Volume 130, Issue 13, 07 April 2009
 ARTICLES

 Theoretical Methods and Algorithms

Simultaneous fitting of a potentialenergy surface and its corresponding force fields using feedforward neural networks
View Description Hide DescriptionAn improved neural network (NN) approach is presented for the simultaneous development of accurate potentialenergy hypersurfaces and corresponding force fields that can be utilized to conduct ab initiomolecular dynamics and Monte Carlo studies on gasphase chemical reactions. The method is termed as combined function derivative approximation (CFDA). The novelty of the CFDA method lies in the fact that although the NN has only a single output neuron that represents potential energy, the network is trained in such a way that the derivatives of the NN output match the gradient of the potentialenergy hypersurface. Accurate force fields can therefore be computed simply by differentiating the network. Both the computed energies and the gradients are then accurately interpolated using the NN. This approach is superior to having the gradients appear in the output layer of the NN because it greatly simplifies the required architecture of the network. The CFDA permits weighting of function fitting relative to gradient fitting. In every test that we have run on six different systems, CFDA training (without a validation set) has produced smaller outofsample testing error than early stopping (with a validation set) or Bayesian regularization (without a validation set). This indicates that CFDA training does a better job of preventing overfitting than the standard methods currently in use. The training data can be obtained using an empirical potential surface or any ab initio method. The accuracy and interpolation power of the method have been tested for the reactiondynamics of using an analytical potential. The results show that the present NN training technique produces more accurate fits to both the potentialenergysurface as well as the corresponding force fields than the previous methods. The fitting and interpolation accuracy is so high that trajectories computed on the NN potential exhibit pointbypoint agreement with corresponding trajectories on the analytic surface.

Sampling conformations in high dimensions using lowdimensional distribution functions
View Description Hide DescriptionWe present an approximation to a molecule’s dimensional conformational probability density function (pdf) in terms of marginal pdfs of highest order , where is much less than . The approximation is constructed as a product of conditional pdfs derived by recursive application of the generalized Kirkwood superposition approximation. Furthermore, an algorithm is presented to sample conformations from the approximate fulldimensional pdf based upon all input marginal pdfs. The sampling algorithm is tested for three small molecule systems by using the algorithm to sample conformations at levels , 2, or 3 and comparing the distributions of sampled conformations with those from the molecular dynamics (MD) simulations. The distributions of conformations sampled at third order resemble the MD distributions rather well and significantly better than those sampled at second or first order. In addition to highlighting the importance of correlations among internal degrees of freedom, these results suggest that loworder correlations suffice to describe most of the conformationalfluctuations of molecules in a thermal environment.

Anisotropic intracule densities and electron correlation in : A quantum Monte Carlo study
View Description Hide DescriptionWe derive efficient quantum Monte Carlo estimators for the anisotropic intracule and extracule densities. These estimators are used in conjunction with an accurate explicitly correlated wave function to investigate the bondlength dependence of electron correlation effects in the groundstate molecule. It is shown that the localized increase in the magnitude of the correlation energy as the bond is stretched is accompanied by highly anisotropiccorrelation effects. In addition, we find a small longrange part of the Coulomb hole, which is present even at the equilibrium bond length.

A hierarchy of potential energy surfaces constructed from energies and energy derivatives calculated on grids
View Description Hide DescriptionIn this work we develop and test a methodology for the generation of Born–Oppenheimer potential energy surfaces (PES) for use in vibrational structure calculations. The method relies on the widely used restrictedmodecoupling expansion of the fully coupled potential surface where only up to or less vibrational coordinates are coupled in the potential. Loworder derivatives of the energy are then used to extrapolate the higher modecoupling potential terms; derivative information is thus used in a convenient way for the evaluation of higher mode couplings avoiding their explicit calculation on multidimensional grids. The formulation, which is a variant of the popular modified Shepard interpolation, is general for any extrapolation of modecoupling terms from mode couplings and can be applied to the energy or any other molecular propertysurface for which derivative information is available. The method depends only on analytical parameterfree weight functions that satisfy important limiting conditions and control the contribution from each direction of extrapolation. The procedure has been applied on a representative set of 13 molecules, and its accuracy has been tested using only gradients and using both gradients and Hessians. The results provide evidence for the importance of higher mode couplings and illustrate the cost efficiency of the proposed approach.

Calculation of orientational correlation functions for free anisotropic rotational diffusion revisited
View Description Hide DescriptionA simple matrix method for evaluating the orientational correlation functions of arbitrary rank pertaining to free noninertial anisotropic rotational diffusion of rigid Brownian particles is presented. The first and secondrank correlation functions are calculated analytically for a diagonal diffusiontensor.

Nonperturbative timeconvolutionless quantum master equation from the path integral approach
View Description Hide DescriptionThe timeconvolutionless quantum master equation is widely used to simulate reduced dynamics of a quantum system coupled to a bath. However, except for several special cases, applications of this equation are based on perturbative calculation of the dissipative tensor, and are limited to the weak systembath coupling regime. In this paper, we derive an exact timeconvolutionless quantum master equation from the path integral approach, which provides a new way to calculate the dissipative tensor nonperturbatively. Application of the new method is demonstrated in the case of an asymmetrical twolevel system linearly coupled to a harmonic bath.

A moment closure method for stochastic reaction networks
View Description Hide DescriptionIn this paper we present a moment closure method for stochastically modeled chemical or biochemical reaction networks. We derive a system of differential equations which describes the dynamics of means and all central moments from a chemical master equation. Truncating the system for the central moments at a certain moment term and using Taylor approximation, we obtain explicit representations of means and covariances and even higher central moments in recursive forms. This enables us to deal with the moments in successive differential equations and use conventional numerical methods for their approximations. Furthermore, we estimate the errors in the means and central moments generated by the approximation method. We also find the moments at equilibrium by solving truncated algebraic equations. We show in examples that numerical solutions based on the moment closure method are accurate and efficient by comparing the results to those of stochastic simulation algorithms.

Optimization of selected molecular orbitals in group basis sets
View Description Hide DescriptionWe derive a local basis equation which may be used to determine the orbitals of a group of electrons in a system when the orbitals of that group are represented by a group basis set, i.e., not the basis set one would normally use but a subset suited to a specific electronic group. The group orbitals determined by the local basis equation minimize the energy of a system when a group basis set is used and the orbitals of other groups are frozen. In contrast, under the constraint of a group basis set, the group orbitals satisfying the Huzinaga equation do not minimize the energy. In a test of the local basis equation on HCl, the group basis set included only 12 of the 21 functions in a basis set one might ordinarily use, but the calculated active orbital energies were within 0.001 hartree of the values obtained by solving the Hartree–Fock–Roothaan (HFR) equation using all 21 basis functions. The total energy found was just 0.003 hartree higher than the HFR value. The errors with the group basis set approximation to the Huzinaga equation were larger by over two orders of magnitude. Similar results were obtained for with the group basis approximation. Retaining more basis functions allows an even higher accuracy as shown by the perfect reproduction of the HFR energy of HCl with 16 out of 21 basis functions in the valence basis set. When the core basis set was also truncated then no additional error was introduced in the calculations performed for HCl with various basis sets. The same calculations with fixed core orbitals taken from isolated heavy atoms added a small error of about . This offers a practical way to calculate wave functions with predetermined fixed core and reduced base valence orbitals at reduced computational costs. The local basis equation can also be used to combine the above approximations with the assignment of local basis sets to groups of localized valence molecular orbitals and to derive a priori localized orbitals. An appropriately chosen localization and basis set assignment allowed a reproduction of the energy of hexane with an error of , while the energy difference between its two conformers was reproduced with a similar accuracy for several combinations of localizations and basis set assignments. These calculations include localized orbitals extending to 4–5 heavy atoms and thus they require to solve reduced dimension secular equations. The dimensions are not expected to increase with increasing system size and thus the local basis equation may find use in linear scaling electronic structure calculations.

Parallel canonical Monte Carlo simulations through sequential updating of particles
View Description Hide DescriptionIn canonical Monte Carlo simulations, sequential updating of particles is equivalent to random updating due to particle indistinguishability. In contrast, in grand canonical Monte Carlo simulations, sequential implementation of the particle transfer steps in a dense grid of distinct points in space improves both the serial and the parallel efficiency of the simulation. The main advantage of sequential updating in parallel canonical Monte Carlo simulations is the reduction in interprocessor communication, which is usually a slow process. In this work, we propose a parallelization method for canonical Monte Carlo simulations via domain decomposition techniques and sequential updating of particles. Each domain is further divided into a middle and two outer sections. Information exchange is required after the completion of the updating of the outer regions. During the updating of the middle section, communication does not occur unless a particle moves out of this section. Results on two and threedimensional LennardJones fluids indicate a nearly perfect improvement in parallel efficiency for large systems.

Modeling vibrational resonance in linear hydrocarbon chain with a mixed quantumclassical method
View Description Hide DescriptionThe quantum dynamics of a vibrational excitation in a linear hydrocarbon model system is studied with a new mixed quantumclassical method. The method is suited to treat manybody systems consisting of a low dimensional quantum primary part coupled to a classical bath. The dynamics of the primary part is governed by the quantum corrected propagator, with the corrections defined in terms of matrix elements of zeroth order propagators. The corrections are taken to the classical limit by introducing the frozen Gaussian approximation for the bath degrees of freedom. The ability of the method to describe dynamics of multidimensional systems has been tested. The results obtained by the method have been compared to previous quantum simulations performed with the quasiadiabatic path integral method.

Bridging fluctuating hydrodynamics and molecular dynamics simulations of fluids
View Description Hide DescriptionA new multiscale coarsegraining (CG) methodology is developed to bridge molecular and hydrodynamic models of a fluid. The hydrodynamic representation considered in this work is based on the equations of fluctuatinghydrodynamics (FH). The essence of this method is a mapping from the position and velocity vectors of a snapshot of a molecular dynamics (MD) simulation to the field variables on Eulerian cells of a hydrodynamic representation. By explicit consideration of the effective lengthscale that characterizes the volume of a molecule, the computed density fluctuations from MD via our mapping procedure have volume dependence that corresponds to a grand canonical ensemble of a cold liquid even when a small cell length (5–10 Å) is used in a hydrodynamic representation. For TIP3P water at 300 K and 1 atm, is found to be 2.4 Å, corresponding to the excluded radius of a water molecule as revealed by its centerofmass radial distribution function. By matching the density fluctuations and autocorrelation functions of momentum fields computed from solving the FH equations with those computed from MD simulation, the sound velocity and shear and bulk viscosities of a CG hydrodynamic model can be determined directly from MD. Furthermore, a novel staggered discretization scheme is developed for solving the FH equations of an isothermal compressive fluid in a three dimensional space with a central difference method. This scheme demonstrates high accuracy in satisfying the fluctuationdissipation theorem. Since the causative relationship between field variables and fluxes is captured, we demonstrate that the staggered discretization scheme also predicts correct physical behaviors in simulating transient fluid flows. The techniques presented in this work may also be employed to design multiscale strategies for modeling complex fluids and macromolecules in solution.

Toward blackboxtype full and reduceddimensional variational (ro)vibrational computations
View Description Hide DescriptionA blackboxtype algorithm is presented for the variational computation of energy levels and wave functions using a (ro)vibrational Hamiltonian expressed in an arbitrarily chosen bodyfixed frame and in any set of internal coordinates of full or reduced vibrational dimensionality. To make the required numerical work feasible, matrix representation of the operators is constructed using a discrete variable representation (DVR). The favorable properties of DVR are exploited in the straightforward and numerically exact inclusion of any representation of the potential and the kinetic energy including the matrix and the extrapotential term. In this algorithm there is no need for an a priori analytic derivation of the kinetic energy operator, as all of its matrix elements at each grid point are computed numerically either in a full or a reduceddimensional model. Due to the simple and straightforward definition of reduceddimensional models within this approach, a fully anharmonic variational treatment of large, otherwise intractable molecular systems becomes available. In the computer code based on the above algorithm, there is no inherent limitation for the maximally coupled number of vibrational degrees of freedom. However, in practice current personal computers allow the treatment of about nine fully coupled vibrational dimensions. Computation of vibrational band origins of full and reduced dimensions showing the advantages and limitations of the algorithm and the related computer code are presented for the water, ammonia, and methane molecules.

An “optimal” spawning algorithm for adaptive basis set expansion in nonadiabatic dynamics
View Description Hide DescriptionThe full multiple spawning (FMS) method has been developed to simulate quantum dynamics in the multistate electronic problem. In FMS, the nuclear wave function is represented in a basis of coupled, frozen Gaussians, and a “spawning” procedure prescribes a means of adaptively increasing the size of this basis in order to capture population transfer between electronic states. Herein we detail a new algorithm for specifying the initial conditions of newly spawned basis functions that minimizes the number of spawned basis functions needed for convergence. “Optimally” spawned basis functions are placed to maximize the coupling between parent and child trajectories at the point of spawning. The method is tested with a twostate, onemode avoided crossing model and a twostate, twomode conical intersection model.

Simulated xray scattering of protein solutions using explicitsolvent models
View Description Hide DescriptionXray solution scattering shows new promise for the study of protein structures, complementing crystallography and nuclear magnetic resonance. In order to realize the full potential of solution scattering, it is necessary to not only improve experimental techniques but also develop accurate and efficient computational schemes to relate atomistic models to measurements. Previous computational methods, based on continuum models of water, have been unable to calculate scattering patterns accurately, especially in the wideangle regime which contains most of the information on the secondary, tertiary, and quaternary structures. Here we present a novel formulation based on the atomistic description of water, in which scattering patterns are calculated from atomic coordinates of protein and water. Without any empirical adjustments, this method produces scattering patterns of unprecedented accuracy in the length scale between 5 and 100 Å, as we demonstrate by comparing simulated and observed scattering patterns for myoglobin and lysozyme.

Theoretical prediction of atomic and electronic structure of neutral clusters
View Description Hide DescriptionIn this paper we found the most stable structures of siliconoxide clusters of by using the genetic algorithm. In this work the genetic algorithm uses a semiempirical energy function, MSINDO, to find the best cluster structures of . The best structures found were further optimized using the density functional theory. We report the stable geometries, binding energies, lowest unoccupied molecular orbitalhighest occupied molecular orbital gap, dissociation energies for the most favorable fragmentation channels and polarizabilities of . For most of the clusters studied here we report structures not previously found using limited search approaches on common structural motifs.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Renner–Teller effect in linear tetraatomic molecules. I. Variational method including couplings between all degrees of freedom on sixdimensional potential energy surfaces
View Description Hide DescriptionFor electronically degenerate states of linear tetraatomic molecules, a new method is developed for the variational treatment of the Renner–Teller and spinorbit couplings. The approach takes into account all rotational and vibrational degrees of freedom, the dominant couplings between the corresponding angular momenta as well as the couplings with the electronic and electron spin angular momenta. The complete rovibrational kinetic energy operator is expressed in Jacobi coordinates, where the rovibrational angular momenta have been replaced by and the spinorbit coupling has been described by the perturbative term . Attention has been paid on the electronic wave functions, which require additional phase for linear tetraatomic molecules. Our implemented rovibrational basis functions and the integration of the different parts of the total Hamiltonian operator are described. This new variational approach is tested on the electronic ground state of for which new sixdimensional potential energy surfaces have been computed using the internally contracted multireference configuration interaction method and the ccpV5Z basis set. The calculated rovibronic energies and their comparisons with previous theoretical and experimental works are presented in the next paper.

Renner–Teller effect in linear tetraatomic molecules. II. Rovibronic levels analysis of the electronic state of
View Description Hide DescriptionThe variational approach detailed in the previous paper (Paper I) for the treatment of the Renner–Teller effect in linear tetraatomic molecules including all degrees of freedom and couplings between angular momenta is applied for . The accurate sixdimensional potential energy surfaces of the electronic state, presented in Paper I is incorporated in the variational treatment in order to obtain all rovibronic levels including the spinorbit coupling for and up to above the global zero point energy. The “pure” stretching levels are calculated up to from the stretching zero point energy. The calculated rovibronic energies are compared with previous theoretical and experimental data. The mean agreement with the zero kinetic energy photoelectron measurements of Tang et al. [J. Chem. Phys.125, 133201 (2006)] is of . The Renner–Teller parameters have been determined at , , , and . A detailed analysis of the rovibronic Hund’s cases is presented and the rotational structures of some vibronic bands recorded by Yang and Mo [J. Phys. Chem. A110, 11001 (2006)] are given.

Experimental and theoretical investigation of threedimensional nitrogendoped aluminum clusters and
View Description Hide DescriptionThe structure and electronic properties of the and clusters were investigated by combined photoelectron spectroscopy and ab initio studies. Congested photoelectron spectra were observed and experimental evidence was obtained for the presence of multiple isomers for . Global minimum searches revealed several structures for with close energies. The calculated vertical detachment energies of the two lowestlying isomers, which are of and symmetry, respectively, were shown to agree well with the experimental data. Unlike the threedimensional structures of and , in which the dopant N atom has a high coordination number of 6, the dopant N atom in the two lowlying isomers of has a lower coordination number of 4 and 5, respectively. The competition between the Al–Al and Al–N interactions are shown to determine the global minimum structures of the dopedaluminum clusters and results in the structural diversity for both and .

Small iron doped carbon clusters: A comparison with early and late firstrow transition metal doped clusters
View Description Hide DescriptionA systematic study of the three lowestlying structures, namely, linear, fan, and cyclic, of neutral clusters on the singlet, triplet, quintet, and septet potential energy surfaces has been carried out. Predictions for their electronic energies, rotational constants, dipole moments, and vibrational frequencies have been made using the B3LYP method in conjunction with the basis set. Triplet or quintet states are found as the lowestlying ones for clusters, and the septet states are found more stable than the singlet ones. The incremental binding energies show an evenodd parity effect, with even clusters being more stable than the odd ones in the linear and fan clusters, whereas a decrease with is found for cyclic ones. The most stable isomers for clusters correspond to a fantype structure for , whereas for cyclic structures are the most stable. Only in the case of the most stable isomer is the linear structure. Comparing the results of clusters with the previously studied (, Ti, V, Co, and Zn) systems, we can see that, as it should be expected, irondopedcarbon clusters present an intermediate behavior between early and late firstrow transition metaldoped clusters.

Imaging the rotationally stateselected product from the predissociation of the state of the NO–Ar van der Waals cluster
View Description Hide DescriptionThe origin of the resonant structures in the spectrum of the predissociative part of the state in the NO–Ar van der Waals cluster has been investigated. We have employed direct excitation to the predissociative part of the NO–Ar state followed by rotational state selective ionization of the NO fragment. Velocity map imaging of the NO ion yields the recoil energy of the rotational stateselected fragment. A substantial contribution of rotational hotbands to the resonant structures is observed. Our data indicate that a centrifugal barrier as the origin of these resonances can be ruled out. We hypothesize that after the NO–Ar cluster is excited to the state sufficient mixing within the rotating cluster takes place as it changes geometry from being T shaped in the state to linear in the state. This mixing allows the low energy and high angular momentum tumbling motion of the initially populated hotbands in the ground state complex to be converted into spinning rotation in the state of the complex. The electronically excited spinning complex falls apart adiabatically producing rotationally excited at the energetic threshold. This interpretation indicates that the resonances can be attributed to some type of vibrational Feshbach resonance. The appearance energy for the formation of is found to be .