength treated by Alastuey and Jancovici (J Phys (France) 42:1, 1981) and by Fantoni and Tellez (J Stat Phys 133:449, 2008). Again thermodynamic consequences provide a particular focus. Then quantum–mechanical assemblies are treated, again with separable many-body interactions. The example chosen is that of an N-body inhomogeneous extended system generated by a one-body potential energy V(r). The focus here is on the diagonal element of the canonical density matrix: the so-called Slater sum S(r, β), related to the pf by pf(b) = òS(r, b)d[(r)\vec]{{\rm pf}(\beta) = \int {S({\bf r}, \beta)}d\vec {r}}, β = (k _B T)⁻¹. The Slater sum S(r, β) can be related exactly, via a partial differential equation, to the one-body potential V(r), for specific choices of V which are cited. The work of Green (J Chem Phys 18:1123, 1950), is referred to for a generalization, but now perturbative, to two-body forces. Finally, to avoid perturbation series, the work concludes with some proposals to allow the treatment of extended assemblies in which regions of long-range ordered magnetism exist in the phase diagram. One of us (Z.D.Z.) has recently proposed a putative pf for a three-dimensional (3D) Ising model, based on two, as yet unproved, conjectures and has pointed out some important thermodynamic consequences of this pf. It would obviously be of considerable interest if such a pf, together with conjectures, could be rigorously proved.     

Pulse radiolysis study of the oxidation of the I<Superscript>−</Superscript> ions with the radical anions Br<Subscript>2</Subscript><Superscript>·−</Superscript> in an aqueous solution: formation and properties of the radical anion BrI<Superscript>·−</Superscript>

The radiation chemical redox transformations in solutions of bromides in the presence of minor additives of iodides were studied by pulse radiolysis. The change in the concentrations of the Br⁻ and I⁻ ions changes the ratio of the formed short-lived radical anions Br₂ ^·−, BrI^·−, and I₂ ^·−. The spectrum of the mixed radical anion BrI^·− contains a broad optical band at 370 nm with ɛ₃₇₀ = 9650 L mol⁻¹ cm⁻¹. The reduction potential of the BrI^·−/Br⁻, I⁻ pair is 1.25 V. The rate constants for the forward and backward reactions Br₂ ^·− + I⁻ ⇌ BrI^·− + Br⁻ are k _f = 4.3·10⁹ and k _r = 1.0·10⁵ L mol⁻¹ s⁻¹, respectively; for the reactions BrI^·− ⇌ Br⁻ + I^·, k _f = 5.7·10⁸ s⁻¹ and k _r = 1.0·10¹⁰ L mol⁻¹ s⁻¹. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1787–1792, September, 2008.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

Topological studies on IRC paths of X+H2→XH+H reactions

Topological properties of potential energy and electronic density distribution on five reaction paths X+H₂→XH+H (X=H, N, HN, H₂C, NC) are investigated at the level of UMP2/6–311G(d,p). It has been found that in the region of the reaction paths studied, where B(r_c)|_s>0 [B(r_c)|_s is the product of ρ(r_c) and ∇²ρ(r_c) at the point of reaction process, i.e., B(r_c)|_s=ρ(r_c)∇² ρ(r_c)] is basically the same as the region of V′(s)<0[V′(s) is the second derivative of potential energy with respect to the reaction coordinate, i.e., V′(s)=d²V/ds²], and the point with maximum B(r_c)|_s is almost coincident with the point of minimum V′(s). It can be concluded from the calculated results that there is a good correlation between the topological properties of potential energy and electronic density distribution along the reaction path. The structure transition state of such collinear reactions may be determined by topological analysis of electronic density. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1167–1174     

Influence of the linking chain length on the recombination kinetics of covalently bonded triplet radical pairs and magnetic field effects

Nanosecond laser flash photolysis technique is used to study the formation and decay kinetics of covalently linked triplet radical pairs (RP) formed after photoinduced electron transfer in the series of 21 zinc porphyrin—chain—viologen (Pph—Sp_n—Vi²⁺) dyads, where the number of atoms (n) in the chain increases from 2 to 138. In poorly viscous polar solvents (acetone, CHCl₃—CH₃OH (1 : 1) mixture), the dependence of the rate constant of RP formation on n can be described by the equation k _e = k _e ⁰ n ^–a at k _e ⁰ = 2.95·10⁸ s^–1 anda = 0.8. In the zero magnetic field, the RP recombination rate constant (k _r(B = 0)) is significantly lower than k _e and ranges from 0.7·10⁶ to 8·10⁶ s^–1. The dependence of k _r(B = 0) on n is extreme. The dependence k _r(B = 0) reaches a maximum at n = 20. In the strong magnetic field (B = 0.21 T), the significant retardation of triplet RP recombination is observed. The chain length has an insignificant effect on k _r(B = 0.21 T), which ranges from 0.3·10⁶ to 0.9·10⁶ s^–1. The regularities found are discussed in terms of the interplay of molecular and spin dynamics.     

DFT studies and AIM analysis of AlN-polycycles

The structure and properties of AlN-polycycles were studied by DFT (density functional theory) method. The results of calculations were obtained at B3LYP/6-311G(d, p) level on model species. Topological parameters such as electron density, its Laplacian, kinetic electron energy density, potential electron energy density, and total electron energy density at the ring critical points (RCP) from Bader’s ‘Atoms in molecules’ (AIM) theory were analyzed in detail. These results indicate a good correlation between ρ(3, +1), G(r), H(r), and V(r) averaged values and hardness of AlN-polycycles. The aromaticity of all molecules has been studied by nucleus-independent chemical shift. There is a linear correlation between ΣNICS(0.0)_molecule values and polarizability.