Volume 136, Issue 10, 14 March 2012

The lowestenergy isomers of Cu_{N}clusters for N = 20–30 are identified using an unbiased search algorithm and density functional theory calculations. The lowenergy structures over this size range are dominated by those based on a 13atom icosahedral (I_{h}) core and a 19atom double icosahedron (DI_{h}) core. A transition in the groundstate isomers from I_{h}based to DI_{h}based structures is predicted overt N = 21–23. We discuss this transition in the broader context of the growth pattern for Cu_{N} over N = 2–30 that features regions of gradual evolution in which atoms successively add to the clustersurface, separated by sudden changes to a different structural organization and more compact shape. These transitions result from a competition between interatomic bonding energy and surface energy. The implications of this growth pattern for the further evolution of copper from microstructure to bulk are discussed.
 ARTICLES

 Theoretical Methods and Algorithms

Benchmarking the performance of timedependent density functional methods
View Description Hide DescriptionThe performance of 24 density functionals, including 14 metageneralized gradient approximation (mGGA) functionals, is assessed for the calculation of vertical excitation energies against an experimental benchmark set comprising 14 small to mediumsized compounds with 101 total excited states. The experimental benchmark set consists of singlet, triplet, valence, and Rydbergexcited states. The globalhybrid (GH) version of the PerdewBurkeErnzerhoff GGA density functional (PBE0) is found to offer the best overall performance with a mean absolute error (MAE) of 0.28 eV. The GHmGGA Minnesota 2006 density functional with 54% HartreeFock exchange (M062X) gives a lower MAE of 0.26 eV, but this functional encounters some convergence problems in the ground state. The local density approximation functional consisting of the Slater exchange and VolkWilkNusair correlation functional (SVWN) outperformed all nonGH GGAs tested. The best pure density functional performance is obtained with the local version of the Minnesota 2006 mGGA density functional (M06L) with an MAE of 0.41 eV.

Resolutions of the Coulomb operator. VI. Computation of auxiliary integrals
View Description Hide DescriptionWe discuss the efficient computation of the auxiliary integrals that arise when resolutions of twoelectron operators (specifically, the Coulomb operator [T. Limpanuparb, A. T. B. Gilbert, and P. M. W. Gill, J. Chem. Theory Comput.7, 830 (2011)10.1021/ct200115t] and the longrange Ewald operator [T. Limpanuparb and P. M. W. Gill, J. Chem. Theory Comput.7, 2353 (2011)10.1021/ct200305n]) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a nearoptimal algorithm for its use.

Monotonically convergent optimization in quantum control using Krotov's method
View Description Hide DescriptionThe nonlinear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.)60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the nonlinear Schrödinger equation[S. E. Sklarz and D. J. Tannor, Phys. Rev. A66, 053619 (2002)]10.1103/PhysRevA.66.053619. Here we show that based on the KonnovKrotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by nonunitary time evolution, nonlinear dependencies of the Hamiltonian on the control, timedependent targets, and optimizationfunctionals that depend to higher than second order on the timeevolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimizationfunctional that is an eighthdegree polynomial in the states. For the “standard” quantum control problem of a convex finaltime functional,linear equations of motion and linear dependency of the Hamiltonian on the field, the secondorder contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first and secondorder algorithms for two examples.

The longrange nonadditive threebody dispersion interactions for the rare gases, alkali, and alkalineearth atoms
View Description Hide DescriptionThe longrange nonadditive threebody dispersion interaction coefficients Z _{111}, Z _{112}, Z _{113}, and Z _{122} are computed for many atomic combinations using standard expressions. The atoms considered include hydrogen, the rare gases, the alkali atoms (up to Rb), and the alkalineearth atoms (up to Sr). The term Z _{111} arising from three mutual dipole interactions is known as the AxilrodTellerMuto coefficient or the DDD (dipoledipoledipole) coefficient. Similarly, the terms Z _{112}, Z _{113}, and Z _{122} arise from the mutual combinations of dipole (1), quadrupole (2), and octupole (3) interactions between atoms and they are sometimes known, respectively, as dipoledipolequadrupole, dipoledipoleoctupole, and dipolequadrupolequadrupole coefficients. Results for the four Z coefficients are given for the homonuclear trimers, for the trimers involving two likeraregas atoms, and for the trimers with all combinations of the H, He, and Li atoms. An exhaustive compilation of all coefficients between all possible atomic combinations is presented as supplementary data.

A formalism for scattering of complex composite structures. I. Applications to branched structures of asymmetric subunits
View Description Hide DescriptionWe present a formalism for the scattering of an arbitrary linear or acyclic branched structure build by joining mutually noninteracting arbitrary functional subunits. The formalism consists of three equations expressing the structuralscattering in terms of three equations expressing the subunit scattering. The structuralscattering expressions allow composite structures to be used as subunits within the formalism itself. This allows the scattering expressions for complex hierarchical structures to be derived with great ease. The formalism is generic in the sense that the scattering due to structural connectivity is completely decoupled from internal structure of the subunits. This allows subunits to be replaced by more complex structures. We illustrate the physical interpretation of the formalism diagrammatically. By applying a selfconsistency requirement, we derive the pair distributions of an ideal flexible polymer subunit. We illustrate the formalism by deriving generic scattering expressions for branched structures such as stars, pompoms, bottlebrushes, and dendrimers build out of asymmetric twofunctional subunits.

Steadystate parameter sensitivity in stochastic modeling via trajectory reweighting
View Description Hide DescriptionParameter sensitivity analysis is a powerful tool in the building and analysis of biochemical network models. For stochastic simulations, parameter sensitivity analysis can be computationally expensive, requiring multiple simulations for perturbed values of the parameters. Here, we use trajectory reweighting to derive a method for computing sensitivity coefficients in stochastic simulations without explicitly perturbing the parameter values, avoiding the need for repeated simulations. The method allows the simultaneous computation of multiple sensitivity coefficients. Our approach recovers results originally obtained by application of the Girsanov measure transform in the general theory of stochastic processes[A. Plyasunov and A. P. Arkin, J. Comput. Phys.221, 724 (2007)10.1016/j.jcp.2006.06.047]. We build on these results to show how the method can be used to compute steadystate sensitivity coefficients from a single simulation run, and we present various efficiency improvements. For models of biochemical signaling networks, the method has a particularly simple implementation. We demonstrate its application to a signaling network showing stochastic focussing and to a bistable genetic switch, and present exact results for models with linear propensity functions.

A DFT+U study of acetylene selective hydrogenation on oxygen defective anatase (101) and rutile (110) TiO_{2} supported Pd_{4} cluster
View Description Hide DescriptionThe reaction mechanisms for selective acetylene hydrogenation on three different supports, Pd_{4} cluster, oxygen defective anatase (101), and rutile (110) titania supported Pd_{4}, cluster are studied using the density functional theory calculations with a Hubbard U correction (DFT+U). The present calculations show that the defect anatase support binds Pd_{4} cluster more strongly than that of rutile titania due to the existence of Ti^{3+} in anatase titania. Consequently, the binding energies of adsorbed species such as acetylene and ethylene on Pd_{4} cluster become weaker on anatase supported catalysts compared to the rutile supported Pd_{4} cluster. Anatase catalyst has higher selectivity of acetylene hydrogenation than rutile catalyst. On the one hand, the activation energies of ethylene formation are similar on the two catalysts, while they vary a lot on ethyl formation. The rutile supported Pd catalyst with lower activation energy is preferable for further hydrogenation. On the other hand, the relatively weak adsorptionenergy of ethylene is gained on anatase surface, which means it is easier for ethylene desorption, hence getting higher selectivity. For further understanding, the energy decomposition method and microkinetic analysis are also introduced.

Nonempirical improvement of PBE and its hybrid PBE0 for general description of molecular properties
View Description Hide DescriptionImposition of the constraint that, for the hydrogen atom, the exchange energy cancels the Coulomb repulsion energy yields a nonempirical reparameterization of the PerdewBurkeErnzerhof (PBE) generalized gradient approximation (GGA) exchangecorrelation energy functional, and of the related PBE hybrid (PBE0). The reparameterization, which leads to an increase of the gradient contribution to the exchange energy with respect to the original PBE functional, is tested through the calculation of heats of formation, ionization potentials,electron affinities,proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and nonhydrogen transfer reactions, bond distances, and harmonic frequencies, for some well known test sets designed to validate energy functionals. The results for the reparameterized PBE GGA, called PBEmol, give substantial improvement over the original PBE in the prediction of the heats of formation, while retaining the quality of the original PBE functional for description of all the other properties considered. The results for the hybrids indicate that, although the PBE0 functional provides a rather good description of these properties, the predictions of the reparameterized functional, called PBEmolβ0, are, except in the case of the ionization potentials, modestly better. Also, the results for PBEmolβ0 are comparable to those of B3LYP. In particular, the mean absolute error for the bond distance test set is 17% lower than the corresponding error for B3LYP. The reparameterization for the pure GGA (PBEmol) differs from that for the hybrid (PBEmolβ0), illustrating that improvement at the GGA level of complexity does not necessarily provide the best GGA for use in a hybrid.

On the pressure calculation for polarizable models in computer simulation
View Description Hide DescriptionWe present a short overview of pressure calculation in molecular dynamics or Monte Carlo simulations. The emphasis is given to polarizable models in order to resolve the controversy caused by the paper of M. J. Louwerse and E. J. Baerends [Chem. Phys. Lett.421, 138 (2006)10.1016/j.cplett.2006.01.087] about pressure calculation in systems with periodic boundaries. We systematically derive expressions for the pressure and show that despite the lack of explicit pairwise additivity, the pressure formula for polarizable models is identical with that of nonpolarizable ones. However, a strict condition for using this formula is that the induced dipole should be in perfect mechanical equilibrium prior to pressure calculation. The perfect convergence of induced dipoles ensures conservation of energy as well. We demonstrate using more cumbersome but exact methods that the derived expressions for the polarizable model of water provide correct numerical results. We also show that the inaccuracy caused by imperfect convergence of the induced dipoles correlates with the inaccuracy of the calculated pressure.
 Advanced Experimental Techniques

Doping helium nanodroplets with high temperature metals: Formation of chromium clusters
View Description Hide DescriptionA new method for stable and continuous doping of superfluid heliumnanodroplets (He_{N}) with highmelting elements such as refractory metals is presented. The method exploits the advantages of electron bombardment heating and avoids stray fields induced by high currents or high frequency fields. It is thus especially suitable for magnetic studies of atoms and clusters in He_{N}. The source is characterized by means of mass spectroscopic investigations of He_{N}doped with chromium atoms and clusters. Source temperatures of up to (1650 ± 50) °C were reached and Cr clusters up to Cr_{9} could be formed in He_{N}.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

B_{14}: An allboron fullerene
View Description Hide DescriptionExperiments revealed that small boron cluster anions and cations are (quasi)planar. For neutral boron cluster, (quasi)planar motifs are also suggested to be global minimum by many theoretical studies, and a structural transformation from quasiplanar to doublering tubular structures occurs at B_{20}. However, a missing opportunity is found for neutral B_{14}, which is a flat cage and more stable than the previous quasiplanar one by high level ab initio calculations. The B_{14} cage has a large HOMOLUMO gap (2.69 eV), and NICS values reveal that it is even more aromatic than the known most aromatic quasiplanar B_{12} and doublering B_{20}, which indicates a closeshell electronic structure.Chemical bondinganalysis given by AdNDP reveals that the B_{14} cage is an allboron fullerene with 18 delocalized σelectrons following the 2(n+1)^{2} rule of spherical aromaticity. The geometry and bonding features of the B_{14} cage are unique denying conversional thinking.

Quantumdynamics study of the cluster: Full dimensional benchmark results on its vibrational states
View Description Hide DescriptionA fulldimensional quantum dynamics study is carried out for the highly fluxional cation on a recent reference potential energy surface by using the multi configuration timedependent Hartree method. With five equivalent light atoms and shallow barriers between various lowlying stationary points on the surface, the spectroscopic characterization of represents a huge challenge for accurate quantum dynamics simulations. The present calculation is the first such a study on this cation, which together with its isotope analogies are of primary importance in the interstellar chemistry. The vibrational ground state properties and several vibrationally excited states corresponding to low vibrational frequency motions, not yet directly observable by the experiment, are presented and analyzed.

Role of the RennerTeller effect after core hole excitation in the dissociation dynamics of carbon dioxide dication
View Description Hide DescriptionThe fragmentation of the doublycharged carbon dioxide molecule is studied after photoexcitation to the C 1s ^{1}2π_{ u } and O 1s ^{1}2π_{ u } states using a multicoincidence ionimaging technique. The bent component of the RennerTeller split states populated in the 1s→ π* resonant excitation at both the carbon and oxygen 1sionization edges opens pathways to potential surfaces in highly bent geometries in the dication. Evidence for a complete deformation of the molecule is found in the coincident detection of C^{+} and ions. The distinct alignment of this fragmentation channel indicates rapid deformation and subsequent fragmentation. Investigation of the complete atomization dynamics in the dication leading to asymmetric charge separation shows that the primary dissociation mechanisms, sequential, concerted, and asynchronous concerted, are correlated to specific fragment kinetic energies. The study shows that the bond angle in fragmentation can extend below 20°.

Accurate quantum wave packet calculations for the F + HCl → Cl + HF reaction on the ground 1^{2} A ^{′} potential energy surface
View Description Hide DescriptionWe present converged exact quantum wave packet calculations of reaction probabilities, integral cross sections, and thermal rate coefficients for the title reaction. Calculations have been carried out on the ground 1^{2} A ^{′} global adiabatic potential energy surface of Deskevich et al. [J. Chem. Phys.124, 224303 (2006)10.1063/1.2192505]. Converged wave packet reaction probabilities at selected values of the total angular momentum up to a partial wave of J = 140 with the HCl reagent initially selected in the v = 0, j = 0–16 rovibrational states have been obtained for the collision energy range from threshold up to 0.8 eV. The present calculations confirm an important enhancement of reactivity with rotational excitation of the HCl molecule. First, accurate integral cross sections and rate constants have been calculated and compared with the available experimental data.

Spectroscopic investigations of ThF and ThF^{+}
View Description Hide DescriptionThe electronic spectra of ThF and ThF^{+} have been examined using laser induced fluorescence and resonant twophoton ionization techniques. The results from highlevel ab initio calculations have been used to guide the assignment of these data. Spectra for ThF show that the molecule has an X ^{2}Δ_{3/2}ground state. The upper spinorbit component, X ^{2}Δ_{5/2} was found at an energy of 2575(15) cm^{−1}. The lowlying states of ThF^{+} were probed using dispersed fluorescence and pulsed field ionization  zero kinetic energy (PFIZEKE) photoelectron spectroscopy. Vibronic progressions belonging to four electronic states were identified. The lowest energy states were clearly ^{1}Σ^{+} and ^{3}Δ_{1}. Although the energy ordering could not be rigorously determined, the evidence favors assignment of ^{1}Σ^{+} as the ground state. The ^{3}Δ_{1} state, of interest for investigation of the electron electric dipole moment, is just 315.0(5) cm^{−1} above the ground state. The PFIZEKE measurements for ThF yielded an ionization energy of 51 581(3) cm^{−1}. Molecular constants show that the vibrational constant increases and the bond length shortens on ionization. This is consistent with removal of a nonbonding Thcentered 6d or 7s electron. Laser excitation of ThF^{+} was used to probe electronically excited states in the range of 19 000–21 500 cm^{−1}.

Fragment momentum distributions obtained from coupled electronnuclear dynamics
View Description Hide DescriptionWe theoretically investigate fragmentation processes induced by femtosecond laser pulses within a model which incorporates electronic and nuclear motion. Singlepulse excitation leads to diffraction patterns in the electron momentum distribution which depend on the nature of the electronic state and also on the nuclear charge distribution. Additional structures appear in the nuclear momentum distribution if two timedelayed pulses produce fragments in the same dissociation channel. It is shown that these functions are modified by the electronic degreeoffreedom. A simultaneous excitation of two different electronic states results in further interferences which are related to electronic wavepacket dynamics on the attosecond timescale.

Vibrationally resolved photoionization of N_{2} near threshold
View Description Hide DescriptionA recently developed velocity map imaging spectrometer has been used to study the photoionization of molecular nitrogen near threshold. The potentialities of the spectrometer have been exploited to measure simultaneously the energy and angular distribution of the photoelectrons corresponding to the residual N_{2} ^{+} X^{ 2}Σ_{g} v = 0–3 ion states. In a single experiment all the experimental observables, i.e., the total and partial cross sections, their branching ratios and the asymmetry parameter of the angular distributions have been determined.

Photoelectron spectroscopy and density functional calculations of CuSi_{n} ^{−} (n = 4–18) clusters
View Description Hide DescriptionWe conducted a combined anion photoelectron spectroscopy and density functional theory study on the structural evolution of copperdoped silicon clusters, CuSi_{n} ^{−} (n = 4–18). Based on the comparison between the experiments and theoretical calculations, CuSi_{12} ^{−} is suggested to be the smallest fully endohedral cluster. The lowlying isomers of CuSi_{n} ^{−} with n ≥ 12 are dominated by endohedral structures, those of CuSi_{n} ^{−} with n < 12 are dominated by exohedral structures. The most stable structure of CuSi_{12} ^{−} is a doublechair endohedral structure with the copper atom sandwiched between two chairstyle Si_{6} rings or, in another word, encapsulated in a distorted Si_{12} hexagonal prism cage. CuSi_{14} ^{−} has an interesting C_{3h} symmetry structure, in which the Si_{14} cage is composed by three fourmembered rings and six fivemembered rings.

Very accurate potential energy curve of the ion
View Description Hide DescriptionA very accurate groundstate potential energy curve (PEC) of the molecule is calculated with 1200 explicitly correlated Gaussian functions with shifted centers in the range between 0.9 and 100 a_{0}. The calculations include the adiabatic corrections determined for the ^{3}He^{4}He^{+}, ^{3} , and ^{4} isotopologues. The absolute accuracy of the PEC is better than 0.05 cm^{−1} and that of the adiabatic corrections is around 0.01 cm^{−1}. The depths of the PECs augmented with the adiabatic corrections for the three isotopologues are: 19 956.708 cm^{−1} for ^{4} , 19 957.054 cm^{−1} for ^{3}He^{4}He^{+}, and 19 957.401 cm^{−1} for ^{3} . The rovibrational energies are also determined. For ^{3}He^{4}He^{+} the computed rovibrational transitions corresponding to the ν = 1–0 band differ from the experiment by less than 0.005 cm^{−1}. For the rovibrational transitions corresponding to the ν = 23–22 band the difference is around 0.012 cm^{−1}. Presently, this represents the best agreement between theory and experiment for .

A photoelectron spectroscopy and ab initio study of B_{21} ^{−}: Negatively charged boron clusters continue to be planar at 21
View Description Hide DescriptionThe structures and chemical bonding of the B_{21} ^{−} cluster have been investigated by a combined photoelectron spectroscopy and ab initio study. The photoelectron spectrum at 193 nm revealed a very high adiabatic electron binding energy of 4.38 eV for B_{21} ^{−} and a congested spectral pattern. Extensive global minimum searches were conducted using two different methods, followed by highlevel calculations of the lowlying isomers. The global minimum of B_{21} ^{−} was found to be a quasiplanar structure with the next lowlying planar isomer only 1.9 kcal/mol higher in energy at the CCSD(T)/6311G* level of theory. The calculated vertical detachment energies for the two isomers were found to be in good agreement with the experimental spectrum, suggesting that they were both present experimentally and contributed to the observed spectrum. Chemical bondinganalyses showed that both isomers consist of a 14atom periphery, which is bonded by classical twocenter twoelectron bonds, and seven interior atoms in the planar structures. A localized twocenter twoelectron bond is found in the interior of the two planar isomers, in addition to delocalized multicenter σ and π bonds. The structures and the delocalized bonding of the two lowest lying isomers of B_{21} ^{−} were found to be similar to those in the two lowest energy isomers in B_{19} ^{−}.