Anharmonic effect on unimolecular reactions with application to the photodissociation of ethylene

The importance of anharmonic effect on dissociation of molecular systems, especially clusters, has been noted. In this paper, we shall present a theoretical approach that can carry out the first principle calculations of anharmonic canonical and microcanonical rate constants of unimolecular reactions within the framework of transition state theory. In the canonical case, it is essential to calculate the partition function of anharmonic oscillators; for convenience, the Morse oscillator potential will be used for demonstration in this paper. In the microcanical case, which involves the calculation of the total number of states for the activated complex and the density of states for the reactant, we make use of the fact that both the total number of states and the density of states can be expressed in the inverse Laplace transformation of the partition functions and that the inverse Laplace transformation can in turn be carried out by using the saddle-point method. We shall also show that using the theoretical approach presented in this paper the total number of states and density of states can be determined from thermodynamic properties and the difference between the method used in this paper and the thermodynamic model used by Krems and Nordholm will be given. To demonstrate the application of our theoretical approach, we chose the photodissociation of ethylene at 157 and 193 nm as an example.     

Why the pore geometry model could affect the uniqueness of the PSD in AC characterization

In this paper we discuss why the pore geometry can affect the unicity of the pore size distribution (PSD) of a given activated carbon (AC) sample, when different probe gases are used in adsorption measures. In order to characterize the solid sample we used grand canonical Monte Carlo simulation and the independent pore model with slit or triangular pore geometry, focusing our analysis on the possibility of representing the adsorptive processes of a triangular pore of defined size by means of a combination of slit pores of different sizes. This representation is tested on experimental adsorption data of N₂ (77 K) on AC samples and acceptable results were obtained. Finally, we have performed a theoretical test, which consisted of analyzing a virtual porous solid with this approach and different probe gases (N₂ at 77 K and CO₂ at 273 K), showing that the differences between the pore representations can cause differences between the solid representations for the adsorptive properties, for these different gases. The analysis presented here can be extended to other pore geometries and other adsorbates, and provide arguments to further explain results presented in our previous paper, which refers to cases when different adsorbates yield different PSDs for a given sample and the same pore geometry model.     

Method for performing LCAO band structure calculations in crystalline solids: Application to rubidium

In this work we present a method for performing LCAO (linear combination of atomic orbitals) band structure calculations (tight binding) in crystalline solids. In the first part of the article we apply group theoretical methods to the establishment of a least‐squares scheme for the calculation of the matrix of the crystal potential: This scheme is based on a well‐defined choice of independent parameters for the Bloch vector‐dependent matrix elements and on the considerations of the symmetries between these independent parameters and their Fourier coefficients. In the second part of this work we deal with the representation of the matrices of the identity operator and of the operator of the kinetic energy by linear combinations in terms of two center integrals: We express these linear combinations by a closed formula, which can be easily programmed on a computer, and we mention a method by which the two center integrals can be evaluated numerically fast and accurately. Finally we apply our theory to the derivation of numerical results: We determine the electronic states and the high‐momentum components of Compton rates in the alkali metal rubidium and we compare the results obtained with those of augmented plane‐wave (APW) calculations. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 212–225, 2000     

Fast construction of assembly trees for molecular graphs

A number of modeling and simulation algorithms using internal coordinates rely on hierarchical representations of molecular systems. Given the potentially complex topologies of molecular systems, though, automatically generating such hierarchical decompositions may be difficult. In this article, we present a fast general algorithm for the complete construction of a hierarchical representation of a molecular system. This two-step algorithm treats the input molecular system as a graph in which vertices represent atoms or pseudo-atoms, and edges represent covalent bonds. The first step contracts all cycles in the input graph. The second step builds an assembly tree from the reduced graph. We analyze the complexity of this algorithm and show that the first step is linear in the number of edges in the input graph, whereas the second one is linear in the number of edges in the graph without cycles, but dependent on the branching factor of the molecular graph. We demonstrate the performance of our algorithm on a set of specifically tailored difficult cases as well as on a large subset of molecular graphs extracted from the protein data bank. In particular, we experimentally show that both steps behave linearly in the number of edges in the input graph (the branching factor is fixed for the second step). Finally, we demonstrate an application of our hierarchy construction algorithm to adaptive torsion-angle molecular mechanics.     

On a new self-consistent-field theory for the canonical ensemble

A significant amount of many-body problems of quantum or classical equilibrium statistical mechanics are conveniently treated at fixed temperature and system size. In this paper, we present a new functional integral approach for solving canonical ensemble problems over the entire coupling range, relying on the method of Gaussian equivalent representation of Efimov and Ganbold. We demonstrate its suitability and competitiveness for performing approximate calculations of thermodynamic and structural quantities on the example of a repulsive potential model, widely used in soft matter theory.     

Real-time linear response for time-dependent density-functional theory

We present a linear-response approach for time-dependent density-functional theories using time-adiabatic functionals. The resulting theory can be performed both in the time and in the frequency domain. The derivation considers an impulsive perturbation after which the Kohn-Sham orbitals develop in time autonomously. The equation describing the evolution is not strictly linear in the wave function representation. Only after going into a symplectic real-spinor representation does the linearity make itself explicit. For performing the numerical integration of the resulting equations, yielding the linear response in time, we develop a modified Chebyshev expansion approach. The frequency domain is easily accessible as well by changing the coefficients of the Chebyshev polynomial, yielding the expansion of a formal symplectic Green's operator.     

On the computational calculation of surface free energies for the disordered semihexagonal reconstructed Au(100) surface

Previously we developed a general method for calculating the free energy of any surface constrained to a distinct surface excess number/density. In this paper we show how to combine a range of such surfaces, whose free energies have been calculated, to produce an ad hoc semigrand canonical ensemble of surfaces from which ensemble surface properties can be calculated, including the ensemble surface free energy. We construct such an ensemble for the disordered Au(100) semihexagonal reconstructed surface using a Glue model potential at 1000 K and calculate the ensemble surface free energy to be 0.088 18 eVA(2). The ensemble average surface lateral density was found to be 1.375 (with respect to the bulk), which is in agreement with previous grand canonical Monte Carlo studies.     

Canonical labels for protein spots of proteomics maps

We consider the problem of canonical labeling for a class of maps, which include proteomics maps, which consist of a set of vertices or protein spots. If this problem is solved and followed, different laboratories studying proteomics maps will arrive at the same numbering of spots, which would facilitate comparisons of data from different sources. In addition, the proposed canonical numberings of protein spots would allow compiling a catalog of proteomics maps just as canonical labeling allows making catalogs graphs, or molecules, and other canonically labeled systems, which would make search for similar sets of maps very efficient. We approach the problem by modifying the algorithm of Jeffrey for graphical representation of DNA based on the chaos game. Graphical representation of DNA as a chaos game map has an important property in that this representation allows one to assign sequential labels to spots in a DNA map. We have modified the approach for sequential labeling of chaos game map representations to graphical representation of any tabular data, such as listing of (x, y) coordinates of protein spots of proteomics maps.     

Mapping a homopolymer onto a model fluid

We describe a linear homopolymer using a grand canonical ensemble formalism, a statistical representation that is very convenient for formal manipulations. We investigate the properties of a system where only next neighbor interactions and an external, confining, field are present and then show how a general pair interaction can be introduced perturbatively, making use of a Mayer expansion. Through a diagrammatic analysis, we shall show how constitutive equations derived for the polymeric system are equivalent to the Ornstein-Zernike and Percus-Yevick equations for a simple fluid and find the implications of such a mapping for the simple situation of Van der Waals mean field model for the fluid.     

Linear-scaling method for calculating nuclear magnetic resonance chemical shifts using gauge-including atomic orbitals within Hartree-Fock and density-functional theory

Details of a new density matrix-based formulation for calculating nuclear magnetic resonance chemical shifts at both Hartree-Fock and density functional theory levels are presented. For systems with a nonvanishing highest occupied molecular orbital-lowest unoccupied molecular orbital gap, the method allows us to reduce the asymptotic scaling order of the computational effort from cubic to linear, so that molecular systems with 1000 and more atoms can be tackled with today's computers. The key feature is a reformulation of the coupled-perturbed self-consistent field (CPSCF) theory in terms of the one-particle density matrix (D-CPSCF), which avoids entirely the use of canonical MOs. By means of a direct solution for the required perturbed density matrices and the adaptation of linear-scaling integral contraction schemes, the overall scaling of the computational effort is reduced to linear. A particular focus of our formulation is to ensure numerical stability when sparse-algebra routines are used to obtain an overall linear-scaling behavior.     

On the response of an ionic liquid to external perturbations and the calculation of shear viscosity

In this article, attention is directed to three related problems: (1) the response of the ionic liquid (IL) 1-hexyl-3-methylimidazolium chloride ([HMIM+][Cl-]) to different external perturbations, (2) the calculation of its shear viscosity, and (3) the investigation of the range of validity of linear response theory for these types of systems. For this purpose, we derive a set of equations linking bulk hydrodynamic predictions with microscopic simulations which are valid when linear response theory is applicable. As far as we are aware, this article reports results from the largest atomistic simulations ever performed on this liquid. Our study shows that even for systems with a box length as large as 0.03 mu the viscosities computed from perturbation frequencies compatible with this box size have not yet reached the bulk hydrodynamic limit. This is in sharp contrast with the case of other solvents such as water in which the hydrodynamic limit can be achieved by using perturbations on a length scale of typical molecular dynamics simulation box sizes. In order to achieve our goals, we comprehensively investigated how the IL relaxed upon weak external perturbations at different wavenumbers. We also studied the steady-state flow created by external shear acceleration fields. The short time behavior of instantaneous velocity profiles was compared with the results of linear response theory. The short time response appears to match the prediction from linear response theory, while the long time response deviates as the external field becomes stronger. From this study, the range on which a perturbation can be considered "weak" in the linear response sense can be established. The relaxation of initial velocity profiles was also examined and correlated to the decay of the transverse-current autocorrelation function. Even though none of our calculations reached the bulk hydrodynamic limit, we are able to make predictions for the shear viscosity of the bulk system at different temperatures which qualitatively agree with experimental data.     

Bonded tableau unitary group approach to the many-electron correlation problem

A spin-free symmetry-adapted valence bond (VB ) state, named bonded tableau (BT ), is deduced from the classical bonded function and labeled by an at most two-column Weyl tableau. The complete set, which is composed of the BT basis or canonical bonded tableau (CBT ), can be constructed from an overcomplete set of BT states. CI CBT and VB CBT are two kinds of complete sets that are constructed in this paper. They can be used, respectively, in the CI and VB theory. It is shown that there is a one-to-one correspondence between the labeling scheme for CI CBT and the Gelfand–Tsetlin (GT ) basis. This relationship enables an efficient generation and compact representation of the BT basis if one desires to use the known global representation scheme for the GT basis. Effective algorithms for the matrix element evaluation of unitary group generators and products of generators between BT states are presented. In the formulation, the action of a generator on a BT state yields another BT state times a coefficient, so that the matrix elements of an arbitrary multiple product of generators are reduced to a calculation of the overlaps between BT states. The evaluation of the overlaps leads to a simple factorization into cycle contributions, whose values are given explicitly and only depend on the length parameters of the cycles. It is hoped that the presented formalism can facilitate the procedures for handling of the many-electron correlation problem.     

Modeling of adsorption and nucleation in infinite cylindrical pores by two-dimensional density functional theory

In this paper, we present an analysis of argon adsorption in cylindrical pores having amorphous silica structure by means of a nonlocal density functional theory (NLDFT). In the modeling, we account for the radial and longitudinal density distributions, which allow us to consider the interface between the liquidlike and vaporlike fluids separated by a hemispherical meniscus in the canonical ensemble. The Helmholtz free energy of the meniscus was determined as a function of pore diameter. The canonical NLDFT simulations show the details of density rearrangement at the vaporlike and liquidlike spinodal points. The limits of stability of the smallest bridge and the smallest bubble were also determined with the canonical NLDFT. The energy of nucleation as a function of the bulk pressure and the pore diameter was determined with the grand canonical NLDFT using an additional external potential field. It was shown that the experimentally observed reversibility of argon adsorption isotherms at its boiling point up to the pore diameter of 4 nm is possible if the potential barrier of 22kT is overcome due to density fluctuations.     

Regarding the validity of the time-dependent Kohn-Sham approach for electron-nuclear dynamics via trajectory surface hopping

The implementation of fewest-switches surface-hopping (FSSH) within time-dependent Kohn-Sham (TDKS) theory [Phys. Rev. Lett. 95, 163001 (2005)] has allowed us to study successfully excited state dynamics involving many electronic states in a variety of molecular and nanoscale systems, including chromophore-semiconductor interfaces, semiconductor and metallic quantum dots, carbon nanotubes and graphene nanoribbons, etc. At the same time, a concern has been raised that the KS orbital basis used in the calculation provides only approximate potential energy surfaces [J. Chem. Phys. 125, 014110 (2006)]. While this approximation does exist in our method, we show here that FSSH-TDKS is a viable option for computationally efficient calculations in large systems with straightforward excited state dynamics. We demonstrate that the potential energy surfaces and nonadiabatic transition probabilities obtained within the TDKS and linear response (LR) time-dependent density functional theories (TDDFT) agree semiquantitatively for three different systems, including an organic chromophore ligating a transition metal, a quantum dot, and a small molecule. Further, in the latter case the FSSH-TDKS procedure generates results that are in line with FSSH implemented within LR-TDDFT. The FSSH-TDKS approach is successful for several reasons. First, single-particle KS excitations often give a good representation of LR excitations. In this regard, DFT compares favorably with the Hartree-Fock theory, for which LR excitations are typically combinations of multiple single-particle excitations. Second, the majority of the FSSH-TDKS applications have been performed with large systems involving simple excitations types. Excitation of a single electron in such systems creates a relatively small perturbation to the total electron density summed over all electrons, and it has a small effect on the nuclear dynamics compared, for instance, with thermal nuclear fluctuations. In such cases an additional, classical-path approximation can be made. Third, typical observables measured in time-resolved experiments involve averaging over many initial conditions. Such averaging tends to cancel out random errors that may be encountered in individual simulated trajectories. Finally, if the flow of energy between electronic and nuclear subsystems is insignificant, the ad hoc FSSH procedure is not required, and a straightforward mean-field, Ehrenfest approach is sufficient. Then, the KS representation provides rigorously a convenient and efficient basis for numerically solving the TDDFT equations of motion.     

Rigorous integral screening for electron correlation methods

We derive rigorous multipole-based integral estimates (MBIE) in order to account for the distance dependence occurring in atomic-orbital (AO) formulations of electron correlation theory, where our focus is on AO-MP2 theory within a Laplace scheme. We find for the exact transformed integral products an extremely early onset of a linear-scaling behavior and a very small number of significant products. To preselect the significant integral products we adapt our MBIE method as rigorous upper bound. In this way it is possible to exploit the favorable scaling behavior observed and to reduce the scaling of estimated products asymptotically to linear, without sacrificing accuracy or reliability. By separating Coulomb- and exchange-type contractions only half-transformed integrals need to be computed. Furthermore, our scheme of rigorously preselecting transformed integral products via MBIE seems to offer particularly interesting perspectives for a direct formation of half- or fully transformed integrals by using multipole expansions and auxiliary basis sets.     

The central cell model: a mesoscopic hopping model for the study of the displacement autocorrelation function

On the mesoscale, the molecular motion in a microporous material can be represented as a sequence of hops between different pore locations and from one pore to the other. On the same scale, the memory effects in the motion of a tagged particle are embedded in the displacement autocorrelation function (DACF), the discrete counterpart of the velocity autocorrelation function (VACF). In this paper, a mesoscopic hopping model, based on a lattice-gas automata dynamics, is presented for the coarse-grained modeling of the DACF in a microporous material under conditions of thermodynamic equilibrium. In our model, that we will refer to as central cell model, the motion of one tagged particle is mimicked through probabilistic hops from one location to the other in a small lattice of cells where all the other particles are indistinguishable; the cells closest to the one containing the tagged particle are simulated explicitly in the canonical ensemble, whereas the border cells are treated as mean-field cells in the grand-canonical ensemble. In the present paper, numerical simulation of the central cell model are shown to provide the same results as a traditional lattice-gas simulation. Along with this a mean-field theory of self-diffusion which incorporates time correlations is discussed.     

On formulas in closed form for Van Vleck expansions

Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so-called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley-Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase-space. Explicit results have been obtained for several two- to four-dimensional problems.