Harmonic Oscillator Approximation Research Articles

Simulations of the emission spectra of fac-tris(2-phenylpyridine) iridium and Duschinsky rotation effects using the Herman–Kluk semiclassical initial value representation method

The phosphorescent emission spectra of fac-tris(2-phenylpyridine) iridium [fac-Ir(ppy)(3)] due to the lowest triplet T(1) and T(2) states are simulated using the harmonic oscillator approximation for the S(0), T(1), and T(2) potential energy surfaces (PESs) and taking the Duschinsky rotation into account. The simulations involve the propagation of 177-dimensional wave packets on the coupled PES according to the Herman-Kluk (HK) semiclassical (SC) initial value representation (IVR) method. The HK SC-IVR method is employed because of its accuracy for the PES with mode mixing and its efficiency in dealing with coupled degrees of freedom for large systems. The simulated emission spectrum due to T(1) reproduces the structures of the emission spectra observed experimentally, while T(2) is found very unlikely to participate in the phosphorescent emission. Although the effect of the Duschinsky mode mixing is small for the T(1) state, neglecting it blueshifts the spectrum due to the T(2) state by 800 cm(-1) and changes the relative intensities, indicating that the importance of the Duschinsky rotation is rather unpredictable and should not be overlooked. The present simulations demonstrate that the simple harmonic oscillator approximation combined with the Duschinsky rotation can adequately describe the photophysics of fac-Ir(ppy)(3) and that the HK SC-IVR method is a powerful tool in studies of this kind.

Theoretical study on S1(1B3u) state electronic structure and absorption spectrum of pyrazine

Making use of a set of quantum chemistry methods, the harmonic potential surfaces of the ground state (S0(1Ag)) and the first (S1(1B3u)) excited state of pyrazine are investigated, and the electronic structures of the two states are characterized. In the present study, the conventional quantum mechanical method, taking account of the Born-Oppenheimer adiabatic approximation, is adopted to simulate the absorption spectrum of S1(1B3u) state of pyrazine. The assignment of main vibronic transitions is made for S1(1B3u) state. It is found that the spectral profile is mainly described by the Franck-Condon progression of totally symmetric mode ν6a. For the five totally symmetric modes, the present calculations show that the frequency differences between the ground and the S1(1B3u) state are small. Therefore the displaced harmonic oscillator approximation along with Franck-Condon transition is used to simulate S1(1B3u) absorption spectra. The distortion effect due to the so-called quadratic coupling is demonstrated to be unimportant for the absorption spectrum, except the coupling mode ν10a. The calculated S1(1B3u) absorption spectrum is in reasonable agreement with the experimental spectra.

Theoretical rate coefficients for the reaction of methyl radical with hydroperoxyl radical and for methylhydroperoxide decomposition

The kinetics of the CH3+HO2 bimolecular reaction and the thermal decomposition of CH3OOH are studied theoretically. Direct variable reaction coordinate transition state theory (VRC-TST), coupled with high level multireference electronic structure calculations, is used to compute capture rates for the CH3+HO2 reaction and to characterize the transition state of the barrierless CH3O+OH product channel. The CH2O+H2O product channel and the CH3+HO2→CH4+O2 reaction are treated using variational transition state theory and the harmonic oscillator and rigid rotor approximations. Pressure dependence and product branching in the bimolecular and decomposition reactions are modeled using master equation simulations. The predicted rate coefficients for the major products channels of the bimolecular reaction, CH3O+OH and CH4+O2, are found to be in excellent agreement with values obtained in two recent modeling studies. The present calculations are also used to obtain rate coefficients for the CH3O+OH association/decomposition reaction.

Classical and Quantum Gibbs Free Energies and Phase Behavior of Water Using Simulation and Cell Theory

A method to calculate the classical and quantum free energy of a liquid from a computer simulation by using cell theory [J. Chem. Phys. 2007, 126, 064504] is tested for liquid water and ice Ih against experiment as a function of temperature. This fast and efficient method reproduces reasonably well the experimental values of entropy, enthalpy, and free energy of a liquid across the supercooled, stable, and superheated range of temperatures considered. There are small differences between classical and quantum results of water at 298 K, necessitating a small correction term to reproduce water's enthalpy of vaporisation. Only at higher temperatures is entropy underestimated by up to 9 J K(-1) mol(-1) as verified by thermodynamic integration calculations. Satisfactory agreement for ice, however, is only obtained by using the quantum formulation. Even then, at higher temperatures, the entropies exceed experiment by up to 15 J K(-1) mol(-1). Further insight into the quantum nature of water is provided by inspecting the temperature dependence of the frequencies. The harmonic approximation is further supported by the harmonic force and torque distributions and the very similar entropies obtained from the force and torque variances. All these results suggest that the single molecule harmonic oscillator approximation for water, although not exact, provides a rapid, insightful, and useful means to evaluate the thermodynamic properties of water from a computer simulation in a way that can account for the quantization of water's energy levels.

Variational Analysis of the Phenyl + O2 and Phenoxy + O Reactions

Variational transition state analysis was performed on the barrierless phenyl + O2 and phenoxy + O association reactions. In addition, we also calculated rate constants for the related vinyl radical (C2H3) + O2 and vinoxy radical (C2H3O) + O reactions and provided rate constant estimates for analogous reactions in substituted aromatic systems. Potential energy scans along the dissociating C-OO and CO-O bonds (with consideration of C-OO internal rotation) were obtained at the O3LYP/6-31G(d) density functional theory level. The CO-O and C-OO bond scission reactions were observed to be barrierless, in both phenyl and vinyl systems. Potential energy wells were scaled by G3B3 reaction enthalpies to obtain accurate activation enthalpies. Frequency calculations were performed for all reactants and products and at points along the potential energy surfaces, allowing us to evaluate thermochemical properties as a function of temperature according to the principles of statistical mechanics and the rigid rotor harmonic oscillator (RRHO) approximation. The low-frequency vibrational modes corresponding to R-OO internal rotation were omitted from the RRHO analysis and replaced with a hindered internal rotor analysis using O3LYP/6-31G(d) rotor potentials. Rate constants were calculated as a function of temperature (300-2000 K) and position from activation entropies and enthalpies, according to canonical transition state theory; these rate constants were minimized with respect to position to obtain variational rate constants as a function of temperature. For the phenyl + O2 reaction, we identified the transition state to be located at a C-OO bond length of between 2.56 and 2.16 A (300-2000 K), while for the phenoxy + O reaction, the transition state was located at a CO-O bond length of 2.00-1.90 A. Variational rate constants were fit to a three-parameter form of the Arrhenius equation, and for the phenyl + O2 association reaction, we found k(T) = 1.860 x 1013T-0.217 exp(0.358/T) (with k in cm3 mol-1 s-1 and T in K); this rate equation provides good agreement with low-temperature experimental measurements of the phenyl + O2 rate constant. Preliminary results were presented for a correlation between activation energy (or reaction enthalpy) and pre-exponential factor for heterolytic O-O bond scission reactions.

How Accurate Are Approximate Methods for Evaluating Partition Functions for Hindered Internal Rotations?

The accuracy of several low-cost methods (harmonic oscillator approximation, CT-Comega, SR-TDPPI-HS, and TDPPI-HS) for calculating one-dimensional hindered rotor (1D-HR) partition functions is assessed for a test set of 644 rotations in 104 organic molecules, using full torsional eigenvalue summation (TES) as a benchmark. For methods requiring full rotational potentials, the effect of the resolution at which the rotational potential was calculated was also assessed. Although lower-cost methods such as Pitzer's Tables are appropriate when potentials can be adequately described by simple cosine curves, these were found to show large errors (as much as 3 orders of magnitude) for non-cosine curve potentials. In those cases, it is found that the TDPPI-HS method in conjunction with a potential compiled at a resolution of 60 degrees offers the best compromise between accuracy and computational expense. It can reproduce the benchmark values of the partition functions for an individual mode to within a factor of 2; its average error is just of a factor of 1.08. The corresponding error in the overall internal rotational partition functions of the molecules studied is less than a factor of 4 in all cases. Excellent cost-effective performance is also offered by CT-Comega, which requires only the geometries, energies, and frequencies of the distinguishable minima in the potential. With this method the geometric mean error in individual partition functions is 1.14, the maximum error is a modest 2.98 and the resulting error in the total 1D-HR partition function of a molecule is less than a factor of 5 in all cases.

Line intensities of the asymptotic asymmetric-top radical HO2 at high temperatures

The total internal partition sums were calculated in the product approximation at temperatures up to 5000 K for the asymptotic asymmetric-top HO2 molecule. The calculations of the rotational partition function and the vibrational partition function were carried out with the rigid-top model and in the harmonic oscillator approximation, respectively. Our values of the total internal partition sums are consistent with the data of HITRAN database with -0.14% at 296 K. Using the calculated partition functions, we have calculated the line intensities of v2 band of HO2 at several high temperatures. The results showed that the calculated line intensities are in very good agreement with those of HITRAN database at temperatures up to 3000 K, which provides a strong support for the calculations of partition functions and line intensities at high temperatures. Then we have extended the calculation to higher temperatures. The simulated spectra of v2 band of the asymptotic asymmetric-top HO2 molecule at 4000 and 5000K are also obtained.

Dimensionality of electronic excitations in organic semiconductors: A dielectric function approach

We present a detailed investigation on the effective dimensionality (associated with the degree of delocalization) of electronic excitations in thin organic films using the dielectric function as obtained from ellipsometry. To this end, we study first the best analytical representation of the optical dielectric function of these materials and compare different approaches found in the literature: (i) the harmonic oscillator approximation, (ii) the standard critical-point model (SCP), (iii) the model dielectric function (MDF), and (iv) the Forouhi-Bloomer model. We use these models to analyze variable angle spectroscopic ellipsometry raw data for a thin poly(9,9-dioctylfluorene) (PFO) film deposited on quartz (taken as an archetypal sample). The superiority of the SCP model for PFO films and a wide range of other spin-coated conjugated polymers (and guest-molecules in polymers) is demonstrated. Moreover, we show how the SCP model can be used to gain physical information on the microscopic structure. As an example, we show that the delocalization of excitons decreases for nonconjugated polymers, such as polymethylmethacrylate and polyimide, while the conjugation length and exciton delocalization are, respectively, enhanced in cases where a planar conformation (e.g., $\ensuremath{\beta}$ phase of PFO) or a high degree of crystallinity [e.g., poly(3-hexylthiophene)] is achieved. As an additional example, we employ the SCP excitonic model to investigate the temperature dependence of the dielectric function of crystalline and glassy PFO films. We propose that the SCP excitonic model should be adopted as the standard choice to model the optical properties of polymer thin films from ellipsometry data.

Vibronic transitions in large molecular systems: Rigorous prescreening conditions for Franck-Condon factors

In this work, rigorous prescreening conditions for the calculation of Franck-Condon factors (FCFs) are derived and implemented. These factors play an important role in numerous applications including the prediction and simulation of vibronic spectra, electron transfer rates, and nonradiative transition probabilities. For larger systems it is crucial to calculate only relevant FCFs, as the computational burden becomes otherwise prohibitive due to the sheer number of Franck-Condon integrals. By exploiting rigorous prescreening criteria one can significantly reduce the computational effort and systematically refine results to the desired target accuracy. In this work, such criteria are derived via the use of sum rules obtained through a coherent state generating function for the FCFs in the harmonic oscillator approximation, following the prescription of Doktorov et al. These sum rules allow efficient and rigorous prescreening prior to the calculation of entire batches of Franck-Condon integrals, reducing the subsequent computational burden. To illustrate the benefit of employing such conditions, they are applied in this work to calculate FC profiles for vibronic spectra of formic acid, thymine, anthracene, and a polycyclic aromatic hydrocarbon derivative containing about 450 vibrational degrees of freedom. Since the prescreening step gives stringent upper and lower bounds for the loss of FC intensity in these spectra arising from the neglected FCFs, the present approach renders most previous a priori selection schemes obsolete and has the potential to complement or even replace other approximate treatments.

Estimation of Gas-Phase Heat Capacities for Hydrofluoroethers with Two Carbon Atoms

Density functional calculations at the B3LYP/6-31g* level of theory are used to predict gas-phase heat capacities as a function of temperature for the two-carbon hydrofluoroethers and dimethyl ether. We use equilibrium thermodynamic calculations to determine weighting factors for different rotameric forms to find an observable heat capacity estimate. Hindered rotor corrections were made to the partition functions for these weightings and for the heat capacity contributions for the modes where the harmonic oscillator approximation was inappropriate. We find close agreement with experimental values for the two hydrofluoroethers where heat capacities have been measured and good agreement over a wide temperature range for dimethyl ether. This agreement indicates that the heat capacities we report for the other seven species are accurate and would represent experimental data if it were available in the open literature. This work populates a needed database for the evaluation of these materials as heat transfer media in the temperature range from 100 to 3000 K.

The rovibrational excitation of H2 induced by H

Cross sections have been computed for rovibrational transitions of H2, induced by collisions with H atoms, using the H–H2 interaction potential calculated recently by Mielke et al and an earlier potential of Boothroyd et al. We compare cross sections derived using a simple harmonic oscillator approximation to the vibrational motion with results obtained using numerically ‘exact’ solutions of the rovibrational eigenvalue equation. We also investigate the convergence of the cross sections with respect to the size of the rovibrational basis set. Convergence is found to be slow, owing to the strength of the collisional coupling between vibrational manifolds.

Ab Initio Thermochemistry and Kinetics for Carbon-Centered Radical Addition and β-Scission Reactions

A quantitative comparison of ab initio calculated rate coefficients using five computational methods and five different approaches of treating hindered internal rotation and tunneling with experimental values of rate coefficients for nine carbon-centered radical additions/beta scissions at 300, 600, and 1000 K is performed. The high-accuracy compound methods, CBS-QB3 and G3B3, and the density functionals, MPW1PW91, BB1K, and BMK, have been evaluated using the following approaches: (i) the harmonic oscillator approximation; (ii) the hindered internal rotor approximation for the internal rotation about the forming/breaking bond in the transition state and product; and the hindered internal rotation approximation combined with (iii) Wigner, (iv) Skodje and Truhlar, and (v) Eckart zero-curvature tunneling corrections. The density functional theory (DFT) based values for beta-scission rate coefficients deviate significantly from the experimental ones at 300 K, and the DFT methods do not accurately predict the equilibrium coefficient. The hindered rotor approximation offers a significant improvement in the agreement with experimental rate coefficients as compared to the harmonic oscillator treatment, especially at higher temperatures. Tunneling correction factors are smaller than 1.40 at 300 K and 1.03 at 1000 K. For both the CBS-QB3 method, including the hindered rotor treatment but excluding tunneling corrections, and the G3B3 method, including hindered rotor and Eckart tunneling corrections, a mean factor of deviation with experimentally observed values of 3 is found.

Introducing the Practical Aspects of Computational Chemistry to Undergraduate Chemistry Students

Educating early undergraduate-level students in theoretical and computational chemistry is difficult for a variety of reasons. Often students are left underexposed to the discipline because of a hesitation to add to an already full course syllabus or because of the lack of a clear connection between standard class material and research-grade quantum chemical software. Here we present a laboratory exercise in which students use standard second-year concepts such as the rigid rotor and harmonic oscillator approximations in conjunction with Gaussian 03 to enforce practical concepts in computational chemistry. Such concepts include the compromise between accuracy and computational time as well as the limitations of a non-correlated approach to certain molecular properties. The lab is designed for a three-hour period to afford a relatively easy inclusion into the undergraduate curriculum without requiring any change to the standard lecture material.

Four-Membered Heterocycles with a Carbon−Heteroatom Exocyclic Double Bond at the 3-Position: Puckering Potential and Thermodynamic Properties

Despite the specific importance of four-membered heterocycles with a carbon-heteroatom double bond at the 3-position in organic and medicinal chemistry, little attention has been given up to now to theoretical computational studies on these molecules. However, the overall geometry, and degree of ring puckering especially, could significantly influence the reactivity and biological properties of these four-membered ring compounds. In this paper, focus is made on the influence of different substituents on the equilibrium geometry, ring puckering potential, and thermodynamic quantities. It was found that these properties are mainly affected by the heteroatom (oxygen, nitrogen, sulfur, phosphorus) contained in the ring skeleton. Moreover, the correct description of the puckering potential with the hindered rotor treatment leads to substantial corrections on the thermodynamic properties in the harmonic oscillator approximation.

Transport Spectroscopy of a Quantum Dot in a Silicon-on-Insulator (SOI) MOSFET

We report transport spectroscopy in a gate-induced quantum dot (QD) based on a silicon-oninsulator (SOI) metal-oxide-semiconductor field-effect-transistor (MOSFET). The differential conductance G measurement in the large voltage region shows small conductance peaks, which reveal single-electron tunnelling through exited quantum states of the QD. The discrete energy levels extracted from the differential conductance characteristics agree well with those estimated from a harmonic oscillator approximation. The separation between energy levels is confirmed by using a potential shape extracted from a 3-dimensional device simulation.

Line intensities of 3000—0200 and 1001—0110 transition bands of 14N216O at high temperature

The total internal partition sums (TIPS) were calculated for nitrous oxide with the product approximation. For rotational partition sums Qrot, the centrifugal distortion corrections are under consideration. The calculation method for the vibrational partition sums Qvib used is the harmonic oscillator approximation. Using experimental transition moment squared and Herman-Waills factor coefficients and the calculated partition functions, we computed the line intensities of 3000—0200 and 1001—0110 transitions of nitrous oxide at several temperatures. The agreement between the calculated line intensities and experimental results and those extrapolated from HITRAN database is fairly good at temperatures up to 3000 K, which shows that the calculation of partition function and line intensity at high temperature is reliable. Furthermore, the line intensities and spectral simulations at the higher temperatures of 4000 and 5000 K are also presented.

Theoretical study of molecular vibrational hyperpolarizability of 4-N-methylstilbazonium salt derivatives

Based on double harmonic oscillator (DHO) approximation, the static vibrational first hyperpolarizabilities,and the IR and Raman spectra of 4-N-methylstilbazonium salt derivatives are calculated using coupled perturbed Hartree-Fock (CPHF) approach. Two vibrational normal modes which dominate the contribution to the vibrational first hyperpolarizabilities are obtained. It is found that the static vibrational first hyperpolarizabilities of title molecules are large and show a good linear relationship with their electronic counterparts. For the first time we give an approach of estimating the vibrational first hyperpolarizabilities using a few normal modes identified from IR and Raman spectral peaks, which the name as the few mode approach. The results indicate that this approach is practical in the estimation of the vibrational first hyperpolarizability.

The equilibrium constant of carbon dioxide dimerization and the lifetime of the dimer

A new method, similar to previous methods applied to hydrogen bonding in aqueous solutions, is developed to calculate the equilibrium constant of weak dimer complexes and the lifetime of the dimer in the gas phase. In contrast to reactions involving covalent bonds, where, for example, statistical mechanics in the rigid-rotator and harmonic-oscillator approximation yields good results, no such methods are established for weak complexes. There are even problems in defining when approaching monomers form a dimer. In the new method a time correlation function is defined and sampled in molecular dynamics simulations, that shows a slow decay corresponding to real dimers and a fast decay corresponding to unstable collisions or nearby passes, yielding a distinct definition for a dimer. The results obtained for carbon dioxide dimerization are compared to results obtained by two other methods using different approximations. A possible application is to predict the dimer concentration in the atmospheres of Mars and Venus, which mainly consist of carbon dioxide.

Theoretical Studies of Phosphorescence Spectra of Tris(2,2‘-bipyridine) Transition Metal Compounds

Phosphorescence spectra of tris(2,2'-bipyridine) metal compounds, [M(bpy)3]n+, where M = Zn(II), Ru(II), Os(II), Rh(III), and Ir(III), were calculated using a harmonic oscillator approximation of adiabatic potential surfaces obtained by density functional theory (DFT). Using the Huang-Rhys (S) factors calculated by theoretical Franck-Condon analysis of T1 and S0 geometries, we successfully reproduced the emission spectra observed under various conditions by nonempirical calculations. The simulations of well-structured spectra of the Zn(II), Rh(III), and Ir(III) compounds confirmed that the emission originated from localized ligand-centered excited states with considerably distorted geometries of C2 symmetry. The spectrum simulation revealed that the phosphorescence state of [Ru(bpy)3]2+ was localized 3MLCT both in a solution and a glass matrix. Furthermore, a highly resolved phosphorescence spectrum observed for [Ru(bpy)3]2+ doped in a [Zn(bpy)3](ClO4)2 crystal was reproduced well using the geometry of the localized 3MLCT by assuming mode-specific broadening of low-frequency intramolecular vibrational modes. The deuterium effects of the electronic origins of the doped crystal observed by Riesen et al. were in excellent agreement with those predicted for the localized 3MLCT. However, the calculated satellite structures of the localized 3MLCT involving bpy-h8 in [Ru(bpy-h8)(3-x)(bpy-d8)x]2+ (x = 1,2) exhibited only the bpy-h8 vibrational modes, inconsistent with the simultaneous appearance of both bpy-h8 and bpy-h8 modes in the observed spectra. A simulation on the basis of the geometry of the delocalized 3MLCT was in reasonable agreement with an unresolved spectrum observed for a neat crystal of [Ru(bpy)3](PF6)2, which is inconsistent with the assignments of localized 3MLCT on the basis of the electronic origins. The inconsistency of the assignment on the basis of the adiabatic model is discussed in terms of vibronic coupling between the localized 3MLCT states. The 3MLCT state in [Os(bpy)3]2+ seems to vary with the environment: a fully localized 3MLCT in a solution, partially localized in a glass matrix, and delocalized in PF6 salts.

Particle transfer and fusion cross-section for super-heavy nuclei in dinuclear system

Within the dinuclear system (DNS) conception, instead of solving the Fokker–Planck equation (FPE) analytically, the master equation is solved numerically to calculate the fusion probability of super-heavy nuclei, so that the harmonic oscillator approximation to the potential energy of the DNS is avoided. The relative motion concerning the energy, the angular momentum and the fragment deformation relaxations is explicitly treated to couple with the diffusion process. The nucleon transition probabilities, which are derived microscopically, are related with the energy dissipation of the relative motion. Thus they are time dependent. Comparing with the analytical solution of FPE at the equilibrium, our time-dependent results preserve more dynamical effects. The calculated evaporation residue cross-sections for one-neutron emission channel of Pb-based reactions are basically in agreement with the known experimental data within one order of magnitude.