Study of approximate coupled cluster methods for first-order static properties

In this paper, we analyse the algebraic structure of the equations for calculating the first order static properties using several approximate versions of Coupled Cluster (CC) methods. In particular, the non-variational and the variational method using a CC wavefunction corresponding to an appropriately defined perturbed Hamiltonian as well as the simple expectation value expression using a CC stationary state are studied under different approximations. Two different models are proposed: (a) use of maximum overlap orbitals where the pertinent approximations are TT ₂, T ⁽¹⁾ T ₂ ⁽¹⁾, (b) use of Hartree-Fock orbitals and T(T ₁+T ₂), T ⁽¹⁾(T ₁ ⁽¹⁾ +T ₂ ⁽¹⁾ ) approximations. It is analytically shown that in both these models certain approximate versions of the methods under purview yield identical results for first order static properties.NCL Communication No. 3725     

Ab initio calculations including electron correlation for the minimum energy path of the (1A1) CH2+(1Σ)H2→ (1A1) CH4 insertion reaction

Ab initio calculations on the SCF level and with the inclusion of valence shell electron correlation in the IEPA–PNO (independent electron pair approximation with pair natural orbitals), the PNO–CI (pair-natural-orbital configuration interaction) and the CEPA–PNO (coupled electron pair approximation with pair natural orbitals) schemes with Gaussian lobe functions of “double zeta quality” have been performed for the minimum energy path of the insertion of singlet (¹A₁) methylene to the (¹Σ)H₂ molecule to yield methane. The energy was minimized on the SCF level to all geometrical parameters for various values of the “approximate” reaction coordinate. The energy along the reaction path decreases monotonically without a barrier and the curves representing the total energy of the system as a function of approximate reaction coordinates obtained at different levels of approximations have the same shape. From the physical point of view three phases of the reaction can be distinguished (chemically two steps) with different geometrical arrangements and different internal geometries of the partners.     

Accurate electrical and spectroscopic properties ofX 1Σ+ BeO from coupled-cluster methods

Summary This paper reports a series of coupled-cluster (CC) calculations through CCSDT on the theoretically challenging ground state of the BeO molecule. Along with CC methods, quadratic configuration interaction (QCI) approximations to CC theory have been used (QCISD and QCISD(T)), which show several dramatic failings. Equilibrium electrical properties (, _xx , and _zz ) and basic spectroscopic properties (r _e, _e,D _e, and infrared intensity (I)) have been computed. Basis set and electron correlation effects are analyzed in order to arrive at accurate values of the dipole moment and polarizability, which are not known experimentally. For the dipole moment, we obtain a value of 6.25 D, with an uncertainty of about 0.1 D. For _xx and _zz , we suggest respective values of 32 and 36 atomic units (a.u.) and error bars of about 1 and 2 a.u. With extended basis sets, the spectroscopic propertiesr _e, _e, andD _e are reproduced to high accuracy, which is the first time this has been achieved for this species byab initio methods. At the highest calculation levels,I is predicted to be very small. AlthoughI has not been measured, some support for this prediction comes from a recent infrared study of BeO-rare gas complexes. The QCI methods are shown to be much more sensitive to basis set, and even with large basis sets yield values of _zz andI which differ from CC results by an order of magnitude and three orders of magnitude, respectively. These differences doubtless arise from the importance of single excitations (T ₁) for this molecule, as several terms involvingT ₁ are neglected in the QCISD approximation compared with CCSD. We also report CC calculations with Brueckner orbitals, which yield results similar to those obtained with restricted Hartree-Fock orbitals.     

Diffusion–Migration Currents on Microelectrodes: Comparing Two Approximate Approaches

Two electroneutrality and constant-field solutions of the diffusion–migration problem in steady-state conditions on microelectrodes, where both approximations admit analytical solutions, are compared. Analytical equations for the potential drop across the diffusion layer ₀ in terms of RT/F and the migration factor Y are obtained for three- and four-component systems containing two and three types of electroactive ions, respectively, and one type of ions that takes no part in the reaction. Both methods yield virtually identical ₀. The migration coefficients at large absolute values of ₀ noticeably differ. The Y vs. ₀ dependences in the two approaches different. The Y values yielded by these methods are close only at |₀| 1. For real electrochemical reactions considered, the electroneutrality condition at the limiting current is fulfilled at electrode radii >1 m and electrolyte concentrations >0.1 mM.     

Collective electron excitations in long molecules with conjugated bonds (cyclic and linear polyenes)

Conclusions The technique used above for calculating electronic excitations is equivalent to the random phase approximation, but permits a clearer understanding of the approximations made. The linearization with respect to Ø in the derivation of the equations for the excited states means that the approximation made is valid only for small changes in the spin (or electron) density in the atoms in the excited states from that in the ground state. This is always the case for fairly small excitation energies. The proposed calculation technique may be used to calculate excitations both in long conjugated molecules and in ordinary molecules just as well as the Pariser-Parr-Pople and random phase approximations [14, 17, 18].We note that another approach was used in [6] to find the energy of the first triplet level in polyenes. In that paper the wave function of the generalized Hartree-Fock approximation was projected onto a singlet (ground) and a triplet state. The latter was treated as a very low triplet excited state. However, as shown in [1, 2], the energies of these (the singlet and triplet) states differ by a quantity that decreases like N^–2 or even faster as N. On the other hand, as shown in [7], the energy of the first triplet excitation should decrease like ₁ 1/N as N. This implies that the interaction between electrons above the generalized Hartree-Fock approximation must be taken into account in order to obtain the first triplet state.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 10, No. 6, pp. 723–731, November–December, 1974.     

Kinetics and mechanisms of oxidation of 1-octene and heptanal by crown ether-solubilized potassium permanganate in non-aqueous solvents

The kinetics of oxidation of 1-octene and heptanal by 18-crown-6-ether-solubilized KMnO₄ in benzene and CH₂Cl₂ have been investigated. In benzene, the oxidation of 1-octene is first order with respect to the oxidant and zero order with respect to the substrate, whereas in CH₂Cl₂ the reaction is first order with respect to both substrate and oxidant. The reaction of heptanal followed different kinetics being first order with respect to both substrate and oxidant, regardless of whether benzene or CH₂Cl₂ was employed as the solvent. The values of activation energy E _a, standard enthalpy H ^*, standard entropy change S ^*, and standard free energy G ^*, for the reaction, are reported. Mechanistic pathways for the studied reactions are also proposed.     

On the Nitroxide Quasi‐Equilibrium in the Alkoxyamine‐Mediated Radical Polymerization of Styrene

Summary: The range of validity of two popular versions of the nitroxide quasi‐equilibrium (NQE) approximation used in the theory of kinetics of alkoxyamine mediated styrene polymerization, are systematically tested by simulation comparing the approximate and exact solutions of the equations describing the system. The validity of the different versions of the NQE approximation is analyzed in terms of the relative magnitude of (dN/dt)/(dP/dt). The approximation with a rigorous NQE, k_c[P][N] = k_d[P N], where P, N and P N are living, nitroxide radicals and dormant species respectively, with kinetic constants k_c and k_d, is found valid only for small values of the equilibrium constant K (10⁻¹¹–10⁻¹² mol · L⁻¹) and its validity is found to depend strongly of the value of K. On the other hand, the relaxed NQE approximation of Fischer and Fukuda, k_c[P][N] = k_d[P N]₀ was found to be remarkably good up to values of K around 10⁻⁸ mol · L⁻¹. This upper bound is numerically found to be 2–3 orders of magnitude smaller than the theoretical one given by Fischer. The relaxed NQE is a better one due to the fact that it never completely neglects dN/dt. It is found that the difference between these approximations lies essentially in the number of significant figures taken for the approximation; still this subtle difference results in dramatic changes in the predicted course of the reaction. Some results confirm previous findings, but a deeper understanding of the physico‐chemical phenomena and their mathematical representation and another viewpoint of the theory is offered. Additionally, experiments and simulations indicate that polymerization rate data alone are not reliable to estimate the value of K, as recently suggested.

Validity of the rigorous nitroxide quasi‐equilibrium assumption as a function of the nitroxide equilibrium constant.     

The van der Waals equation: analytical and approximate solutions

The thermodynamic properties, enthalpy of vaporization, entropy, Helmholtz function, Gibbs function, but especially the heat capacity at constant volume of a van der Waals gas (and liquid) at the phase transition are examined in two different limit approximations. The first limit approximation is at the near-critical temperatures, i.e., for T/T _c → 1, where T _c is the critical temperature, the other limit approximation is at the near-zero temperatures, T→ 0. In these limits, the analytical equations for liquid and gas concentrations at saturated conditions were obtained. Although the heat capacities at constant volume of a van der Waals gas and liquid do not depend on the volume, they have different values and their change during the phase transition was calculated. It should be noticed that for real substances the equations obtained at the near-zero temperature are only valid for T > T _{triple point} and T ≪ T _c , which means that found equations can be used only for substances with T _{triple point} ≪ T _c .     

Boundary phase stability and critical phenomena in higher order solid solution systems II

The thermodynamic treatment of double-pseudobinary solutions of the type (A _xB_y) _{r o}(M _uN_v) _{s o} presented in a preceding publication was extended to include the conditions defining the critical points for the asymmetric case (r _os _o) and approximations for the spinodal and binodal surface near the critical solution point.Closed solutions for the coordinates of the critical points were obtained only for systems with ideal mixing behavior, and the isothermal binodal and spinodal near the critical solution point in such systems are adequately approximated by circles and ellipses, respectively. An axes ratio of is nearly independent of the relative sublattice abundance and the major effect of changes in the ratios _o/r _o is a rotation of the binodal by an angle tg .The principal features of nonideal regular systems for temperatures close to the critical solution point are described by expressions derived from small term expansions of the conditional equations, but generalizations are not possible to the same extent as for the case with ideal boundary systems. The results are discussed and the application of the equations demonstrated on model examples.Mit 6 Abbildungen     

A new approach to the determination of the statistical segment length of wormlike polymers

The Stockmayer-Fixman-Burchard (SFB) and the Dondos-Benoit (DB) equations have been applied to determine the unperturbed dimensions parameterK of wormlike polymers. An empirical relation between the Flory's constant and the Mark-Houwink-Sakurada (MHS) exponenta has been proposed. The values found by this equation are lower than the value 2.5×10²³ used in the case of flexible polymers and this deviation is attributed to the influence of the draining effect. From theK value and the so calculated value of , we calculate the Kuhn statistical segment length of wormlike polymers. The obtained — for a great number of wormlike polymers — statistical segment lengths are almost the same as these calculated by the Yamakawa-Fujii and the Bohdanecky methods. The molecular mass regions in which the SFB, the DB, and the MHS equations are valid are explored. A criterion for the distinction between flexible and wormlike polymers is proposed based on the way of approach to the power law.     

Zur konstruktion von energiehyperflächen

An analytical approximation for the hypersurface ? (pr-approximation) is given using the two- and three-center-functions p_λμ and r, which is valid for any number N of atoms and is invariant with respect to the number of centers in its analytical structure (rule of construction). This is valid too if N is reduced either by the association of atoms or by transitions to infinity. p_λμ can be fixed by two-center-systems. But r is still free except for the fulfilling of some simple requirements. The rule of construction proposed for ? is an example for using so-called analytical computers.     

Simultaneous laser absorption measurements of CN and OH in a shock tube study of HCN + OH → products

Simultaneous, quantitative, narrow-line laser absorption measurements of CN time-histories at 388.444 nm and OH time-histories at 306.687 nm have been made in incident and reflected shock wave experiments using dilute mixtures of nitiric acid (HNO₃) and HCN in argon. The thermal decomposition of HNO₃ serves as a rapid source of OH upon shock-heating, and the OH subsequently reacts predominantly with the HCN in the test gas mixture. The rate coefficient for the reaction was determined in the temperature range 1120–1960 K via detailed kinetics modeling of the simultaneously acquired CN and OH measurements. These data are in good agreement with lower temperature measurements of the rate of the reverse reaction (?1a) when recent values of the heats of formation of CN and HCN are used. The expression valid for temperatures 500 to 2000 K, effectively represents the experimental measurements. The estimated uncertainty of the expression for k_1a is ±30%, based on the experimental uncertainties of the individual rate coefficient studies. Analysis of the decay region of the experimental OH time-histories yielded the total rate coefficient k₁ (all product channels) for the reaction of HCN with OH for temperatures ranging from 1490 to 1950 K. These measurements are consistent with a previous theoretical analysis of the three primary addition-isomerization-dissociation processes for the HCN + OH reaction at combustion temperatures when the contribution to k₁ from reaction (1a) is included. © 1995 John Wiley & Sons, Inc.     

Nichtstationäre Kinetik bei SN1, SN2, E1, E2 und E1cb-Mechanismen

For the S_N1, S_N2, E₁, E₂, AND E₁cb mechanisms exact solutions of the kinetic equations are compared with solutions obtained under the hypothesis of stationarity. Exact integrals are calculated numerically for a set of the essential reaction rate constants. Comparison with the stationary state solution demonstrates that both types of solutions are approximately equal in many cases if the transient particle concentration remains low, whereas the conventional requirement | \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop x\limits^. $\end{document} | = 0 is unimportant. For approximation of exact kinetics by stationary kinetics, however, neither condition is sufficient nor necessary. Practical criterions for recognition of deviations from stationary kinetics are given.     

Schrödinger Equation Solutions for the Central Field Power Potential Energy I. V(r) = V 0(r/a 0)2ν−2, ν ≥ 1

The solution of a generalized non-relativistic Schrödinger equation with radial potential energy V(r)=V ₀(r/a ₀)^2–2 is presented. After reviewing the general properties of the radial ordinary differential equation, power series solutions are developed. The Green's function is constructed, its trace and the trace of its first iteration are calculated, and the ability of the traces to provide upper and lower bounds for the ground eigenvalue is examined. In addition, WKB-like solutions for the eigenvalues and eigenfunctions are derived. The approximation method yields valid eigenvalues for large quantum numbers (Rydberg states).     

Thermodynamic quantities for the ionization of nitric acid in aqueous solution from 250 to 319°C

The aqueous reaction, HNO₃(aq)=H⁺+NO ₃ ^– was studied as a function of ionic strength I at 250, 275, 300 and 319°C using a flow calorimeter and the equilibrium constant K and enthalpy change (H) at I=0 were determined. Using these experimental values, equations describing logK, H, the entropy change S and the heat capacity change C _p of reaction at I=0 and temperatures from 250 to 319°C were derived. The increasing importance of ion association as temperature rises was discussed. The use of an equation containing identical numbers of positive and identical numbers of negative charges on both sides of the equal sign (isocoulombic reaction principle) was applied to the logK values reported here and to those determined by others. The resulting plot of logK for the isocoulombic reaction vs. 1/T was fairly linear which supports the postulate that the principle is a useful technique for the extrapolation of logK values from low to high temperatures.Presented at the Second International Symposium on Chemistry in High Temperature Water, Provo, UT, August 1991.     

The perturbation calculation of van der Waals potentials

Summary The unsymmetrized perturbation theory for interaction potentials is reformulated in such a way that the overlap and exchange effects can be taken into account in a satisfactory and conceptually simple way. This formulation, known as the generalized Heitler-London theory, its shown to be valid regardless of the ultimate limit to which the polarization approximation converges. Within the framework of this theory, the van der Waals potential of the triplet H₂(³ _u ) state is calculated and shown to be in excellent agreement with the exactab initio results. Both the exchange energy and the polarization energy are obtained from a perturbation calculation.     

Ionic interactions in solutions. III. Derivation of the ″associated″ electrolyte formulation from the Onsager treatment of conductance

The system of equations of the Onsager treatment of conductance is rederived in a systematic way by the use of several sets of hierarchy equations generalizing to other quantities the classical Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of internal force. The monitoring term of the Onsager continuity equation obtained is compared to former theoretical studies. It is inferred in particular that the perturbation on the anion-cation pair distribution for a symmetrical binary electrolyte is proportional not only to the external fieldX but also to the conductance coefficientf _A=1+X/X+ _el/ ₀. This result, called the echo effect, leads to a new formulation of molar conductance which shows a great similarity to the empirical conductance function obtained by grafting the chemical model of Bjerrum onto the conductance equations derived on the basis of the Debye approximations as proposed formerly by the author from experimental observations. The result allows a simple generalization of the conductance equation to any direct short-range anion-cation energy-potential model.     

Bond dissociation energies and the Hammett correlation,Part 1: Remotely substituted aromatic compounds and vitamin E

Four hundred and seventy nine experimental values of X? Z bond dissociation energies (BDEs) of para‐, ortho‐, meta‐, and multisubstituted aromatic compounds were reproduced by the following simple equations: Here X represents a wide variety of univalent atoms or groups; Y is one or several various remote substituents; and Z represents an oxygen, nitrogen, carbon, or sulfur atom. DH⁰(X? ZC₅H₆) is a known reference/anchor point for a series of X? Z bonds. E_sr is the remote conjugation energy between the substituents and the reaction center (broken bond), and can be expressed as a Hammett‐type correlation. σ⁺(Y) is the Brown–Okamoto constant of a substituent Y or the sum of several substituent constants. The Hammett slope or reaction constant ρ is a linear function of the difference between the covalent potentials V_x(Z) and V_x(X). It has been found that and The set of equations are very powerful for the prediction of BDEs, chemical reactivity, and reaction center (or active site) in antioxidants, nutrients, pharmaceuticals, toxicants, carcinogens, and explosives. © 2002 Wiley Periodicals, Inc. Int J Chem Kinet 34: 453–466, 2002     

Kinetics of the oxidation of cyclohexene by thallium(III) acetate

The kinetics of the reaction by which thallium(III) acetate oxidizes cyclohexene in glacial acetic acid medium, has been studied by UV spectrophotometric observation at 30°C. The consumption of thallium(III) acetate follows a second-order rate law exhibiting first-order dependence on each of thallium(III) acetate and cyclohexene; however, the first-order dependence on cyclohexene disappears at high cyclohexene concentrations as pseudo-first-order conditions prevail above 0.2 M cyclohexene. A steady-state model of the following form is proposed: where Tl, Cy, and Com are units of Thallium(III) acetate, cyclohexene, and a reaction complex. The value of k₂ has been evaluated as 0.00027 and (k_?1 + k₂) as 0.0385k₁. For low thallium(III) acetate concentrations the reaction kinetics follow the rate law: where α = the excess concentration of cyclohexene over thallium(III) triacetate. For thallium(III) acetate concentrations above 0.02 M, double salt formation of thallium(III) acetate with product thallium(I) acetate removes thallium(III) acetate from the reaction and a modified rate law is observed. Runge–Kutta numerical solutions to the differential equations provide confirmation that the rate expressions are valid in predicting the observed concentrations of thallium(III) acetate.