On the determination of partial differential cross sections for photodetachment and photoionization processes producing polyatomic molecules with electronic states coupled by conical intersections

A formalism is derived for the computation of partial differential cross sections for electron photodetachment and photoionization processes that leave the residual or target molecule in electronic states that are strongly coupled by conical intersections. Because the electronic states of the target are nonadiabatically coupled, the standard adiabatic states approach of solving the electronic Schro?dinger equation for the detached electron at fixed nuclear geometries and then vibrationally averaging must be fundamentally modified. We use a Lippmann-Schwinger equation based approach, which leads naturally to a partitioning of the transition amplitude into a Dyson orbital like part plus a scattering correction. The requisite Green's function is that developed in our previous paper for the direct determination of total integral cross sections. The method takes proper account of electron exchange, possible nonorthogonality of the orbital describing the detached electron, and nonadiabatic effects in the product molecule. The Green's function is constructed in an L(2) basis using complex scaling techniques. The accurate treatment of nonadiabatic effects in the residual molecule is accomplished using the multimode vibronic coupling model. For photodetachment, an approximate approach, which is less computationally demanding, is suggested.     

Proposed standard inelastic differential electron scattering cross sections; Excitation of the 21 P level in He

Differential and integral cross section data for electron-impact excitation of the 2¹ P level in He have been critically reviewed. Experimental and theoretical results have been compared and a set of differential cross sections at 20° scattering angle in the 25 to 500 eV impact energy range has been deduced based on all available information. It is proposed that this set of data represents the most accurate inelastic differential cross sections available at the present time and could be used as a secondary standard for normalization of cross sections.     

Absolute partial cross sections and kinetic energy analysis for the electron impact ionization of ethylene

We present absolute partial electron impact ionization cross sections for ethylene in the electron energy range between threshold and 1000 eV measured with a two sector field double focusing mass spectrometer. Ion kinetic energy distribution functions have been measured at all electron energies by applying a deflection field method. Multiplication of the measured relative cross sections by the appropriately determined discrimination factors lead to accurate relative partial cross sections. Normalization of the sum of the relative partial cross sections to an absolute total cross section gives absolute partial cross section values. The initial kinetic energy distributions of several fragment ions show the presence of two or more contributions that exhibit different electron energy dependencies. Differential cross sections with respect to the initial kinetic energy of the ions are provided and are related to specific ion production channels. The electron threshold energies for the direct and numerous other dissociative ionization channels are determined by quantum chemical calculation and these allow the determination of the total kinetic energy release and the electron energy loss for the most prominent dissociative ionization channels.     

Transport coefficients and cross sections for electrons in water vapour: comparison of cross section sets using an improved Boltzmann equation solution

This paper revisits the issues surrounding computation of electron transport properties in water vapour as a function of E/n(0) (the ratio of the applied electric field to the water vapour number density) up to 1200 Td. We solve the Boltzmann equation using an improved version of the code of Ness and Robson [Phys. Rev. A 38, 1446 (1988)], facilitating the calculation of transport coefficients to a considerably higher degree of accuracy. This allows a correspondingly more discriminating test of the various electron-water vapour cross section sets proposed by a number of authors, which has become an important issue as such sets are now being applied to study electron driven processes in atmospheric phenomena [P. Thorn, L. Campbell, and M. Brunger, PMC Physics B 2, 1 (2009)] and in modeling charged particle tracks in matter [A. Munoz, F. Blanco, G. Garcia, P. A. Thorn, M. J. Brunger, J. P. Sullivan, and S. J. Buckman, Int. J. Mass Spectrom. 277, 175 (2008)].     

Numerical solution of a partial differential equation system describing chemical kinetics and diffusion in a cell with the aid of compartmentalization

To build a kinetic model of a cell with diffusion one has to solve a coupled nonlinear partial differential equation system consisting of several hundred equations. (Several hundred chemical components undergoing several hundred reactions.) To solve this formidable mathematical problem the division of the model cell into compartments (most biochemical reactions take place in a certain part of the cell) was suggested.¹ Solving the differential equation system in one compartment, the results can be used as input at other compartments until mutually consistent solutions are achieved. To test this suggestion 10 coupled chemical reactions with diffusion were investigated in a model that contains three compartments. The results in the case of pure diffusion are in excellent agreement with and without compartmentalization. After this the full problem was treated by compartmentalization using for the solution of the differential equation system a discretization of the concentrations as functions of space and time and the Newton–Raphson iterative procedure. The results obtained give reasonable space and time dependence for the concentrations of all 10 components.     

On the design of experiments for determining ternary mixture free energies from static light scattering data using a nonlinear partial differential equation

We mathematically design sets of static light scattering experiments to provide for model-independent measurements of ternary liquid mixing free energies to a desired level of accuracy. A parabolic partial differential equation (PDE), linearized from the full nonlinear PDE [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)], describes how data noise affects the free energies to be inferred. The linearized PDE creates a net of spacelike characteristic curves and orthogonal, timelike curves in the composition triangle, and this net governs diffusion of information coming from light scattering measurements to the free energy. Free energy perturbations induced by a light scattering perturbation diffuse along the characteristic curves and towards their concave sides, with a diffusivity that is proportional to the local characteristic curvature radius. Consequently, static light scattering can determine mixing free energies in regions with convex characteristic curve boundaries, given suitable boundary data. The dielectric coefficient is a Lyapunov function for the dynamical system whose trajectories are PDE characteristics. Information diffusion is heterogeneous and system-dependent in the composition triangle, since the characteristics depend on molecular interactions and are tangent to liquid-liquid phase separation coexistence loci at critical points. We find scaling relations that link free energy accuracy, total measurement time, the number of samples, and the interpolation method, and identify the key quantitative tradeoffs between devoting time to measuring more samples, or fewer samples more accurately. For each total measurement time there are optimal sample numbers beyond which more will not improve free energy accuracy. We estimate the degree to which many-point interpolation and optimized measurement concentrations can improve accuracy and save time. For a modest light scattering setup, a sample calculation shows that less than two minutes of measurement time is, in principle, sufficient to determine the dimensionless mixing free energy of a non-associating ternary mixture to within an integrated error norm of 0.003. These findings establish a quantitative framework for designing light scattering experiments to determine the Gibbs free energy of ternary liquid mixtures.     

Exact solution (within a double-zeta basis set) of the schrodinger electronic equation for water

Using recently developed theoretical techniques it has been possible to achieve an exact variational solution of the Schrödinger equation within a modest basis set or one-eleltron functions. The full configuration interaction for this system included a total of 256473 ¹A₁ spin-and space-adapted configurations. Comparison with many-body perturbation theory proves to be quite interesting.     

Accurate quantum mechanical calculations of differential and integral cross sections and rate constant for the O+OH reaction using an ab initio potential energy surface

The authors report accurate quantum mechanical studies of the O+OH reaction on the improved Xu-Xie-Zhang-Lin-Guo potential energy surface. The differential cross section was obtained at several energies near the reaction threshold using a time-independent method. The dominant forward and backward peaks in the angular distribution are consistent with a complex-forming mechanism, which is also confirmed by the extensive rotational excitation in the O2 product. However, the asymmetry of these peaks suggests a significant nonstatistical component. The initial state (upsilon i=0, j i=0) specified integral cross section, which was calculated up to 1.15 eV of collision energy using the Chebyshev wave packet method, shows no energy threshold and decreases with the increasing collision energy, consistent with the barrierless nature of the reaction. The resulting rate constant exhibits a negative temperature dependence for T>100 K and decays as the temperature is lowered, in qualitative agreement with available experimental data.     

Calculated electron impact cross sections for the K-shell ionization of Fe,Co, Mn,Ti, Zn,Nb, and Mo atoms using the DM formalism

We used the Deutsch-Märk (DM) formalism to calculate atomic K-shell electron impact ionization cross sections for the elements Fe, Co, Mn, Ti, Zn, Nb, and Mo. The calculated K-shell ionization cross sections are compared with recently measured K-shell ionization cross sections. Good to satisfactory agreement was found for all atoms with the exception of Ti. Moreover, when compared to other available K-shell ionization cross sections for these atoms, calculated using other theoretical methods and semiempirical formulae, the predictions of the DM formalism achieve a level of agreement with experimental data that is as good or better than the predictions from the other methods.     

Theory of variational calculation with a scaling correct moment functional to solve the electronic schrödinger equation directly for ground state one‐electron density and electronic energy

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a nonlinear, approximate density functional of a three spatial dimension one‐electron density for an N electron system which is tractable in practice, is a long‐desired goal in electronic structure calculation. In a seminal work, Parr et al. (Phys. Rev. A 1997, 55, 1792) suggested a well behaving density functional in power series with respect to density scaling within the orbital‐free framework for kinetic and repulsion energy of electrons. The updated literature on this subject is listed, reviewed, and summarized. Using this series with some modifications, a good density functional approximation is analyzed and solved via the Lagrange multiplier device. (We call the attention that the introduction of a Lagrangian multiplier to ensure normalization is a new element in this part of the related, general theory.) Its relation to Hartree–Fock (HF) and Kohn–Sham (KS) formalism is also analyzed for the goal to replace all the analytical Gaussian based two and four center integrals (∫g_i( r ₁)g_k( r ₂)rd r ₁d r ₂, etc.) to estimate electron‐electron interactions with cheaper numerical integration. The KS method needs the numerical integration anyway for correlation estimation. © 2012 Wiley Periodicals, Inc.     

On a partial differential equation method for determining the free energies and coexisting phase compositions of ternary mixtures from light scattering data

In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.     

Multielement x-ray fluorescence analysis of ores and products of their processing using the standard background method with a modified universal equation

A new algorithm is proposed for x-ray fluorescence analysis by the standard background method using a modified universal equation common for all analytes. In the new version of the universal equation, complicating factors can be accounted for by introducing additional correcting members into the analytical parameter. The developed procedure allows the analyst to give up numerous constraint equations, simplify the calibration of spectrometers, and improve the rapidity of the analysis of multielement ores and products of their processing without a drop in the accuracy of the results.     

An evolved explanation for the molecular geometry and electronic structure of diphenyl-substituted cyclic trimethylenemethane in the ground state: a nearly planar conformation with a considerably localized electronic state

[structures: see text] We reinvestigated the molecular geometry and electronic structure of the diphenyl-substituted, five-membered cyclic trimethylenemethane (TMM) diradical (Berson's TMM, 3**) using UV/VIS absorption and emission spectroscopy combined with density functional theory (DFT) and time-dependent (TD)-DFT calculations. Two intense absorption bands, A and B, with lambda(ab) at 298 and 328 nm, respectively, a weak absorption band C, with lambda(ab) at 472 nm, and an intense emission band D, with lambda(em) at 491 nm, were observed for 3**. By comparing the spectrum of 3** with those of the 1,1-diphenylethyl (7*) and cyclopent-2-en-1-yl (9*) radicals, it was found that bands B, C, and D originated from the diphenylmethyl radical moiety (subunit I), while band A should most likely be assigned to an electronic transition related to an interaction between subunit I and residual subunit II, the cyclopentenyl radical moiety. An UB3LYP/cc-pVDZ calculation indicated that, in the ground state, the two unpaired electrons of 3** are mainly localized in subunits I and II, respectively, and the interaction between them is inefficient, despite the nearly planar conformation (theta = +23.5 degrees). Furthermore, a TD-UB3LYP/cc-pVDZ calculation suggested that absorption band A is assigned to an electronic transition involved with enhancement of the electron density of the C-2-C-3 bond. Substituent effects on the absorption and emission spectra of 3** using 11** and 13** support the conclusion based on the experiments and calculations. Therefore, we propose an evolved explanation for the molecular geometry and electronic structure of the ground state of 3** in a low-temperature matrix, a nearly planar conformation with a considerably localized electronic state, which alone accounts for the spectroscopic characteristics.     

Elucidating the vibrational spectra of hydrogen-bonded aggregates in solution: electronic structure calculations with implicit solvent and first-principles molecular dynamics simulations with explicit solvent for 1-hexanol in n-hexane

Fourier transform infrared spectroscopy is a popular method for the experimental investigation of hydrogen-bonded aggregates, but linking spectral information to microscopic information on aggregate size distribution and aggregate architecture is an arduous task. Static electronic structure calculations with an implicit solvent model, Car-Parrinello molecular dynamics (CPMD) using the Becke-Lee-Yang-Parr (BLYP) exchange and correlation energy functionals and classical molecular dynamics simulations for the all-atom version of the optimized parameters for liquid simulations (OPLS-AA) force field were carried out for an ensemble of 1-hexanol aggregates solvated in n-hexane. The initial configurations for these calculations were size-selected from a distribution of aggregates obtained from a large-scale Monte Carlo simulation. The vibrational spectra computed from the static electronic structure calculations for monomers and dimers and from the CPMD simulations for aggregates up to pentamers demonstrate the extent of the contribution of dangling or nondonating hydroxyl groups found in linear and branched aggregates to the "monomeric" peak. Furthermore, the computed spectra show that there is no simple relationship between peak shift and aggregate size nor architecture, but the effect of hydrogen-bond cooperativity is shown to differentiate polymer-like (cooperative) and dimer-like (noncooperative) hydrogen bonds in the vibrational spectrum. In contrast to the static electronic structure calculations and the CPMD simulations, the classical molecular dynamics simulations greatly underestimate the vibrational peak shift due to hydrogen bonding.     

Explicit form of Pauli potential for direct derivation of pair density from a two-particle differential equation for the quintet state of four electrons with harmonic interparticle interactions

Recently, an analytic two-particle density matrix (2DM) has been derived for the quintet state of four electrons interacting via two-body harmonic forces. Here we use this 2DM to extract the exact pair density $\Gamma (\mathbf{r}_1,\mathbf{r}_2)$ . This is then employed in the known two-particle partial differential equation for the pair density amplitude to extract the Pauli potential $v_P(\mathbf{r}_1,\mathbf{r}_2)$ for this quintet state.     

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 1. Introductory exploration of the finite-difference adaptive moving grid solution of the one-dimensional fast homogeneous reaction-diffusion problem with a reaction layer

An effective finite difference solution of extremely fast homogeneous reaction-diffusion problems in electrochemical kinetics might well be achieved by using dynamically adaptive grid techniques, instead of considering limiting cases resulting from steady- state or equilibrium assumptions and traditional calculations on fixed grids. A simple adaptive moving-grid strategy has been applied to the example of linear potential scan voltammetry and a catalytic mechanism in one-dimensional geometry. Although not entirely satisfactory at present, the strategy permits homogeneous rate constants greater by as much as 20 orders of magnitude than the maximum possible values in corresponding fixed-grid calculations using the same number of space grid points at not much greater computational cost.