Volume 130, Issue 5, 07 February 2009
 ARTICLES

 Theoretical Methods and Algorithms

Higherorder explicitly correlated coupledcluster methods
View Description Hide DescriptionEfficient computer codes for the explicitly correlated coupledcluster (CCR12 or F12) methods with up to triple (CCSDTR12) and quadruple excitations (CCSDTQR12), which take account of the spin, Abelian pointgroup, and indexpermutation symmetries and are based on complete diagrammatic equations, have been implemented with the aid of the computerized symbolic algebraSMITH. Together with the explicitly correlated coupledcluster singles and doubles (CCSDR12) method reported earlier [T. Shiozaki et al., J. Chem. Phys.129, 071101 (2008)], they form a hierarchy of systematic approximations that converge very rapidly toward the exact solutions of the polyatomic Schrödinger equations with respect to both the highest excitation rank and basisset size. Using the Slatertype function as a correlation function, a CCR12 method can provide the augccpV5Zquality results of the conventional CC method of the same excitation rank using only the augccpVTZ basis set. Combining these CCR12 methods with the gridbased, numerical Hartree–Fock equation solver [T. Shiozaki and S. Hirata, Phys. Rev. A76, 040503(R) (2007)], the solutions (eigenvalues) of the Schrödinger equations of neon, boron hydride, hydrogen fluoride, and water at their equilibrium geometries have been obtained as , , , and , respectively, without resorting to completebasisset extrapolations. These absolute total energies or the corresponding correlationenergies agree within the quoted uncertainty with the accurate, nonrelativistic, Born–Oppenheimer values derived experimentally and/or computationally.

Nonequilibrium work relations for systems subject to mechanical and thermal changes
View Description Hide DescriptionGeneralized forms of the Crooks fluctuation theorem are derived for nonHamiltonian and Hamiltonian systems subject to both mechanical and thermal changes. Almost identical derivations are provided for the two cases under rather general assumptions. The basic one is that the probability distribution is a stationary solution of the Liouville equation for fixed values of mechanical control parameters applied to collective variables of the system and for fixed temperature. Generalized expressions for several nonequilibrium work relations derivable from the Crooks fluctuation theorem, such as the Jarzynski equality, pathensemble averages for systems driven far from equilibrium, Bennett acceptance ratio, and two workbased potential of mean force estimators, are also derived. Although this list is not complete, the extension to other related work theorems is straightforward. The application of the methodology is illustrated for two representative cases, namely, for systems evolving with isochoricisokinetic and isothermalisobaric equations of motion.

How to make thermodynamic perturbation theory to be suitable for low temperature?
View Description Hide DescriptionLow temperature unsuitability is a problem plaguing thermodynamic perturbation theory (TPT) for years. Present investigation indicates that the low temperature predicament can be overcome by employing as reference system a nonhard sphere potential which incorporates one part of the attractive ingredient in a potential function of interest. In combination with a recently proposed TPT [S. Zhou, J. Chem. Phys.125, 144518 (2006)] based on a expansion ( being coupling parameter), the new perturbation strategy is employed to predict for several model potentials. It is shown that the new perturbation strategy can very accurately predict various thermodynamic properties even if the potential range is extremely short and hence the temperature of interest is very low and current theoretical formalisms seriously deteriorate or critically fail to predict even the existence of the critical point. Extensive comparison with existing liquid state theories and available computer simulation data discloses a superiority of the present TPT to two Ornstein–Zerniketype integral equationtheories, i.e., hierarchical reference theory and selfconsistent Ornstein–Zernike approximation.

Simplified CCSD(T)F12 methods: Theory and benchmarks
View Description Hide DescriptionThe simple and efficient CCSD(T) approximations we proposed in a recent communication [T. B. Adler, G. Knizia, and H.J. Werner, J. Chem. Phys.127, 221106 (2007)] are explained in more detail and extended to openshell systems. Extensive benchmark calculations are presented, which demonstrate great improvements in basis set convergence for a wide variety of applications. These include reactionenergies of both open and closedshell reactions, atomization energies,electron affinities,ionization potentials, equilibrium geometries, and harmonic vibrational frequencies. For all these quantities, results better than the AV5Z quality are obtained already with AVTZ basis sets, and usually AVDZ treatments reach at least the conventional AVQZ quality. For larger molecules, the additional cost for these improvements is only a few percent of the time for a standard CCSD(T) calculation. For the first time ever, total reactionenergies with chemical accuracy are obtained using valencedoublebasis sets.

Accurate calculation of vibrational frequencies using explicitly correlated coupledcluster theory
View Description Hide DescriptionThe recently proposed explicitly correlated approximations [T. B. Adler, G. Knizia, and H.J. Werner, J. Chem. Phys.127, 221106 (2007)] are applied to compute equilibrium structures and harmonic as well as anharmonic vibrational frequencies for , HCN, , , , , , , and the transisomer of 1,2. Using augccpVTZ basis sets, the CCSD(T)F12a equilibrium geometries and harmonic vibrational frequencies are in very close agreement with CCSD(T)/augccpV5Z values. The anharmonic frequencies are evaluated using vibrational selfconsistent field and vibrational configuration interaction methods based on automatically generated potential energy surfaces. The mean absolute deviation of the CCSD(T)F12a/augccpVTZ anharmonic frequencies from experimental values amounts to only .

Local explicitly correlated secondorder perturbation theory for the accurate treatment of large molecules
View Description Hide DescriptionA local explicitly correlated LMP2F12 method is described that can be applied to large molecules. The steep scaling of computer time with molecular size is reduced by the use of local approximations, the scaling with respect to the basis set size per atom is improved by density fitting, and the slow convergence of the correlation energy with orbital basis size is much accelerated by the introduction of terms into the wave function that explicitly depend on the interelectronic distance. The local approximations lead to almost linear scaling of the computational effort with molecular size without much affecting the accuracy. At the same time, the domain error of conventional LMP2 is removed in LMP2F12. LMP2F12 calculations on molecules of chemical interest involving up to 80 atoms, 200 correlated electrons, and 2600 contracted Gaussiantype orbitals, as well as several reactions of large biochemical molecules are reported.

A semiclassical model for the calculation of nonadiabatic transition probabilities for classically forbidden transitions
View Description Hide DescriptionA semiclassical surface hopping model is presented for the calculation of nonadiabatic transition probabilities for the case in which the avoided crossing point is in the classically forbidden regions. The exact potentials and coupling are replaced with simple functional forms that are fitted to the values, evaluated at the turning point in the classical motion, of the Born–Oppenheimer potentials, the nonadiabatic coupling, and their first few derivatives. For the onedimensional model considered, reasonably accurate results for transition probabilities are obtained down to around . The possible extension of this model to many dimensional problems is discussed. The fact that the model requires only information at the turning point, a point that the trajectories encounter would be a significant advantage in many dimensional problems over Landau–Zener type models, which require information at the avoided crossing seam, which is in the forbidden region where the trajectories do not go.

Combining the nuclearelectronic orbital approach with vibronic coupling theory: Calculation of the tunneling splitting for malonaldehyde
View Description Hide DescriptionThe nuclearelectronic orbital (NEO) method is combined with vibronic coupling theory to calculate hydrogen tunneling splittings in polyatomic molecules. In this NEOvibronic coupling approach, the transferring proton and all electrons are treated quantum mechanically at the NEO level, and the other nuclei are treated quantum mechanically using vibronic coupling theory. The dynamics of the molecule are described by a vibronic Hamiltonian in a diabatic basis of two localized nuclearelectronic states for the electrons and transferring proton. This ab initio approach is computationally practical and efficient for relatively large molecules, and the accuracy can be improved systematically. The NEOvibronic coupling approach is used to calculate the hydrogen tunneling splitting for malonaldehyde. The calculated tunneling splitting of is in excellent agreement with the experimental value of . This approach also enables the identification of the dominant modes coupled to the transferring hydrogen motion and provides insight into their roles in the hydrogen tunneling process.

Layered discrete variable representations and their application within the multiconfigurational timedependent Hartree approach
View Description Hide DescriptionThe multiconfigurational timedependent Hartree (MCTDH) approach facilitates multidimensional quantum dynamics calculations by employing a layered representation: in the upper layer of the representation the multidimensional wave function is expanded in a basis of timedependent singleparticle functions, in the lower layer then these singleparticle functions are represented in a larger timeindependent basis. This two layer approach can be generalized to a multilayer MCTDH representation to further increase the numerical efficiency. In recent work [U. Manthe, J. Chem. Phys.128, 164116 (2008)], a quadrature scheme has been introduced which facilitates multilayer MCTDH calculations for Hamiltonians with general potential energy functions. Based on timedependent discrete variable representation (DVR) grids corresponding to the singleparticle function bases the scheme employs the correlation DVR (CDVR) approach. The present work shows that serious problems arise when this original multilayer CDVR approach is applied for accurate calculations of vibrational states. A solution to these problems is presented which uses a revised scheme to define the timedependent DVR grids entering into the CDVR scheme. Applications studying the vibrational states of demonstrate that the resulting revised multilayer MCTDH/CDVR approach yields smoothly convergent and accurate results.

Controlling multidimensional offresonantRaman and infrared vibrational spectroscopy by finite pulse band shapes
View Description Hide DescriptionClosed expressions are derived which incorporate pulse shaping effects in femtosecond nonlinear optical signals involving various combinations of temporally wellseparated vibrationally resonant infrared and electronically offresonant Raman pulses. Combinations of broadband and narrow band pulses that yield multidimensional extensions of coherent antiStokes Raman and sum frequency generationspectroscopy are presented.

Representation independent algorithms for molecular response calculations in timedependent selfconsistent field theories
View Description Hide DescriptionFour different numerical algorithms suitable for a linear scaling implementation of timedependent Hartree–Fock and Kohn–Sham selfconsistent field theories are examined. We compare the performance of modified Lanczos, Arooldi, Davidson, and Rayleigh quotient iterative procedures to solve the randomphase approximation (RPA) (nonHermitian) and Tamm–Dancoff approximation (TDA) (Hermitian) eigenvalueequations in the molecular orbitalfree framework. Semiempirical Hamiltonian models are used to numerically benchmark algorithms for the computation of excited states of realistic molecular systems (conjugated polymers and carbon nanotubes). Convergence behavior and stability are tested with respect to a numerical noise imposed to simulate linear scaling conditions. The results single out the most suitable procedures for linear scaling largescale timedependent perturbation theory calculations of electronic excitations.

A longrangecorrected density functional that performs well for both groundstate properties and timedependent density functional theory excitation energies, including chargetransfer excited states
View Description Hide DescriptionWe introduce a hybrid density functional that asymptotically incorporates full Hartree–Fock exchange, based on the longrangecorrected exchangehole model of Henderson et al. [J. Chem. Phys.128, 194105 (2008)]. The performance of this functional, for groundstateproperties and for vertical excitation energies within timedependent density functional theory, is systematically evaluated, and optimal values are determined for the rangeseparation parameter, , and for the fraction of shortrange Hartree–Fock exchange. We denote the new functional as , since it reduces to the standard PBEh hybrid functional (also known as PBE0 or PBE1PBE) for a certain choice of its two parameters. Upon optimization of these parameters against a set of ground and excitedstate benchmarks, the functional fulfills three important requirements: (i) It outperforms the PBEh hybrid functional for groundstate atomization energies and reaction barrier heights; (ii) it yields statistical errors comparable to PBEh for valence excitation energies in both small and mediumsized molecules; and (iii) its performance for chargetransfer excitations is comparable to its performance for valence excitations. , with the parameters determined herein, is the first density functional that satisfies all three criteria. Notably, shortrange Hartree–Fock exchange appears to be necessary in order to obtain accurate groundstateproperties and vertical excitation energies using the same value of .

Quasichemical theory with a soft cutoff
View Description Hide DescriptionIn view of the wide success of molecular quasichemical theory of liquids, this paper develops the softcutoff version of that theory. This development allows molecular dynamics simulations to be used for the calculation of solvation free energy, whereas the hardcutoff version of the theory needs Monte Carlo simulations. This development also shows how fluids composed of molecules with smooth repulsive interactions can be treated analogously to the molecularfield theory of the hardsphere fluid. In the treatment of liquid water, quasichemical theory with softcutoff conditioning does not change the fundamental convergence characteristics of the theory using hardcutoff conditioning. In fact, hard cutoffs are found here to work better than softer ones in that case.

Optical limiting for microsecond pulses
View Description Hide DescriptionWe present a dynamical theory of nonlinear absorption and propagation of laser pulses with duration in the microsecond time domain. The general theory is applied to fullerene because of its good optical limitingproperties, namely, a rather low ground stateabsorption and a strong triplettriplet absorption. It is shown that sequential absorption involving strong triplettriplet transitions is the major mechanism of nonlinear absorption. The intrinsic hierarchy of time scales makes an adiabatic solution of the coupled rate equations valid, which therefore can be reduced to a single dynamical equation for the ground state population. The slow evolution of this population is defined by an effective rate of population transfer to the triplet state and by the pulse duration. The propagation effect plays an important role in the optical power limiting performance. The intensity of the field as well as the population of the triplet state decreases during the pulse propagation, and a weakened nonlinear sequential twophoton absorption is followed by a linear onephoton absorption which gradually becomes the dominating process. The competition between these qualitatively different processes depends on the field intensity, the length of the absorber, and the concentration. The pulse propagation is studied by solving numerically the twodimensional paraxial field equation together with the effective rate equation for the ground state population.

Closure of quantum hydrodynamic moment equations
View Description Hide DescriptionThe hydrodynamic formulation of mixed quantum states involves a hierarchy of coupled equations of motion for the momentum moments of the Wigner function. In this work a closure scheme for the hierarchy is developed. The closure scheme uses information contained in the lower known moments to expand the Wigner phasespace distribution function in a Gauss–Hermite orthonormal basis. The higher moment required to terminate the hierarchy is then easily obtained from the reconstructed approximate Wigner function by a straightforward integration over the momentum space. Application of the moment closure scheme is demonstrated for the dissipative and nondissipative dynamics of two different systems: (i) doublewell potential, (ii) periodic potential.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Exploring the dynamics of reactions of oxygen atoms in states and with ethene at collision energy
View Description Hide DescriptionIn a crossed molecularbeam apparatus, we reacted atomic O in states and with ethene at collision energy . Employing two mixtures, and , as discharge media allowed us to generate two sources of oxygen atoms that have the same mean velocity but different ratios of populations, 0.0017 and 0.035. We identified six reactions and recorded timeofflight spectra of products , , and as a function of laboratory angle. Reaction has a fraction of energy release in translation, and product has a maximal probability at scattering angle of 140°. For reaction, , and the angular distribution of product shows a backward preference. For reaction, , and the angular distribution of product has a slight preference for a sideways direction. In contrast, reaction has and an angular distribution with forward and backward peaking and symmetry. Reactions have and 0.08, respectively, and angular distributions with forward and backward peaking and nearly symmetric. The reactivity of with ethene is ca. 38 and 90 times that of for channels to eliminate and , respectively. For reactions of , the branching ratio for elimination of 2H is ca. 3.3 times that for elimination of .

Electron momentum spectroscopy study of Jahn–Teller effect in cyclopropane
View Description Hide DescriptionThe binding energy spectra of cyclopropane in energy range of 9–19 eV have been measured by the high resolution (, ) spectrometer with 0.5 eV energy resolution. The individual electron momentum distributions for the outer valence orbitals including the two Jahn–Teller splitting components for the highest occupied molecular orbital have been obtained. Compared with the results of high level quantum chemistry calculations, the observed different intensities of the experimental momentum profiles for the two Jahn–Teller splitting components have been phenomenologically explained, showing the influences caused by the change in molecular geometry at the instant of ionization from doubly degenerate orbital on the electron momentum distributions. In addition, the controversial ordering of the two outer valence orbitals and has also been assigned unambiguously.

A new method for determining absorption cross sections out of initially excited vibrational states
View Description Hide DescriptionA first experimental demonstration, combining the methods of vibrationally mediated photodissociation (VMP) and ionizationloss stimulated Raman spectroscopy (ILSRS) for measuring cross sections for dissociation of vibrationally excited levels is reported. The action spectrum obtained in the VMP of methylamine exhibits enhancement of the H photofragment yield as a result of initial vibrational excitation and the ILSRS monitors the fraction of molecules being excited. The partial cross sections for H production out of the sampled vibrational states and the extent of mode selectivity were thus determined.

Photoelectron imaging and theoretical investigation of bimetallic and cluster anions
View Description Hide DescriptionWe present the results of photoelectron velocitymap imaging experiments for the photodetachment of small negatively charged (, ), and clusters at 527 nm. The photoelectron images reveal new features along with their angular distributions in the photoelectron spectra of these clusters. We report the vertical detachment energies of the observed multiple electronic bands and their respective anisotropy parameters for the and clusters derived from the photoelectron images. Experiments on the clusters reveal that the electron affinity increases with the number of Ga atoms from to 2. The cluster is found to be stable, both because of its even electron number and the high electron affinity of . The measured photoelectron angular distributions of the and clusters are dependent on both the orbital symmetry and electron kinetic energies. Densityfunctional theory calculations employing the generalized gradient approximation for the exchangecorrelation potential were performed on these clusters to determine their atomic and electronic structures. From the theoretical calculations, we find that the , and (anionic), and , , and (neutral) clusters are unusually stable. The stability of the anionic and neutral clusters is attributed to an evenodd effect, with clusters having an even number of electrons presenting a larger gain in energy through the addition of a Ga atom to the preceding size compared to odd electron systems. The stability of the neutral cluster is rationalized as being similar to , an allmetal aromatic cluster.

Quasiclassical trajectory calculations analyzing the role of vibrational and translational energy in the reaction
View Description Hide DescriptionAn exhaustive statetostate dynamics study was performed to analyze the effects of vibrational excitation and translational energy on the dynamics of the gasphase reaction, which are connected to such issues as bond selectivity, mode selectivity, and Polanyi’s rules. This reaction can evolve along two channels: Dabstraction, , and Habstraction, . Quasiclassical trajectory calculations were performed on an analytical potential energy surface previously developed by our group. Vibrational excitation of the C–D or C–H mode of favors slightly the Dabstraction over the Habstraction, indicating that this reaction does not exhibit bond selectivity and suggesting a breakdown of the spectator model. For Dabstraction, the vibrational excitation of the nonreactive C–H stretch mode is partially retained in the products, and for Habstraction, the excitation of the nonreactive C–D stretch mode is also partially retained in the products, indicating that this reaction exhibits mode selectivity only partially. Moreover, the independent excitation of the C–H symmetric or asymmetric stretch modes leads to reactions with similar (practically identical) reaction cross sections and product scattering distributions, discarding bond selectivity and mode selectivity for this reaction. Finally, for this “early transition state” reaction, vibrational energy is more effective in driving the reaction than an equivalent amount of energy in translation, indicating that the application of the Polanyi rules that are well established in atomdiatom reactions is neither straightforward nor always valid in polyatomic reactions. All these results were interpreted on the basis of strong coupling between modes along the reaction path, a behavior which seems to be more of the general tendency than the exception in polyatomic reactions.