Volume 111, Issue 19, 15 November 1999
 COMMUNICATIONS


Relationship between bipolar moments and moleculeframe polarization parameters in Doppler photofragment spectroscopy
View Description Hide DescriptionIn terms of the molecularframe polarization parameters an equation is derived that describes the shape of a photofragment Doppler profile as a function of the three angles Γ, Δ, and Φ that specify the photolysis and probe laser polarizations about the detection axis. This expression is specialized to linearly polarized photolysis and probe laser beams. For the particular value of the angle between the probe laser polarization and the detection axis, this equation can be reduced to the form of wellknown laboratoryframe expressions that use the bipolar moment formalism introduced by Dixon. Comparison of these forms shows the equivalence of the two formalisms and gives the relationships between the bipolar moments and the moleculeframe parameters. We show that linear combinations of the bipolar moments completely describe photofragment polarization in the molecular frame and possess distinct quantum mechanical significance. In particular, it is shown that the coherent contribution to the photofragment alignment is proportional to the linear combination

Absorption spectroscopy on single molecules in solids
View Description Hide DescriptionAbsorption signals of single terrylene molecules in nhexadecane and naphthalene crystals were recorded at liquidhelium temperatures. The method is based upon rf Stark effect modulation in the megahertz range. The electric rf field strength was applied by means of interdigitating electrodes with 18 μm spacing. Signaltonoise ratios better than 10 were obtained with approximately 300 ms integration time. The measured line shapes depend on the relative contributions of the linear and the quadratic Stark shift.
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 THEORETICAL METHODS AND ALGORITHMS


The multiconfiguration timedependent Hartree method generalized to the propagation of density operators
View Description Hide DescriptionThe multiconfiguration timedependent Hartree (MCTDH) method is formulated for density operators and applied to their numerical propagation. We introduce two types of MCTDH density operators which are expanded in different kinds of socalled singleparticle density operators. The latter may either be hermitian, or else represent ket–bra products of socalled singleparticle functions. For both types of MCTDH expansions of density operators we derive equations of motion employing the Dirac–Frenkel/MacLachlan variational principle. Further an alternative set of equations of motion for the second type of density operators is proposed, which is not based on a variational principle but derived by taking partial traces. We thus obtain three sensible approaches within the framework of the MCTDH method which differ in their performance and properties. We investigate these approaches and their properties analytically and numerically. Our numerical results refer to a model of vibroniccoupling dynamics in the pyrazine molecule representing coupled electronic states with four vibrational modes and two and three electronic states respectively. We analyze the closedsystem dynamics for this model with temperaturedependent initial states. The influence of temperature on state populations, on correlation functions and on absorption spectra is discussed. We assess the numerical performance of two of the three approaches and find that both can be very efficiently applied to investigate the type of systems studied here.

Ab initio classical trajectories on the Born–Oppenheimer surface: Updating methods for Hessianbased integrators
View Description Hide DescriptionFor the integration of the classical equations of motion in the Born–Oppenheimer approach, each time the energy and gradient of the potential energy surface are needed, a properly converged wave function is calculated. If Hessians (second derivatives) can be calculated, significantly larger steps can be taken in the numerical integration of the equations of motion without loss of accuracy. Even larger steps can be taken with a Hessianbased predictor–corrector algorithm. Since updated Hessians are used successfully in quasiNewton methods for geometry optimization, it should be possible to improve the performance of trajectory calculations using updated Hessians. The Murtagh–Sargent (MS) update, the Powellsymmetric–Broyden (PSB) update and Bofill’s update (a weighted combination of MS and PSB) were tested, and Bofill’s update was found to be the best. Slightly smaller step sizes were needed with Hessian updating to maintain good conservation of the energy, but this was more than compensated by the reduction in total computational cost. An overall factor of 3 in speedup was obtained for trajectories of systems containing 4 to 6 heavy atoms computed at the HF/321G level.

Fast computation of analytical second derivatives with effective core potentials: Application to and
View Description Hide DescriptionAn improved method is described for the computation of integrals involving effective core potentials. The improved method provides better scalability to higher angular momenta as well as improved speed. The new method is also applied to the determination of the minimum energy structures of and main group analogs of the compounds (known as metcars). Relative energies, geometries, and vibrational frequencies are reported for several novel structures.

A simple scheme for the direct calculation of ionization potentials with coupledcluster theory that exploits established excitation energy methods
View Description Hide DescriptionVertical ionization potentials can be obtained from existing computer programs for the highlevel treatment of excited states by simply including a continuum orbital in the basis set. Exploiting this feature easily allows final state energies for ionized states to be calculated at several previously untested levels of theory that go beyond the equationofmotion coupledcluster singles and doubles model. Values obtained for CO, and with the most theoretically complete approximations studied here (those based on the CCSDT3 and CC3 parametrizations of the neutral ground state) are in excellent agreement with experiment when a large basis set is used.

The mapping of the conditional pair density onto the electron density
View Description Hide DescriptionThis paper shows that the Fermi hole of a reference electron can be so strongly localized to a given region of space, as to cause the conditional pair density for samespin electrons to approach the oneelectron spin density outside the region of localization and for a closedshell system, the conditional pair density for both spins will approach the total density. Correspondingly, the Laplacian of the conditional pair density, whose local concentrations indicate the positions where the density of the remaining electrons are most likely to be found for a fixed position of a reference pair, approaches the Laplacian of the density. The Laplacian of the conditional pair density generated by a sampling of pair space by an α,β pair of reference electrons, exhibits a homeomorphism with the Laplacian of the electron density. This homeomorphism approaches an isomorphic mapping of one field onto the other, as the reference electron pair becomes increasingly localized to a given region of space. Thus the local charge concentrations (CCs) displayed by the Laplacian of the electron density, the local maxima in signify the presence of regions of partial pair condensation, regions with greater than average probabilities of occupation by a single pair of electrons, as has been previously surmized on empirical grounds. This paper establishes a mapping of the essential aspects of electron pairing, determined in sixdimensional space, onto the threedimensional space of the electron density. The properties of the conditional pair density enable one to determine which CCs of are coupled and represent the same localized pair of electrons. It is found that the pattern and properties of the electron localization domains predicted by the Laplacian of the conditional pair density differ in important aspects from those predicted by ELF, the electron localization function.

The accurate calculation and prediction of the bond dissociation energies in a series of hydrocarbons using the IMOMO (integrated molecular orbital+molecular orbital) methods
View Description Hide DescriptionThe IMOMO (integrated molecular orbital+molecular orbital) method was used to accurately calculate and compare with the experiment for the singlebond C–H and C–C bond dissociation energies of a series of hydrocarbons, where is H or while the largest considered is 1,1diphenylethyl, While the geometries and zero point vibrational energies were obtained at the hybrid density function (B3LYP/631G) level for the real system, a small system, or was used as the “model” in the IMOMO energy calculation, for which a high level method is used. Of a large number of IMOMO combinations tested, the combination of the modified Gaussian2 method (G2MSr) with the restricted openshell secondorder Møller–Plesset perturbation method (ROMP2), the IMOMO(G2MSr:ROMP2/631G(d)) method, yields the best results, and can be used for bond dissociation energy predictions of very large molecules. Finally, the IMOMO(G2MSr:ROMP2/631G(d)) method was used to predict the C–H bond dissociation energy in and the C–C bond dissociation energy in neither of which is available experimentally. These predicted values are 75.9 and 64.1 kcal/mol, respectively, which are smaller than any other C–H and BDE studied in this paper.

Iterative and direct methods employing distributed approximating functionals for the reconstruction of a potential energy surface from its sampled values
View Description Hide DescriptionThe reconstruction of a function from knowing only its values on a finite set of grid points, that is the construction of an analytical approximation reproducing the function with good accuracy everywhere within the sampled volume, is an important problem in all branches of sciences. One such problem in chemical physics is the determination of an analytical representation of Born–Oppenheimer potential energy surfaces by ab initio calculations which give the value of the potential at a finite set of grid points in configuration space. This article describes the rudiments of iterative and direct methods of potential surfacereconstruction. The major new results are the derivation, numerical demonstration, and interpretation of a reconstruction formula. The reconstruction formula derived approximates the unknown function, say V, by linear combination of functions obtained by discretizing the continuous distributed approximating functional (DAF) approximation of V over the grid of sampling. The simplest of contracted and ordinary HermiteDAFs are shown to be sufficient for reconstruction. The linear combination coefficients can be obtained either iteratively or directly by finding the minimal norm leastsquares solution of a linear system of equations. Several numerical examples of reconstructing functions of one and two variables, and very different shape are given. The examples demonstrate the robustness, high accuracy, as well as the caveats of the proposed method. As to the mathematical foundation of the method, it is shown that the reconstruction formula can be interpreted as, and in fact is, frame expansion. By recognizing the relevance of frames in determining analytical approximation to potential energy surfaces, an extremely rich and beautiful toolbox of mathematics has come to our disposal. Thus, the simple reconstruction method derived in this paper can be refined, extended, and improved in numerous ways.

Comparison study of the prediction of Raman intensities using electronic structure methods
View Description Hide DescriptionRaman intensities have been computed for a series of test molecules and ) using Hartree–Fock, secondorder Mo/ller–Plesset perturbation theory (MP2), and density functional theory, including local, gradientcorrected, and hybrid methods (SVWN, BLYP and B3LYP, and MPW1PW91) to evaluate their relative performance. Comparisons were made with three different basis sets: Sadlej, and augccpVTZ. The quality of basis set used was found to be the most important factor in achieving quantitative results. The medium sized Sadlej basis provided excellent quantitative Raman intensities, comparable to those obtained with the much larger augccpVTZ basis set. Harmonic vibrational frequencies computed with the Sadlej basis set were in good agreement with experimental fundamentals. For the quantitative prediction of vibrational Raman spectra, the Sadlej basis set is an excellent compromise between computational cost and quality of results.

Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion
View Description Hide DescriptionRecurrence relations are derived for constructing rotation matrices between complex spherical harmonics directly as polynomials of the elements of the generating rotation matrix, bypassing the intermediary of any parameters such as Euler angles. The connection to the rotation matrices for real spherical harmonics is made explicit. The recurrence formulas furnish a simple, efficient, and numerically stable evaluation procedure for the real and complex representations of the rotation group. The advantages over the Wigner formulas are documented. The results are relevant for directing atomic orbitals as well as multipoles.

Exact conditions on physically realizable correlation functions of random media
View Description Hide DescriptionAlgorithms have been developed recently to construct realizations of random media with specified statistical correlation functions. There is a need for the formulation of exact conditions on the correlation functions in order to ensure that hypothetical correlation functions are physically realizable. Here we obtain positivity conditions on certain integrals of the autocorrelation function of dimensional statistically homogeneous media and of statistically isotropic media. These integral conditions are then applied to test various classes of autocorrelation functions. Finally, we note some integral conditions on the threepoint correlation function.
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 GAS PHASE DYNAMICS AND STRUCTURE: SPECTROSCOPY, MOLECULAR INTERACTIONS, SCATTERING, AND PHOTOCHEMISTRY


Anion photoelectron spectroscopy of
View Description Hide DescriptionVibrationally resolved 355 and 266 nm anion photoelectron spectra of are presented. Photodetachment to two electronic states of linear B–N–B is observed and, aided by electronic structure calculations, assigned to the and transitions. The electron affinity of is and the term energy is Observation of excitations involving uneven quanta of the antisymmetric stretching mode indicates a breakdown of the Franck–Condon (FC) approximation and results from Herzberg–Teller vibronic coupling between the and states involving the mode. Measurement of the angular dependence of the photodetached electrons serves as a sensitive probe for the identification of these FC forbidden transitions. A linear vibronic coupling model qualitatively reproduces the perturbed potentials of the and states. Artifactual symmetry breaking along the coordinate is observed in the ab initiowave functions for the neutral ground state up to the coupledcluster level of theory, even when Brueckner orbitals are used. No evidence is found for an energetically lowlying cyclic state of which has been invoked in the assignment of the matrix infrared spectrum of However, the matrix infrared data agrees well with the peak spacing observed in the photoelectron spectra and reassigned to the linear ground state.

Guided ion beam investigation of the reaction bond activation and C–C bond formation
View Description Hide DescriptionWe have investigated six different endothermic channels in the reaction of ions with neutral CO. For each ionic product we have measured the kinetic energy dependence of the integral cross section and inferred the neutral products by the reaction energetics. The onset of the process producing O, and CO, has been identified by a feature of the integral cross section located at about 8.5 eV. Measurements of the product isotopic ratio suggest that originates from both the ion and the neutral CO molecule. For the reaction channels producing and respectively, measurements of the reaction thresholds allow us to estimate the heats of formation of these two ionic products, and These values are in good agreement with recent independent estimations. Finally, we reevaluated the dissociation energy of

Dopplerfree optical–optical double resonance polarization spectroscopy of the and states
View Description Hide DescriptionHigh resolution spectra of the and transitions of the molecule have been measured with the technique of Dopplerfree optical–optical double resonance polarizationspectroscopy (OODRPS). Molecular constants of the and levels have been determined, and potential energy curves constructed by the RKR method. The RKR potential of the state was found to have a distortion at outer wall, which originates from an avoided crossing of two states. The perturbations between the and levels were found from the energy shifts of the rotational levels. The magnitude of the nonadiabaticinteraction between the and levels, was evaluated to be 2.2 cm^{−1} by a least squares fitting to the energy shifts of the levels. The line intensities were observed to change dramatically around the maximum energy shift. These intensity anomalies are interpreted as an interference effect, which occurs when two interacting levels have comparable transition moments. A remarkable line broadening was observed for the transitions to the levels, and it was identified as originating from the predissociation to atoms. The dissociation energies of the and states have been determined to be 4217.4±0.8, 2021.5±0.8, and 1050.0±0.8 cm^{−1}, respectively.

Tricapped tetrahedral A structural determination by resonance Raman spectroscopy and density functional theory
View Description Hide DescriptionThe resonance Raman spectrum of mass selected out of a cation beam of sputteredsilver, neutralized, and codeposited with solid Ar, is presented. By comparing the observed spectrum with one calculated using density functional theory one concludes the structure of the silver septamer to be a tricapped tetrahedron. A partial resonance Raman spectrum of is also included. Both spectra are dominated by totally symmetric “breathing” modes at (as is also true for [T. L. Haslett et al., J. Chem. Phys. 108, 3453 (1998)]). This frequency is close to the Debye frequency of solid silver, implying that the nature of the chemical bond in these silver clusters already approximates closely that which exists in bulk silver.

Ultrafast structural deformation of in intense laser fields studied by massresolved momentum imaging
View Description Hide DescriptionThe ultrafast structural deformation of in an intense laser field (1.0 PW/cm^{2}) is studied by massresolved momentum imaging (MRMI) of the and fragment ions produced from through the Coulomb explosion processes, The N–O distance just before the Coulomb explosion is elongated significantly from that in the electronic ground state, and it monotonically increases from 1.7 to 2.1 Å as z increases from 4 to 9. The ∠O–N–O bond angle increases toward a linear configuration as a function of z, which is interpreted in terms of the formation of the lightdressed potential energy surfaces. The twobody fragmentation pathways to produce and are also investigated by the MRMI measurements to derive the extent of the asymmetrical bond elongation of one of the two N–O bonds.

Rotationally resolved pulsed field ionization photoelectron study of in the energy range of 13.98–21.92 eV
View Description Hide DescriptionWe have obtained rotationally resolved pulsed field ionization–photoelectron (PFIPE) spectra of CO in the energy range of 13.98–21.92 eV, covering the ionization transitions The PFIPE bands for 24, and 28–39) obtained here represent the first rotationally resolved spectroscopic data for these states. The highresolution features observed in the PFIPE spectra allow the identification of vibrational bands for the 14, 15, 17, 18, 21, 24, 25, 29–31, 33, 35–37, and 39) states, which strongly overlap with prominent vibrational bands of the states. The simulation using the Buckingham–Orr–Sichel model has provided accurate molecular constants for including ionization energies, vibrational constants ( and ), and rotational constants [ Enhancement of rotational branches, attributable to fieldinduced rotational autoionization, was clearly discernible in PFIPE bands for 11, and 12). Significant local enhancements due to nearresonance autoionization were observed for low (<10) PFIPE bands of where the density of interloper Rydberg states converging to higher ionic levels is high as manifested in the photoion spectrum. The observation of a long vibrational progression in the Franck–Condon gap region, where strong autoionization states are absent, is consistent with the suggestion that highnRydberg states converging to highly excited vibrational levels of are partially populated via direct excitation to a repulsive neutral state. The relatively minor band intensity variation observed for high PFIPE bands is also in accord with the direct excitation model. Since ±1, ±2, and ±3 rotational branches are observed in the PFIPE spectra, we conclude that the ejected photoelectrons are restricted to angular momentum continuum states

Quantum scattering calculations for between 1–600 K in comparison with pressure broadening, shift, and time resolved double resonance experiments
View Description Hide DescriptionWe have performed quantum scattering calculations to predict pressure broadening, pressure shift, and inelastic depopulation crosssections for the rotational transitions and of the molecule in collision with helium atoms over a temperature range from 1 to 600 K. The calculated crosssections are compared with experimental values obtained by millimeter wavespectroscopic techniques and the collisional cooling method. We observe good agreement between theory and experiment over the temperature region from 20 to 600 K, but increasing differences below 20 K. Possible reasons for the deviations at lower temperatures are discussed. The calculations also illustrate the contribution of elastic collisions to the pressure broadening crosssections.

Timedependent density functional study on the electronic excitation energies of polycyclic aromatic hydrocarbon radical cations of naphthalene, anthracene, pyrene, and perylene
View Description Hide DescriptionTimedependent density functional theory (TDDFT) and its modification, the Tamm–Dancoff approximation to TDDFT, are employed to calculate the electronic excitation energies and oscillator strengths for a series of polycyclic aromatic hydrocarbon radical cations. For the radical cations of naphthalene and anthracene, TDDFT using the Becke–Lee–Yang–Parr functional and the 631G^{*} ^{*} basis set provides the excitation energies that are roughly within 0.3 eV of the experimental data. The assignments of the electron transitions proposed by TDDFT accord with the previous assignments made by accurate ab initio calculations, except that TDDFT indicates the existence of a few additional transitions of π^{*}←σ character among the several lowlying transitions. The calculated energies for these π^{*}←σ transitions are found to be consistent with the onset of a σ electron ionizationmanifold in the photoelectron spectra. For the pyrene radical cation, TDDFT supports the previous assignments made by semiempirical calculations, whereas for the perylene radical cation, TDDFT suggests the energy ordering of the three lowestlying excited states be changed from those of the semiempirical results.
