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Here, we provide a detailed account of novel experiments employing electron-ion coincidence imaging to discriminate chiral molecules. The full three-dimensional angular scattering distribution of electrons is measured after photoexcitation with either left or right circular polarized light. The experiment is performed using a simplified photoelectron-photoion coincidence imaging setup employing only a single particle imaging detector. Results are reported applying this technique to enantiomers of the chiral molecule camphor after three-photon ionization by circularly polarized femtosecond laser pulses at 400 nm and 380 nm. The electron-ion coincidence imaging provides the photoelectron spectrum of mass-selected ions that are observed in the time-of-flight mass spectra. The coincident photoelectron spectra of the parent camphor ion and the various fragment ions are the same, so it can be concluded that fragmentation of camphor happens after ionization. We discuss the forwardbackward asymmetry in the photoelectron angular distribution which is expressed in Legendre polynomials with moments up to order six. Furthermore, we present a method, similar to one-photon electron circular dichroism, to quantify the strength of the chiral electron asymmetry in a single parameter. The circular dichroism in the photoelectron angular distribution of camphor is measured to be 8% at 400 nm. The electron circular dichroism using femtosecond multiphoton excitation is of opposite sign and about 60% larger than the electron dichroism observed before in near-threshold one-photon ionization with synchrotron excitation. We interpret our multiphoton ionization as being resonant at the two-photon level with the 3s and 3p Rydberg states of camphor. Theoretical calculations are presented that model the photoelectron angular distribution from a prealigned camphor molecule using density functional theory and continuum multiple scattering X alpha photoelectron scattering calculations. Qualitative agreement is observed between the experimental results and the theoretical calculations of the Legendre moments representing the angular distribution for the two enantiomers. The electron-ion coincidence technique using multiphoton ionization opens new directions in table-top analytical mass-spectrometric applications of mixtures of chiral molecules. [ABSTRACT FROM AUTHOR]

A method in time domain is proposed to investigate resonance Raman spectra of absorbed molecules on semiconductor surfaces. The charge transfer at the molecule-surface interface is incorporated with the use of an Anderson-Newns type Hamiltonian, where the surface continuum state is dealt with an expansion of Legendre polynomials for fast numerical convergence. From a model test, it is found that the intensities of Raman modes in the sole molecule generally decrease as the molecule-surface interaction is switched on, except that the energy gaps between the molecular excited state and the bottom of the band are at special values. New Raman peaks which are not observed in the sole molecule, however, appear and are greatly enhanced. The enhancement depends on the electronic coupling and the energy gap. It is also highly sensitive to the mode-specific reorganization energy in the charge transfer state, and a thousand times enhancement can be obtained at a certain reorganization energy. The corresponding electron dynamics is revealed by the population decay from the absorbed molecule. [ABSTRACT FROM AUTHOR]

We present a new basis set as an alternative to Legendre polynomials for the variational treatment of bending vibrational degrees of freedom in order to highly reduce the number of basis functions. This basis set is inspired from the harmonic oscillator eigenfunctions but is defined for a bending angle in the range θ∈[0:π]. The aim is to bring the basis functions closer to the final (ro)vibronic wave functions nature. Our methodology is extended to complicated potential energy surfaces, such as quasilinearity or multiequilibrium geometries, by using several free parameters in the basis functions. These parameters allow several density maxima, linear or not, around which the basis functions will be mainly located. Divergences at linearity in integral computations are resolved as generalized Legendre polynomials. All integral computations required for the evaluation of molecular Hamiltonian matrix elements are given for both discrete variable representation and finite basis representation. Convergence tests for the low energy vibronic states of HCCH++, HCCH+, and HCCS are presented. [ABSTRACT FROM AUTHOR]

The dynamical properties of a complex system incorporate contributions from the diverse components from which it is constituted. To study this relationship in a multicomponent system, relaxation times based on rotation autocorrelation functions in molecular dynamics simulations were analyzed for molecules in two sets of unmodified and polymer-modified model asphalt/bitumen systems over 298–473 K. The model asphalt systems were proposed previously to approximate the chemical and mechanical properties of real asphalts. Relaxations were modeled using a modified Kaulrausch–Williams–Watts function and were based on the third Legendre polynomial of normal vector time correlation functions for aromatic species (asphaltene, polar aromatic, naphthene aromatic). Both the end-to-end vector and the longest axis eigenvector of the radius of gyration matrix were used for time correlation functions of chain molecules (C22, polystyrene). Decreases in temperature induced large increases in relaxation time consistent with the Vogel–Fulcher–Tammann equation. The presence of a polymer slowed the decay of each correlation function to some extent. The product of relaxation time and diffusion coefficient revealed qualitative differences between larger and smaller molecules in the same system. These relaxation mechanisms remained coupled for small molecules, while the larger asphaltene and polymer molecules revealed significant slowdowns in rotation compared to translational diffusion at lower temperatures. Smaller values of the stretched exponential parameter β for asphaltenes compared to smaller molecules suggested a broader range of relaxation times and were consistent with this distinction. Difficulties in converging polymer chain relaxation times are discussed in terms of fluctuations in the magnitude and orientation of the end-to-end vector and chain axis eigenvector. Viscosity results suggested by the Debye–Stokes–Einstein relationship are consistent with trends shown in the literature for true bitumen systems. [ABSTRACT FROM AUTHOR]

Self-avoiding polymers in strictly two-dimensional (d=2) melts are investigated by means of molecular dynamics simulation of a standard bead-spring model with chain lengths ranging up to N=2048. The chains adopt compact configurations of typical size R(N)∼Nν with ν=1/d. The precise measurement of various distributions of internal chain distances allows a direct test of the contact exponents Θ0=3/8, Θ1=1/2, and Θ2=3/4 predicted by Duplantier. Due to the segregation of the chains the ratio of end-to-end distance Re(N) and gyration radius Rg(N) becomes Re2(N)/Rg2(N)≈5.3<6 for N>100 and the chains are more spherical than Gaussian phantom chains. The second Legendre polynomial P2(s) of the bond vectors decays as P2(s)∼1/s1+νΘ2, thus measuring the return probability of the chain after s steps. The irregular chain contours are shown to be characterized by a perimeter length L(N)∼R(N)dp of fractal line dimension dp=d-Θ2=5/4. In agreement with the generalized Porod scattering of compact objects with fractal contour, the Kratky representation of the intramolecular structure factor F(q) reveals a strong nonmonotonous behavior with qdF(q)∼1/(qR(N))Θ2 in the intermediate regime of the wave vector q. This may allow to confirm the predicted contour fractality in a real experiment. [ABSTRACT FROM AUTHOR]

The density matrix ρ(r,r′) of a spherically symmetric system can be expanded as a Fourier–Legendre series of Legendre polynomials Pl(cos θ=r·r′/rr′). Application is here made to harmonically trapped electron pairs (i.e., Moshinsky’s and Hooke’s atoms), for which exact wavefunctions are known, and to the helium atom, using a near-exact wavefunction. In the present approach, generic closed form expressions are derived for the series coefficients of ρ(r,r′). The series expansions are shown to converge rapidly in each case, with respect to both the electron number and the kinetic energy. In practice, a two-term expansion accounts for most of the correlation effects, so that the correlated density matrices of the atoms at issue are essentially a linear functions of Pl(cos θ)=cos θ. For example, in the case of Hooke’s atom, a two-term expansion takes in 99.9% of the electrons and 99.6% of the kinetic energy. The correlated density matrices obtained are finally compared to their determinantal counterparts, using a simplified representation of the density matrix ρ(r,r′), suggested by the Legendre expansion. Interestingly, two-particle correlation is shown to impact the angular delocalization of each electron, in the one-particle space spanned by the r and r′ variables. [ABSTRACT FROM AUTHOR]

Dynamical properties of short freely jointed and freely rotating chains are studied using molecular dynamics simulations. These results are combined with those of previous studies, and the degree of rheological complexity of the two models is assessed. New results are based on an improved analysis procedure of the rotational relaxation of the second Legendre polynomials of the end-to-end vector in terms of the Kohlrausch–Williams–Watts (KWW) function. Increased accuracy permits the variation of the KWW stretching exponent β to be tracked over a wide range of state points. The smoothness of β as a function of packing fraction η is a testimony both to the accuracy of the analytical methods and the appropriateness of (η0-η) as a measure of the distance to the ideal glass transition at η0. Relatively direct comparison is made with experiment by viewing β as a function of the KWW relaxation time τKWW. The simulation results are found to be typical of small molecular glass formers. Several manifestations of rheological complexity are considered. First, the proportionality of α-relaxation times is explored by the comparison of translational to rotational motion (i.e., the Debye–Stokes–Einstein relation), of motion on different length scales (i.e., the Stokes–Einstein relation), and of rotational motion at intermediate times to that at long time. Second, the range of time-temperature superposition master curve behavior is assessed. Third, the variation of β across state points is tracked. Although no particulate model of a liquid is rigorously rheologically simple, we find freely jointed chains closely approximated this idealization, while freely rotating chains display distinctly complex dynamical features. [ABSTRACT FROM AUTHOR]

The rotational dynamics of chemically similar systems based on freely jointed and freely rotating chains are studied. The second Legendre polynomial of vectors along chain backbones is used to investigate the rotational dynamics at different length scales. In a previous study, it was demonstrated that the additional bond-angle constraint in the freely rotating case noticeably perturbs the character of the translational relaxation away from that of the freely jointed system. Here, it is shown that differences are also apparent in the two systems’ rotational dynamics. The relaxation of the end-to-end vector is found to display a long time, single-exponential tail and a stretched exponential region at intermediate times. The stretching exponents β are found to be 0.75±0.02 for the freely jointed case and 0.68±0.02 for the freely rotating case. For both system types, time-packing-fraction superposition is seen to hold on the end-to-end length scale. In addition, for both systems, the rotational relaxation times are shown to be proportional to the translational relaxation times, demonstrating that the Debye-Stokes-Einstein law holds. The second Legendre polynomial of the bond vector is used to probe relaxation behavior at short length scales. For the freely rotating case, the end-to-end relaxation times scale differently than the bond relaxation times, implying that the behavior is non-Stokes-Einstein, and that time-packing-fraction superposition does not hold across length scales for this system. For the freely jointed case, end-to-endrelaxation times do scale with bond relaxation times, and both Stokes-Einstein and time-packing-fraction-across-length-scales superposition are obeyed. [ABSTRACT FROM AUTHOR]

The problem of the nonlinear dielectric relaxation of complex liquids is tackled in the context of the Cole-Davidson [J. Chem. Phys. 19, 1484 (1951)] model. By using an appropriate time derivative of noninteger order, an infinite hierarchy of differential-recurrence relations for the moments (expectation values of the Legendre polynomials) is obtained. The solution is established for the stationary regime of an ensemble of polar and symmetric-top molecules acted on by a strong dc bias electric field superimposed on a weak ac electric field. The results for the first three nonlinear harmonic components of the electric susceptibility are analytically established and illustrated with the help of Argand diagrams for the nonlinear dielectric increment and three-dimensional dispersion and absorption spectra for the second and the third harmonic components as a function of the anomalous exponent β≤=1, the value of which gives rise to skewed arcs (Argand plots) and asymmetric shapes (loss spectra) in the high-frequency domain. [ABSTRACT FROM AUTHOR]

We present extensive equilibrium and out-of-equilibrium molecular-dynamics simulations of a liquid of symmetric dumbbell molecules, for constant packing fraction, as a function of temperature and molecular elongation. We compute diffusion constants as well as odd and even orientational correlators. The notations odd and even refer to the parity of the order l of the corresponding Legendre l polynomial, evaluated for the orientation of the molecular axis relative to its initial position. Rotational degrees of freedom of order l are arrested if, in the long-time limit, the corresponding orientational l correlator does not decay to zero. It is found that for large elongations translational and rotational degrees of freedom freeze at the same temperature. For small elongations only the even rotational degrees of freedom remain coupled to translational motions and arrest at a finite common temperature. On the contrary, the odd rotational degrees of freedom remain ergodic at all investigated temperatures. Hence, in the translationally arrested state, each molecule remains trapped in the cage formed by its neighboring molecules, but is able to perform 180° rotations, which lead to relaxation only for the odd orientational correlators. The temperature dependence of the characteristic time of these residual rotations is well described by an Arrhenius law. Finally, we discuss the evidence in favor of the presence of the type-A transition for the odd rotational degrees of freedom, as predicted by mode-coupling theory for small molecular elongations. This transition is distinct from the type-B transition, associated with the arrest of the translational and even rotational degrees of freedom for small elongations, and with all degrees of freedom for large elongations. Odd orientational correlators are computed for small elongations at very low temperatures in the translationally arrested state. The obtained results suggest that hopping events restore the ergodicity of the odd rotational degrees of freedom at temperatures far below the A transition. [ABSTRACT FROM AUTHOR]

The internal structure of spherical colloidal monolayers of charged particles is studied here, both by means of Monte Carlo computer simulations and of an integral equation approach based on the application of the Ornstein–Zernike equation for spherical surfaces. The latter is complemented with a relatively fast and accurate numerical method for its solution, obtained by expanding the corresponding correlation functions in series of Legendre polynomials. It is found that the density correlations among the particles within the monolayer have some special features that differentiate them from the corresponding bulk corrections in open spaces. In particular, for a sufficiently small radius of the spherical monolayer, the distribution of particles around a particle fixed at one of the poles exhibits a peak at the opposite pole which is noticeably larger than the peaks immediately before it. It is also shown here that the introduction of a simple functional form with one adjustable parameter for the bridge function greatly enhances the fit between the theoretical approach and the simulation data. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The first two orders of bridge diagrams for the f-bond expansion and the h-bond expansion are calculated for a binary mixture of Lennard-Jones spheres. The method used follows the Legendre polynomial integration methods outlined in the first two papers of this series. As for the pure fluid cases, the thermodynamic results which follow from these methods are found to be in reasonable agreement with the simulation result. Analysis of the thermodynamic and structural results in comparison to the best current bridge function approximations indicate that accurate descriptions of higher order mixtures will require methods beyond the current mean field treatments which are of utility in simple fluids. The methods given are unfortunately not computationally convenient at highest order; however, the lower order diagrams are both accessible and give reasonable numerical results. [ABSTRACT FROM AUTHOR]

We present a catalogue of diffusion coefficients and reorientational angle distribution (RAD) for various periodic surfaces, such as I-WP, F-RD, S, and S1 nodal surfaces; cylindrical structures like simple, undulated, and spiral cylinders, and a three-dimensional interconnected-rod structures. The results are obtained on the basis of a simulation algorithm for a diffusion on a surface given by the general equation φ(r)=0 [Holyst et al., Phys Rev. E 60, 302 (1999)]. I-WP, S, and S1 surfaces have a spherelike RAD, while F-RD has a cubelike RAD. The average of the second Legendre polynomial with RAD function for all nodal surfaces, except the F-RD nodal surface, decays exponentially with time for short times. The decay time is related to the Euler characteristic and the area per unit cell of a surface. This analytical formula, first proposed by B. Halle, S. Ljunggren, and S. Lidin in J. Chem. Phys. 97, 1401 (1992), is checked here on nodal surfaces, and its range of validity is determined. RAD function approaches its stationary limit exponentially with time. We determine the time to reach stationary state for all surfaces. In the case of the value of the effective diffusion coefficient the mean curvature and a connectivity between parts of surfaces have the main influence on it. The surfaces with low mean curvature at every point of the surface are characterized by high-diffusion coefficient. However if a surface has globally low mean curvature with large regions of nonzero mean curvature (negative and positive) the effective diffusion coefficient is low, as for example, in the case of undulated cylinders. Increasing the connectivity, at fixed curvatures, increases the diffusion coefficient. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]