Systematic Gaussian basis-set limit using completeness-optimized primitive sets. A case for magnetic properties

We discuss the connection between the completeness of a basis set, measured by the completeness profiles introduced by Chong (Can J Chem 1995, 73, 79) at a certain exponent interval, and the possibility of reproducing molecular properties that arise either in the region close to the atomic nuclei or in the valence region. We present a scheme for generating completeness-optimized Gaussian basis sets, in which a preselected range of exponents is covered to an arbitrary accuracy. This is done by requiring Gaussian functions, the exponents of which are selected without reference to the atomic structure, to span the range with completeness profile as close to unity as wanted with as few functions as possible. The initial exponent range can be chosen suitable for calculations of molecular energetics or other valence-like properties. By extending the exponent range, properties requiring augmentation of the basis at a given angular momentum value and/or in a given distance range from the nucleus may be straightforwardly and systematically treated. In this scheme a universal, element-independent exponent set is generated in an automated way. The relation of basis-set completeness and performance in the calculation of magnetizability, nuclear magnetic shielding, and spin-spin coupling is tested with the completeness-optimized primitive sets and literature basis sets.     

The equation of state for an interface during the adsorption of organic substances on an electrode

The state of the interface between a metal and a solution of an electrolyte containing a neutral surfactant was investigated using a method alternative to the traditional thermodynamic approach. The method was based on the concept that there was a stability limit of a surfactant on an electrode, and the corresponding state could be described in terms of the catastrophe theory. The surface pressure was approximated by the Whitney polynomial in powers of the de Donder parameter (completeness of adsorption) with the coefficients depending on the chemical potential and polarization of the interface. The equation of state and the equation for the stability limit were obtained from the condition of zero first and second derivatives. These equations correctly described the results of electrocapillary measurements in the spirit of the law of corresponding states. The correlation between surface pressure maxima and critical stability potentials predicted by the theory was substantiated by the electrocapillary measurements data provided that the inflexions of surface pressure curves calculated from the electrocapillary data were related to the limiting stability at which the competing forces are balanced during the adsorption of surfactants. A simple equation for surface pressure was suggested in the form of a function of the state of thermodynamic parameters and completeness of adsorption. This function described the state of a surfactant at the interface. Equilibrium equations were derived for the state of a surfactant and the spinodal.     

热力学方程在计算热力学函数中的应用

热力学方程是包含热力学基本方程、热力学状态方程以及吉布斯-亥姆霍兹方程在内的广泛的一组方程。文中对目前物理化学教材中很少提及的其他热力学状态方程进行了系统的总结和推导,同时强调理论的系统性和完整性在教学中的重要性,并在此基础上导出了物理化学教材中很少见到的吉布斯-亥姆霍兹第一方程及其在表面化学中的应用。在物理化学教学中需要特别强调特征变量的重要性,以及特征变量和化学反应条件之间的内在关系,并注意与交叉学科物理专业的热力学和统计物理学的衔接。     

Correlations between two sets of angular relation equations

It is shown here that the angular relation equations between direct and reciprocal vectors are very similar to the angular relation equations in Euler’s theorem. These two sets of equations are usually treated separately as unrelated equations in different fields. In this careful study, the connection between the two sets of angular equations is revealed by considering the cosine rule for the spherical triangle. It is found that understanding of the correlation is hindered by the facts that the same variables are defined differently and different symbols are used to represent them in the two fields. Understanding the connection between different concepts is not only stimulating and beneficial, but also a fundamental tool in innovation and research, and has historical significance. The background of the work presented here contains elements of many scientific disciplines. This work illustrates the common ground of two theories usually considered separately and is therefore of benefit not only for its own sake but also to illustrate a general principle that a theory relevant to one discipline can often be used in another. The paper works with chemistry related concepts using mathematical methodologies unfamiliar to the usual audience of mainstream experimental and theoretical chemists.     

Volumetric and refractive properties of binary mixtures containing 1,3-dioxolane and isomeric chlorobutanes

The relation between refractive index deviations and excess volumes for binary mixtures formed by a cyclic ether and a haloalkane has been tested using several methods: refractive index mixing rules and equations of state. Refractive index deviations, excess volumes and molar refractions have been calculated from experimental data of refractive indices and densities at two temperatures 298.15 and 313.15 K. Results obtained have been discussed in terms of intermolecular interactions. Refractive indices were compared with those predicted by several mixing rules. Excess volumes have also been correlated using several cubic equations of state and finally a relation between parameter b from equations of state and molar refraction has been provided.     

Generalization of membrane reflection coefficients for nonideal,nonisothermal, multicomponent systems with external forces and viscous flow

The notion of reflection coefficients is generalized from dilute ideal solutions to apply to virtually any kind of solution and any kind of membrane whose properties are not affected by the solution. The crucial points in the generalization are the selection of a suitable definition of partial osmotic pressure and the inclusion of separative viscous flow as a transport mechanism (necessary to obtain the correct semipermeability limit). The latter can lead to loss or concealment of Onsager reciprocity, so that the reflection coefficients for volume flow and for solute fluxes are not necessarily equal. Two choices of reference state are presented: the traditional choice of zero reflection coefficient for solvent volume flow, and a more symmetric choice of an average reflection coefficient equal to zero. Several examples are worked out for binary and ternary solutions and compared with results from experiments and from model calculations. Thermal gradients are included for completeness, but play no essential role in the reflection coefficients. The development is given entirely in terms of differential equations of transport, and problems and inconsistencies associated with the use of finite-difference equations are briefly discussed.     

Comment on the recent controversy over the theory of chirality functions

Derflinger,et al., have criticized the concept of qualitative completeness on the grounds that a chirality function may be qualitatively complete and still vanish identically for certain nonracemic, non-isomeric mixtures. It is pointed out that the functions considered by them are ones in which the full chirality function splits into parts which can be thought of as contributions from “effective fragments” of the molecule, and that their mixtures are indeed racemic in these fragments. The function can be augmented in a very simple way so as not to vanish for such mixtures, and the idea of qualitative completeness is in no way disturbed by this process.     

System eigenmobilities in zone electrophoresis: A general moving-boundary approach

System zones in capillary zone electrophoresis represent an important topic, very interesting from the theoretical point but also important for practice. This paper is aimed at contributing to the understanding of system zones as one of the very fundamental properties of electrophoretic systems, by developing an alternative approach to the so far used vector-matrix model of calculation of system mobilities (system eigenmobilities). The presented model is based on the solution of the differential form of the moving-boundary equation. The result for acid-base systems is a single algebraic equation valid universally for a zone comprising any number of constituents (mono- or polyhydric strong or weak acids or bases and/or amphoteric compounds). The value of the described solution against previous models consists in its explicit form, expressing the system eigenmobility of a homogeneous zone of given composition as a function of only known quantities. The obtained equation is shown to be the common source of various simplified equations obtained in the past for particular simple systems. The applicability of the simplified equations is discussed in terms of completeness of the results (number of output system eigenmobilities). For non-buffered systems, the occurrence of a previously unreported non-zero value of system eigenmobility is discussed that is equal to the arithmetic average of mobilities of the solvent ions.     

Neural Wave Representation in Early Vision

The neural wave representation which describes spatiotemporal information in early vision is proposed in terms of extended Gabor functions in this paper. Its reasonability, completeness and uncertainty in spatiotemporal domains and their frequency domains were investigated. The ranges of these parameters in this representation were estimated on electrophysiological and psychophysical data. We derived the partial differential equations which the neural wave satisfied. The general significance of 3D neural wave representation has been explored.     

Influence of irradiation on oxygen-containing compounds in the presence of olefins

The relation between the structure of olefins and oxygen-containing compounds and their reactivity in radical addition, substitution and isomerization processes under γ-irradiation in a wide temperature range was examined. Correlation equations for the determination of elementary rate constants and their Arrhenius parameters are given. The compensation effect is discussed. A mechanism of product formation is proposed and the corresponding kinetic equations are derived from the steady-state principle.     

Approximate relativistic Hartree-Fock equations and their solution within a minimum basis set of slater-type functions

Relativistic closed-shell atoms are treated by the use of a specific approximation for the small component of the one-electron Dirac spinors. It is assumed that the large and the small component are interconnected by a parameter-dependent relation which is formally analogous to that of the one-electron system. Subject to this constraint, the total energy is varied with respect to the large components. The resulting eigenvalue equations for the large components contain only regular potential terms and reduce to the familiar Hartree-Fock equations in the limit of infinite velocity of light. Analytical solution of these approximate relativistic Hartree-Fock equations is achieved using a minimum basis set of Slater-type functions for the expansion of the radial part of the large components. Total relativistic energies, orbital energies, orbital exponents and mean radii are calculated for the ground states of He, Be, Ne, Mg, Ar, Kr Xe and Cu⁺. Dedicated to Prof. O. E. Polansky on occasion of his 60th birthday.     

Koopmans' theorem in the ROHF method: canonical form for the Hartree-Fock Hamiltonian

Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.     

Completeness and linear independence of basis sets used in quantum chemistry

Infinite sets of functions in Hilbert space are characterized by their completeness properties and the extent of linear independence. Different measures of linear independence such as orthonormality, Gram's determinant, the special measure of linear independence, and the asymptotic dimension are related to each other and with the degrees of completeness such as overcompleteness, exact completeness, and incompleteness as far as possible.     

Point excesses in the theory of ordinary and micellar solutions

Point excesses of substances and thermodynamic properties and the role they play in the solvation and binding of counterions in solutions of electrolytes, including micellar systems, are analyzed. A complete system of fundamental thermodynamic equations for point excesses is formulated. Statistical mechanics equations that relate point excesses of substances to the electrochemical potentials and concentrations of components are derived. For ionic micellar systems, a relation between point excesses and charges and concentrations of ions and micelles is obtained. The results are substantiated by direct calculations of point excesses with the use of the Debye-Hückel method.     

The Coincidence of Mass Integrals g(α)

The idea of the generation of thermokinetic models on the basis of the analysis of kinetic equations (isothermal conditions) and thermokinetic equations (dynamic conditions) is presented. The method resembles that used in polisothermal conditions, which consists in analysis of the relation of the equilibrium conversion degree vs. temperature. The interpretation of the coincidence of mass integrals g(α) in the relation α vs. temperature has been attempted. This revised version was published online in August 2006 with corrections to the Cover Date.     

Spin projected extended Hartree-Fock equations

The (spin projected) extended Hartree-Fock equations are derived and discussed in the general (“many λ_i”) case. The derivation is based on the corresponding generalized Brillouin theorem, substituting in its expression the spin projected DODS Slater determinant as a linear combination of determinants. The expressions obtained for the elements of the different matrices ε^cd (c = a or b, and d = a or b) occurring in the equations are analyzed. The transformation of the equations into the form of a pseudoeigenvalue problem and the LCAO form of the equations are given, too. Finally the relation to the Goddard's GF equations is discussed in detail.     

Fluctuation-dissipation relations for polymer systems. I. The molecular weight dependence of the viscosity

Generalised Langevin equations are used to describe the motion of interacting polymer molecules. In these equations the many-body aspects of the problem are incorporated into generalised friction functions and random forces. The so-called second fluctuation-dissipation relation gives a general relation between these two quantities and enables us to relate the spatial correlations present in the environment to a normal mode dependent friction coefficient and force constant. We then show how the various regimes of molecular weight dependence of the viscosity can be understood in a fairly general and non-specific way in terms of the spatial correlations of the random forces representing the rest of the system of polymer and solvent molecules. In particular a molecular weight dependence of M³ is shown to be a general feature of a spatially corelated environment.     

Dilated Weyl theory applied to a set of N-coupled singular second-order differential equations

Weyl's theory for a set of N-coupled singular second-order differential equations is analyzed in relation to S-matrix theory and a dilated version is presented. Applications of this theory to two single channel scattering model problems and a two-channel model problem are given. Some implications of the present theory are discussed.     

Order parameters for macromolecules: application to multiscale simulation

Order parameters (OPs) characterizing the nanoscale features of macromolecules are presented. They are generated in a general fashion so that they do not need to be redesigned with each new application. They evolve on time scales much longer than 10(-14) s typical for individual atomic collisions/vibrations. The list of OPs can be automatically increased, and completeness can be determined via a correlation analysis. They serve as the basis of a multiscale analysis that starts with the N-atom Liouville equation and yields rigorous Smoluchowski/Langevin equations of stochastic OP dynamics. Such OPs and the multiscale analysis imply computational algorithms that we demonstrate in an application to ribonucleic acid structural dynamics for 50 ns.     

Vibrational absorption spectra from vibrational coupled cluster damped linear response functions calculated using an asymmetric Lanczos algorithm

We report the theory and implementation of vibrational coupled cluster (VCC) damped response functions. From the imaginary part of the damped VCC response function the absorption as function of frequency can be obtained, requiring formally the solution of the now complex VCC response equations. The absorption spectrum can in this formulation be seen as a matrix function of the characteristic VCC Jacobian response matrix. The asymmetric matrix version of the Lanczos method is used to generate a tridiagonal representation of the VCC response Jacobian. Solving the complex response equations in the relevant Lanczos space provides a method for calculating the VCC damped response functions and thereby subsequently the absorption spectra. The convergence behaviour of the algorithm is discussed theoretically and tested for different levels of completeness of the VCC expansion. Comparison is made with results from the recently reported [P. Seidler, M. B. Hansen, W. Gyo?rffy, D. Toffoli, and O. Christiansen, J. Chem. Phys. 132, 164105 (2010)] vibrational configuration interaction damped response function calculated using a symmetric Lanczos algorithm. Calculations of IR spectra of oxazole, cyclopropene, and uracil illustrate the usefulness of the new VCC based method.