Volume 129, Issue 4, 28 July 2008
 ARTICLES

 Theoretical Methods and Algorithms

Intermediate Hamiltonian Fockspace multireference coupledcluster method with full triples for calculation of excitation energies
View Description Hide DescriptionThe intermediate Hamiltonian multireference coupledcluster (CC) method with singles, doubles, and triples within the excited (1,1) sector of Fock space (FS) is implemented and formulated to calculate excitation energies (EEs). Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FSCC equations, coupled to an efficient factorization strategy, relatively large basis sets and model spaces are employed permitting basis set converged comparisons of the calculated vertical EEs, which can be compared to the experimental data for the and CO molecules. The issue of chargetransfer separability is also addressed.

Electron spinspin coupling from multireference configuration interaction wave functions
View Description Hide DescriptionWe present the implementation of twoelectron spinspin coupling as a quasidegenerate perturbative treatment of the Breit–Pauli spinspin Hamiltonian. The evaluation is based on a multireference CI treatment and constitutes one of the first efforts in the calculation of this effect within a highly sophisticated consideration of both nondynamical and dynamical correlation. The extension of existing schemes for efficient calculation, in particular, of the spincoupling elements necessitated some involved derivations, the outline of which is presented herein. Application of the program to calculations of diagonal as well as offdiagonal spincoupling elements is illustrated with the test cases and NH.

Extension of linearscaling divideandconquerbased correlation method to coupled cluster theory with singles and doubles excitations
View Description Hide DescriptionThis paper describes the extension of the linearscaling divideandconquer (DC)based correlation method to the coupled cluster with singles and doubles excitations (CCSD) theory. In this DCCCSD method, the CCSD equations are solved for all subsystems including their buffer regions with the use of the subsystem orbitals, which are obtained by the DC–Hartree–Fock method. Then, the correlation energy of the total system is evaluated by summing up the subsystem contributions other than the buffer regions by the energy density analysis technique. Numerical applications demonstrate that the present DCCCSD gives highly accurate results with drastically less computational costs with regard to the required computer memory, scratchdisk capacity, and calculation time.

Mixed quantumclassical equilibrium: Surface hopping
View Description Hide DescriptionWe reexamine the analysis of the equilibrium limits of the fewest switches surface hopping algorithm for mixed quantumclassical dynamics. In contrast with previously reported results, we show that surface hopping does not, in general, exactly yield Boltzmann equilibrium, but that in practice the observed deviations are quite small. We also demonstrate that surface hopping does approach the exact equilibrium distribution in both the limits of small adiabatic splitting and/or strong nonadiabatic coupling. We verify these analytical results with numerical simulations for a simple twolevel quantum system connected to a bath of classical particles.

Timedependent representability on lattice systems
View Description Hide DescriptionWe study the mapping between timedependent densities and potentials for noninteracting electronic systems on lattices. As discovered recently by Baer [J. Chem. Phys.128, 044103 (2008)], there exist wellbehaved timedependent density functions on lattices which cannot be associated with any real timedependent potential. This breakdown of timedependent representability can be tracked down to problems with the continuity equation which arise from discretization of the kineticenergy operator. Examples are given for lattices with two points and with points, and implications for practical numerical applications of timedependent densityfunctional theory are discussed. In the continuum limit, timedependent noninteracting representability is restored.

Inverse molecular design in a tightbinding framework
View Description Hide DescriptionThe number of chemical species of modest molecular weight that can be accessed with known synthetic methods is astronomical. An open challenge is to explore this space in a manner that will enable the discovery of molecular species and materials with optimized properties. Recently, an inverse molecular design strategy, the linear combination of atomic potentials (LCAP) approach [J. Am. Chem. Soc. 128, 3228 (2006)] was developed to optimize electronic polarizabilities and first hyperpolarizabilities. Here, using a simple tightbinding (TB) approach, we show that continuous optimization can be carried out on the LCAP surface successfully to explore vast chemical libraries of to extended aromatic compounds. We show that the TBLCAP optimization is not only effective in locating globally optimal structures based on their electronic polarizabilities and first hyperpolarizabilities, but also is straightforwardly extended to optimize transitiondipole moments and HOMOLUMO energy gaps. This approach finds optimal structures among candidates with about 40 individual molecular property calculations. As such, for structurally similar molecular candidates, the TBLCAP approach may provide an effective means to identify structures with optimal properties.

Extended Hückel tightbinding approach to electronic excitations
View Description Hide DescriptionIn this work, we propose the application of a selfconsistent extended Hückel tightbinding (EHTB) method in the computation of the absorption optical spectrum of molecules within the linear response time dependent density functional formalism. The EHTB approach is presented as an approximation to the Kohn–Sham energy functional. The method is applied to the computation of excitation energies and oscillator strengths of benzene, pyridine, naphthalene, diazines, and the fullerenes:, , and . The very good agreement with experimental data is very encouraging and suggests the possibility of using the EHTB as a computational efficient and reliable tool to study optical properties of a wide variety of molecular systems.

Approximation scheme for master equations: Variational approach to multivariate case
View Description Hide DescriptionWe study an approximation scheme based on a second quantization method for a chemical master equation. Small systems, such as cells, could not be studied by the traditional rate equation approach because fluctuation effects are very large in such small systems. Although a Fokker–Planck equation obtained by the system size expansion includes the fluctuation effects, it needs large computational costs for complicated chemical reaction systems. In addition, discrete characteristics of the original master equation are neglected in the system size expansion scheme. It has been shown that the use of the second quantization description and a variational method achieves tremendous reduction in the dimensionality of the master equation approximately, without loss of the discrete characteristics. Here, we propose a new scheme for the choice of variational functions, which is applicable to multivariate cases. It is revealed that the new scheme gives better numerical results than old ones and the computational cost increases only slightly.

Applicability of hybrid density functional theory methods to calculation of molecular hyperpolarizability
View Description Hide DescriptionThe donor/acceptor (D/A) substituted conjugated organic molecules possess extremely fast nonlinear optical (NLO) response time that is purely electronic in origin. This makes them promising candidates for optoelectronic applications. In the present study, we utilized four hybrid density functionals (B3LYP, B972, PBE0, BMK), Hartree–Fock, and second order Møller–Plesset correlation energy correction, truncated at secondorder (MP2) methods with different basis sets to estimate molecular first hyperpolarizability of D/Asubstituted benzenes and stilbenes (, OH, , ; , CN). The results of density functional theory(DFT) calculations are compared to those of MP2 method and to the experimental data. We addressed the following questions: (1) the accurate techniques to compare calculated results to each other and to experiment, (2) the choice of the basis set, (3) the effect of molecular planarity, and (4) the choice of the method. Comparison of the absolute values of hyperpolarizabilities obtained computationally and experimentally is complicated by the ambiguities in conventions and reference values used by different experimental groups. A much more tangible way is to compare the ratios of ’s for two (or more) given molecules of interest that were calculated at the same level of theory and measured at the same laboratory using the same conventions and reference values. Coincidentally, it is the relative hyperpolarizabilities rather than absolute ones that are of importance in the rational molecular design of effective NLO materials. This design includes prediction of the most promising candidates from particular homologous series, which are to be synthesized and used for further investigation. In order to accomplish this goal, semiquantitative level of accuracy is usually sufficient. Augmentation of the basis set with polarization and diffuse functions changes by 20%; however, further extension of the basis set does not have significant effect. Thus, we recommend basis set. We also show that the use of planar geometry constraints for the molecules, which can somewhat deviate from planarity in the gas phase, leads to sufficient accuracy (with an error less than 10%) of predicted values. For all the molecules studied, MP2 values are in better agreement with experiment, while DFT hybrid methods overestimate values. BMK functional gives the best agreement with experiment, with systematic overestimation close to the factor of 1.4. We propose to use the scaled BMK results for prediction of molecular hyperpolarizability at semiquantitative level of accuracy.

Orbital energies and negative electron affinities from density functional theory: Insight from the integer discontinuity
View Description Hide DescriptionOrbital energies in Kohn–Sham density functional theory(DFT) are investigated, paying attention to the role of the integer discontinuity in the exact exchangecorrelation potential. A series of closedshell molecules are considered, comprising some that vertically bind an excess electron and others that do not. Highlevel ab initio electron densities are used to calculate accurate orbital energy differences, , between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO), using the same potential for both. They are combined with accurate vertical ionization potentials,, and electron affinities,, to determine accurate “average” orbital energies. These are the orbital energies associated with an exchangecorrelation potential that averages over a constant jump in the accurate potential, of magnitude , as given by the discontinuity analysis. Local functional HOMO energies are shown to be almost an order of magnitude closer to these average values than to , with typical discrepancies of just 0.02 a.u. For systems that do not bind an excess electron, this level of agreement is only achieved when is set equal to the negative experimental affinity from electron transmission spectroscopy (ETS); it degrades notably when the zero ground state affinity is instead used. Analogous observations are made for the local functional LUMO energies, although the need to use the ETS affinities is less pronounced for systems where the ETS values are very negative. The application of an asymptotic correction recovers the preference, leading to positive LUMO energies (but bound orbitals) for these systems, consistent with the behavior of the average energies. The asymptotically corrected LUMO energies typically agree with the average values to within 0.02 a.u., comparable to that observed with the HOMOs. The study provides numerical support for the view that local functionals exhibit a nearaverage behavior based on a constant jump of magnitude . It illustrates why a recently proposed DFT expression involving local functional frontier orbital energies and ionization potential yields reasonable estimates of negative ETS affinities and is consistent with earlier work on the failure of DFT for chargetransferexcited states. The nearaverage behavior of the exchangecorrelation potential is explicitly illustrated for selected systems. The nature of hybrid functional orbital energies is also mentioned, and the results of the study are discussed in terms of the variation in electronic energy as a function of electron number. The nature of DFT orbital energies is of great importance in chemistry; this study contributes to the understanding of these quantities.

Parametrization of analytic interatomic potential functions using neural networks
View Description Hide DescriptionA generalized method that permits the parameters of an arbitrary empirical potential to be efficiently and accurately fitted to a database is presented. The method permits the values of a subset of the potential parameters to be considered as general functions of the internal coordinates that define the instantaneous configuration of the system. The parameters in this subset are computed by a generalized neural network (NN) with one or more hidden layers and an input vector with at least elements, where is the number of atoms in the system. The Levenberg–Marquardt algorithm is employed to efficiently affect the optimization of the weights and biases of the NN as well as all other potential parameters being treated as constants rather than as functions of the input coordinates. In order to effect this minimization, the usual Jacobian employed in NN operations is modified to include the Jacobian of the computed errors with respect to the parameters of the potential function. The total Jacobian employed in each epoch of minimization is the concatenation of two Jacobians, one containing derivatives of the errors with respect to the weights and biases of the network, and the other with respect to the constant parameters of the potential function. The method provides three principal advantages. First, it obviates the problem of selecting the form of the functional dependence of the parameters upon the system’s coordinates by employing a NN. If this network contains a sufficient number of neurons, it will automatically find something close to the best functional form. This is the case since Hornik et al., [Neural Networks2, 359 (1989)] have shown that twolayer NNs with sigmoid transfer functions in the first hidden layer and linear functions in the output layer are universal approximators for analytic functions. Second, the entire fitting procedure is automated so that excellent fits are obtained rapidly with little human effort. Third, the method provides a procedure to avoid local minima in the multidimensional parameter hyperspace. As an illustrative example, the general method has been applied to the specific case of fitting the ab initio energies of clusters that are observed in a molecular dynamics (MD) simulation of the machining of a silicon workpiece. The energies of the configurations obtained in the MD calculations are computed using the B3LYP procedure with a basis set. The final ab initiodatabase, which comprises the density functional theory energies of 10 202 clusters, is fitted to an empirical Tersoff potential containing nine adjustable parameters, two of which are allowed to be the functions of the configuration. The fitting error averaged over all 10 202 points is . This result is comparable to the accuracy achieved by more general fitting methods that do not rely on an assumed functional form for the potential surface.

Minimax approximation for the decomposition of energy denominators in Laplacetransformed Møller–Plesset perturbation theories
View Description Hide DescriptionWe implement the minimax approximation for the decomposition of energy denominators in Laplacetransformed Møller–Plesset perturbation theories. The best approximation is defined by minimizing the Chebyshev norm of the quadrature error. The application to the Laplacetransformed second order perturbation theory clearly shows that the present method is much more accurate than other numerical quadratures. It is also shown that the error in the energy decays almost exponentially with respect to the number of quadrature points.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Energy transfer of highly vibrationally excited 2methylnaphthalene: Methylation effects
View Description Hide DescriptionThe methylation effects in the energy transfer between Kr atoms and highly vibrationally excited 2methylnaphthalene in the triplet state were investigated using crossedbeam/timesliced velocitymap ion imaging at a translational collision energy of . Comparison of the energy transfer between naphthalene and 2methylnaphthalene shows that the difference in total collisional cross section and the difference in energy transferprobability density functions are small. The ratio of the total cross sections is : . The energy transferprobability density function shows that naphthalene has a little larger probability at small energy transfer,, and 2methylnaphthalene has a little larger probability at large energy transfer,. However, these differences are close to our experimental uncertainty. No significant difference in the probability of very large energy transfer, such as supercollisions defined arbitrarily as , was observed. The possible methylation effects due to the subsequent successive collisions were discussed.

Energyswitching potential energy surface for the water molecule revisited: A highly accurate singledsheeted form
View Description Hide DescriptionA global ab initiopotential energy surface is proposed for the water molecule by energyswitching/merging a highly accurate isotopedependent local potential function reported by Polyansky et al. [Science299, 539 (2003)] with a global form of the manybody expansion type suitably adapted to account explicitly for the dynamical correlation and parametrized from extensive accurate multireference configuration interaction energies extrapolated to the complete basis set limit. The new function mimics also the complicated crossing that arises at linear geometries of the water molecule.

Understanding the rate of spinforbidden thermolysis of and
View Description Hide DescriptionThe pyrolysis of the simplest azides and has been studied computationally. Nitrogen extrusion leads to the production of NH or . The azides have singlet ground states but the nitrenes and NH have triplet ground states. The competition between spinallowed decomposition to the excited state singlet nitrenes and the spinforbidden loss is explored using accurate electronic structure methods (CASSCF/ccpVTZ and MRAQCC/ccpVTZ) as well as statistical rate theories. Nonadiabatic rate theories are used for the dissociation leading to the triplet nitrenes. For , formation is predicted to dominate at low energy, and the calculated rate constant agrees very well with energyresolved experimental measurements. Under thermal conditions, however, the singlet and triplet pathways are predicted to occur competitively, with the spinallowed product increasingly favored at higher temperatures. For thermolysis, spinallowed dissociation to form should largely dominate at all temperatures, with spinforbidden formation of almost negligible. Singlet methyl nitrene is very unstable and should rearrange to immediately upon formation, and the latter species may lose competitively with vibrational cooling, depending on temperature and pressure.

Electric deflection studies on lead clusters
View Description Hide DescriptionThe dielectric response to an inhomogeneous electric field has been investigated for clusters within a molecular beam experiment. The experiments give clear evidence that leadclusters with 12, 14, and 18 atoms possess permanent dipole moments. For these cluster sizes, the permanent electric dipole moments strongly determine the response to the electric field, leading to a significantly increased apparent polarizability. An adiabatic polarization mechanism allows a semiquantitative explanation of the observed susceptibility anomalies. The beam profiles of most of the leadclusters with , 14, and 18 also display a small broadening induced by the electric field, indicating permanent dipole moments of about /atom. Nearly constant dipole moments per atom for larger leadclusters manifest in a linear increase in the polarizability per atom. Also, for leadclusters such as , which do exhibit almost no measurablebeam broadening, the polarizabilties are increased compared to the bulk value. This could be partially explained by the electronic structure of the leadclusters but might be also a consequence of quenched permanent dipole moments because for highly flexible clusters only an increased beam deflection, but no broadening, will be observed.

Carbon chains of type : A theoretical study of potential interstellar anions
View Description Hide DescriptionLinear anions of type , which are expected to be good candidates for experimental investigation by microwave spectroscopy and radio astronomy, were studied by means of the coupled cluster variant CCSD(T). Making use of corrections taken over from and , accurate equilibrium structures ( accuracy in bond lengths) have been established for all five anions. The electric dipole moments increase strongly with increasing chain length. For , a very large equilibrium dipole moment of (with respect to centerofmass coordinate system, negative end of dipole at terminal carbon site) is predicted. The lowest vertical detachment energies, leading to states of the radicals for and and to states in the case of the larger anions, are calculated to lie in the range of . The groundstate rotational and quartic centrifugal distortion constants of are predicted to be and , respectively. All anions studied appear to be fairly normal semirigid linear molecules. Throughout, good agreement with available matrix isolation IR spectroscopic data is obtained and many predictions of spectroscopic properties are made.

An extended multireference study of the electronic states of parabenzyne
View Description Hide DescriptionA stateaveraged, multireference complete active space (CAS) approach was used for the determination of the vertical excitation energies of valence and Rydberg states of parabenzyne. Orbitals were generated with a 10 and 32state averaged multiconfigurational selfconsistent field approach. Electron correlation was included using multireference configuration interaction with singles and doubles, including the Pople correction for size extensivity, multireference averaged quadratic coupled cluster (MRAQCC), and MRAQCC based on linear response theory. There is a very high density of electronic states in this diradical system—there are more than 17 states within of the ground state including two Rydberg states. All excitations, except , are from the system to the system. Of the 32 states characterized, 15 were multiconfigurational, including the ground state, providing further evidence for the necessity of a multireference approach for benzyne. The vertical singlettriplet splitting was also characterized using a twostate averaged approach. A CAS(2,2) calculation was shown to be inadequate due to interaction with the orbitals.

On the doublet states of the potassium trimer
View Description Hide DescriptionThe potassium trimer is investigated in its lowest electronic doublet states, employing several highlevel ab initio methods (coupled cluster with single, double, and noniterative triple excitations, multiconfiguration selfconsistent field, and multireference Rayleigh–Schrödinger perturbation theory of second order). Onedimensional cuts through the lowest 12 electronic states at symmetry give insight in the complex electronic structure of the trimer, showing several (pseudo)Jahn–Teller distortions that involve two or three excited states. Contour plots of the involved molecular orbitals are shown to prove the validity of the shell model frequently used for a qualitative description of metallic clusters.

An IR study of cluster ions in the region
View Description Hide DescriptionInfrared photodissociation (IRPD) spectra of carbon dioxide cluster ions, with , are measured in the region. IR bands assignable to solvent molecules are observed at positions close to the vibrational frequencies of neutral [1290 and ( and ), , and 3610 and ( and )]. The ion core in shows several IR bands in the 1200–1350, 2100–2200, and regions. On the basis of previous IR studies in solid Ne and quantum chemical calculations, these bands are ascribed to the ion, which has a semicovalent bond between the components. The number of the bands and the bandwidth of the IRPD spectra drastically change with an increase in the cluster size up to , which is ascribed to the symmetry change of by the solvation of molecules and a full occupation of the first solvation shell at .