On finding the rate constants of linear and nonlinear systems

In lowest approximation, a certain chemical reaction is described by a system of first-order linear differential equations with unknown constant coefficients. One can therefore write down an expression for the state of the system at time t, and from this find the endpoint of the reaction in terms of the initial state and the rate constants. The relative values of some rate constants can then be estimated from experimental data. A better approximation in which the differential equations are nonlinear is also considered, and it turns out that because of symmetry in the reaction, the relationship between the final state and the ratios of the rate constants is unchanged. Although the differential equations now appear much less tractable, the problem of relating the rate constants to the endpoint of the reaction can be formulated and solved in terms of probabilities. The results illustrate an important property of reaction schemes in which some of the steps are reversible. More generally, this is a property of differential equations: provided that they continue to satisfy certain linear constraints, the parameters of a linear system of ordinary differential equations can vary without affecting the asymptotic solution.     

On the Validity of the Steady-state Approximation in Non-isothermal Kinetics. Part III

The authors continue their considerations concerning the validity of the steady-state approximation in non-isothermal kinetics. A sequence of two first-order consecutive reactions with an active intermediate was subjected to kinetic analysis by numerical solution of the corresponding differential kinetic equations for a number of particular cases. The results demonstrated that the rate of change of concentration of the active intermediate is negligibly small if the assumption made in the isothermal case is also accepted for the non-isothermal case, i.e. k ₂(T(t))>> k ₁(T(t)). This revised version was published online in July 2006 with corrections to the Cover Date.     

The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Kinetics and mechanism of decarboxylation of aspartic acid with ninhydrin

The kinetic study of the decarboxylation of aspartic acid has been carried out at various [ninhydrin], [H⁺] and at different temperature ranging from 60–95°C. The reaction follows an irreversible first-order reaction path under pseudo first-order kinetic conditions. The variation of pseudo first-order rate constant (k_obs) with ninhydrin concentration was found to be in agreement with equation 1/k_obs = B₁ + B₂/[Ninhydrin]. One mol of carbondioxide evolved from decarboxylation of α-COOH and second mol of carbondioxide comes from the decarboxylation of β-keto acid which is an intermediate and formed during the course of ninhydrin and aspartic acid reaction. On the basis of the observed data, a possible mechanism has been proposed.     

Physical and rheological characterization of poly(aryloxyphosphazene) copolymers

A series of poly(aryloxyphosphazene)s was prepared with phenoxy and p-ethylphenoxy substituents in various ratios. The thermal, morphological, and rheological properties of this series of polymers were studied by differential thermal analysis, x-ray analysis, and rheometrics mechanical spectroscopy. Thermal analysis showed that the resulting polymers follow melting-point-depression- and glass-transition-temperature-composition relationships expected of random copolymers. The lower first-order transition temperature [T(1)] disappears near equimolar substitution while the higher first-order transition temperature (T_m) persists over the whole range of compositions. X-ray analysis revealed that crystalline order in the polymer chain direction is destroyed by nearly equimolar substitution but lateral order remains. The rheological characterization of a copolymer with nearly equimolar substitution showed that the polymer is in a pseudocrosslinked state indicating the existence of chain-to-chain interactions. When a small amount of an antioxidant (6-dodecyl-1,2-dihydro-2,2,4-trimethylquinoline) is added, the lateral order of the copolymer is destroyed, and its rubbery plateau modulus decreases by several fold.     

Effect of scaling of a quantum mechanical force field on the frequencies and forms of molecular vibrations

The effect of scaling of an ab initio quantum mechanical force field on the frequencies and forms of normal vibrations are studied in terms of first- and second-order perturbation theory. Scaling the force constant matrix according to Pulay using certain assumptions in first-order perturbation theory is equivalent to scaling vibration frequencies and does not modify the form of vibrations. In this case, the second-order corrections to the frequencies and forms of vibrations become zero. The first-order perturbation theory formulas are used to verify the assumptions by calculating the frequencies and matrices of transition to perturbed forms of vibrations of ethane, propane, ethylene, cyclopropene, and isobutene molecules from quantum mechanical force fields found with the 6-31G basis set. It is shown that the vibration frequencies calculated by the formulas of first-order perturbation theory are in good agreement with exact values; the matrix of transition to perturbed eigenvectors is rarefied, with only ≈1% of its elements being markedly nonzero. Moscow State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 2, pp. 210–216, March–April, 1998. This work was supported by RFFR grant No. 96-03-34085.     

Numerical simulation of the kinetics of ozone decomposition in an aqueous solution

The kinetic scheme for ozone decomposition in an aqueous solution in a wide pH interval was proposed on the basis of the known literature data. The apparent first-order rate constant for ozone decomposition at pH 1–14 was calculated. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1151–1156, June, 2008.     

Electrostatic Interaction between Two Dissimilar Spheres with Constant Surface Charge Density

Explicit exact analytic expressions are obtained in the form of infinite series for the potential energy of the electrostatic interaction for the system of two dissimilar hard spheres with constant surface charge density in an electrolyte solution on the basis of the linearized Poisson-Boltzmann equation. The effects of the particle polarization, that is, the internal fields induced within tim interacting spheres, which are found to be of the order of instead of 1/κa (where κ is the Debye-Hückel parameter and a is the sphere radius), are taken into account. As in the case of the interaction at constant surface potential, the zeroth-order approximation to the interaction energy corresponds to the interaction energy that would be obtained if both spheres were ion-penetrable spheres ("soft" spheres) and to that obtained by the linear superposition approximation. The first-order approximation corresponds to the interaction energy that would be obtained if either sphere were a soft sphere, with the other being a hard sphere with constant surface charge density. The first-order correction term can be interpreted as the image interaction between the soft sphere and its image with respect to the hard sphere.     

Density matrices for correlated Gaussians: Helium and dipositronium

Formulas are derived for the density matrices belonging to an n-particle wave function built on the basis of single-center explicitly correlated Gaussian basis functions. An explicit formula for the first-order density matrix, P(r₁, r′₁), is obtained for computing the probability distribution P(r₁, r₁). Other formulas are derived for matrix elements of the first-order density operator P on a basis of single-particle Gaussian orbitals so that natural orbitals (NOs) can be expressed in such a basis. The method is illustrated for the case of the ground state of the helium atom using the 16-term (geminal) wave function by Singer and Longstaff (E = −2.90233 au) and a set of even-tempered Gaussian orbitals. The resulting natural orbitals compare favorably with natural orbitals from Cl expansions. The method is also applied to our 20 term (trimal) wave function for the ground state of dipositronium (E = −0.51560 au). Analysis is made in this case for pair correlation functions of both the electron-electron and the positron-electron pairs; results include the radial distributions of these pairs and their relative angular momentum. © 1996 John Wiley & Sons, Inc.     

Extended wave packet dynamics; exact solution for collinear atom,diatomic molecule scattering

The semiclassical wave packet dynamics method of Heller is extended to provide a formally exact theory of quantum mechanical motion for multidimensional anharmonic systems by introducing a complete, orthonormal, time-dependent basis of generalized oscillator functions. The exact wavefunction is expressed in terms of this basis and the expansions are shown to develop according to linear, coupled first-order differential equations. Application to collinear inelastic atom-diatomic molecule scattering demonstrates the feasibility and convergence of the new method.     

Gauge invariant gaussian orbitals and the ab initio calculation of diamagnetic susceptibility for molecules

It is shown that gauge terms can be introduced into the Gaussian functions used as the basis functions for an ab initio calculation of the energy of a molecule in the presence of a uniform magnetic field so that all the integrals become independent of the origin of the vector potential. The perturbation treatment of the diamagnetic susceptibility is considered in the molecular orbital approximation. The results show that the susceptibility can be calculated using only the unperturbed orbitals and their first-order corrections. All the integrals that arise can be expressed in terms of known functions.     

粒子隧道几率的一种新形式

基于二阶线型微分方程的多个任意阶转向点，建立了粒子进入或跳出势垒或势井时状态波函数的连结公式，导出了粒子穿过势垒的隧道几率公式，并利用严格的数学方法讨论了粒子进入势井时的量子化条件。该公式可合理地还原到一阶转向点情况，并与早期利用一阶转向点近似所推导出的公式相一致。     

Non-local relation between kinetic and exchange energy densities in Hartree–Fock theory

A non-local generalization K( r, r' ) of the kinetic energy t( r ) such that t( r ) = ∫K( r, r' ) dr' is defined using the idempotency property of the Hartree–Fock first-order density matrix. This is, in turn, related by means of an explicit differential equation to the non-local exchange energy density X( r, r' ). The relationship is illustrated for a couple of examples: with the Fermi-hole in a uniform electron gas, of importance in the local density version of density functional theory, and with inhomogeneous electron systems.     

Comparative study of the molecular electrostatic potential obtained from different wavefunctions. Reliability of the semiempirical MNDO wavefunction

A systematic analysis of the molecular electrostatic potential (MEP) is presented. This study has been performed with a twofold purpose: first, to study the MEP dependence with regard to the quality of the basis set used to compute the ab initio SCF wavefunction and second, to develop and to assess a new strategy for computing isoelectrostatic potential maps using the semiempirical MNDO wavefunction. The only differences between this procedure and the ab initio SCF MEP computation lie in the freezing of the inner electrons and in the origin of the first-order density matrix. The statistical analysis of MEPs computed for a large number of molecules from MNDO wavefunction and ab initio SCF wavefunctions obtained using STO-3G, 4-31G, 6-31G, 4-31G*, 6-31G*, and 6-31G** basis sets points out the ability of any wavefunction to reproduce the general topological characteristics of the MEP surfaces. Nevertheless, split-valence basis sets including polarization functions are necessary to obtain accurate MEP minimum energy values. MNDO wavefunction tends to overestimate the MEP minima depth by a constant factor and shows an excellent ability to reflect the relative variation of MEP minima energies derived from a rather sophisticated (6-31G*) basis set, lacking of the shortcomings detected in the semiempirical CNDO approximation.     

Accurate nonrelativistic energies for 2Po states of the Li isoelectronic series

With large Hylleraas-configuration interaction (CI) basis sets highly accurate upper bounds for the lowest ²P^o states of the Li isoelectronic series up to Ne are given. The corresponding Hamiltonian H and the operator Σ_{i < j}, ▿_i ▿_j are transformed into nonorthogonal coordinates expressed in interparticle and angular terms. The evaluation of the occurring integrals is reduced to the calculation of well-known auxiliary integrals. Furthermore some expectation values and isotope energies are calculated. The isotope energies are obtained using perturbation theory in first-order approximation. © 1997 John Wiley & Sons, Inc.     

Thermal and kinetic study of the ferroelectric phase transition in deuterated triglycine selenate

The specific heat and the enthalpy variation of a highly deuterated crystal of ferroelectric triglycine selenate have been measured around its first-order phase transition using the technique square modulated differential thermal analysis (SMDTA). The low temperature variation rate has allowed analyzing the kinetics of the phase transition. Due to an internal crack in the sample, the transition is carried out in two steps and an intermediate region where the transition is blocked and both phases coexist without transformation has been found. The latent heat on cooling (L _c=1.32±0.02 J g^–1) is higher than on heating (L _h=1.08±0.02 J g^–1) due to the thermal hysteresis and the great difference between the specific heat in both phases. Nevertheless, the enthalpy balance is fulfilled on heating and on cooling.     

An analysis of the zero differential overlap approximation. Towards an improved semiempirical MO method beyond it

Summary Some systematic errors of the zero differential overlap (ZDO) approximation in semiempirical molecular orbital (MO) methods are discussed. In electron methods, a power series expansion of the inverse square rootS ^–1/2 of the overlap matrix and application of the Mulliken approximation to the two-electron integrals show that the ZDO Hamiltonian coincides with the Hamiltonian obtained by explicit performance of the Löwdin transformation up to first-order terms of diatomic overlap densities. Higher than first-order terms lead to a systematic up-shift of the canonical MO energies. Although a power series expansion ofS ^–1/2 is no longer possible in all-valence-electron methods, the MO levels resulting from the ZDO approximation are also systematically placed at too low energies, especially the low-lying occupied and the virtual MOs. A method based on explicit performance of the Löwdin transformation and retaining the simplicity of the ZDO approach for the calculation of Fock matrix elements is developed. The parameters of this method are obtained by very simple manipulations of the original ZDO parameters. Numerical calculations show that a considerable improvement of the MO energy spectrum in the inner valence region can be obtained in this way     

Elimination of the diagonalization bottleneck in parallel Direct-SCF methods

Summary It is shown that the matrix diagonalization bottleneck associated with thesequential O(N _BFN ³ ) diagonalization of the fock matrix within each iteration of the Direct-SCF procedure may be eliminated, and replaced instead with a combination ofparallel O(N _BFN ^<4 ) andsequential O(N _Sub ³ ) steps. For large basis sets, the relation N_Sub N_BFN between the dimension of the expansion subspace and the number of basis functions leads to a method of wave-function optimization in which the sequential bottleneck is eliminated. As a side benefit, the second-order iterative procedure on which this method is based displays superior convergence properties, and provides greater insight into the behavior of the energy with respect to orbital variations, than the traditional first-order, fixed-point, iterative approaches. The implementation of this method may be incorporated into essentially any existing Direct-SCF program with only minimal, and localized, changes.     

Thermophysical properties of azomethine ether main-chain liquid crystalline polymers at pressures to 200 MPa

This article reports on an experimental investigation of the equation of state and the transition behavior of main-chain thermotropic liquid crystalline polymers over a wide temperature range, and at pressures to 200 MPa. The materials studied were a series of azomethine ether polymers. A varying number n (= 4, 7, 8, 9, 10 and 11) of methylene spacer units in the backbone provided systematic variation of the structure. Experimental techniques used included high-pressure dilatometry (PVT measurements) to 200 MPa, high-pressure differential thermal analysis, also to 200 MPa, and conventional (atmospheric-pressure) differential scanning calorimetry (DSC). The equation of state of the materials can be well represented by the Tait equation in distinct regions, separated by a glass transition, T_g(P), a first-order transition to a nematic state, T_k-n(P), and a first-order transition to an isotropic melt state T_c(P). The atmospheric pressure values of T_k-n and T_c decreased with increasing number of spacer units and showed a clear odd-even effect. T_g and T_k-n both increased with pressure. The pressure dependence of T_c could not be observed due to the onset of degradation in the same temperature region. On isobaric cooling at 3°C/min, the crystallization from the nematic state occurred a few tens of degrees below T_k-n. This supercooling was independent of pressure for some materials, while for others it increased with increasing pressure. The values of the enthalpy and entropy associated with the first-order transition into the nematic state were lower than those of typical isotropic polymers at their melting transitions. The transition enthalpy did not have any systematic variation with increasing number of spacer units. Values of the transition enthalpy calculated from the Ciapeyron equation did not always agree with the values measured by DSC. This may be due to the two-phase nature of the low-temperature state. At the transition to the isotropic state, the transition enthalpy at P = 0 decreased with n and showed an odd-even effect. © 1994 John Wiley & Sons, Inc.