New method of calculating idempotent density matrices and the total energy of many-electron systems

We have developed a method of successive approximations for the direct calculation of the density matrix and total energy of a many-electron system. The method can be used when the energy is a linear functional of the density matrix and the density matrix itself satisfies the idempotent conditions. A mathematical proof is given of the convergence of the successive approximations to the exact solution.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 26, No. 1, pp. 1–5, January–February, 1990.The author thanks H. M. Mestechkina and I. V. Abarenkov for useful discussions.     

Adiabatic and self-consistent-field approximations for coupled vibrations: A simple two-mode comparison

A simple coupled harmonic oscillator model is used to test the SCF and adiabatic approximations for coupled-mode systems. Both approximations are quite good; both fail when the unperturbed frequencies of the two modes approach degeneracy. While for two-mode systems the approximations are roughly comparable, for real molecules, the more easily applied SCF is preferable. Semiclassical quantization may be used with both approximations.     

Range of validity of distorted wave Born and impulse approximations for (e, 2e)

The formulation of approximations for the calculation of the (e, 2e) reaction on atoms is considered. The range of validity of the two common working approximations, the distorted wave Born and impulse approximations, is illustrated. A feasible approximation that treats the reaction in more detail is derived.     

Kinetic analysis of solid-state reactions: Evaluation of approximations to temperature integral and their applications

The temperature integral, which has no exact analytical solution, is involved in the analysis of the experiment data obtained under nonisothermal conditions. Some approximations for the temperature integral have been proposed in the literature for the determination of the kinetic parameters, in particular the activation energy. Those approximations are classified into two categories, that is, exponential and rational approximations. The precision of them for estimating the temperature integral was evaluated within a certain continuous range rather than at several discrete points. Some applications of the approximations in the kinetic methods were presented. The relative errors of the activation energy and pre-exponential factor with four rational approximations by employing model-fitting method were calculated. The relative errors of the activation energy for a series of conversion rate with four rational and four exponential approximations by employing linear integral isoconversional methods were evaluated.     

Dynamic Mobility of Particles with Thick Double Layers in a Nondilute Suspension

The dynamic mobility of a nondilute suspension of spherical particles is investigated in the case where the thickness of the electrical double layer around each particle is comparable to the particle radius. A formula is obtained for the O(φ) correction in a random suspension of particles with volume fraction φ, involving an integral over the dynamic mobility of a pair of spheres. This formula is then evaluated using both analytical approximations and numerical results previously obtained for the pair mobilities and valid for low surface potentials. The effect of double-layer thickness on the O(φ) coefficient is most pronounced at low frequencies, and lessens once the hydrodynamic penetration depth is smaller than the particle radius. Various approximations are considered that use the O(φ) result to predict the dynamic mobility in concentrated suspensions, and at high frequencies these approximations are shown to give results qualitatively different from those of recent cell models. Copyright 2000 Academic Press.     

Fourier analysis,correlation functions and nonadiabatic electron transfer: Wavepackets and exact representations

Summary We investigate the validity of several common approximations in the analysis of nonadiabatic intramolecular electron transfer rate constants. Utilizing the Fourier representation of the golden rule form, we study the evolution of the vibrational correlation function that represents the density-of-states-weighted Franck-Condon factor. In particular, we test the validity of the perturbation theoretic golden rule form and of the Gaussian wavepacket representation for the vibrational wavefunctions against numerically exact quantum mechanical propagations. Although specific cases are found in which both of these break down, for a wide range of conditions (including anharmonic behavior and frequency changes), both the Gaussian wavepacket representation and the golden rule are excellent approximations.     

Practical approximation to reaction probabilities from selected initial quantum states

A method for the economical of reasonable approximations to the probability of reaction from specified internal states of reactants is proposed for elementary exchange reactions. It requires the solution of the close coupling equations in the reactant channel only and the imposition of boundary conditions on a hypersurface at the transition state. A numerical comparison with the complete quantum calculation for the planar H + H₂ reaction is presented and the agreement is shown to be good.     

Global analysis of composite particles

The theory of vibrations of a composite particle when vibrational amplitudes are not constrained to be small according to the Eckart conditions is developed using the methods of differential topology. A global classical Hamiltonian appropriate for this system is given, and for the case of the molecular vibration–rotation problem, it is transformed into a global quantum Hamiltonian operator. It is shown that the zeroth-order term in the global Hamiltonian operator is identical to the Wilson–Howard Hamiltonian; higher-order terms are shown to give successively better approximations to the large amplitude problem. Generalized Eckart conditions are derived for the global classical Hamiltonian; the quantum equivalent of these conditions along with the quantum equivalent of the Eckart conditions are given. The spectrum of the global Hamiltonian operator is discussed and it is shown that the calculation of the vibration–rotation energy states of the system reduces to the same straight-forward procedure, the solution of a secular determinant, as was carried out for the Wilson–Howard Hamiltonian at a later time by Nielsen.     

Numerical and analytical studies of the electrical conductivity of a concentrated colloidal suspension

In the past few years, different models and analytical approximations have been developed facing the problem of the electrical conductivity of a concentrated colloidal suspension, according to the cell-model concept. Most of them make use of the Kuwabara cell model to account for hydrodynamic particle-particle interactions, but they differ in the choice of electrostatic boundary conditions at the outer surface of the cell. Most analytical and numerical studies have been developed using two different sets of boundary conditions of the Neumann or Dirichlet type for the electrical potential, ionic concentrations or electrochemical potentials at that outer surface. In this contribution, we study and compare numerical conductivity predictions with results obtained using different analytical formulas valid for arbitrary zeta potentials and thin double layers for each of the two common sets of boundary conditions referred to above. The conductivity will be analyzed as a function of particle volume fraction, phi, zeta potential, zeta, and electrokinetic radius, kappaa (kappa(-1) is the double layer thickness, and a is the radius of the particle). A comparison with some experimental conductivity results in the literature is also given. We demonstrate in this work that the two analytical conductivity formulas, which are mainly based on Neumann- and Dirichlet-type boundary conditions for the electrochemical potential, predict values of the conductivity very close to their corresponding numerical results for the same boundary conditions, whatever the suspension or solution parameters, under the assumption of thin double layers where these approximations are valid. Furthermore, both analytical conductivity equations fulfill the Maxwell limit for uncharged nonconductive spheres, which coincides with the limit kappaa --> infinity. However, some experimental data will show that the Neumann, either numerical or analytical, approach is unable to make predictions in agreement with experiments, unlike the Dirichlet approach which correctly predicts the experimental conductivity results. In consequence, a deeper study has been performed with numerical and analytical predictions based on Dirichlet-type boundary conditions.     

Relation of the field-theoretical method for molecular scattering to the distorted wave formalism

The results of the field-theoretical method for molecular scattering are obtained without the formalism of field theory. Given the Feshbach optical potential the exact transition potential is obtained on the basis of the distorted wave method. A series of approximations to these two operators is presented which allow one to get the results obtained previously by successive approximations in the field-theoretical method.     

New Approximate Model for Nonlinear Adsorption and Concentration Dependent Surface Diffusion in a Single Particle

A new approximate model for nonlinear adsorption (Langmuir model) and concentration dependent surface diffusion (HIO model) in a single particle was derived, based on a parabolic concentration profile assumption for the summation of the gas and adsorbed phases. The surface diffusivity was approximated with the adsorbed phase concentration evaluated at the surface of the particle, as the average of the adsorbed phase concentration, and as the average of the first two approximations. Overall, the approximate model based on the average of the first two approximations compared the best with the exact solution for a wide variety of systems and conditions.     

Comparison of perturbative and multiconfigurational electron propagator methods

Ionization energies below 20 eV of 10 molecules calculated with electron propagator techniques employing Hartree-Fock orbitals and multiconfigurational self-consistent field orbitals are compared. Diagonal and nondiagonal self-energy approximations are used in the perturbative formalism. Three diagonal methods based on second- and third-order self-energy terms, all known as the outer valence Green's function, are discussed. A procedure for selecting the most reliable of these three versions for a given calculation is tested. Results with a polarized, triple ζ basis produce root mean square errors with respect to experiment of approximately 0.3 eV. Use of the selection procedure has a slight influence on the quality of the results. A related, nondiagonal method, known as ADC (3), performs infinite-order summations on several types of self-energy contributions, is complete through third-order, and produces similar accuracy. These results are compared to ionization energies calculated with the multiconfigurational spin-tensor electron propagator method. Complete active space wave functions or close approximations constitute the reference states. Simple field operators and transfer operators pertaining to the active space define the operator manifold. With the same basis sets, these methods produce ionization energies with accuracy that is comparable to that of the perturbative techniques. © 1996 John Wiley & Sons, Inc.     

Influence of longitudinal relaxation time on double modulation ESR spectra

Extremely narrow lines, observed by the double modulation ESR technique in systems with an inhomogeneously broadened spectrum, are explained in terms of higher-order processes in non-linear conditions. Under proper approximations the shape of these lines is accounted for by the longitudinal rather than the transverse relaxation time.     

Approximate charge density localization of molecular orbitals

A NDO approximate procedure based on the indirect intrinsic ab initio localization method of von Niessen is developed. It is shown that only when the NDO approximations are introduced at the two electron level, expressions are obtained which are the charge density counterpart of those found in the approximate energy localization methods. The results of these two methods are quite similar both in the CNDO and INDO approximations. The indeterminacies observed in the CNDO localization for unsaturated systems and for molecules with two or three lone pairs on the same atom, are removed by localizing up to an INDO level. The approximate charge density localization is however computationally much easier than the approximate energy localization method and should be more appropriate in LMO studies of large organic molecules.     

A theory of molecules in molecules

A theory of molecules in molecules is presented, which permits the computation of the wave function of a molecule from the wave functions of fragment molecules by transferring some of the localized molecular orbitals of the fragments and recalculating the orbitals in the region of interaction. A projection operator is used to obtain orthogonality of the orbitals to be determined to the transferred and fixed orbitals. Additional approximations allow the reduction of the dimension of the matrices to be diagonalized and the neglect of a part of the basic integrals, which can lead to a considerable saving in the computation time. The justification of these approximations will be investigated for the case of the molecules Be-Be, Li₂-Li₂, and for the calculation of the rotational barrier in C₂H₆.     

New exact relations for improving the exchange and correlation potentials

For the purpose of improving present approximations to the exchange and correlation potentials, newly derived properties of the exact exchange and correlation potentials are summarized. Present approximations are not expected to generally satisfy these properties. The summarized properties include relations at the Fermi level, low-density requirements, and a new density functional formula for computing ionization energies. © 1995 John Wiley & Sons, Inc.