Rovibrational Partition Functions Research Articles

Path-dependent thermal rate constants: The case of the combustion reaction between methyl hexanoate and hydrogen atom

This work uses the multipath canonical variational theory with small curvature tunneling (MP-CVT/SCT) to calculate the thermal rate constants for the hydrogen abstraction reaction on the α carbon of methyl hexanoate by H-atom. In the conformational search, 233 indistinguishable conformers for MHEX and TSα were located. In the MP-CVT/SCT approach, the recrossing and quantum tunneling effects were incorporated into the rate constants by constructing the individual reaction paths from the jth transition state. The efficiency of truncation on the multistructural rovibrational partition functions and reactive paths and their impact on the thermal rate constants were also evaluated. The torsion over the ▪ group (ϕ5) produces unstable structures, and the reactive paths associated with them do not contribute to the magnitude of MP-CVT/SCT rate coefficients. Additionally, we concluded that rate constants generated from small sets of conformations have a precision of up to 90%, compared to considering all paths.

The thermodynamic properties of diatomic gases. A rigorous analytical approach

The paper reports a closed-form analytical representation of the thermodynamic functions of diatomic molecules in the ground electronic state. For this purpose, analytical formulas for vibrational and rotational partition functions are derived. A rigorous method is proposed for factorizing the vibrational and rotational contributions to the ro-vibrational partition function using an average vibrational quantum number. The behavior of thermodynamic functions as a function of temperature is discussed for carbon monoxide as a test system. The reliability of the analytical method is verified by comparison with experimental data and with the direct summation method, which uses theoretical values of energy levels.

EFEITOS DAS MÚLTIPLAS ESTRUTURAS E DA ANARMONICIDADE TORCIONAL PARA AS REAÇÕES DE ABSTRAÇÃO DE HIDROGÊNIO NO BUTANOATO DE METILA PELO RADICAL H

EFFECTS OF MULTIPLE STRUCTURES AND TORSIONAL ANHARMONICITY FOR THE HYDROGEN ABSTRACTION REACTIONS OF THE THMETHYL BUTANOATE BY H RADICAL. Thermal rate constants for the hydrogen abstraction reactions of the methyl butanoate (MB) by H radical were estimated by applying the multistructural canonical variational theory with small-curvature tunneling (CVT/SCT). The conformational search was performed at MPWB1K/6-31+G(d,p) level of theory and 68 distinguishable conformers for MB and transition state located. The multistructural and torsional anharmonicity effects were corrected through the rovibrational partition functions calculated with the multistructural method based on a coupled torsional potential, MS-T(C). The MS-CVT/SCT thermal rate constants are shown in good concordance with previous combustion models. After fitting the rate constants in a four-parameters equation, the activation energy showed a temperature dependence. The product branching ratios indicate a preference for (Rα) until 500 K. Above this temperature, the contribution of other reactions becomes relevant, especially for (Rβ).

Multistructural partition function truncation and its effect on the thermal rate constants.

Thermal rate constants for the hydrogen abstraction reaction of methyl pentanoate were calculated using the multistructural canonical variational theory with small-curvature tunneling (MS-CVT/SCT). The conformational search for the stationary points generated by these reactions was performed with an algorithm that combines systematic and stochastic searches at dual-level. At the high-level (MPWB1K/6-31+G(d,p)), 244 geometries for methyl pentanoate and transition states were found. A systematic estimative of the efficiency of the truncation of multistructural rovibrational partition functions and its effects on thermal rate constants was carried out. In this analysis, we observed that rotation about -OCH3 group (ϕ4) of the ester generates unstable structures with little contribution to the magnitude of the partition function. We also estimate that the truncation of the partition functions in a small set of conformations can produce deviations up to 75% compared to converged theoretical results.

Prenol as a Next-Generation Biofuel or Additive: A Comprehension of the Hydrogen Abstraction Reactions by a H Atom.

Thermal rate coefficients for the hydrogen abstraction reactions of prenol (3-methyl-2-butenol) by a hydrogen atom were calculated with the multipath canonical variational theory with small-curvature tunneling (MP-CVT/SCT). The conformational search was performed with a dual-level approach, and the multistructural torsional anharmonicity effects were corrected through the rovibrational partition function calculated with the multistructural method based on a coupled torsional potential (MS-T(C)). This methodology allows us to estimate the thermal rate constants in the temperature range of 200-2500 K and fit them into two analytical expressions. Differences between the number of conformations on the torsional potential energy surfaces for prenol and the transition state decrease the thermal rate constants for the H-abstraction at the α carbon. An opposite behavior was detected for the abstractions on the δ site. The product branching ratios were calculated using single-structure and multipath approaches. The product distributions from the former are shown to be inadequate for studying the mechanism under combustion conditions. The values estimated from MP-CVT/SCT rate coefficients indicated that the radicals from (Rα) and (Rδ)/(Rδ') are formed in considerable amounts. These species are fundamental in comprehending the inhibition and promotion of the autoignition phenomena.

Thermodynamic relations and ro-vibrational energy levels of the improved Pöschl–Teller oscillator for diatomic molecules

In this paper, the formalism of improved quantization rule is used to obtain analytical expression for bound state ro-vibrational energies of the improved Pöschl–Teller oscillator (IPTO). In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is used. Expression for analytical normalized eigenfunctions is obtained by ansatz solution technique. The equation obtained for ro-vibrational energy is then used to derive relations for some important thermodynamic functions for the IPTO; these include ro-vibrational: partition function, Helmholtz free energy, mean energy, entropy and specific heat capacity. Using the spectroscopic data of RbH (X1Σ+) diatomic molecule, pure vibrational state energies for the IPTO are calculated. The results obtained are in good agreement with experimental Rydberg–Klein–Rees data and are better than energies of the RbH (X1Σ+) diatomic molecule obtained via improved generalized Pöschl–Teller potential. Studies have also shown that the ro-vibrational partition function increases monotonically with increase in either temperature (T) or upper bound vibrational quantum number (λ). On the other hand, increase in any of T or λ results in monotonic decrease in ro-vibrational Helmholtz free energy of the system.

RO-VIBRATIONAL PARTITION FUNCTION AND MEAN THERMAL ENERGY OF THE IMPROVED WEI OSCILLATOR

In this paper, the improved Wei oscillator has been used to model the experimental Rydberg-Klein-Rees data of the X2 Σg+ state of N2+ diatomic ions. Average absolute deviation from the dissociation energy of 0.3211% and mean absolute percentage deviation of 0.6107% were obtained. These results are quite satisfactory since they are within error requirement rate of less than 1% of the Lippincott’s criterion. Using an existing equation in the literature for bound state ro-vibrational energy, expressions for ro-vibrational partition function and mean thermal energy were derived for the improved Wei oscillator within the context of classical physics. The formulas obtained for ro-vibrational partition function and mean thermal energy were subsequently applied to the spectroscopic data of N2+ (X2 Σg+) diatomic ions. Studies have revealed that the partition function of the system decreases monotonically with decrease in temperature and increases with increase in upper bound vibrational quantum number. On the other hand, the mean thermal energies of the diatomic ions show an initial sharp decrease when the temperature is decreased and afterwards remains fairly stable as the temperature is further lowered. The results obtained in this work may find suitable applications in astrophysics were potential energy functions are required to model experimentally determined potential energy data with high precision. The work may also be useful in many other areas of physics which include: chemical physics, molecular physics, atomic physics and solid-state physics

Rigorous factorization method of the vibrational and rotational contributions to the ro-vibrational partition function

The study reports an accurate and new closed-form representation of the ro-vibrational partition function, using a small amount of spectroscopic constants of the ground electronic state. A rigorous factorization method is proposed for the vibrational and rotational contributions to the partition function using an average vibrational quantum number.

An accurate closed-form expression for the rovibrational partition function of diatomic molecules

The rovibrational partition function is expressed in terms of analytic functions using summation formula. It is derived for a double series, which represents the rovibrational partition function in quantum statistical mechanics. With the summation formula obtained, the rovibrational partition function of isotonic oscillator is calculated in an analytic form and its temperature dependence is analyzed carefully.

On the Complementarity of the Harmonic Oscillator Model and the Classical Wigner-Kirkwood Corrected Partition Functions of Diatomic Molecules.

The vibrational and rovibrational partition functions of diatomic molecules are considered in the regime of intermediate temperatures. The low temperatures are those at which the harmonic oscillator approximation is appropriate, and the high temperatures are those at which classical partition function (with Wigner–Kirkwood correction) is applicable. The complementarity of the harmonic oscillator and classical integration over the phase space approaches is investigated for the CO and H molecules showing that those two approaches are complementary in the sense that they smoothly overlap.

Vibrational Partition Function for the Multitemperature Theories of High-Temperature Flows of Gases and Plasmas.

The vibrational partition function is calculated using the classical method of integration over the whole phase space. The calculations were done for the ground electronic state of a carbon monoxide molecule. The main focus is on temperature in the range 5000-20 000 K, which is common in hypersonic flows of gases and plasmas. The method presented here, because of the exclusion of the noninteracting part of canonical partition function according to the ideas of T.L. Hill, is applicable at temperatures of tens of thousands of Kelvins, where the standard expression for the vibrational partition function fails. At lower temperatures (here 1000-6000 K), the correct quantum results can be obtained with the help of Wigner-Kirkwood expansion. The influence of vibrations on the rotational partition function by bond-length elongation is examined, and the results are compared with the exact ro-vibrational partition function.

The role of rotation-vibration coupling in symmetric and asymmetric isotopomers of ozone.

A theoretical framework and a computer code (SpectrumSDT) are developed for accurate calculations of coupled rotational-vibrational states in triatomic molecules using hyper-spherical coordinates and taking into account the Coriolis coupling effect. Concise final formulas are derived for the construction of the Hamiltonian matrix using an efficient combination of the variational basis representation and discrete variable representation methods with locally optimized basis sets and grids. First, the new code is tested by comparing its results with those of the APH3D program of Kendrick et al. [Kendrick, Pack, Walker, and Hayes, J. Chem. Phys. 110, 6673 (1999)]. Then, accurate calculations of the rovibrational spectra are carried out for doubly substituted symmetric (18O16O18O) and asymmetric (18O18O16O) ozone isotopomers for the total angular momentum up to J = 5. Together with similar data recently reported for the singly substituted symmetric (16O18O16O) and asymmetric (16O16O18O) ozone isotopomers, these calculations quantify the role of the Coriolis coupling effect in the large mass-independent isotopic enrichment of ozone, observed in both laboratory experiments and the atmosphere of the Earth. It is found that the Coriolis effect in ozone is relatively small, as evidenced by deviations of its rotational constants from the symmetric-top-rotor behavior, magnitudes of parity splittings (Λ-doubling), and ratios of rovibrational partition functions for asymmetric vs symmetric ozone molecules. It is concluded that all of these characteristics are influenced by the isotopic masses as much as they are influenced by the overall symmetry of the molecule. It is therefore unlikely that the Coriolis coupling effect could be responsible for symmetry-driven mass-independent fractionation of oxygen isotopes in ozone.

Impact of nonconvergence and various approximations of the partition function on the molecular column densities in the interstellar medium

We emphasize that the completeness of the partition function, that is, the use of a converged partition function at the typical temperature range of the survey, is very important to decrease the uncertainty on this quantity and thus to derive reliable interstellar molecular densities. In that context, we show how the use of different approximations for the rovibrational partition function together with some interpolation and/or extrapolation procedures may affect the estimate of the interstellar molecular column density. For that purpose, we apply the partition function calculations to astronomical observations performed with the IRAM-30 m telescope towards the NGC 7538–IRS1 source of two N-bearing molecules: isocyanic acid (HNCO, a quasilinear molecule) and methyl cyanide (CH3CN, a symmetric top molecule). The case of methyl formate (HCOOCH3), which is an asymmetric top O-bearing molecule containing an internal rotor is also discussed. Our analysis shows that the use of different partition function approximations leads to relative differences in the resulting column densities in the range 9–43%. Thus, we expect this work to be relevant for surveys of sources with temperatures higher than 300 K and to observations in the infrared.

Anharmonic Rovibrational Partition Functions at High Temperatures: Tests of Reduced-Dimensional Models for Systems with up to Three Fluxional Modes.

Curvilinear coordinate Monte Carlo phase space integration and a series of full-dimensional fitted potential energy surfaces are used to study the effectiveness of reduced-dimensional models for predicting rovibrational anharmonicity at high temperatures. Fully coupled and fully anharmonic, but classical, rovibrational partition functions Q are computed for 14 species with two or three fluxional modes (inversions or torsions) and as many as 30 degrees of freedom. These results are converted to semiclassical anharmonicity correction factors f and are analyzed alongside results obtained previously for 22 species with up to two fluxional modes. As expected, fluxional species show considerable variation in f at high temperatures; f is as small as 0.2 for acetone and is as large as 9 for methylene glycol at 2500 K. This set of full-dimensional results is used to test the accuracy of reduced-dimensional models where fluxional modes are treated as coupled to one another but as separable from the remaining nonfluxional modes. For most systems, we find that such an approximation is accurate at high temperatures, with average errors in Q of just ∼25%. For some systems, however, larger errors are found, and these are attributed to strong coupling of the fluxional modes to one or more nonfluxional modes. In particular, we identify strong coupling to low-frequency bends for some systems, and we show that by comparing curvilinear and rectilinear harmonic frequencies for the fluxional modes, we can estimate the effect of this coupling on rovibrational anharmonicity. We also quantify the accuracy of the more severe but common assumption of treating fluxional modes as separable from one another, that is, as sets of uncoupled one-dimensional inversions and torsions. This approach can work well for methyl and alkyl rotors, but it is shown to have errors as large as a factor of 7 at high temperatures for more complex systems. Finally, we note that while the present analysis focuses on the treatment of fluxional modes, the collective anharmonicity correction associated with the more numerous nonfluxional modes, although simpler to describe, comprises a significant fraction of the overall anharmonicity.

Direct Computation of the Quantum Partition Function by Path-Integral Nested Sampling.

In the present work we introduce a computational approach to the absolute rovibrational quantum partition function using the path-integral formalism of quantum mechanics in combination with the nested sampling technique. The numerical applicability of path-integral nested sampling is demonstrated for small molecules of spectroscopic interest. The computational cost of the method is determined by the evaluation time of a point on the potential energy surface (PES). For efficient PES implementations, the path-integral nested sampling method can be a viable alternative to the direct-Boltzmann-summation technique of variationally computed rovibrational energies, especially for medium-sized molecules and at elevated temperatures.

Q2DTor: A program to treat torsional anharmonicity through coupled pair torsions in flexible molecules

The Q 2DTor program (Quantum 2-Dimensional Torsions) is designed to calculate accurate rotational–vibrational partition functions (also called rovibrational partition functions) and thermodynamic functions for molecular systems having two [1] or more torsions. Systems with more than two torsions can also be studied by treating the torsions by pairs. The program searches for all the torsional conformers and evaluates the rovibrational partition function using the multi-structural harmonic oscillator (MS-HO) approximation and the extended two-dimensional torsion (E2DT) approximation. The latter incorporates full coupling of the two torsions by means of the two-dimensional non-separable (2D-NS) approximation [2], and it also includes their influence on the remaining degrees of freedom. The program also calculates the ideal gas-phase standard-state thermodynamic functions at the requested temperatures. Twenty molecules have been used to test Q 2DTor. Program summaryProgram Title:Q2DTorProgram Files doi:http://dx.doi.org/10.17632/wbgchgc2kp.1Licensing provisions: GNU GPL v3Programming language: Python 2.7Nature of problem: Calculation of accurate partition functions and thermodynamic functions in molecular systems involving two torsional modes. Torsional anharmonicity is treated quantically and includes full coupling in the kinetic and potential energies between the torsions and between the torsions and the rest of the degrees of freedom.Solution method: The program uses the variational method to solve the Schrödinger equation of a two-dimensional torsional potential using Fourier series. All of the remaining degrees of freedom (non-torsional) are incorporated through a projected (the torsional modes are removed) rigid-rotator harmonic-oscillator partition function which is calculated at every torsional stationary point and that is allowed to vary with the torsional motion. The integration of the rovibrational partition function over the torsional space leads to a mixed quantum-classical vibrational partition function, which is transformed into a full quantum partition function by including the quantum contribution due to the torsions. For the evaluation of the integral, the rovibrational partition function at nonstationary points is carried out through a Delaunay triangulation procedure using the calculated rovibrational partition functions at the stationary points as nodes.Additional comments including Restrictions and Unusual features: The program is limited to two coupled torsional modes.

Anharmonic Rovibrational Partition Functions for Fluxional Species at High Temperatures via Monte Carlo Phase Space Integrals.

Monte Carlo phase space integration (MCPSI) is used to compute full dimensional and fully anharmonic, but classical, rovibrational partition functions for 22 small- and medium-sized molecules and radicals. Several of the species considered here feature multiple minima and low-frequency nonlocal motions, and efficiently sampling these systems is facilitated using curvilinear (stretch, bend, and torsion) coordinates. The curvilinear coordinate MCPSI method is demonstrated to be applicable to the treatment of fluxional species with complex rovibrational structures and as many as 21 fully coupled rovibrational degrees of freedom. Trends in the computed anharmonicity corrections are discussed. For many systems, rovibrational anharmonicities at elevated temperatures are shown to vary consistently with the number of degrees of freedom and with temperature once rovibrational coupling and torsional anharmonicity are accounted for. Larger corrections are found for systems with complex vibrational structures, such as systems with multiple large-amplitude modes and/or multiple minima.

Partition functions of thermally dissociating diatomic molecules and related momentum problem

The anharmonicity and ro-vibrational coupling in ro-vibrational partition functions of diatomic molecules are analyzed for the high temperatures of the thermal dissociation regime.The numerically exact partition functions and thermal energies are calculated. At the high temperatures the proper integration of momenta is important if the partition function of the molecule, understood as bounded system, is to be obtained. The problem of proper treatment of momentum is crucial for correctness of high temperature molecular simulations as the decomposition of simulated molecule have to be avoided; the analysis of O2, H2+, and NH3 molecules allows to show importance of βDe value.

Rovibrational energies, partition functions and equilibrium fractionation of the CO2 isotopologues

Rovibrational energy levels, partition functions and relative abundances of the stable isotopologues of CO2 in gas phase at equilibrium are calculated using an empirical Morse-cosine potential energy surface (PES) refined by fitting to the updated pure (l2=0) vibrational frequencies observed for the main 12C16O2 isotopologue. The rovibrational energy levels are calculated variationally using a system of optimized hyperspherical normal coordinates, and from these the vibrational terms Gv and rotational constants Bv of the isotopologues are determined. The refined potential surface is shown to be clearly superior to the original potential surface, with the former reproducing the observed values of the spectroscopic constants Gv and Bv with accuracies of about 0.1cm−1 and 0.00020cm−1, respectively, for levels with l2≥0 up to 10,000cm−1 above the ground state. The internal partition functions of the isotopologues are calculated by approximated direct summation over the rovibrational energies and compared with both previous partition sums and values obtained from analytical expressions based on the harmonic oscillator and rigid rotor models. The partition functions calculated by approximated direct summation are then used to determine the abundances of the CO2 isotopologues at thermodynamic equilibrium using the method developed by Wang et al. [74]. Significant variations in the relative abundances of some of the CO2 multiple substituted isotopologues at terrestrial temperatures with respect to those provided by the classical harmonic-based Urey theory are found, which may be of relevance in geochemical processes.

Free energy calculations for a flexible water model

In this work, we consider the problem of calculating the classical free energies of liquids and solids for molecular models with intramolecular flexibility. We show that thermodynamic integration from the fully-interacting solid of interest to a Debye crystal reference state, with anisotropic harmonic interactions derived from the Hessian of the original crystal, provides a straightforward route to calculating the Gibbs free energy of the solid. To calculate the molecular liquid free energy, it is essential to correctly account for contributions from both intermolecular and intramolecular motion; we employ thermodynamic integration to a Lennard-Jones reference fluid, coupled with direct evaluation of the molecular ro-vibrational partition function. These approaches are used to study the low-pressure classical phase diagram of the flexible q-TIP4P/F water model. We find that, while the experimental ice-I/liquid and ice-III/liquid coexistence lines are described reasonably well by this model, the ice-II phase is predicted to be metastable. In light of this finding, we go on to examine how the coupling between intramolecular flexibility and intermolecular interactions influences the computed phase diagram by comparing our results with those of the underlying rigid-body water model.