Vibrational Partition Function Research Articles

Thermodynamic properties for a combination of Varshni and Coulomb potentials

A Varshni potential and a Coulomb potential are combined as a single potential to produce energy levels. The vibrational thermodynamic properties such as the enthalpy, Gibbs free energy and entropy are obtained via the vibrational partition function and its derivatives calculated from energy levels of the combined potentials. The vibrational thermodynamic properties of four subset potentials of the mother potential are also studied as special cases. The result of this study shows that the features of the thermodynamic properties as a function of temperature for the mother potential and the Varshni potential as one of the subset potentials, exhibit the same characteristics while the thermodynamic properties of the remaining three subset potentials exhibit the same features. The features of these three subset potentials follow the trends of many molecular potentials for some molecules. However, the Gibbs free energy of both the mother potential and the subset potentials exhibits the same variation with temperature.

Theoretical prediction of molar entropy of modified shifted Morse potential for gaseous molecules

The modified shifted Morse potential is a molecular potential that describes the molecular vibration and atomic interaction. This potential has not received much report as one of the useful potential function. In this study, the molar entropy of a modified shifted Morse is obtained via the vibrational partition function, rotational partition function and translational partition function. Numerical results are obtained for iodine molecule (I2), silicon carbide (SiC), carbon phosphide (CP) and fluorine molecule (F2) for temperatures of 200 K to 6000 K. The predicted results are compared with the experimental results obtained from the National Institute of Standards and Technology (NIST) data base. The predicted values aligned with the experimental results for the four molecules. The average absolute percentage deviation obtained for each molecule is less than one, which shows a good representation of the modified shifted Morse potential for thermodynamic functions.

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.

Information theory and thermodynamic properties of diatomic molecules using molecular potential.

Owing to the devise applications of molecules in industries, the bound state solution of the non-relativistic wave equation with a molecular potential function has been obtained in a closed-form using the Nikiforov-Uvarov method. The solutions of the bound state are then applied to study the information-theoretic measures such as the one-dimensional Shannon and Renyi entropic densities. The expectation values for the position and momentum spaces were obtained to verify the Heisenberg's uncertainty principle. Utilizing the energy spectrum equation, the thermodynamic vibrational partition function is obtained via the Poisson summation. Other thermodynamic function variations with absolute temperature have been obtained numerically for four diatomic molecules (H2, N2, O2, and HF) using Maple 18 software. The Shannon global entropic sum inequality has also been verified. The Renyi sum for constrained index parameters satisfies the global entropic inequality. The thermodynamic properties of the four molecules are similar and conform to works reported in the existing literature. The obtained vibrational energies are in fair agreement with the ones obtained using other forms of potential energy. The result further indicates that the lowest bounds for the Shannon, Renyi, and Heisenberg inequalities are ground states phenomena.

Thermal Responses and the Energy Spectral of Diatomic Molecules Using Nikiforov–Uvarov Methodology

The parametric Nikiforov–Uvarov approach and the Greene–Aldrich approximation scheme were used to achieve approximate analytical solutions to the Schrödinger equation, involving an interaction of the modified deformed Hylleraas potential mixed linearly with the improved Frost–Musulin diatomic molecular potential. For each ℓ-state, the energy spectra and normalized wave functions were generated from the hypergeometric function in the closed form. The thermal properties of such a system, including the vibrational partition function, vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity, and vibrational entropy, were then calculated for the selected diatomic molecules using their experimental spectroscopic parameters. Furthermore, the peculiar conditions of this potential were evaluated, and their energy eigenvalues were calculated for the purpose of comparison. The acquired results were found to be in reasonable agreement with those reported in the literature.

Thermodynamic properties for some diatomic molecules with the q-deformed hyperbolic barrier potential

In this paper, we obtain analytical vibrational partition function with recent energy spectrum of the deformed hyperbolic barrier potential. We derive different thermodynamic properties for the , , , , and diatomic molecules. In addition, we obtain the total molar entropy for the two last molecules. Our results are in excellent agreement with the experimental data.

Structure, spectroscopic signatures, and formation of hydroxy-azirine: a potential interstellar prebiotic molecule

ABSTRACT Hydroxy-azirine (C2H3NO) is a -OH derivative of azirine (C2H3N), molecule that has been the subject of several unfruitful searches in space. Hydroxy-azirine is an isomer of the detected prebiotic species methyl isocyanate, CH3NCO, and glycolonitrile, HOCH2CN, as well as the yet undetected imine acetaldehyde, NHCHCHO. However, the lack of preliminary spectroscopic data on hydroxy-azirine has prevented its astronomical search. The aim of this study is to provide high-level theoretical spectroscopic signatures of the most stable hydroxy-azirine isomers to enable their eventual interstellar search. A total of 12 isomers have been characterized for hydroxy-azirine and their isomerization processes have been analysed at the CCSD(T)-F12/cc-pVTZ-F12 level. The most stable structures are 3-hydroxy-2H-azirine (I) and 2-hydroxy-2H-azirine (II) in their syn- and anti-configurations, which are suggested as the most relevant candidates for laboratory and interstellar detection. To ease their identification by means of rotational spectroscopy, we report a set of the required spectroscopic parameters using state-of-the-art composite and coupled-cluster approaches. For astronomical purposes, we provide a complete line list for I-syn and I-antihydroxy-azirine up to 50 GHz, which takes the hyperfine structure into account, and will be essential to hunt for these interstellar candidates experimentally. In addition, anharmonic vibrational frequencies and intensities are reported to predict a trustworthy vibrational spectra and to estimate the vibrational partition function. Finally, we analyse the possibility of formation of hydroxy-azirine from the reaction of azirine with the hydroxyl radical in the gas-phase and on the surface of ices, finding for the latter a feasible formation route under interstellar conditions.

Asymmetry-enhanced 59Co NMR thermometry in Co(iii) complexes.

Design strategies for molecular thermometers by magnetic resonance are essential for enabling new noninvasive means of temperature mapping for disease diagnoses and treatments. Herein we demonstrate a new design strategy for thermometry based on chemical control of the vibrational partition function. To do so, we performed variable-temperature 59Co NMR investigations of four air-stable Co(iii) complexes: Co(accp)3 (1), Co(bzac)3 (2), Co(tBu2-acac)3 (3), and Co(acac)3 (4) (accp = 2-acetylcyclopentanonate; bzac = benzoylacetonate; tBu2-acac = 2,2,6,6-tetramethyl-3,5-heptanedionate and acac = acetylacetonate). We discovered 59Co chemical shift temperature sensitivity (Δδ/ΔT) values of 3.50(2), 3.39(3), 1.63(3), and 2.83(1) ppm °C-1 for 1-4, respectively, at 100 mM concentration. The values observed for 1 and 2 are new records for sensitivity for low-spin Co(iii) complexes. We propose that the observed heightened sensitivities for 1 and 2 are intimately tied to the asymmetry of the accp and bzac ligands versus the acac and tBu2-acac ligands, which enables a larger number of low-energy Raman-active vibrational modes to contribute to the observed Δδ/ΔT values.

Temperature and pressure dependent rate constants of the reactions of OH• with cyclopentene from variational TST and SS-QRRK methods.

In this work, the pressure- and temperature-dependent reaction rate constants for the hydrogen abstraction and addition of hydroxyl radicals to the unsaturated cyclopentene were studied. Geometries and vibrational frequencies of reactants, products, and transition states were calculated using density functional theory, with single-point energy corrections determined at the domain-based local pair natural orbital-coupled-cluster single double triple/cc-pVTZ-F12 level. The high-pressure limit rate constants were calculated using the canonical variational transition state theory with the small-curvature tunneling approximation. The vibrational partition functions were corrected by the effects of torsional and ring-puckering anharmonicities of the transition states and cyclopentene, respectively. Variational effects are shown to be relevant for all the hydrogen abstraction reactions. The increasing of the rate constants by tunneling is significant at temperatures below 500K. The pressure dependence on the rate constants of the addition of OH• to cyclopentene was calculated using the system-specific quantum Rice-Ramsperger-Kassel model. The high-pressure limit rate constants decrease with increasing temperature in the range 250-1000K. The falloff behavior was studied at several temperatures with pressures varying between 10-3 and 103 bar. At temperatures below 500K, the effect of the pressure on the addition rate constant is very modest. However, at temperatures around and above 1000K, taking pressure into account is mandatory for an accurate rate constant calculation. Branching ratio analyses reveal that the addition reaction dominates at temperatures below 500K, decreasing rapidly at higher temperatures. Arrhenius parameters are provided for all reactions and pressure dependent Arrhenius parameters are given for the addition of OH• to cyclopentene.

Computing thermodynamic properties of some diatomic molecules using a q-deformed Scarf-like potential

This paper computes the vibrational partition function of some diatomic molecules using an improved energy spectrum obtained recently with the q-deformed hyperbolic Scarf-like potential. Compared to the experimental energy curves, this potential proved to represent well the interaction between the atoms in the studied molecules. The related thermodynamic functions are derived for the , ScI and molecules. Results demonstrated that the computed molecular entropy is in very good agreement with the experimental data for a large range of temperature values. In addition, results are given for the ScI molecule for which no experimental entropy values exist. Our results can serve as a reference to experimental studies in chemical physics.

ULYSSES: An Efficient and Easy to Use Semiempirical Library for C+.

We introduce ULYSSES, a user-friendly and robust C++ library for semiempirical quantum chemical calculations. In its current version, ULYSSES is equipped with a large set of different semiempirical models, most of which are based on the Neglect of Diatomic Differential Overlap (NDDO) approximation. Empirical corrections for dispersion and hydrogen bonding are available for most methods, so that higher quality is achieved in the calculation of energies of nonbonded complexes. The library is furthermore equipped with geometry optimization, as well as modules for calculating molecular properties of general interest. Ideal gas thermodynamics is available and allows single structure as well as conformer (multistructure) averaged properties to be calculated. We offer the possibility to use several vibrational partition functions according to the nature of interactions being studied: for covalent systems, the traditional harmonic oscillator approximation is available; for nonbonded complexes, we systematically extended the partition function proposed by Grimme for all thermodynamic functions. The library is also capable of running Born-Oppenheimer molecular dynamics.

Partition sums for non-local thermodynamic equilibrium conditions for nine molecules of importance in planetary atmospheres

Internal vibrational and rotational partition functions are calculated for nine molecules that are abundant in the terrestrial atmosphere and some are also observed in planetary and exoplanet atmospheres. These molecules are often observed in non-local thermodynamic equilibrium (NLTE) conditions. Calculations are made for all isotopologues of these molecules for which data are available, generally for the temperature range T = 1–5000 K. The methods of calculation of the vibrational and rotational partition functions, Qv and Qr, respectively, the convergence, and the uncertainties for each isotopologue are discussed. Finally, a FORTRAN program, NLTE_TIPS_2021.FOR, to recall the NLTE partition sums is discussed.

Thermal properties of diatomic molecules with multi parameter exponential-type potential

In this work, we have calculated the thermal properties of H2 and O2 diatomic molecules with multi parameter exponential type potential within the framework of the statistical mechanics. In this regard, using the improved energy spectrum, we have determined the vibrational partition function obtained recently via the path integral formalism. Based on obtained partition function, we find thermodynamic properties of diatomic molecules such as Gibbs free energy, enthalpy and specific heat in constant pressure by the Poisson summation formalism. Also, we have validated our results with experimental data and our results show that there is a good agreement between them. Finally, the average absolute deviations of the calculated data from the experimental data are obtained.

Calculation of improved enthalpy and entropy of vaporization by a modified partition function in quantum cluster equilibrium theory.

In this work, we present an altered partition function that leads to an improved calculation of the enthalpy and entropy of vaporization in the framework of quantum cluster equilibrium theory. The changes are based on a previously suggested modification [S. Grimme, Chem. Eur. J. 18, 9955-9964 (2012)] of the molecular entropy calculation in the gas phase. Here, the low energy vibrational frequencies in the vibrational partition function are treated as hindered rotations instead of vibrations. The new scheme is tested on a set of nine organic solvents for the calculation of the enthalpy and entropy of vaporization. The enthalpies and entropies of vaporization show improvements from 6.5 error to 3.3 kJ mol-1 deviation to experiment and from 28.4 error to 13.5J mol-1K-1 deviation to experiment, respectively. The effect of the corrected partition function is visible in the different populations of clusters, which become physically more meaningful in that larger clusters are higher populated in the liquid phase and the gas phase is mainly populated by the monomers. Furthermore, the corrected partition function also overcomes technical difficulties and leads to an increased stability of the calculations in regard to the size of the cluster set.

Chemically Accurate Vibrational Free Energies of Adsorption from Density Functional Theory Molecular Dynamics: Alkanes in Zeolites.

We present a methodology to compute, at reduced computational cost, Gibbs free energies, enthalpies, and entropies of adsorption from molecular dynamics. We calculate vibrational partition functions from vibrational energies, which we obtain from the vibrational density of states by projection on the normal modes. The use of a set of well-chosen reference structures along the trajectories accounts for the anharmonicities of the modes. For the adsorption of methane, ethane, and propane in the H-CHA zeolite, we limit our treatment to a set of vibrational modes localized at the adsorption site (zeolitic OH group) and the alkane molecule interacting with it. Only two short trajectories (1–20 ps) are required to reach convergence (<1 kJ/mol) for the thermodynamic functions. The mean absolute deviations from the experimentally measured values are 2.6, 2.8, and 4.7 kJ/mol for the Gibbs free energy, the enthalpy, and the entropy term (−TΔS), respectively. In particular, the entropy terms show a major improvement compared to the harmonic approximation and almost reach the accuracy of the previous use of anharmonic frequencies obtained with curvilinear distortions of individual modes. The thermodynamic functions so obtained follow the trend of the experimental values for methane, ethane, and propane, and the Gibbs free energy of adsorption at experimental conditions is correctly predicted to change from positive for methane (5.9 kJ/mol) to negative for ethane (−4.8 kJ/mol) and propane (−7.1 kJ/mol).

Approximate energy spectra and statistical mechanical functions of some diatomic molecular hydrides

In this study, we have investigated the statistical mechanical properties of the Varshni potential model for some diatomic molecular hydrides via the Euler–Maclaurin formula. This was done using the approximate analytical energy eigenvalues, which were obtained by solving the radial Schrödinger equation with the Greene–Aldrich approximation and suitable coordinate transformation schemes. The effect of high temperatures and upper bound vibration quantum number on the vibrational partition function and other thermodynamic functions of the selected diatomic molecular hydrides were studied. We also show that these effects on the thermodynamic functions considered were similar for all the diatomic molecular hydrides selected.

ANALYTICAL SOLUTION OF SCHRÖDINGER EQUATION IN BISPHERICAL COORDINATE FOR MOBIUS SQUARE PLUS MODIFIED YUKAWA POTENTIAL USING NIKIFOROV UVAROV FUNCTIONAL ANALYSIS (NUFA) METHOD WITH ITS THERMODYNAMICS PROPERTIES FOR DIATOMIC MOLECULES

Background: The analytical solution of the Schrödinger equation in bispherical coordinates has attracted a great deal of interest for theoretical physics researchers in the branch of quantum physics. The energy and wave function are solutions of the Schrödinger equation which are very important because it contains all necessary information regarding the behavior of quantum systems. Aim: This study aimed to obtain energy, radial wave functions and thermodynamic properties for diatomic molecules from the radial part of the Schrödinger equation in bispherical coordinates for the modified Mobius square plus Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Methods: The variable separation method was applied to reduce the Schrodinger equation in bispherical coordinates to the radial part and angular part Schrodinger equation. The Schrodinger equation of the radial part in bispherical coordinates was solved using the Nikiforov Uvarov Functional Analysis (NUFA) method to obtain the energy equation and radial wave function. Furthermore, the vibrational partition function 𝑍 was obtained from the energy equation. The vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆 were obtained from the vibrational partition function 𝑍. Results and Discussion: The results showed that the increase of parameters of 𝑛 and 𝛼 caused the decrease of energy, but the increase of parameters of 𝐿 and 𝑚0 caused the increase of energy. The radial quantum number 𝑛 and the potential range 𝛼 had the most effect to the wave functions. The parameters 𝑛𝑚𝑎𝑥, 𝑇, and 𝛼 had effect to the vibrational partition function 𝑍, vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆. Conclusions: From the results of this study, it can be concluded that the energy, radial wave function, and thermodynamic properties for diatomic molecules have been obtained using the Nikiforov Uvarov Functional Analysis (NUFA) method.

The Synchrotron-based Far-infrared Spectrum of Glycolaldehyde

Abstract Glycolaldehyde (GA) has been observed toward several different sources, with a broad range of rotational temperatures (8–300 K). At the high end, the temperature is comparable to the energy of the lowest vibrational states of GA, making the vibrational contribution to the partition function significant. Here, we report an analysis of the high-resolution far-infrared spectrum of GA, which features a plethora of well-resolved lines from 170–430 cm−1 (13–5 THz). We focus on the three fundamental vibrational bands in this range, i.e., the symmetric ν 12 bend at 282 cm−1, and the asymmetric ν 17 and ν 18 torsions at 360 and 208 cm−1, respectively. We assigned 23,266 transitions to 13,999 lines within these bands, which, when combined with the previously reported microwave and millimeter-wave spectra, allowed for refinement of the vibrationally excited rotational constants, and accurate determination of their band origins. Additionally, the assignment of a number of lines in several hot bands that are significantly populated at 300 K allowed for determination of their band origins. The rotational constants reported here should be useful in searches of vibrationally excited GA toward warm sources, and the accurately determined band origins allow for refinement of the vibrational partition function, and therefore column density, for a given excitation temperature.

Solutions of the Schrödinger equation and thermodynamic properties of a combined potential

The solution of the radial Schrödinger equation was obtained using the methodology of supersymmetric approach with a combination of modified generalized Pöschl-Teller potential and inversely quadratic Yukawa potential model. The non-relativistic ro-vibrational energy spectra and the corresponding wave functions were obtained and numerical results were generated for some states. The variation of energy of the combined potential and the subsets potentials with the screening parameter for various quantum number were graphically studied. The effect of the potential parameters on the energy for different states was also studied numerically. For more usefulness and applications of the work, the vibrational partition function and the various thermal properties like mean energy, Helmholtz energy, heat capacity and entropy were calculated. The behaviour of the thermodynamic properties with respect to temperature change for various quantum number and maximum quantum states were examined in detail. The temperature has positive effect on all the thermal properties except the free energy.

Statistical analysis and information theory of screened Kratzer–Hellmann potential model

In this research, the radial Schrödinger equation for a newly proposed screened Kratzer–Hellmann potential model was studied via the conventional Nikiforov–Uvarov method. The approximate bound state solution of the Schrödinger equation was obtained using the Greene–Aldrich approximation in addition to the normalized eigenfunction for the new potential model, both analytically and numerically. These results were employed to evaluate the rotational–vibrational partition function and other thermodynamic properties for the screened Kratzer–Hellmann potential. The results obtained have been graphically discussed. Also, the normalized eigenfunction has been used to calculate some information-theoretic measures including Shannon entropy and Fisher information for low-lying states in both position and momentum spaces numerically. The Shannon entropy results obtained agreed with the Bialynicki-Birula and Mycielski inequality, while the Fisher information results obtained agreed with the Stam, Cramér–Rao inequality. Also, alternating increasing and decreasing localization across the screening parameter for both eigenstates was observed.