Volume 122, Issue 18, 08 May 2005
 ARTICLES

 Theoretical Methods and Algorithms

On the acceleration of the convergence of singular operators in Gaussian basis sets
View Description Hide DescriptionGaussian type wave functions do not reproduce the interparticle cusps which result in a slow convergence of the expectation values of the operators involved in calculations of the relativistic and QED energy corrections. Methods correcting this deficiency are the main topic discussed in this paper. Benchmark expectation values of the singular operators for several fewelectron systems are presented.

Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems
View Description Hide DescriptionWe address the problem of eliminating fast reaction kinetics in stochastic biochemical systems by employing a quasiequilibrium approximation. We build on two previous methodologies developed by [Haseltine and Rawlings, J. Chem. Phys.117, 6959 (2002)] and by [Rao and Arkin, J. Chem. Phys.118, 4999 (2003)]. By following Haseltine and Rawlings, we use the numbers of occurrences of the underlying reactions to characterize the state of a biochemical system. We consider systems that can be effectively partitioned into two distinct subsystems, one that comprises “slow” reactions and one that comprises “fast” reactions. We show that when the probabilities of occurrence of the slow reactions depend at most linearly on the states of the fast reactions, we can effectively eliminate the fast reactions by modifying the probabilities of occurrence of the slow reactions. This modification requires computation of the mean states of the fast reactions, conditioned on the states of the slow reactions. By assuming that within consecutive occurrences of slow reactions, the fast reactions rapidly reach equilibrium, we show that the conditional state means of the fast reactions satisfy a system of at most quadratic equations, subject to linear inequality constraints. We present three examples which allow analytical calculations that clearly illustrate the mathematical steps underlying the proposed approximation and demonstrate the accuracy and effectiveness of our method.

Instantaneous dynamics and quantum control fields: Principle and numerical applications
View Description Hide DescriptionThe relation between laser pulses serving the purpose of controlling elementary molecular processes and the instantaneous dynamics of the perturbed system is investigated. The application of the conditions assuring a controlled change of the expectation value of an observable directly links the internal motion to the external perturbation. Several numerical applications document that the derived control fields are efficient and can be interpreted clearly on physical grounds.

Calculation of Franck–Condon factors including anharmonicity: Simulation of the band in the photoelectron spectrum of ethylene
View Description Hide DescriptionOur new simple method for calculating accurate Franck–Condon factors including nondiagonal (i.e., modemode) anharmonic coupling is used to simulate the band in the photoelectron spectrum. An improved vibrational basis set truncation algorithm, which permits very efficient computations, is employed. Because the torsional mode is highly anharmonic it is separated from the other modes and treated exactly. All other modes are treated through the secondorder perturbation theory. The perturbationtheory corrections are significant and lead to a good agreement with experiment, although the separability assumption for torsion causes the results to be not as good as those for . A variational formulation to overcome this circumstance, and deal with large anharmonicities in general, is suggested.

An efficient approach for ab initio energy calculation of biopolymers
View Description Hide DescriptionWe present a new method for efficient totalenergy calculation of biopolymers using the densitymatrix (DM) scheme based on the molecular fractionation with conjugate caps (MFCC) approach. In this MFCCDM method, a biopolymer such as a protein is partitioned into properly capped fragments whose density matrices are calculated by conventional ab initio methods which are then assembled to construct the full system density matrix. The assembled full density matrix is then employed to calculate the total energy and dipole moment of the protein using Hartree–Fock or densityfunctional theory methods. Using this MFCCDM method, the selfconsistentfield procedure for solving the full Hamiltonian problem is avoided and an efficient approach for ab initioenergy calculation of biopolymers is achieved. Two implementations of the approach are presented in this paper. Systematic numerical studies are carried out on a series of extended polyglycines and excellent results are obtained.

Gradientbased direct normalmode analysis
View Description Hide DescriptionA formulation of a direct, iterative method for obtaining the lowest eigenvalues and eigenvectors of a Hessian matrix is presented. Similar to the iterative schemes in electronic structure configuration interaction calculations (methods due to Lanczos, Davidson, and others), the massweighted Hessian matrix is not constructed explicitly; instead, its operation on a basis vector (a direction in the Cartesian configuration space of the atoms) is computed based on the principles of dynamical equations of motion. By noting that the time derivative of the gradient vector in the harmonic force field is related to the particles’ momenta via , a Hessianvector product can be computed on the fly by finite differentiation of the gradient along the direction specified by the vector. Thus, only two evaluations of the gradient are required per Davidsonlike iteration per root, which leads to a linear scaling behavior of the computational effort with the number of atoms (provided that the evaluation of the gradient scales linearly). Preliminary results are presented for a 27 000atom nanodroplet.

Ionization potentials and electron affinities in the Perdew–Zunger selfinteraction corrected densityfunctional theory
View Description Hide DescriptionUsing a selfconsistent implementation of the Perdew–Zunger selfinteraction corrected (PZSIC) densityfunctional theory, we have calculated ionization potentials (IP) and electron affinities (EA) of first and secondrow atoms and a set of small molecules. Several exchangecorrelation functionals were tested. IPs and EAs were obtained by two methods: as the difference in selfconsistent field (SCF) energies of neutrals and ions and as negatives of highestoccupied orbital energies. We found that, except for local spindensity approximation, PZSIC worsens IPs and EAs. On the other hand, PZSIC brings orbital eigenvalues into much better agreement with electron removal energies. The Perdew–Zunger SIC seems to overcorrect manyelectron systems; for molecules it performs worse than for atoms. We also discuss several common approximations to PZSIC such as spherical averaging of orbital densities in atoms.

First principles local pseudopotential for silver: Towards orbitalfree densityfunctional theory for transition metals
View Description Hide DescriptionOrbitalfree densityfunctional theory (OFDFT) with modern kineticenergy density functionals (KEDFs) is a linear scaling technique that accurately describes nearlyfreeelectronlike (main group) metals. In an attempt towards extending OFDFT to transition metals, here we consider whether OFDFT can be used effectively to study Ag, a metal with a localized shell. OFDFT has two approximations: use of a KEDF and local pseudopotentials (LPSs). This paper reports construction of a reasonably accurate LPS for Ag by means of inversion of the Kohn–Sham (KS) DFT equations in a bulk crystal environment. The accuracy of this LPS is determined within KSDFT (where the exact noninteracting kinetic energy is employed) by comparing its predictions of bulk properties to those obtained from a conventional (orbitalbased) nonlocal pseudopotential (NLPS). We find that the static bulk properties of fcc and hcp Ag predicted within KSDFT using this LPS compare fairly well to those predicted by an NLPS. With the transferability of the LPS established, we then use this LPS in OFDFT, where several approximate KEDFs were tested. We find that a combination of the Thomas–Fermi and von Weizsäcker functionals produces better densities than those from the linearresponsebased Wang–Teter KEDF. However, the equations of state obtained from both KEDFs in OFDFT contain unacceptably large errors. The lack of accurate KEDFs remains the final barrier to extending OFDFT to treat transition metals.

Computational strategies for a fourcomponent Dirac–Kohn–Sham program: Implementation and first applications
View Description Hide DescriptionAn implementation of the generalized gradient approximation within the fourcomponent formulation of relativistic densityfunctional theory using spinor basis sets is presented. This approach is based on the direct evaluation of the relativistic density and its gradient from the spinor amplitudes and gradients without explicit reference to the total density matrix. This proves to be a particularly efficient scheme, with an intrinsic computational cost that scales linearly with the number of spinor basis functions. In order to validate this new implementation, incorporated in the parallel version of the program BERTHA, a detailed study of the diatomic system CsAu is also reported. The spectroscopic constants , and and the dipole moment have been calculated and compared with the best available theoretical and experimental data. The sensitivity of our results to the details of the numerical schemes used to evaluate the matrix elements is analyzed in detail. Also presented is a comparative study of molecular properties in the alkali auride series which have been obtained using several standard nonrelativistic density functionals.

Computer simulations of polymer chain structure and dynamics on a hypersphere in fourspace
View Description Hide DescriptionThere is a rapidly growing interest in performing computer simulations in a closed space, avoiding periodic boundary conditions. To extend the range of potential systems to include also macromolecules, we describe an algorithm for computer simulations of polymer chain molecules on , a hypersphere in four dimensions. In particular, we show how to generate initial conformations with a bond angle distribution given by the persistence length of the chain and how to calculate the bending forces for a molecule moving on . Furthermore, we discuss how to describe the shape of a macromolecule on , by deriving the radius of gyration tensor in this nonEuclidean space. The results from both Monte Carlo and Brownian dynamics simulations in the infinite dilution limit show that the results on and in coincide, both with respect to the size and shape as well as for the diffusion coefficient. All data on can also be described by master curves by suitable scaling by the corresponding values in . We thus show how to extend the use of spherical boundary conditions, which are most effective for calculating electrostatic forces, to polymer chain molecules, making it possible to perform simulations on also for polyelectrolyte systems.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Second virial coefficients of asymmetric top molecules
View Description Hide DescriptionA short selfcontained derivation is given for the second virial coefficient of a gas consisting of identical interacting asymmetric rigid rotors. The resulting expression is correct through . First, the canonical partition function is derived by means of an expansion of due to Friedmann [Adv. Chem. Phys.4, 225 (1962)]. The present work applies angular momentum operators and known facts from angular momentumtheory. It is considerably more accessible than Friedmann’s exposition, which is not based on angular momentum operators, but instead on explicit derivatives with respect to Euler angles. The partition function obtained from the expansion is applied to the derivation of an expression for that is identical in appearance to the expression for symmetric rotors of T Pack [J. Chem. Phys.78, 7217 (1983)]. The final equation in this work is valid for rigid rotors of any symmetry.

Microwave and millimeterwave spectroscopy of the openshell van der Waals complex
View Description Hide DescriptionPure rotational transitions of a rare gas atom–reactive openshell triatom van der Waals complex have been observed by Fourier transform microwave spectroscopy. The transitions observed are of type with and 1. Furthermore, by monitoring the change of the free induction decay signal of the type transitions, type transitions have been observed by a double resonance technique in the region 18–49 GHz. All these transitions provide us precise molecular constants. The structure of has been determined by fixing the structure of the monomer. The determined structure is planar and almost shaped, where the argon atom is slightly shifted to the hydrogen atom of . The experimental data supplemented by highlevel ab initio calculations indicate that the van der Waals bond of is relatively rigid. On the other hand, effects on the unpaired electron distribution by the complex formation are found to be fairly small, since the fine and hyperfine constants of are well explained by those of the monomer.

Classical and quantum mechanics of diatomic molecules in tilted fields
View Description Hide DescriptionWe investigate the classical and quantum mechanics of diatomic molecules in noncollinear (tilted) static electric and nonresonant linearly polarized laser fields. The classical diatomic in tilted fields is a nonintegrable system, and we study the phase space structure for physically relevant parameter regimes for the molecule KCl. While exhibiting lowenergy (pendular) and highenergy (freerotor) integrable limits, the rotor in tilted fields shows chaotic dynamics at intermediate energies, and the degree of classical chaos can be tuned by changing the tilt angle. We examine the quantum mechanics of rotors in tilted fields. Energylevel correlation diagrams are computed, and the presence of avoided crossings quantified by the study of nearestneighbor spacing distributions as a function of energy and tilting angle. Finally, we examine the influence of classical periodic orbits on rotor wave functions. Many wave functions in the tilted field case are found to be highly nonseparable in spherical polar coordinates. Localization of wave functions in the vicinity of classical periodic orbits, both stable and unstable, is observed for many states.

Wave packets in a bifurcating region of an energy landscape: DielsAlder dimerization of cyclopentadiene
View Description Hide DescriptionQuantum dynamics in a valley ridge inflection (VRI) point region is analyzed in the case of the DielsAlder endodimerization of cyclopentadiene pointed out recently by [Caramella et al., J. Am. Chem. Soc.124, 1130 (2002)]. The VRI point is located along the reaction path connecting the bispericyclic symmetrical transition structure put in evidence by Caramella et al. and the transition state of the Cope rearrangement. Dynamics is carried out by using constrained Hamiltonian methodology. The active coordinates are the first formed C–C bond length and the difference between the two other C–C bond lengths which achieve the dimerization as or adducts. A twodimensional (2D) minimumenergy surface have been computed at the Becke 3 Lee–Yong–Parr∕ level. The energy landscape can be classified as an uphill ridgepitchfork VRI bifurcation according to a recent classification of bifurcation events [W. Quapp, J. Mol. Struct.695–696, 95 (2004)]. Dynamics does not describe the thermal reaction but concerns wave packets which could be prepared by pulse reagents, i.e., by coherent control. We analyze how the shape and initial location on the ground potentialenergy surface are linked to the synchronous or asynchronous mechanism of the final step after the first transition state. We use a onedimensional model of optimum control theory to check the feasibility of such a coherent preparation. The wavepacket evolution in the VRI domain is well explained by semiclassical predictions even with the negative curvature of the unstable ridge. Finally, a crude model of dissipation has been introduced to test the stability of the 2D predictions.

NonBorn–Oppenheimer molecular structure and oneparticle densities for
View Description Hide DescriptionWe show that the nonadiabatic (nonBorn–Oppenheimer) ground state of a threenuclei system can be effectively calculated with the use of an explicitly correlated Gaussian basis set with floating centers. Sample calculations performed for the system with various basis set sizes show good convergence with respect to both the total energy and the expectation values of the internuclear distances (molecular geometry), the distances between the nuclei and the electrons, and between the electrons. We also provide a derivation of the formulas for oneparticle density calculations and some density plots showing the spatial distribution of the nuclear and electronic densities.

Hydrogenbonded acetic acid dimers: Anharmonic coupling and linear infrared spectra studied with densityfunctional theory
View Description Hide DescriptionAnharmonic vibrational force field calculations provide a quantitative understanding of the width and substructure of the linear IRabsorption spectrum of the O–H stretching mode in acetic acid dimers and . Anharmonic coupling of the highfrequency mode to fingerprint and lowfrequency modes is included resulting in 11 and 9dimensional vibrational Hamiltonians. A sixthorder force field covering up to threebody interactions is used. Force constants are calculated by fitting onedimensional potentialenergy surfaces and a finite difference procedure applying densityfunctional theory [Becke 3 Lee–Yang–Parr 6–311+G(d,p)]. It is demonstrated that both anharmonic coupling to lowfrequency modes as well as Fermi resonance coupling with fingerprint modes are important mechanisms explaining the line shape of the O–H stretching IRabsorption band in acetic acid dimers.

Potential surfaces and dynamics of the reaction
View Description Hide DescriptionWe present global potential energy surfaces for the three lowest triplet states in collisions and present results of classical dynamics calculations on the reaction using these surfaces. The surfaces are splinebased fits of fixed geometry ab initio calculations at the completeactivespace selfconsistent field+secondorder perturbation theory (CASSCF+MP2) level with a / one electron basis set. Computed rate constants compare well to measurements in the 1000–2500 K range using these surfaces. We also compute the total, rovibrationally resolved, and differential angular cross sections at fixed collision velocities from near threshold at collision energy) to (collision energy), and we compare these computed cross sections to available spacebased and laboratory data. A major finding of the present work is that above collision energy rovibrationally excited products are a significant and perhaps dominant contributor to the observed 1–5 μ spectral emission from collisions. Another important result is that products are formed in two distinct rovibrational distributions. The “active” OH products are formed with the reagent O atom, and their rovibrational distributions are extremely hot. The remaining “spectator” OH is relatively rovibrationally cold. For the active OH, rotational energy is dominant at all collision velocities, but the opposite holds for the spectator OH. Summed over both OH products, below collision energy, vibration dominates the OH internal energy, and above rotation is greater than vibrational energy. As the collision energy increases, energy is diverted from vibration to mostly translational energy. We note that the present fitted surfaces can also be used to investigate direct collisional excitation of by and also collisions.

Fourier transform microwave spectroscopy of vinyldiacetylene, vinyltriacetylene, and vinylcyanodiacetylene
View Description Hide DescriptionThe rotational spectra of the three carbon chain molecules vinyldiacetylene (hex1ene3,5diyne, ), vinyltriacetylene (oct1ene3,5,7triyne, ), and its cyano analog vinylcyanodiacetylene (1cyanohex5ene1,3diyne, ) have been observed for the first time by Fourier transformmicrowave spectroscopy of a supersonic molecular beam. The molecules were observed as products of an electrical discharge through selected precursor mixtures: ethylene/diacetylene and vinylacetylene/diacetylene for the pure hydrocarbon molecules and vinylacetylene/cyanoacetylene for vinylcyanodiacetylene. The measurements yield precise sets of rotational constants that compare very well with theoretical constants obtained by quantum chemical calculations at the B3LYP/ccpVTZ level of theory. Since these three carbon chains are similar in structure and composition to known astronomical molecules and because of their significant polarity, all three are candidates for radio astronomical detection.

Ab initio investigation of the autoionization process : Potential energy curves and autoionization widths, ionization cross sections, and electron energy spectra
View Description Hide DescriptionMultireference configuration interaction (MRCI) calculations have been performed for the collision complex. Feshbach projection based on orbital occupancy defines the entrance channel resonance states and provides their potential energy curves as well as resonancecontinuum coupling matrix elements, which are turned into an autoionization width function by Stieltjes imaging. Coupled cluster calculations with singles, doubles, and pertubative triples [CCSD(T)] give the exit channel potential of . The core is treated by a scalarrelativistic effective core potential, reparametrized to reproduce experimental excitation and ionizationenergies.Spinorbit interaction is included for the open shell. The nuclear motion is treated within the local complex potential approximation. Ionization occurs for 85% and 98% of the symmetry allowed close collisions. Calculated ionization cross sections show good agreement with experimental data. The difference potential of the collision complex is remarkably flat down to internuclear separations of and leads to very sharp peaks in theoretical electron energy spectra for single collision energies. After accounting for the experimental energy distribution and the resolution function of the spectrometer, a very satisfying agreement with experimental electron energy spectra is found, including subtle differences due to spinorbit coupling. Theoretical input appears indispensable for an analysis of the measured data in terms of potential energy curves and autoionization width functions.

Accurate, twostate ab initio study of the ground and firstexcited states of , including exact treatment of all Born–Oppenheimer correction terms
View Description Hide DescriptionBorn–Oppenheimer (BO) potentials for the ground and firstexcited electronic states of are determined using high level ab initio techniques for internuclear separations of 1.2–100 bohrs and accurately fit to analytical functions. In the present formulation, the BO potentials are nuclear mass independent, and the corresponding BO approximation is obtained by ignoring four terms of the full rovibronic Hamiltonian. These four Born–Oppenheimer correction (BOC) terms are as follows: (1) mass polarization, (2) electronic orbital angular momentum, (3) first derivative with respect to , and (4) second derivative with respect to . In order to enable an exact rovibronic calculation, each of the four BOC terms are computed as a function of , for the two electronic states and for their coupling, without any approximation or use of empirical parameters. Each of the BOC terms is found to make a contribution to the total energy over at least some portion of the range of investigated. Interestingly, the most significant coupling contribution arises from the electronic orbital angular momentum term, which is evidently computed for the first time in this work. Although several BOC curves exhibit a nontrivial dependence on , all are accurately fit to analytical functions. The resulting functions, together with the BO potentials, are used to compute exact rovibronic energy levels for , and . Comparison to available high quality experimental data indicates that the present BOC potentials provide the most accurate representation currently available of both the low and highlying levels of the ground electronic state and the bound levels of the excited state.