Spatial shape of electron delocalization: structure of the laplacian of the negative exchange-correlation density

The Laplacian of the negative exchange-correlation density (with respect to coordinate r(2)), nabla<(r)2>(2)[-Gamma(sigma1)(sigma2)(XC) (r(1),r(2))] = nabla(r)2(2)X(sigma1)(sigma2)(r(1),r(2)), is proposed as an instrument for the analysis of electron delocalization in real space. It determines local concentrations in the amount of electrons that are delocalized from a reference point r(1) over space. Integration of the reference coordinate r(1) over an atomic basin Omega(n) gives the function nabla(2)X(sigma1)(sigma2)(Omega(n);r), which contains detailed information about the spatial shape of the delocalization that originates from an atom in a molecule. Its isosurface representations are richly structured and resemble molecular orbitals in their complexity and partly also in their shape. The sum over all nabla(2)X(sigma1)(sigma2)(Omega(n);r) functions of a molecule equals the Laplacian of the electron density nabla(2)rho(r), for which it provides a meaningful partitioning into atomic contributions.     

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.     

Deuterium isotope effect on the atomic alignment dependence in the reaction of oriented Ar (3P2) with (CH3CN)2 and (CD3CN)2 dimers

The effect of atomic alignment on CN (B2Sigma+) formation has been studied in the reaction of oriented Ar (3P2) with (CX3CN)2 (X = H, D). The reaction cross-section for each magnetic M'(J) substate in the collision frame sigma|M'(J)|(H(D),d) relative to the cross-section sigma0(H,m) in the CH(3)CN reaction was determined to be sigma0(H,d)/sigma|1|(H,d)/sigma|2|(H,d)/sigma0(D,d)/sigma|1|(D,d):/sigma|2|(D,d)= 0.87/1.00/0.98/1.58/1.93/1.78. A notable deuterium isotope effect was observed. In contrast with the monomer reactions, a significant decrease of sigma0(H(D),d) relative to the other cross-sections of sigma|M'(J)|(H(D),d) was observed.     

Conformational structure of gaseous 3-chloropropanoyl chloride by electron diffraction, normal coordinate analysis, and ab initio molecular orbital, and density functional theory calculations

The molecular and conformational structures of 3-chloropropanoyl chloride (CH(2)Cl-CH(2)-C(=O)Cl) have been studied by using gas-phase electron diffraction (GED) data obtained at 22 degrees C (295 K) and ab initio molecular orbital (MO) and density functional theory (DFT) calculations up to the levels of MP4(SDQ) and B3LYP using larger basis sets. Normal coordinate calculations (NCA) taking into account nonlinear vibrational effects were also used in the analyses. The title compound may have up to four low-energy conformers in the gas phase, labeled according to the position of each of the two chlorine atoms in relation to the CCC propanoyl backbone, labeling the carbonyl chlorine torsion angle first: AA, AG, GG, and GA; where A is anti (ideal C-C-C-Cl torsion angle of approximately 180 degrees) and G is gauche (ideal C-C-C-Cl torsion angle of approximately 60 degrees). It has been judged from the experimental GED data and the theoretical calculations, as well as from previously published infrared (IR) studies on the molecule in both the liquid phase and in argon-trapped matrices at 10 K, that the gas phase consists of a mixture of at least three conformers: AA (most stable), AG, and GG, with the possibility of a smaller contribution (<10%) from the higher-energy GA form. The GA conformer cannot be ruled out by the GED experimental data. Relevant structural parameter values obtained from the GED least-squares refinements, with calculated ab initio MO MP2/6-31+G(2d,p) values used as constraints, were as follows (AA values with estimated 2sigma uncertainties): Bond lengths (r(h1)): r(C-C(=O)) = 1.505(4) A, r(C-CH(2)Cl) = 1.520(4) A, r(C=O) = 1.197(4) A, r(C(=O)-Cl) = 1.789(3) A, and r(C-Cl) = 1.782(3) A. Bond angles (angle(h1)): angle CCC = 111.5(11) degrees , angle CCO = 127.0(5) degrees, angle CC(O)Cl = 112.5(3) degrees, and angle CCCl = 110.3(3) degrees. Torsion angles (phi(C-C) = phi(ClCCC)): for AA, phi(1)(C-C(O)) = phi(2)(C-CH(2)Cl) = 180 degrees (assumed for true C(s) symmetry); for AG, phi(1)(C-C(O)) = -140(5) degrees, phi(2)(C-CH(2)Cl) = 76(13) degrees; for GG, phi(1)(C-C(O)) = 46(8) degrees, phi(2)(C-CH(2)Cl) = 77(14) degrees; for GA, phi(1)(C-C(O)) = 67.9 degrees (assumed), phi(2)(C-CH(2)Cl) = 177.8 degrees (assumed). The non-AA conformers all have chiral C(1) symmetry with twice the statistical weight (multiplicity) of C(s). The MP2/6-31+G(2d,p) calculated composition (%) based on the zero-point energy (ZPE) corrected energy differences, and the statistical weights for conformers: AA/AG/GG/GA = 28/35/28/9 was assumed in the final GED refinement. The more recent literature concerning the title molecule, as well as for several related molecules, has been examined and a survey has been attempted in the present article. The new experimental results for 3-chloropropanoyl chloride are discussed and compared with the previously published findings.     

Significant change of alignment effect by dimer formation in the dissociative energy transfer reaction of Ar(3P2)+(N2O)n and (H2O)n

An alignment effect in the dissociative energy transfer reaction of Ar((3)P(2))+(X(2)O)(n)(X=N,H) was directly measured using an oriented Ar((3)P(2),M(J)=2) beam. The chemiluminescence intensity of N(2)(B,(3)Pi(g)) for (N(2)O)(n) and OH(A,(2)Sigma(+)) for (H(2)O)(n) was measured as a function of the magnetic orientation field direction in the collision frame. The relative reaction cross section for each magnetic substate in the collision frame, sigma(M(J) (') ), was determined. In both the reaction systems, it is observed that the dimer formation significantly enhances the alignment effect and decreases the reactivity, especially for sigma|1| and sigma|2|. A significant contribution of rank 4 moment is recognized in the dimer reaction.     

Determination of the chiral indices (n,m) of carbon nanotubes by electron diffraction

The atomic structure of a carbon nanotube can be defined by the chiral indices, (n,m), that specify its perimeter vector (chiral vector), with which the diameter and helicity are also determined. The fine electron beam available in a modern Transmission Electron Microscope (TEM) offers a unique and powerful probe to reveal the atomic structure of individual nanotubes. This article covers two aspects related to the use of the electron probe in the TEM for the study of carbon nanotubes: (i) to express the electron diffraction intensity distribution in the electron diffraction patterns of carbon nanotubes and (ii) to obtain the chiral indices (n,m) of carbon nanotubes from their electron diffraction patterns. For a nanotube of given chiral indices (n,m), the electron scattering amplitude from the carbon nanotube can be expressed analytically in closed form using the helical diffraction theory, from which its electron diffraction pattern can be calculated and understood. The reverse problem, i.e., assignment of the chiral indices (n,m) of a carbon nanotube from its electron diffraction pattern, is approached from the relationship between the electron diffraction intensity distribution and the chiral indices (n,m). The first method is to obtain indiscriminately the chiral indices (n,m) by reading directly the intensity distribution on the three principal layer lines, l(1), l(2), and l(3), which have intensities proportional to the square of the Bessel functions of orders m, n, and n + m: I(l1) proportional, variant |J(m) (pidR)|(2), I(l2) proportional, variant |J(n) (pidR)|(2), and I(l3) proportional, variant |J(n+m) (pidR)|(2). The second method is to obtain and use the ratio of the indices n/m = (2D(1)-D(2))/(2D(2)-D(1)) in which D(1) and D(2) are the spacings of principal layer lines l(1) and l(2), respectively. Examples of using these methods are also illustrated in the determination of chiral indices of isolated individual single-walled carbon nanotubes, a bundle of single-walled carbon nanotubes, and multi-walled carbon nanotubes.     

Hammett equation and generalized Pauling's electronegativity equation

Substituent interaction energy (SIE) was defined as the energy change of the isodesmic reaction X-spacer-Y + H-spacer-H --> X-spacer-H + H-spacer-Y. It was found that this SIE followed a simple equation, SIE(X,Y) = -ksigma(X)sigma(Y), where k was a constant dependent on the system and sigma was a certain scale of electronic substituent constant. It was demonstrated that the equation was applicable to disubstituted bicyclo[2.2.2]octanes, benzenes, ethylenes, butadienes, and hexatrienes. It was also demonstrated that Hammett's equation was a derivative form of the above equation. Furthermore, it was found that when spacer = nil the above equation was mathematically the same as Pauling's electronegativity equation. Thus it was shown that Hammett's equation was a derivative form of the generalized Pauling's electronegativity equation and that a generalized Pauling's electronegativity equation could be utilized for diverse X-spacer-Y systems. In addition, the total electronic substituent effects were successfully separated into field/inductive and resonance effects in the equation SIE(X,Y) = -k(1)F(X)F(Y) - k(2)R(X)R(Y) - k(3)(F(X)R(Y) + R(X)F(Y)). The existence of the cross term (i.e., F(X)R(Y) and R(X)F(Y)) suggested that the field/inductive effect was not orthogonal to the resonance effect because the field/inductive effect from one substituent interacted with the resonance effect from the other. Further studies on multi-substituted systems suggested that the electronic substituent effects should be pairwise and additive. Hence, the SIE in a multi-substituted system could be described using the equation SIE(X1, X2, ..., Xn) = Sigma(n-1)(i=1)Sigma(n)(j=i+1)k(ij)sigma(X)isigma(X)j.     

Theoretical investigation of the large nonlinear optical properties of (HCN)n clusters with Li atom

Two new classes of (HCN)(n)...Li and Li...(HCN)(n) (n = 1, 2, 3) clusters with the electride characteristic are formed in theory by the metal Li atom attaching to the (HCN)(n) (n = 1, 2, 3) clusters. Because of the interaction between the Li atom and the (HCN)(n) part, the 2s valence electron of the Li atom becomes a loosely bound excess electron. Our high-level ab initio calculations show that these new clusters with the excess electron have large first hyperpolarizabilities, for example, beta(0) = -15,258 au for (HCN)...Li and beta(0) = -3401 au for Li...(HCN) at the QCISD/6-311++G(3df,3pd) level (only beta(0) = -2.8 au for HCN monomer(26)). Obviously, the excess electron from the Li atom plays a crucial role in the large first hyperpolarizabilities of these clusters. The beta(0) value of (HCN)(n)...Li (beta(0) > 10(4) au, from sigma --> pi* transition) is larger than that of Li...(HCN)(n) (beta(0) > 10(3) au, from sigma --> sigma* transition) for n = 1, 2, or 3. In addition, two interesting rules have been observed. They are that |beta(0)| decreases with lengthening of the HCN chain for (HCN)(n)...Li clusters and that |beta(0)| increases as n increases for Li...(HCN)(n) clusters. In this paper, we discuss two classes of clusters that are highly similar to the electride structure model, of which the structural characteristics are that alkali metal atoms ionize to form cations and trapped electrons under the action of other polar molecules. Thus, the investigation on the large first hyperpolarizabilities of (HCN)(n)...Li and Li...(HCN)(n) (n = 1, 2, 3) may prompt one to study the unusual nonlinear optical responses of some electrides.     

Atoms-in-molecules analysis of extended hypervalent five-center, six-electron (5c-6e) C(2)Z(2)O interactions at the 1,8,9-positions of anthraquinone and 9-methoxyanthracene systems

To clarify the nature of five-center, six-electron (5c-6e) C(2)Z(2)O interactions, atoms-in-molecules (AIM) analysis has been applied to an anthraquinone, 1,8-(MeZ)(2)ATQ (1 (Z=Se), 2 (Z=S), and 3 (Z=O)), and a 9-methoxyanthracene system, 9-MeO-1,8-(MeZ)(2)ATC (4 (Z=Se), 5 (Z=S), and 6 (Z=O)), as well as 1-(MeZ)ATQ (7 (Z=Se), 8 (Z=S), and 9 (Z=O)) and 9-MeO-1-(MeZ)ATC (10 (Z=Se), 11 (Z=S), and 12 (Z=O)). The total electronic energy density (H(b)(r(c))) at the bond critical points (BCPs), an appropriate index for weak interactions, has been examined for 5c-6e C(2)Z(2)O and 3c-4e CZO interactions of the n(p)(O)sigma*(Z--C) type in 1-12. Some hydrogen-bonded adducts were also re-examined for convenience of comparison. The total electronic energy densities varied in the following order: OO (3: H(b)(r(c))=0.0028 au)=OO (6: 0.0028 au)>OO (9: 0.0025 au)> or =NNHF (0.0024 au)> or =OO (12: 0.0023 au)>H(2)OHOH (0.0015 au)>SO (8: 0.0013 au)=SO (2: 0.0013 au)> or =SO (11: 0.0012 au)=SO (5: 0.0012 au)>HFHF (0.0008 au)=SeO (10: 0.0008 au)=SeO (4: 0.0008 au)> or =SeO (1: 0.0007 au)> or =SeO (7: 0.0006 au)>HCNHF (-0.0013 au). H(b)(r(c)) values for SO were predicted to be smaller than the hydrogen bond of H(2)OHOH and H(b)(r(c)) values for SeO are very close to or slightly smaller than that for HFHF in both the ATQ and 9-MeOATC systems. In the case of Z=Se and S, H(b)(r(c)) values for 5c-6e C(2)Z(2)O interactions are essentially equal to those for 3c-4e CZO if Z is the same. The results demonstrate that two n(p)(O)sigma*(Z--C) 3c-4e interactions effectively connect through the central n(p)(O) orbital to form the extended hypervalent 5c-6e system of the sigma*(C--Z)n(p)(O)sigma*(Z--C) type for Z=Se and S in both systems. Natural bond orbital (NBO) analysis revealed that n(s)(O) also contributes to some extent. The electron charge densities at the BCPs, NBO analysis, and the total energies calculated for 1-12, together with the structural changes in the PhSe derivatives, support the above discussion.     

Effects of molecular structure on the stability of a thermotropic liquid crystal. Gas electron diffraction study of the molecular structure of phenyl benzoate.

As a model of the core of molecules forming liquid crystals, the molecular structure of phenyl benzoate (Ph-C(=O)-O-Ph) at 409 K was determined by gas electron diffraction, and the relationship between the gas-phase structures of model compounds and the nematic-to-liquid transition temperatures was studied. Structural constraints were obtained from RHF/6-31G ab initio calculations. Vibrational mean amplitudes and shrinkage corrections were calculated from the harmonic force constants given by normal coordinate analysis. Thermal vibrations were treated as small-amplitude motions, except for the phenyl torsion, which was treated as a large-amplitude motion. The potential function for torsion was assumed to be V(phi(1),phi(2)) = V(12)(1 - cos 2phi(1))/2 + V(14)(1 - cos 4phi(1))/2 + V(22)(1 - cos 2phi(2))/2, where phi(1) and phi(2) denote the torsional angles around the C-Ph and O-Ph bonds, respectively. The potential constants (V(ij)()/kcal mol(-)(1)) and the principal structure parameters (r(g)/A, angle(alpha)/deg) with the estimated limits of error (3sigma) are as follows: V(12) = -1.3 (assumed); V(14) = -0.5(9); V(22) = 3.5(15); r(C=O) = 1.208(4); r(C(=O)-O) = 1.362(6); r(C(=O)-O) - r(O-C) = -0.044 (assumed); r(C(=O)-C) = 1.478(10); = 1.396(1); angleOCO = 124.2(13); angleO=CC = 127.3(12); angleCOC = 121.4(22); ( angleOCC(cis) - angleOCC(trans))/2 = 3.0(15); ( angleC(=O)CC(cis) - angleC(=O)CC(trans))/2 = 4.8(17), where < > means an average value and C-C(cis) and C-C(trans) bonds are cis and trans to the C(=O)-O bond, respectively. The torsional angle around the O-Ph bond was determined to be 64(+26,-12) degrees. An apparent correlation was found between the contributions of the cores to the clearing point of liquid crystals and the gas-phase structures of model compounds of the cores of mesogens, i.e., phenyl benzoate, trans-azobenzene (t-AB), N-benzylideneaniline, N-benzylideneaniline N-oxide (NBANO), trans-azoxybenzene (t-AXB), and trans-stilbene. The structures of t-AB, NBANO, and t-AXB have been obtained by our research group.     

Molecular structure and conformational composition of 2-chloro-1-phenylethanone, ClH2C-C(=O)Ph, obtained using gas-phase electron-diffraction data and results from theoretical calculations

The structure and conformation of 2-chloro-1-phenylethanone, ClH(2)C-C(=O)Ph (phenacyl chloride), have been determined by gas-phase electron diffraction (GED), augmented by results from ab initio molecular orbital calculations, employing the second-order M?ller-Plesset (MP2) level of theory and the 6-311+G(d) basis set. The molecules may exist as a mixture of different conformers with the C-Cl bond either syn (torsion angle phi = 0 degrees ) or gauche to the carbonyl bond. At 179 degrees C, the majority of the molecules (90 +/- 11%) have the gauche conformation (phi = 112(3) degrees). Torsion is also possible about the C-Ph single bond. Both experimental and theoretical data indicated, however, that the phenyl ring is coplanar or nearly coplanar with the carbonyl group. The results for the principal distances (r(g)) and angles (angle(alpha)) for the gauche conformer from a combined GED/ab initio study (with estimated 2sigma uncertainties) are the following: r(C-C)(phenyl) = 1.394(2) (average value) A, r(C(phenyl)-C(carbonyl)) = 1.484(5) A, r(C(carbonyl)-C(alkyl)) = 1.513(5) A, r(C-Cl) = 1.790(5) A, r(C=O) = 1.218(6) A, r(C-H)(phenyl) = 1.087(9) (average value) A, r(C-H)(alkyl) = 1.090(9) A (average value), angle C(phenyl)-C=O = 119.5(9) degrees, angle C(phenyl)-C(carbonyl)-C(alkyl) = 119.2(10) degrees, angle C-C-Cl = 109.8(12) degrees, angle C(2)-C(1)-C(carbonyl) = 122.8(15) degrees, angle C-C(alkyl)-H = 111.2 degrees (ab initio value).     

Simple systematization of vibrational excitation cross-section calculations for resonant electron-molecule scattering in the boomerang and impulse models

Vibrational excitation (nu(f)<--nu(i)) cross-sections sigma(nu(f)<--nu(i) )(E) in resonant e-N(2) and e-H(2) scattering are calculated from transition matrix elements T(nu(f),nu(i) )(E) obtained using Fourier transform of the cross correlation function , where psi(nu(i))(R,t) approximately =e(-iH(A(2))-(R)t/h phi(nu(i))(R) with time evolution under the influence of the resonance anionic Hamiltonian H(A(2) (-))(A(2) (-)=N(2)(-)/H(2) (-)) implemented using Lanczos and fast Fourier transforms. The target (A(2)) vibrational eigenfunctions phi(nu(i))(R) and phi(nu(f))(R) are calculated using Fourier grid Hamiltonian method applied to potential energy (PE) curves of the neutral target. Application of this simple systematization to calculate vibrational structure in e-N(2) and e-H(2) scattering cross-sections provides mechanistic insights into features underlying presence/absence of structure in e-N(2) and e-H(2) scattering cross-sections. The results obtained with approximate PE curves are in reasonable agreement with experimental/calculated cross-section profiles, and cross correlation functions provide a simple demarcation between the boomerang and impulse models.     

Electronic spectrum of UO2(2+) and [UO2Cl4]2- calculated with time-dependent density functional theory

The electronic spectra of UO(2) (2+) and [UO(2)Cl(4)](2-) are calculated with a recently proposed relativistic time-dependent density functional theory method based on the two-component zeroth-order regular approximation for the inclusion of spin-orbit coupling and a noncollinear exchange-correlation functional. All excitations out of the bonding sigma(u) (+) orbital into the nonbonding delta(u) or phi(u) orbitals for UO(2) (2+) and the corresponding excitations for [UO(2)Cl(4)](2-) are considered. Scalar relativistic vertical excitation energies are compared to values from previous calculations with the CASPT2 method. Two-component adiabatic excitation energies, U-O equilibrium distances, and symmetric stretching frequencies are compared to CASPT2 and combined configuration-interaction and spin-orbit coupling results, as well as to experimental data. The composition of the excited states in terms of the spin-orbit free states is analyzed. The results point to a significant effect of the chlorine ligands on the electronic spectrum, thereby confirming the CASPT2 results: The excitation energies are shifted and a different luminescent state is found.     

Geometric and electronic structure of the heme-peroxo-copper complex [(F8TPP)FeIII-(O22-)-CuII(TMPA)](ClO4)

The geometric and electronic structure of the untethered heme-peroxo-copper model complex [(F(8)TPP)Fe(III)-(O(2)(2)(-))-Cu(II)(TMPA)](ClO(4)) (1) has been investigated using Cu and Fe K-edge EXAFS spectroscopy and density functional theory calculations in order to describe its geometric and electronic structure. The Fe and Cu K-edge EXAFS data were fit with a Cu...Fe distance of approximately 3.72 A. Spin-unrestricted DFT calculations for the S(T) = 2 spin state were performed on [(P)Fe(III)-(O(2)(2)(-))-Cu(II)(TMPA)](+) as a model of 1. The peroxo unit is bound end-on to the copper, and side-on to the high-spin iron, for an overall mu-eta(1):eta(2) coordination mode. The calculated Cu...Fe distance is approximately 0.3 A longer than that observed experimentally. Reoptimization of [(P)Fe(III)-(O(2)(2)(-))-Cu(II)(TMPA)](+) with a 3.7 A Cu...Fe constrained distance results in a similar energy and structure that retains the overall mu-eta(1):eta(2)-peroxo coordination mode. The primary bonding interaction between the copper and the peroxide involves electron donation into the half-occupied Cu d(z)2 orbital from the peroxide pi(sigma) orbital. In the case of the Fe(III)-peroxide eta(2) bond, the two major components arise from the donor interactions of the peroxide pi*(sigma) and pi*(v) orbitals with the Fe d(xz) and d(xy) orbitals, which give rise to sigma and delta bonds, respectively. The pi*(sigma) interaction with both the half-occupied d(z)2 orbital on the copper (eta(1)) and the d(xz) orbital on the iron (eta(2)), provides an effective superexchange pathway for strong antiferromagnetic coupling between the metal centers.     

Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory

The conventional analysis of Perdew and Levy, and Sham and Schlu?ter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((?ρ(s)(r('),r))/?N)(v(s) ), at integers. (3) Consistent with the Perdew-Levy-Sham-Schlu?ter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons.     

Spectroscopy and bonding in side-on and end-on Cu2(S2) cores: comparison to peroxide analogues

Spectroscopic methods combined with density functional calculations were used to study the disulfide-Cu(II) bonding interactions in the side-on micro -eta(2):eta(2)-bridged Cu(2)(S(2)) complex, [[Cu(II)[HB(3,5-Pr(i)(2)pz)(3)]](2)(S(2))], and the end-on trans- micro -1,2-bridged Cu(2)(S(2)) complex, [[Cu(II)(TMPA)](2)(S(2))](2+), in correlation to their peroxide structural analogues. Resonance Raman shows weaker S-S bonds and stronger Cu-S bonds in the disulfide complexes relative to the O-O and Cu-O bonds in the peroxide analogues. The weaker S-S bonds come from the more limited interaction between the S 3p orbitals relative to that of the O 2s/p hybrid orbitals. The stronger Cu-S bonds result from the more covalent Cu-disulfide interactions relative to the Cu-peroxide interactions. This is consistent with the higher energy of the disulfide valence level relative to that of the peroxide. The ground states of the side-on Cu(2)(S(2))/Cu(2)(O(2)) complexes are more covalent than those of the end-on Cu(2)(S(2))/Cu(2)(O(2)) complexes. This derives from the larger sigma-donor interactions in the side-on micro -eta(2):eta(2) structure, which has four Cu-disulfide/peroxide bonds, relative to the end-on trans- micro -1,2 structure, which forms two bonds to the Cu. The larger disulfide/peroxide sigma-donor interactions in the side-on complexes are reflected in their more intense higher energy disulfide/peroxide to Cu charge transfer transitions in the absorption spectra. The large ground-state covalencies of the side-on complexes result in significant nuclear distortions in the ligand-to-metal charge transfer excited states, which give rise to the strong resonance Raman enhancements of the metal-ligand and intraligand vibrations. Particularly, the large covalency of the Cu-disulfide interaction in the side-on Cu(2)(S(2)) complex leads to a different rR enhancement profile, relative to the peroxide analogues, reflecting a S-S bond distortion in the opposite directions in the disulfide/peroxide pi(sigma) to Cu charge transfer excited states. A ligand sigma back-bonding interaction exists only in the side-on complexes, and there is more sigma mixing in the side-on Cu(2)(S(2)) complex than in the side-on Cu(2)(O(2)) complex. This sigma back-bonding is shown to significantly weaken the S-S/O-O bond relative to that of the analogous end-on complex, leading to the low nu(S)(-)(S)/nu(O)(-)(O) vibrational frequencies observed in the resonance Raman spectra of the side-on complexes.     

Finite-size scaling analysis on the phase transition of a ferromagnetic polymer chain model

The finite-size scaling analysis method is applied to study the phase transition of a self-avoiding walking polymer chain with spatial nearest-neighbor ferromagnetic Ising interaction on the simple cubic lattice. Assuming the scaling M2(T,n) = n(-2beta/nu)[phi0 + phi1n(1/nu)(T-T(c)) + O(n(2/nu)(T-T(c))2)] with the square magnetization M2 as the order parameter and the chain length n as the size, we estimate the second-order phase-transition temperature T(c) = 1.784 J/k(B) and critical exponents 2beta/nu approximately 0.668 and nu approximately 1.0. The self-diffusion constant and the chain dimensions (R2) and (S2) do not obey such a scaling law.     

Physical properties of soft repulsive particle fluids

Molecular dynamics computer simulation has been applied to inverse power or soft-sphere fluids, in which the particles interact through the soft-sphere pair potential, phi(r) = epsilon(sigma/r)(n), where n measures the steepness or stiffness of the potential, and epsilon and sigma are a characteristic energy and distance, respectively. The focus of the study is on very soft particles with n values down to 4 considered, at densities up to and along the fluid-solid co-existence density. It is shown that in the soft-particle limit the local structure is dominated by the lengthscale associated with the average nearest neighbour distance of a random structure, which is proportional, variantrho(-1/3) and increasingly only very weakly dependent on n. This scaling is also manifest in the behaviour of the average energy per particle with density. The self-diffusion coefficient and shear viscosity are computed along the fluid-solid co-existence line as a function of n, for the first time. The product Deta(s) steadily increases with softness for n < 10, whereas the modified Stokes-Einstein relationship of Zwanzig, Deta(s)/rho(1/3), where rho is the number density, is within statistics constant over the same softness range. This is consistent with our observation that the static properties are determined by a characteristic lengthscale (i.e., l) which is proportional, variantrho(-1/3) in the soft-particle limit. The high frequency elastic moduli of these fluids are examined, which reveals that the mechanical properties become more 'rubbery' as the particles get softer.     

The interaction of water and dibromine in the gas phase: an investigation of the complex H(2)O...Br(2) by rotational spectroscopy and ab initio calculations

The ground-state rotational spectra of eight isotopomers of a complex formed by water and dibromine in the gas phase were observed by pulsed-jet, Fourier transform microwave spectroscopy. The spectroscopic constants B(0), C(0), delta(J), delta(JK), chi(aa)(Br(x)) (x=i for inner, o for outer), [chi(bb)(Br(x))-chi(cc)(Br(x))] and M(bb)(Br(x)) were determined for H(2)O...(79)Br(79)Br, H(2)O...(81)Br(79)Br, H(2)O...(79)Br(81)Br, H(2)O...(81)Br(81)Br, D(2)O...(79)Br(81)Br and D(2)O...(81)Br(81)Br. For the isotopomers HDO...(79)Br(81)Br and HDO...(81)Br(81)Br, only (B(0) + C(0))/2, delta(J), the chi(aa)(Br(x)) and M(bb)(Br(x)) were determinable. The spectroscopic constants were interpreted on the basis of several models of the complex to give information about its geometry, binding strength and the extent of electronic rearrangement on complex formation. The molecule H(2)O...Br(2) has C(s) symmetry with a pyramidal configuration at O. The zero-point effective quantities r(O...Br(i))=2.8506(1) A and phi(0)=46.8(1), where phi is the angle between the C(2) axis of H(2)O and the O...Br-Br internuclear axis, were obtained under the assumption of monomer geometries unchanged by complexation. Ab initio calculations, carried out at the aug-cc-pVDZ/MP2 level of theory, gave the equilibrium values r(e)(O...Br(i))=2.7908 A and phi(e)=45.7 degrees and confirmed the collinearity of the O...Br-Br nuclei. The potential energy function V(phi), also determined ab initio, showed that the wavenumber required for inversion of the configuration at O in the zero-point state is only 9 cm(-1). By interpreting the Br nuclear quadrupole coupling constants, the fractions delta(O-->Br(i))=0.004(5) and delta (Br(i)-->Br(o))=0.050(2) of an electron were determined to be transferred from O to Br(i) and Br(i) to Br(o), respectively, when the complex is formed. The complex is relatively weak, as indicated by the small value k(sigma)=9.8(2) N m(-1) of the intermolecular stretching force constant obtained from delta(J). A comparison of the properties, similarly determined, of H(2)O...F(2), H(2)O...Cl(2), H(2)O...Br(2), H(2)O...BrCl, H(2)O...ClF and H(2)O...ICl is presented.     

FT-IR spectroscopic investigation of some Hofmann type complexes: M(1-phenylpiperazine)2Ni(CN)4 (M=Ni, Co, Cd, Pd or Mn)

New Hofmann type complexes in the form of M(pp)(2)Ni(CN)(4) (where pp=1-phenylpiperazine and M=Ni, Co, Cd, Pd or Mn) have been prepared in powder form and their infrared spectra have been reported in the range of (4000-400) cm(-1). The thermal behaviours of these complexes have been investigated by differential thermal analysis (DTA) and thermo-gravimetric analysis (TGA). Ni(pp)(2)Ni(CN)(4) complex has been examined via transmission electron microscope (TEM). The results suggest that these compounds are similar in structure to Hofmann type complexes and their structures consist of polymeric layers |M-Ni(CN)(4)|(infinity) with the pp molecule bounded to the metal atom (M).