Volume 113, Issue 22, 08 December 2000
 ARTICLES

 Theoretical Methods and Algorithms

Intermediate Hamiltonian Fockspace coupledcluster method: Excitation energies of barium and radium
View Description Hide DescriptionAn intermediate Hamiltonian Fockspace coupled cluster method is introduced, based on the formalism developed by Malrieu and coworkers in the context of perturbation theory. The method is designed to make possible the use of large P spaces while avoiding convergence problems traceable to intruder states, which often beset multireference coupled cluster schemes. The essence of the method is the partitioning of P into a main and an intermediate serving as buffer, with concomitant definition of two types of wave and excitation operators. Application to atomic barium and radium yields converged results for a large number of states not accessible by traditional Fockspace coupled cluster. Moreover, states calculated by both methods exhibit better accuracy (by a factor of 2–5) in the intermediate Hamiltonian approach. Energies are given for lowlying states of Ra which have not been observed experimentally.

Overdamped Brownian motion in periodic symmetric potentials
View Description Hide DescriptionThe dynamics of an overdamped Brownian particle in the field of a onedimensional symmetric periodic potential have been studied by numerical solution of the Smoluchowski diffusionequation and the Langevin equation using the Brownian Dynamics method. The parameter controls the shape and height of the potential barrier, which ranges from a sinusoidal spatial dependence for low barrier heights (α small) to a near deltafunction appearance for barrier heights tending to infinity (α very large). Both the mean square displacement (MSD) and the probability density where denotes the initial position, have been calculated. The MSD over a wide time domain has been obtained for a number of values of α. The exact asymptotic form of the diffusion coefficient has been exploited to obtain an accurate representation for at long times. The function, changes its form in the range with the appearance of a “plateau” which signals a transition in the particle’s Brownian dynamics from a weakly hindered (but continuous) mechanism to essentially jump diffusion. In the limit each well of becomes similar to the classical square well (SW), which we have revisited as it provides a valuable limiting case for at An effective “attraction” of the probability density towards the SW walls is observed for offcenter initial starting positions, and it is suggested that this could explain an observed change in the analytic form of the SW MSD, at long times. Two approximate analytic forms for at short times have been derived. The relaxation of the Brownian particle distribution in the initialwell of has been studied.

Reorientational angle distribution and diffusion coefficient for nodal and cylindrical surfaces
View Description Hide DescriptionWe present a catalogue of diffusion coefficients and reorientational angle distribution (RAD) for various periodic surfaces, such as IWP, FRD, S, and S1 nodal surfaces; cylindrical structures like simple, undulated, and spiral cylinders, and a threedimensional interconnectedrod structures. The results are obtained on the basis of a simulation algorithm for a diffusion on a surface given by the general equation [Hołyst et al., Phys Rev. E 60, 302 (1999)]. IWP, S, and S1 surfaces have a spherelike RAD, while FRD has a cubelike RAD. The average of the second Legendre polynomial with RAD function for all nodal surfaces, except the FRD 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 highdiffusion 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.

Densities of states in gases, liquids, and solids
View Description Hide DescriptionFrom heat capacity data one can calculate a set of energy moments for any system. Using the maximumentropy method, these energy moments can be used to construct an approximate energy probability distribution which, in turn, can be used to calculate the density of states as a function of energy. The density of states is a temperature independent function from which the free energy, internal energy, entropy, and heat capacity for the system can be obtained plus the energy probability distribution and its potential at any temperature. We compare the behavior of the density of states for various substances in a single phase and as one goes through different types of phase transition.

Electric field effects on the ionicneutral curve crossing of alkali halide molecules
View Description Hide DescriptionThe weakly avoided crossing between the two lowest electronic states of a series of alkali halide molecules has been studied by means of the recently reported multistate complete active space secondorder perturbation theory, MS–CASPT2, method. For a large enough basis set and a complete active space selfconsistent field that includes part of the radial and angular correlation of the outermost halide electrons, the calculated crossing distance is in very good agreement with that predicted from the Rittner empirical potential. The study of the relevant parameters corresponding to the crossing region on these molecules has been extended to include the effect of a uniform electric field and a generalization of the empirical Rittner formula that includes the electric field effects is presented. The predictions made by the MS–CASPT2 method are also in agreement with those derived from the generalized Rittner potential. Finally, the possible implications of the present work on electron transfer processes at metal electrodes are discussed.

Basis set approach to the quantum dissipative dynamics: Application of the multiconfiguration timedependent Hartree method to the spinboson problem
View Description Hide DescriptionThe feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spinboson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration timedependent Hartree context, is used to solve the timedependent Schrödinger equations by explicitly including all the degrees of freedom Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of timedependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.

An improved ab initio relativistic zerothorder regular approximation correct to order
View Description Hide DescriptionThe equations of the original ab initio scalarrelativistic zerothorder regular approximation (ZORA) and the infiniteorder regular approximation (IORA) are expanded in orders of It is shown that previous ZORA/IORA implementations in ab initio quantum chemistry programs were not correct to order but contained imperfections leading to fictitious selfinteractions. These errors can be avoided by adding exchangetype terms (coupling the large and small components) to the relativistic ZORA correction to the Hamiltonian, yielding improved ab initio relativistic zeroth and infiniteorder regular approximations that are correct to order The new methods have been tested numerically by computing the total energies, orbital energies, and static electric dipole polarizabilities of the rare gas atoms He through Xe.

Reduced multireference coupled cluster method: Rovibrational spectra of
View Description Hide DescriptionThe reduced multireference (RMR) coupled cluster method with singles and doubles (CCSD), relying on four and eightdimensional reference spaces, is employed to generate the potential energy curve for the nitrogen molecule, using both the correlationconsistent polarized valencetriplezeta (ccpVTZ) and atomic natural orbital basis sets. The resulting potentials are then used to compute the rovibrational term values, which are compared with the available experimentally determined values. Moreover, a direct comparison with the measuredspectral line frequencies, for both the fundamental rotational Raman band and the vibrationally excited Q branches, is also carried out. Comparisons with the single reference CCSD, as well as with the corresponding multireference configuration interaction results that serve as the external source for RMR CCSD, are also made. An excellent performance of RMR CCSD, and its systematic improvement with the increasing dimension of the reference space employed, is demonstrated for the ccpVTZ basis set. The fundamental Raman band frequencies are computed with an error of less than 1 while for the vibrationally excited bands a systematic deviation of about 7 is found.

Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points
View Description Hide DescriptionAn improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based densityfunctional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surface, and dissociative adsorption of on an Ir(111) surface.

Loworder scaling local electron correlation methods. III. Linear scaling local perturbative triples correction (T)
View Description Hide DescriptionA new method for the perturbative calculation of the correlationenergy due to connected triple excitations (T) in the framework of local coupled clustertheory is presented, for which all computational resources scale linearly with molecular size. One notable complication in the formalism for connected triples introduced by the local approach is the nondiagonality of the Fock matrix in the localized MO (LMO) and projected AO (PAO) basis, which leads to couplings between individual triples amplitudes via the internal–internal and external–external blocks of the Fock matrix, respectively. Further complications and couplings arise due to the nonorthogonality of the PAOs. While the couplings via the external–external block can easily be dealt with, this is more difficult for the internal–internal couplings. In a previous paper we already published preliminary results of an approximation of the method, which neglects these internal–internal couplings entirely and recovers about 97% of the total local triples correlationenergy. In the present work we implemented the “full” local (T) method, which involves the iterative solution of a system of linear equations for the triples amplitudes to take the internal–internal couplings fully into account. Moreover, a further variant of the method was implemented, which approximates the internal–internal couplings at the level of firstorder perturbation theory with respect to the offdiagonal elements of the Fock matrix in LMO basis, thus avoiding the need for an iterative solution of the triples equations and storage of the triples amplitudes. The latter variant reliably recovers more than 99% of the full local triples energy. Test calculations with more than 1000 basis functions and over 300 correlated electrons are presented, showing a speedup of about relative to the estimated time of a corresponding conventional (T) calculation.

Excited state localization in organic molecules consisting of conjugated and nonconjugated segments
View Description Hide DescriptionWe investigate, both experimentally and theoretically, a series of novel molecules consisting of conjugated segments (such as stilbene, naphthylene, and anthrylene) that are separated from each other by nonconjugated bridges. Excited statelocalization effects are studied theoretically by postHartree–Fock calculations—taking into account electron correlationeffects. In this context, we compute the electronhole twoparticle wave functions for the prominent excited states and discuss the nature of the molecular orbitals involved in their description. We also investigate geometry relaxation effects following the electronic excitations in order to locate the regions where the strongest rearrangement of the electron density occurs. These conceptionally different approaches (relying also on different semiempirical Hamilton operators and configuration interaction techniques) yield consistent results regarding the localization of the excitations and thus prove helpful to determine the nature of the lowest excited states in such multichromophoric systems. Knowing the exact nature of the different states observed in the experimental absorption and luminescenceexcitationspectra allows for selective excitations of the different segments of the molecules. When performing siteselective spectroscopy, we find that in all the materials the emission originates from the transition, independent of the excitation wavelengths. This points to an efficient intramolecular energy transfer that occurs in spite of the broken conjugation between the molecular building blocks.

New τdependent correlation functional combined with a modified Becke exchange
View Description Hide DescriptionA new correlation functional is derived within the Kohn–Sham (KS) Density Functional Theory(DFT) involving the electron kinetic energy density and the Laplacian of the electron density as key nonlocal variables. The derivation is based on a direct resolution of the adiabatic connection formula and using an analogy with the local thermodynamic approach in DFT, following the Lap3 theory developed previously. Compared to the latter, the new functional involves higher order dependent energy terms in a form suggesting a possible resummation procedure that could be used for further development. It is combined with the nonlocal exchange functional of Becke, by modifying the latter in an empirical fashion to achieve better synchronization between the two energy components. The resulting exchangecorrelation scheme (named “Bmτ1”) is validated on several test systems known as difficult for DFT, at least at the Local Spin Density and Generalized Gradient Approximation levels. The recent nonempirical hybrid scheme PBE1PBE (“PBE0”) is included in the comparative tests as a parameterfree benchmark for the hybrid HFKS DFT approach. Improved results for relative energies, activation barriers and equilibrium geometries are obtained with the Bmτ1 functional, particularly concerning aromatic compounds, systems with weak hydrogen bonds, proton transfer processes and transitionmetal carbonyls.

Time dependent quantum propagation in phase space
View Description Hide DescriptionNumerical solutions of the quantum timedependent integrodifferential Schrödinger equation in a coherent state Husimi representation are investigated. Discretization leads to propagation on a grid of nonorthogonal coherent states without the need to invert an overlap matrix, with the further advantage of a sparse Hamiltonian matrix. Applications are made to the evolution of a Gaussian wave packet in a Morse potential. Propagation on a static rectangular grid is fast and accurate. Results are also presented for a moving rectangular grid, guided at its center by a mean classical path, and for a classically guided moving grid of individual coherent states taken from a Monte Carlo ensemble.

Linear scaling computation of the Fock matrix. V. Hierarchical Cubature for numerical integration of the exchangecorrelation matrix
View Description Hide DescriptionHierarchical cubature is a new method for achieving linear scaling computation of the exchangecorrelation matrix central to Density Functional Theory. Hierarchical cubature combines a kdimensional generalization of the binary search tree with adaptive numerical integration involving an entirely Cartesian grid. Hierarchical cubature overcomes strong variations in the electron density associated with nuclear cusps through multiresolution rather than sphericalpolar coordinate transformations. This unique Cartesian representation allows use of the exact integration error during grid construction, supporting rangequeries that exploit locality of the Cartesian Gaussian based electron density. Convergence is controlled by which bounds the local integration error of the electron density. An early onset of linear scaling is observed for RB3LYP/631G * * calculations on water clusters, commencing at and persisting with decreasing values of Comparison with nuclear weight schemes suggests that the new method is competitive on the basis of grid points per atom. Systematic convergence of the RPBE0/631G* binding curve is demonstrated with respect to
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Angular distributions of scattered electrons from gaseous benzene molecules
View Description Hide DescriptionRecent measurements of the angular distributions for scatteredelectrons in gaseous benzene have produced elastic (rotationally summed) differential cross sections over a broad range of angular values and of collision energies. The present calculations examine the corresponding computed quantities over the same energy and angular values and compare them with the measurements. The necessary exchange interaction is included both exactly and via approximate local models in order to test the reliability of the local model potentials for such delicate quantum dynamics features as angular distributions. Very good overall agreement is found between measurements and the calculations presented here. The local model potentials are found to provide good agreement with the exact exchange calculations at energies above 6.0 eV.

Resonant twophoton ionization and laser induced fluorescence spectroscopy of jetcooled adenine
View Description Hide DescriptionElectronic spectra of the jetcooled DNA base adenine were obtained by the resonant twophotonionization (R2PI) and the laser induced fluorescence(LIF) techniques. The 0–0 band to the lowest electronically excited state was found to be located at 35 503 cm^{−1}. Wellresolved vibronic structures were observed up to 1100 cm^{−1} above the 0–0 level, followed by a slow rise of broad structureless absorption. The lowest electronic state was proposed to be of character, which lies ∼600 cm^{−1} below the onset of the state. The broad absorption was attributed to the extensive vibronic mixing between the state and the highlying state.

Inhomogeneous speed effects on vibrational line profiles in ternary mixtures
View Description Hide DescriptionA study of speed inhomogeneous broadening of the hydrogen vibrational line profiles in the collisional regime for ternary mixtures is reported. The line of in and mixtures is investigated by high resolution stimulated Raman spectroscopy for various concentrations and temperatures. A model, successfully used for binary mixtures, is extended to ternary mixtures. An excellent agreement is obtained between theory and experiment for by using the collisional parameters previously obtained from binary mixtures study. For and collisions play a “hardening” effect in the soft speed memory mechanism for or The present experimental results allow us, via the frame of our model, to determine the “hardness” parameter values for (inaccessible from binary mixtures data) and to get an accurate description of the spectral line shape. This study should be useful for hydrogen coherent antiStokes Raman spectroscopy thermometry in flames at high pressure through its possible extension to the system.

Ab initio molecular dynamics studies of the photodissociation of formaldehyde, Direct classical trajectory calculations by MP2 and density functional theory
View Description Hide DescriptionThe dynamics of photodissociation have been studied by classical trajectory calculations at the and levels of theory. The trajectories were calculated directly from the electronic structure computations without first fitting a global potential energy surface. A Hessian based method with updating was used to integrate the trajectories. The translational energy distribution of the products is in better agreement with experiment than the previous Hartree–Fock direct trajectory calculations, since the MP2 and density functional methods reproduce the barrier height better. The MP2 and density functional calculations give very good descriptions of the product rotational state distributions and the CO vibrational state populations. The MP2 method yields a very good representation of the vibrational state populations, whereas the density functional methods give too little vibrational excitation and Hartree–Fock produces too much. This can be attributed to the difference in the potential energy release that accompanies the formation of the bond.

Intramolecular energy transfer in highly vibrationally excited methanol. IV. Spectroscopy and dynamics of
View Description Hide DescriptionRotationally resolved spectra in the region of reveal strong vibrational splittings, which by analogy to we assign to a resonance between and Accordingly, the vibrational dynamics on a subpicosecond time scale are similar for the two isotopomers. Comparison of the secondary structure of the firstorder states resulting from this strong resonance shows a distinct difference in the two isotopomers. While the species exhibits sharp secondary structure for the lower energy band resulting from weak coupling to the remaining bath of dark states, the species shows an equally complex pattern of couplings for both firstorder states. The difference between the two isotopic species arises from the relative position of key dark background states. Despite a vibrational density of states of 100 per cm^{−1}, only a small number of states seem to determine the secondary structure, and the difference in the positions of these states relative to the firstorder states results in a difference in the vibrational dynamics on a picosecond time scale. What one might consider as statistical intramolecular energy transfer appears to occur on significantly longer time scales.

Photodissociation of at 121.6 nm: A statetostate dynamical picture
View Description Hide DescriptionPhotodissociation dynamics of at 121.6 nm have been studied using the H atom Rydberg “tagging” timeofflight technique and by quasiclassical trajectory(QCT) calculations. Product kinetic energy distributions and angular distributions have been measured. From these distributions, rovibronic distributions of the OH radical product as well as the state resolved angular anisotropy parameters were determined. The dissociation energy is determined to be Two clear alternations in the rotational distribution have been observed, with each alternation corresponding to an oscillation in the anisotropy distribution. These oscillations had been attributed to the dynamical interference between the two conical intersection pathways. Further theoretical modeling in this work strongly supports this argument. Very highly vibrationally excited products (up to have also been observed. These are ascribed to interconversion of H–O–H bending (H–H vibration) and O–H vibration in O–H–H geometries. The effect of parent rotational excitation on the product state distribution and anisotropy distribution was observed for the first time. Experimental results also show clear evidence for the triple dissociation channel, Accurate branching ratios of different product channels have been determined. Results of detailed QCT calculations agree well with the experimental results in this work.