Computation of auxiliary functions in STO molecular integrals up to arbitrary accuracy. I. evaluation of incomplete gamma function En(x) by forward recursion

A new inequality and an approximate relation of incomplete Gamma function E_n(x) are derived in this paper. Applying both to simplify the error analysis on the recursive algorithm of E_n(x), we analytically derived a criterion about relative error. The criterion decides whether the forward recursive relation can be used to compute {E_n(x), n = 0, 1, ?, N} with an acceptable loss of significant figures. Further analytical exploration leads to another criterion that decides whether the group of E_n(x) can be evaluated with an acceptable absolute error by the same methods. Both criteria are not only superior to those given in previous work by other authors, but also strongly supported by our numerical experiments. A strategy is illustrated to explain how to apply both criteria to evaluate molecular integral over Slater-type orbitals. © 1993 John Wiley & Sons, Inc.     

Wavelength sensitivity of the photoinduced reaction in polycarbonate

Bisphenol A polycarbonate (PC) was irradiated with monochromatic light of wavelengths 260, 280, 300, 320, 340, 400, and 500 nm by use of the Okazaki large spectrograph (OLS). The quantum yield of main-chain scission (?_cs), efficiency of photo-Fries rearrangement (E_r), and effects of wavelength on ?_cs and E_r were investigated. It was found that photodegradation and photo-Fries rearrangement of PC took place by the irradiation of 260–300 nm light, but did not by the irradiation at λ ≧ 320 nm. The ?_cs has a maximum value in the case of the irradiation with 260 nm light, while E_r was found to have a maximum value by the irradiation of 280 nm light. © 1993 John Wiley & Sons, Inc.     

Theory of the expansion of wave functions in a gaussian basis

The convergence properties of the expansions of (a) the function 1/r and (b) the function exp(-αr) in an even-tempered basis of Gaussians are studied analytically. The starting points are the Gaussian integral representations of 1/r and exp(?αr). One arrives at an expansion in a finite number of Gaussians in three steps: (1) a restriction of the integration domain, (2) a variable transformation, and (3) discretization of the integral. The cutoff error goes in both cases essentially as exp(?ah), and the discretization error, as exp(?b/h). The minimum overall error is reached for the β-parameter of an even-tempered basis β ∽ exp(c/√n), where n is the dimension of the basis, and the error itself decreases as ? ∽ exp(?d√n). Different optimum basis parameters are obtained depending on which quantity one wants to minimize, e.g., the error of the energy expectation value, the distance in Hilbert space, the variance of the energy, or the density at the nucleus. © 1994 John Wiley & Sons, Inc.     

Methyl 2‐O‐β‐d‐glucopyranosyl‐α‐l‐rhamnopyranoside

The overall conformation of the title compound, C₁₃H₂₄O₁₀, is described by the glycosidic torsion angles ?_H (H1_g—C1_g—O2_r—C2_r) and ψ_H (C1_g—O2_r—C2_r—H2_r), which have values of 13.6 and 16.1°, respectively. The former is significantly different from the value predicted by consideration of the exo‐anomeric effect (?_H～ 60°) and from that in solution (?_H～ 50°), as determined previously by NMR spectroscopy. An intramolecular O3_r—H?O2_g hydrogen bond may help to stabilize the conformation in the solid state. The orientation of the hydroxy­methyl group of the glucose residue is gauche–gauche, with a torsion angle ω (O5_g—C5_g—C6_g—O6_g) of ?70.4 (4)°. Both pyranose rings are in their expected chair conformations, i.e.⁴C₁ for d ‐glucose and ¹C₄ for l ‐rhamnose.     

Electrophilic Alkylations of Vinylsilanes: A Comparison of α‐ and β‐Silyl Effects

Kinetics of the reactions of benzhydrylium ions (Aryl₂CH⁺) with the vinylsilanes H₂C?C(CH₃)(SiR₃), H₂C?C(Ph)(SiR₃), and (E)‐PhCH?CHSiMe₃ have been measured photometrically in dichloromethane solution at 20 °C. All reactions follow second‐order kinetics, and the second‐order rate constants correlate linearly with the electrophilicity parameters E of the benzhydrylium ions, thus allowing us to include vinylsilanes in the benzhydrylium‐based nucleophilicity scale. The vinylsilane H₂C?C(CH₃)(SiMe₃), which is attacked by electrophiles at the CH₂ group, reacts one order of magnitude faster than propene, indicating that α‐silyl‐stabilization of the intermediate carbenium ion is significantly weaker than α‐methyl stabilization because H₂C?C(CH₃)₂ is 10³ times more reactive than propene. trans‐β‐(Trimethylsilyl)styrene, which is attacked by electrophiles at the silylated position, is even somewhat less reactive than styrene, showing that the hyperconjugative stabilization of the developing carbocation by the β‐silyl effect is not yet effective in the transition state. As a result, replacement of vinylic hydrogen atoms by SiMe₃ groups affect the nucleophilic reactivities of the corresponding C?C bonds only slightly, and vinylsilanes are significantly less nucleophilic than structurally related allylsilanes.     

Orbital functional theory of linear response and excitation

Orbital functional theory (OFT) is based on a rule that determines a single‐determinant reference state Φ for any exact N‐electron eigenstate Ψ. An OFT model postulates an explicit correlation energy functional E_c of occupied orbital functions {?_i} and occupation numbers {n_i}. The orbital Euler–Lagrange equations are analogous to Kohn–Sham equations, but do not in general contain local potential functions. Time‐dependent Hartree–Fock theory is generalized in OFT to a formally exact linear response theory that includes electronic correlation. In the exchange‐only limit, the theory reduces to the random‐phase approximation of many‐body theory. The formalism determines excitation energies. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Effect of the Media on the Quantum Yield of Singlet Oxygen (O2(1Δg)) Production by 9H‐Fluoren‐9‐one: Solvents and Solvent Mixtures

We have investigated the effect of a series of 18 solvents and mixtures of solvents on the production of singlet molecular oxygen (O₂(¹Δ_g), denoted as ¹O₂) by 9H‐fluoren‐9‐one (FLU). The normalized empirical parameter E derived from E_T(30) has been chosen as a measure of solvent polarity using Reichardt's betaine dyes. Quantum yields of ¹O₂ production (Φ_Δ) decrease with increasing solvent polarity and protic character as a consequence of the decrease of the quantum yield of intersystem crossing (Φ_ISC). Values of Φ_Δ of unity have been found in alkanes. In nonprotic solvents of increasing polarity, Φ_ISC and, therefore, Φ_Δ decrease due to solvent‐induced changes in the energy levels of singlet and triplet excited states of FLU. This compound is a poor ¹O₂ sensitizer in protic solvents, because hydrogen bonding considerably increases the rate of internal conversion from the singlet excited state, thus diminishing Φ_Δ to values much lower than those in nonprotic solvents of similar polarity. In mixtures of cyclohexane and alcohols, preferential solvation of FLU by the protic solvent leads to a fast decrease of Φ_Δ upon addition of increasing amounts of the latter.     

Liquid density,viscosity and spectroscopic studies of binary mixture of tetraethylene glycol + 1,2-ethanediamine for CO2 capture

Thermophysical properties for binary mixture of tetraethylene glycol (T₄EG) (1) + 1,2-ethanediamine (EDA) (2), a potential scrubbing solution for the absorption of CO₂, are very important as well as lacking in the literatures. This work reports densities and viscosities over the entire concentration range for the binary mixture at T = (293.15-318.15) K under atmospheric pressure. According to the experimental density and viscosity values, the mixtures’ excess molar volume (V_m^E), absolute viscosity deviation (?η), excess free energies of activation (?G*^E), apparent molar volumes, partial molar volumes and isobaric thermal expansion coefficient were calculated, respectively. Meanwhile, the V_m^E, ?η and ?G*^E values were fitted by a Redlich–Kister equation to obtain coefficients. To further study, the Fourier transform infrared, UV-Vis and fluorescence spectra of T₄EG + EDA mixtures with various concentrations were measured, and the intermolecular interaction of T₄EG with EDA was also discussed as the formation of –OCH₂CH₂O–H···N(H₂)CH₂CH₂(H₂)N···.     

The electron correlation cusp

The Kais function is an exact solution of the Schrödinger equation for a pair of electrons trapped in a parabolic potential well with r ₁₂ ^?1 electron-electron interaction. Partial wave analysis (PWA) of the Kais function yields E _L = E + C₁(L + \-C ^?1 ₂)^?3 + O(L ^?5) where E is the exact energy and E _L the energy of a renormalized finite sum of partial waves omitting all waves with angular momentum ? > L. Slight rearrangement of an earlier result by Hill shows that the corresponding full CI energy differs from E _L only by terms of order O(L ^?5) with FCI values of C ₁ and \-C ^?1 ₂ identical to PWA values. The dimensionless \-C ₂ parameter is weakly dependent upon the size of the physical system. Its value is 0.788 for the Kais function, and 0.893 for the less diffuse helium atom, and approaches \-C ₂→ 1 in the limit of an infinitely compact charge distribution. The ?th energy increment satisfies an approximate virial theorem which becomes exact in the high ? limit. This analysis, formulated to facilitate use of the Maple system for symbolic computing, lays the mathematical ground work for subsequent studies of the electron correlation cusp problem. The direction of future papers in this series is outlined.     

Regiospecific Formation and Unusual Optical Properties of 2,5‐Bis(arylethynyl)rhodacyclopentadienes: A New Class of Luminescent Organometallics

A series of 2,5‐bis(arylethynyl)rhodacyclopentadienes has been prepared by a rare example of regiospecific reductive coupling of 1,4‐(p‐R‐phenyl)‐1,3‐butadiynes (R?H, Me, OMe, SMe, NMe₂, CF₃, CO₂Me, CN, NO₂, ?C?C‐(p‐C₆H₄?NHex₂), ?C?C?(p‐C₆H₄?CO₂Oct)) at [RhX(PMe₃)₄] ( 1 ) (X=?C?C?SiMe₃ ( a ), ?C?C‐(p‐C₆H₄?NMe₂) ( b ), ?C?C?C?C?(p‐C₆H₄?NPh₂) ( c ) or ?C?C?{p‐C₆H₄‐C?C?(p‐C₆H₄‐N(C₆H₁₃)₂)} ( d ) or Me ( e )), giving the 2,5‐bis(arylethynyl) isomer exclusively. The rhodacyclopentadienes bearing a methyl ligand in the equatorial plane (compound 1 e ) have been converted into their chloro analogues by reaction with HCl etherate. The rhodacycles thus obtained are stable to air and moisture in the solid state and the acceptor‐substituted compounds are even stable to air and moisture in solution. The photophysical properties of the rhodacyclopentadienes are highly unusual in that they exhibit, exclusively, fluorescence between 500–800 nm from the S₁ state, with quantum yields of Φ=0.01–0.18 and short lifetimes (τ=0.45–8.20 ns). The triplet state formation (Φ_ISC=0.57 for 2 a ) is exceptionally slow, occurring on the nanosecond timescale. This is unexpected, because the Rh atom should normally facilitate intersystem crossing within femto‐ to picoseconds, leading to phosphorescence from the T₁ state. This work therefore highlights that in some transition‐metal complexes, the heavy atom can play a more subtle role in controlling the photophysical behavior than is commonly appreciated.     

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V _xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r,?r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V _xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V _x (r) determined solely by the HF ground state density for the Be atom.     

On optimization procedures based upon the atomic thomas-fermi energy

For the total atomic Thomas-Fermi (TF) energy many expressions in terms of the kinetic and potential energy contributions can be given. Thirty of these expressions exhibit either a maximum or a minimum if some variational approximation to the TF function is used. Three of these expressions, to note E, G, and J (see text) have been used in an optimization procedure, in which four two-parameter and two three-parameter approximate TF functions have been considered. One-parameter functions cannot be optimized, as the one parameter must be fixed to ensure proper normalization. It is found that optimization of E and G give reasonable and similar results, whereas the results of optimization of J are generally not very impressive. Where possible, expectation values of the type 〈rⁿ〉 (with n = ?1, 1, 2, and 3) have been calculated from ten approximate TF functions. A new estimate of the exact atomic TF energy, as well as of the derivative of the TF function at the origin, has been obtained.     

Magnesium aluminium chromite

The cation distribution in a natural magnesium aluminium chromite spinel (cubic, space group Fdm), Al_0.41Cr_1.42Fe_0.65Mg_0.4O₄, was determined by electron‐microprobe analysis, Mössbauer spectroscopy and single‐crystal X‐ray analysis. Several structural models of the octahedral and tetrahedral cation distributions were tested; the most probable is (Mg,Al,Fe)[Al,Cr,Fe,Φ_0.19]O, where (…) and […] represent the tetrahedral and octahedral sites, respectively, and Φ represents a vacancy.     

Melting temperatures of poly(2,6-dimethyl-1,4-phenylene oxide) in methylene chloride solutions: Thermodynamic analysis

Melting temperatures were observed visually for poly-(2,6-dimethyl-1,4-phenylene oxide) in methylene chloride at nine concentrations (polymer weight fractions ranging from 0.0042 to 0.2362). The data were analyzed upon the assumptions that ΔH_u and ΔS_u, the molar heat and entropy of fusion per polymer unit, are constant over the temperature range studied, and that a Flory-Huggins chemical potential expression with a concentration-independent pair interaction parameter, χ₁ = (0.5 + ψ₁) + ψ₁Φ/T, satisfactorily describes the polymer unit activity in the binary solutions. Computation gave ΔH_u = 1404 cal/mole of units (therefore Δh = 11.7 cal/g), ψ₁ = ?0.569₁, and Φ = 342.₄°K. The effect of using various combinations of data points upon the values of these three parameters, as determined by least-squares linear regression treatment of the melting temperature expression, is indicated.     

N.m.r. studies of the keto–enol tautomerism of acylthioacetamides

Acetylthioacetamides exist as different keto and enol isomers in chloroform solutions. The keto form with intramolecular hydrogen bonding between the NH and the carbonyl group is the dominant keto isomer. On the other hand the enol forms with intramolecular hydrogen bonding between the OH and the thioketo group are the dominant enol isomers in the temperature range 60°C to ?60°C. The thermodynamic data of the keto-enol equilibria were obtained by measuring the intensities of appropriate high resolution proton signals as a function of temperature. At low temperatures all lines characteristic of the enol forms are doubled in the N-phenyl-substituted derivatives because the rotation of the NH? C₆H₅ group around the C? N bond becomes slow and the chemical shifts characteristic of the E and Z isomers are different. We estimated approximate thermodynamic data of the E/Z equilibrium in some of the compounds. The changes of the line shape as well as the chemical shifts as a function of temperature indicate the presence of various additional exchange processes. In order to obtain further information we performed curve fittings of the chemical shifts of one acetylthioacetanilide and of a series of monothio-β-diketones (studied in another paper) assuming a fast two site exchange process. On the basis of the results obtained a reaction scheme for N-substituted acylthioacetanilides in solution is proposed.     

The N‐Arylamino Conjugation Effect in the Photochemistry of Fluorescent Protein Chromophores and Aminostilbenes

To understand the nonradiative decay mechanism of fluorescent protein chromophores in solutions, a systematic comparison of a series of (Z)‐4‐(N‐arylamino)benzylidene‐2,3‐imidazolinones (ABDIs: 2P , 2PP , 2OM , and 2OMB ) and the corresponding trans‐4‐(N‐arylamino)‐4′‐cyanostilbenes (ACSs: 1P , 1PP , 1OM , and 1OMB ) was performed. We have previously shown that the parameter Φ_f+2 Φ_tc, in which Φ_f and Φ_tc are the quantum yields of fluorescence and trans→cis photoisomerization, respectively, is an effective probe for evaluating the contribution of twisted intramolecular charge transfer (TICT) states in the excited decays of trans‐aminostilbenes, including the push–pull ACSs. One of the criteria for postulating the presence of a TICT state is Φ_f+2 Φ_tc?1.0, because its formation is decoupled with the C?C bond (τ) torsion pathway and its decay is generally nonradiative. Our results show that the same concept also applies to ABDIs 2 with the parameter Φ_f+2 Φ_ZE in which Φ_ZE is the quantum yield of Z→E photoisomerization. We conclude that the τ torsion rather than the C? C bond (φ) torsion is responsible for the nonradiative decays of ABDIs 2 in aprotic solvents (hexane, THF, acetonitrile). The phenyl‐arylamino C? N bond (ω) torsion that leads to a nonradiative TICT state is important only for 2OM in THF and acetonitrile. If the solvent is protic (methanol and 10–20 % H₂O in THF), a new nonradiative decay channel is present for ABDIs 2 , but not for ACSs 1 . It is attributed to internal conversion (IC) induced by solvent (donor)–solute (acceptor) hydrogen‐bonding (HB) interactions. The possible HB modes and the concept of τ torsion‐coupled proton transfer are also discussed.     

Excess properties and spectroscopic studies of binary system 1,4-butanediol + water at T = (293.15, 298.15, 303.15, 308.15, 313.15 and 318.15) K

Density and dynamic viscosity data were measured over the whole concentration range for the binary system 1,4-butanediol (1) + water (2) at T = (293.15, 298.15, 303.15, 308.15, 313.15, and 318.15) K as a function of composition under atmospheric pressure. Based on density and dynamic viscosity data, excess molar density (ρ^E), dynamic viscosity deviation (Δν) and excess molar volume (V_m^E) were calculated. From the dynamic viscosity data, excess Gibbs energies (ΔG*^E), Gibbs free energy of activation of viscous flow (ΔG*), enthalpy of activation for viscous flow (ΔH*) and entropy of activation for viscous flow (ΔS*) were also calculated. The ρ^E, V_m^E, Δν and ΔG*^E values were correlated by a Redlich?Kister-type function to obtain the coefficients and to estimate the standard deviations between the experimental and calculated quantities. Based on FTIR and UV spectral results, the intermolecular interaction of 1,4-butanediol with H₂O was discussed.     

Approximate lower bounds of the Weinstein and Temple variety

By using the Weinstein interval or coupling the Temple lower bound to a variational upper bound one can in principle construct an error bar about the ground‐state energy of an electronic system. Unfortunately there are theoretical and calculational issues which complicate this endeavor so that at best only an upper bound to the electronic energy has been practical in systems with more than a few electrons. The calculational issue is the complexity of 〈H²〉 which is necessary in the Temple or Weinstein approach. In this work we provide a way to approximate the 〈H²〉 to any desired accuracy using much simpler 〈H〉‐like information so that the lower bound calculations are more practical. The helium atom is used as a testing ground in which we obtain approximate error bars for the ground‐state energy of [?2.904230, ?2.903721] hartree using the variational energy with the Temple lower bound and [?2.919098, ?2.888344] hartree for the Weinstein interval. For comparison, the slightly larger error bars using the exact value of 〈H²〉 are: [?2.904358, ?2.903721] hartree and [?2.919765, ?2.887677] hartree, respectively. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

On the convergence of the Møller-Plesset perturbation series

Møller-Plesset perturbation energies, up to as much as 48th order, have been calculated for H₂O (RHF and UHF framework) and NH₂ (UHF framework) atC _2v geometries (r _e , 1.5r _e and 2r _e ). Atr _e , the RHF and UHF series rapidly converge, but at2r _e , the RHF series converges erratically with an energy at 43rd order within 10^–7 hartree of the exact value whereas the UHF series converges smoothly, but very slowly, and for H₂O has an error in excess of 10^–5 hartree at 48th order. The significance of these results is discussed.