Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal–vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn–Sham density functional theory (KS‐DFT) exchange‐correlation potential energy v_xc(r) in terms of the irreducible electron self‐energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange‐correlation potential energy is derived to be v_xc(z → ∞) = ?α_KS,xc/z, and its exchange and correlation components to be v_x(z → ∞) = ?α_KS,x/z and v_c(z → ∞) = ?α_KS,c/z, respectively. The analytical expressions for the coefficients α_KS,xc and α_KS,x show them to be dependent on the bulk‐metal Wigner–Seitz radius and the barrier height at the surface. The coefficient α_KS,c = 1/4 is determined in the plasmon‐pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of v_xc(z) in the vacuum region is image‐potential‐like but not the commonly accepted one of ?1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q‐DFT) shows that both the Pauli W_x(z → ∞) and lowest‐order correlation‐kinetic W(z → ∞) components of the exchange potential energy v_x(z → ∞), and the Coulomb W_c(z → ∞) and higher‐order correlation‐kinetic components of the correlation potential energy v_c(z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation‐kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS‐DFT is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Density functional theory study of geometry and stability of small Zrn (n = 2–10) clusters

Geometric structures, electronic properties, and stabilities of small Zr_n and Zr (n = 2–10) clusters have been investigated using density functional theory with effective core potential LanL2DZ basis set. For both neutral and charged systems, several isomers and different multiplicities were studied to determine the lowest energy structures. Many most stable states with high symmetry were found for small Zr_n clusters. The most stable structures and symmetries of Zr clusters are the same as the neutral ones except n = 4 and 7. We found that the clusters with n > 3 possess highly compact structures. The clusters are inclined to form the caged‐liked geometry containing pentagonal structures for n > 8, which is in favor of energy. From the formation energy and second‐order energy difference, we obtained that 2‐, 5‐, 7‐atoms of neutral and 4‐, 7‐atoms cationic clusters are the magic numbers. Furthermore, the highest occupied molecular orbital‐lowest unoccupied molecular orbital gaps display that the Zr₃, Zr₆, Zr, and Zr are more stable in chemical stability. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Theoretical study of one‐electron bonds in a series of high‐spin lithium–beryllium–hydrogen clusters: “Valence shell single‐electron repulsion” rule and electron localization function analysis

A series of high‐spin clusters containing Li, H, and Be in which the valence shell molecular orbitals (MOs) are occupied by a single electron has been characterized using ab initio and density functional theory (DFT) calculations. A first type (⁵Li₂, ⁿ⁺¹LiH_n+ (n = 2–5), ⁸Li₂H) possesses only one electron pair in the lowest MO, with bond energies of ～3 kcal/mol. In a second type, all the MOs are singly occupied, which results in highly excited species that nevertheless constitute a marked minimum on their potential energy surface (PES). Thus, it is possible to design a larger panel of structures (⁸LiBe, ⁷Li₂, ⁸Li, ⁴LiH⁺, ⁶BeH, ⁿ⁺³LiH (n = 3, 4), ⁿ⁺²LiH (n = 4–6), ⁸Li₂H, ⁹Li₂H, ²²Li₃Be₃ and ²²Li₆H), single‐electron equivalent to doublet “classical” molecules ranging from CO to C₆H₆. The geometrical structure is studied in relation to the valence shell single‐electron repulsion (VSEPR) theory and the electron localization function (ELF) is analyzed, revealing a striking similarity with the corresponding structure having paired electrons. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Reaction mechanism of cysteine proteases model compound HSH with diketone inhibitor PhCOCOCH3−nXn, (X = F,Cl, n = 0, 1, 2)

Caspases are a family of cysteine proteases, which play a crucial role in apoptosis and inflammation. The reaction mechanisms involving the cysteine proteases model compound HSH with diketone (PhCOCOCH_3‐nX_n, (X = F, Cl, n = 0, 1, 2) substrate have been studied using B3LYP/6‐311+G* level of density functional theory method. The harmonic vibrational frequencies were calculated at the same level of theory used for the characterization of stationary points and zero‐point vibrational energy corrections. The condensed Fukui functions have been calculated to find the favorable reactive site for the electrophilic (f), nucleophilic (f), and radical (f) attacks in the reactants. The transition states were connected with reactants, intermediate, and products, and the minimum energy paths have been confirmed through intrinsic reaction coordinate calculation. The potential energy barrier between each step of the reactions has been calculated to find the most favorable reaction path. The binding nature of cysteine model compound with diketone substrate has been studied through the interaction energies, bond lengths, electron density, natural bond orbital, and atoms in molecules theory analysis. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Ab initio characterization of size dependence of electronic spectra for linear anionic carbon clusters C(n) (-) (n = 4-17)

In this article, we determine the ground‐state equilibrium geometries of the linear anionic carbon clusters C (n = 4–17) by means of the density functional theory B3LYP, CAM‐B3LYP, and coupled cluster CCSD(T) calculations, as well as their electronic spectra obtained by the multireference second‐order perturbation theory CASPT2 method. These studies indicate that these linear anions possess doublet ²∏_g or ²∏_u ground state, and the even‐numbered clusters are generally acetylenic, whereas the odd‐numbered ones are essentially cumulenic. The energy differences, electron affinities, and incremental binding energies of C chains all exhibit a notable tread of parity alternation, with n‐even chains being more stable than n‐odd ones. In addition, the predicted vertical excitation energies from the ground state to four low‐lying excited states are in reasonably good agreement with the available experimental observations, and the calculations for the higher excited electronic transitions can provide accurate information for the experimentalists and spectroscopists. Interestingly, the absorption wavelengths of the 1²∏_u/g ← X²∏_g/u transitions of the n‐even clusters show a nonlinear trend of exponential growth, whereas those of the n‐odd counterparts are found to obey a linear relationship as a function of the chain size, as shown experimentally. Moreover, the absorption wavelengths of the transitions to the higher excited states of C series have the similar linear size dependence as well. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

An Ab initio theoretical investigation on the geometrical and electronic structures of BAun−/0 (n = 1–4) clusters

An ab initio theoretical investigation on the geometrical and electronic structures and photoelectron spectroscopies (PES) of BAu_n^?/0 (n = 1–4) auroboranes has been performed in this work. Density functional theory and coupled cluster method (CCSD(T)) calculations indicate that BAu (n = 1–4) clusters with n‐Au terminals possess similar geometrical structures and bonding patterns with the corresponding boron hydrides BH. The PES spectra of BAu (n = 1–4) anions have been simulated computationally to facilitate their future experimental characterizations. In this series, the T_d BAu anion appears to be unique and particularly interesting: it possesses a perfect tetrahedral geometry and has the highest vertical electron detachment energy (VDE = 3.69 eV), largest HOMO‐LUMO gap (ΔE_gap = 3.0 eV), and the highest first excitation energy (E_ex = 2.18 eV). The possibility to use the tetrahedral BAu unit as the building block of Li⁺[BAu₄]^? ion‐pair and other [BAu₄]^?‐containing inorganic solids is discussed. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Leading‐order behavior of the correlation energy in the uniform electron gas

We show that, in the high‐density limit, restricted Møller‐Plesset (RMP) perturbation theory yields E = π^?2(1 ? ln 2) ln r_s + O(r) for the correlation energy per electron in the uniform electron gas, where r_s is the Seitz radius. This contradicts an earlier derivation which yielded E = O(ln|ln r_s|). The reason for the discrepancy is explained. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

A theoretical study of electronic and vibrational properties of neutral,cationic, and anionic B24 clusters

The equilibrium geometries, electronic and vibrational properties, and static polarizability of B₂₄, B, and B clusters are reported here. First‐principles calculations based on density functional theory predict the staggered double‐ring configuration to be the ground state for B₂₄, B, and B, in contrast to the quasi‐planar structure observed in small neutral and ionized B_n clusters with n ≤ 15. Furthermore, the (4 × B₆) tubular structure is found to be relatively stable in comparison to the 3D cage structure. The presence of delocalized π and multicentered σ bonds appears to be the cause of the stability of the double‐ring and tubular isomers. For the ground state of B₂₄, the lower and upper bound of the electron affinity is 2.67 and 2.81 eV, respectively, and the vertical ionization potential is 6.88 eV. Analysis of the frequency spectrum of the double‐ring and tubular isomers reveals the characteristic vibrational modes typically observed in carbon nanotubes. The corresponding IR spectrum also reflects the presence of some of these characteristic modes in the neutral and ionized B₂₄, suggesting that double‐ring and tubular structures can be considered as the building blocks of boron nanotubes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Geometries and electronic structures of W4 and W clusters

Geometries and electronic structures of W₄ and W clusters were studied by use of density functional methods B3LYP, B3P86, B3PW91, BHLYP, BLYP, and MPW1PW91. The calculated results indicate that the three‐dimensional structure of singlet state with either D_2d symmetry (B3LYP, B3P86, B3PW91, BLYP, and MPW1PW91) or C_2v symmetry (BHLYP) is the ground state for the W₄ cluster. For the W cluster, the doublet state is preferred, and the most stable structure is also 3D with either D_2d symmetry (B3LYP, B3PW91, BHLYP, BLYP) or C_2v symmetry (B3P86 and MPW1PW91). The calculated electron affinity at B3P86 gives the best performance compared with experiment. For the dissociation channel, W + W₃ is suggested to be the possible route for the W₄ cluster. For the W cluster, W + W is the most likely route for dissociation, in agreement with experiment. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Crystallization studies of poly(trimethylene terephthalate) using thermal analysis and far‐infrared spectroscopy

Crystallization of poly(trimethylene terephthalate) (PTT) by annealing was examined using density measurement, differential scanning calorimetry, and far‐infrared spectroscopy (FIR). Crystallinity, measured by density, increased slowly up to the T_a of 185 °C and increases rapidly once T_a exceeds 185 °C. It was found that thermally induced crystallization is mainly temperature‐dependent above T_a = 185 °C and temperature‐ and time‐dependent below T_a = 60 °C. Two melting transitions, T and T, were observed for those samples annealed above 120 °C. No significant change in T was observed as a function of T_a while T showed strong dependency on T_a. Digital subtraction of the amorphous contribution from the semicrystalline FIR spectra provided characteristic spectra of amorphous and crystalline PTT. The bands at 373, 282, and 92 cm^?1 were assigned to the crystalline phase, while the bands at 525, 406, and 351 cm^?1 were attributed to the amorphous phase. It was shown that FIR spectroscopy can be used as a means to estimate the degree of crystallinity of PTT. The band ratio of 373 and 501 cm^?1 was plotted against crystallinity measured by density and reasonably good correlation was obtained. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1675–1682, 2007     

Synthesis and characterizations of well‐defined branched polymers with AB2 branches by combination of RAFT polymerization and ROP as well as ATRP

A well‐defined branched copolymer with PLLA‐b‐PS₂ branches was prepared by combination of reversible addition‐fragmentation transfer (RAFT) polymerization, ring‐opening polymerization (ROP), and atom transfer radical polymerization (ATRP). The RAFT copolymerization of methyl acrylate (MA) and hydroxyethyl acrylate (HEA) yielded poly(MA‐co‐HEA), which was used as macro initiator in the successive ROP polymerization of LLA. After divergent reaction of poly(MA‐co‐HEA)‐g‐PLLAOH with divergent agent, the macro initiator, poly(MA‐co‐HEA)‐g‐PLLABr₂ was formed in high conversion. The following ATRP of styrene (St) produced the target polymer, poly(MA‐co‐HEA)‐g‐(PLLA‐b‐PS₂). The structures, molecular weight, and molecular weight distribution of the intermediates and the target polymers obtained from every step were confirmed by their ¹H NMR and GPC measurements. DSC results show one T = 3 °C for the poly(MA‐co‐HEA), T = ?5 °C, T= 122 °C, and T = 157 °C for the branched copolymers (poly(MA‐co‐HEA)‐g‐PLLA), and T = 51 °C, T = 116 °C, and T = 162 °C for poly(MA‐co‐HEA)‐g‐(PLLA‐b‐PS₂). © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 549–560, 2006     

Convergence accelerator approach for the evaluation of some three‐electron integrals containing explicit rij factors

A simplified analysis is presented for the evaluation of the three‐electron one‐center integrals of the form ∫rrrrrred r ₁d r ₂d r ₃, for the cases i, j, k, ≥−2, l=−2, m≥−1, n≥−1. These integrals arise in the calculation of lower bounds for energy levels and certain relativistic corrections to the energy when Hylleraas‐type basis sets are employed. Convergence accelerator techniques are employed to obtain a reasonable number of digits of precision, without excessive CPU requirements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 93–99, 1999     

He 2++ molecular ion in a strong time-dependent magnetic field: a current-density functional study

The He molecular ion exposed to a strong ultrashort time‐dependent (TD) magnetic field of the order of 10⁹ G is investigated through a quantum fluid dynamics (QFD) and current‐density functional theory (CDFT) based approach using vector exchange‐correlation (XC) potential and energy density functional that depend not only on the electronic charge‐density but also on the current density. The TD‐QFD‐CDFT computations are performed in a parallel internuclear‐axis and magnetic field‐axis configuration at the field‐free equilibrium internuclear separation R = 1.3 au with the field‐strength varying between 0 and 10¹¹ G. The TD behavior of the exchange‐ and correlation energy of the He is analyzed and compared with that obtained using a [B‐TD‐QFD‐density functional theory (DFT)] approach based on the conventional TD‐DFT under similar computational constraints but using only scalar XC potential and energy density functional dependent on the electronic charge‐density alone. The CDFT based approach yields TD exchange‐ and correlation energy and TD electronic charge‐density significantly different from that obtained using the conventional TD‐DFT based approach, particularly, at typical magnetic field strengths and during a typical time period of the TD field. This peculiar behavior of the CDFT‐based approach is traced to the TD current‐density dependent vector XC potential, which can induce nonadiabatic effects causing retardation of the oscillating electronic charge density. Such dissipative electron dynamics of the He molecular ion is elucidated by treating electronic charge density as an electron‐“fluid” in the terminology of QFD. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Functionalization of an ethylene–propylene copolymer with allyl(3‐isocyanate‐4‐tolyl) carbamate

An ethylene–propylene copolymer (EPM) was functionalized with an iso cyanate‐bearing unsaturated monomer, allyl(3‐isocyanate‐4‐tolyl) carbamate (TAI), with dicumyl peroxide as an initiator in a xylene solution. Fourier transform infrared (FTIR) was used to confirm the formation of EPM‐g‐TAI. The peak at 2273 cm^?1, characteristic of ? NCO groups in EPM‐g‐TAI, revealed evidence of grafting. The grafting degree was determined with both chemical titration and FTIR. The grafting degree could be adjusted, and the maximum was over 6 wt % without any gelation. The molar mass distribution of EPM‐g‐TAI was narrower than that of EPM. The rheological behavior of both EPM‐g‐TAI and EPM was investigated with a rotational rheometer. The apparent viscosity of EPM‐g‐TAI was higher than that of EPM and increased with an increasing grafting degree of TAI. Surface analysis by contact‐angle measurements showed that contact angles of EPM‐g‐TAI samples to a given polar liquid decreased with an increasing grafting degree of TAI. We also obtained the dispersion component of the surface free energy (γ), the polar component of the surface free energy (γ), and the total surface free energy (γ_S = γ + γ) of the grafted EPM. These parameters increased with the enhancement of the grafting degree, which gave us a quantitative estimation of the polar contribution of the grafted TAI to the total surface free energy of EPM‐g‐TAI. The adhesive property of EPM‐g‐TAI with aluminum foil was studied. The peeling strength between EPM‐g‐TAI plate and aluminum foil increased dramatically with an increasing content of grafted TAI in EPM‐g‐TAI. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 387–402, 2003     

Bridging gold: B‐Au‐B three‐center‐two‐electron bonds in electron‐deficient B2Aun−/0 (n = 1, 3, 5) and mixed analogues

A systematic density functional theory and wave function theory investigation on the geometrical and electronic structures of the electron‐deficient diboron aurides B₂Au (n = 1, 3, 5) and their mixed analogues B₂H_mAu (m + n = 3, 5) has been performed in this work. Ab initio theoretical evidences strongly suggest that bridging gold atoms exist in the ground states of C_2v B₂Au^?(¹A₁), C₂ B₂Au(¹A), C_2v B₂Au₃(²B₁), C_2v B₂Au(¹A₁), and C_s B₂Au₅(²A″), which all prove to possess a B? Au? B three‐center‐two‐electron (3c‐2e) bond. For B₂H_mAu (m + n = 3, 5) mixed anions, bridging B? Au? B units appear to be favored in energy over bridging B? H? B, as demonstrated by the fact that the Au‐bridged C_2v B₂H₂Au^? (¹A₁), C_s B₂HAu (¹A′), and C₁ B₂HAu (¹A) lie clearly lower than their H‐bridged counterparts C_s B₂H₂Au^? (¹A′), C₂ B₂HAu (¹A), and C_2v B₂HAu (¹A₁), respectively. Orbital analyses indicate that Au 6s makes about 92–96% contribution to the Au‐based orbitals in these B‐Au‐B 3c‐2e interactions, whereas Au 5d contributes 8–4%. The adiabatic and vertical detachment energies of the concerned anions have been calculated to facilitate their future experimental characterizations. The results obtained in this work establish an interesting 3c‐2e bonding model (B? Au? B) for electron‐deficient systems in which Au 6s plays a major role with non‐negligible contribution from Au 5d. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

A Flory–Huggins thermodynamic approach for predicting sorption equilibrium in ternary polymer systems

The Flory–Huggins theory as modified by Pouchlý has been applied to calculate preferential (λ) and total (Y) sorption coefficients for a ternary polymer system. The ternary interaction function (?₁?₂?₃G_T(u₁, ?₃)) is described as the product of three independent binary functions. This expression allows prediction of λ and Y from binary interaction parameters χ, χ, g, g, and g₁₂(?₁₀). Three ternary polymer systems are used to check the validity of the expression. Moreover for polymer systems in which the parameters g and/or g are unknown, a procedure to evaluate them has been developed and verified on systems for which sufficient experimental information is available.     

Generalized population analysis of three‐center two‐electron bonding

Modern valence bond theory, in its spin‐coupled (SC) form, is used to assess the utility of generalized population analysis, using correlated density matrices, for identifying patterns of three‐center two‐electron bonding in B₂H₆, B₃H, and CH₂Li₂. The effect on the relevant three‐center indices of including electron correlation, via the SC method, is assessed by means of comparison with analogous Hartree–Fock values. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Theoretical study on AlnO2 (n = 1–10) clusters and O2 adsorption on the Al(111) surface

The structural properties of neutral and ionic Al_nO₂ (n = 1–10) clusters have been systematically investigated using the density functional method B3LYP with a standard 6‐311+G(d) basis set. The calculated results show that in the Al_nO, Al_nO₂, and Al_nO (n ≥ 3) clusters, O atoms tend to penetrate into the aluminum clusters with some Al atoms moving outward. The binding energies and natural charges populations indicate that the oxygen‐etching is generally stronger in the order Al < Al_n < Al for n < 3, and Al > Al_n > Al for n ≥ 3. To further understand the mechanism of interaction between Al and O₂, the adsorption of O₂ on the Al(111) surface was studied using the density functional theory with plane wave pseudopotential method. The calculated results are consistent with the experimental observation that the O₂ molecule would dissociate on the Al(111) surface and be adsorbed in adjacent hollow sites, forming a local structure of Al₃O–Al₃O. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Linear and nonlinear feature of electronic excitation energy in carbon chains HC2n+1H and HC2nH

Density functional theory has been used to investigate the geometries, bonding, and vibrational frequencies of HC_2nH (n = 1–13) and HC_2n+1H (n = 2–12). Vertical excitation energies for the X¹Σ → 1¹Σ transition of HC_2nH (n = 1–5) and for the X³Σ → 1³Σ transition of HC_2n+1H (n = 2–5) have been calculated by the time‐dependent density functional theory and ab initio second‐order multiconfiguration perturbation method, respectively. On the basis of the present calculations, explicit expressions for the size dependence of excitation energy in linear polyynes HC_2n+1H and HC_2n+1H are suggested. Such analytical λ ? n relationships show good agreement with experimental observations. Theoretical investigations of relevant excited states demonstrate that distinct linear and nonlinear spectroscopic features in such polyynes can be ascribed to similarity and difference in bonding between the ground and excited states in HC_2n+1H and HC_2nH. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004