Improvement to the hypervirial theorem and Kramer's rule for the calculation of central potential matrix elements

Ameliorated recurrence relations for the calculation of matrix elements of central potential wavefunctions are presented. These were obtained by using the hypervirial theorem with V(r) three-dimensional potentials and f (r) arbitrary functions. This procedure leads to a generalization of the usual l = l′ diagonal three-dimensional and l == 0 one-dimensional hypervirial relations of first and second class. The use of this kind of generalization to the calculation of r^k integrals, allows one to obtain all off-diagonal recursion formulas for any V(r). Besides, for hydrogenic wavefunctions one gets to equations that reduce to the usual Kramer's rule as a particular diagonal case. The proposed approach can be straightforwardly extended to obtain recurrence relations for the calculation of two center integrals.     

Effective utilization of off-diagonal hypervirial relations considering diagonal hypervirial relation: Harmonic oscillator case

The effective utilization of hypervirial relations is scrutinized to improve the approximate excited-state functions in the harmonic oscillator system. A new method is presented which simultaneously employs the off-diagonal hypervirial relations with the diagonal hypervirial relation. In order to use these relations effectively, the following points are pointed out: (i) the presented method is useful to get better reasonable results for the excitation energies and the state functions; (ii) the ground state given must satisfy the virial theorem; (iii) in the hypervirial operator used here as x^m, the smaller integers of m's present better results; and (iv) the employment of the comparatively small number of trial basis functions of the type exp (?γ|x|) is sufficient for reproducing the exact excited state. Especially among them, condition (ii) plays an important role. Applying all the proposals to the first and the second excited states, one gets a highly improved excitation energy, state function, and other physical quantities (e.g., transition moment and oscillator strength). The presented method is also found to be more effective than the employment of only the off-diagonal hypervirial relations or the method of the scaling operation.     

Kratzer potential algebraic representation and matrix elements recurrence formulae

Generalized recurrence relations for the calculation of multipole matrix elements for Kratzer potential wave functions are obtained operationally. These formulas have been determined by using a non-analytical procedure based on the algebraic representation of the Kratzer eigenfunctions along with the usual ladder properties and commutation relations. For that, the creation and annihilation operators are adequately derived by means of an alternative approach to the factorization method and the exact expressions for matrix elements are achieved with the aid of a relationship between the ladder operators associated with the bra and theket. The proposed algebraic approach as well as the formulas for the calculation of matrix elements thus derived are quite simple and direct when compared with other alternative expressions already obtained analytically or pseudo-algebraically by means of the hypervirial theorem commutator algebra.     

Kinetics of the gas‐phase reactions of chlorine atoms with CH2F2, CH3CCl3, and CF3CFH2 over the temperature range 253–553 K

Relative rate techniques were used to study the title reactions in 930–1200 mbar of N₂ diluent. The reaction rate coefficients measured in the present work are summarized by the expressions k(Cl + CH₂F₂) = 1.19 × 10^?17 T² exp(?1023/T) cm³ molecule^?1 s^?1 (253–553 K), k(Cl + CH₃CCl₃) = 2.41 × 10^?12 exp(?1630/T) cm³ molecule^?1 s^?1 (253–313 K), and k(Cl + CF₃CFH₂) = 1.27 × 10^?12 exp(?2019/T) cm³ molecule^?1 s^?1 (253–313 K). Results are discussed with respect to the literature data. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 401–406, 2009     

Hypervirial theorem and matrix elements for the Coulomb potential

It is well known that hydrogen like atoms may be represented as Morse oscillators. Our aim in this article was to show how this result and the hypervirial theorem lead to a computational method with which the matrix elements for the Coulomb potential can be easily calculated. Also, a closed analytical expression for 〈n₂l₂|r^k|n₁l₁〉 is obtained. © 1995 John Wiley & Sons, Inc.     

Variational RRKM theory calculation of thermal rate constant for carbon—hydrogen bond fission reaction of nitro benzene

The present work provides quantitative results for the rate of unimolecular carbon-hydrogen bond fission reaction of benzene and nitro benzene at elevated temperatures up to 2000 K. The potential energy surface for each C-H (in the ortho, meta, and para sites) bond fission reaction of nitro benzene was investigated by ab initio calculations. The geometry and vibrational frequencies of the species involved in this process were optimized at the MP2 level of theory, using the cc-pvdz basis set. Since C-H bond fission channel is barrier less reaction, we have used variational RRKM theory to predict rate constants. By means of calculated rate constant at the different temperatures, the activation energy and exponential factor were determined. The Arrhenius expression for C-H bond fission reaction of nitro benzene on the ortho, meta and para sites are k(T) = 2.1 × 10¹⁷exp(?56575.98/T), k(T) = 2.1 × 10¹⁷exp(?57587.45/T), and k(T) = 3.3 × 10¹⁶exp(?57594.79/T) respectively. The Arrhenius expression for C-H bond fission reaction of benzene is k(T) = 2 × 10¹⁸exp(?59343.48.18/T). The effect of NO₂ group, location of hydrogen atoms on the substituted benzene ring, reaction degeneracy, benzene ring resonance and tunneling effect on the rate expression have been discussed.     

Standard reactions for comparative rate studies: Experiments on the dehydrochlorination reactions of 2‐chloropropane,chlorocyclopentane, and chlorocyclohexane

Single pulse shock tube studies of the thermal dehydrochlorination reactions (chlorocyclopentane → cyclopentene + HCl) and (chlorocyclohexane → cyclohexene + HCl) at temperatures of 843–1021 K and pressures of 1.4–2.4 bar have been carried out using the comparative rate technique. Rate constants have been measured relative to (2‐chloropropane → propene + HCl) and the decyclization reactions of cyclohexene, 4‐methylcyclohexene, and 4‐vinylcyclohexene. Absolute rate constants have been derived using k(cyclohexene → ethene + butadiene) = 1.4 × 10¹⁵ exp(?33,500/T) s^?1. These data provide a self‐consistent temperature scale of use in the comparison of chemical systems studied with different temperature standards. A combined analysis of the present results with the literature data from lower temperature static studies leads to

• k(2‐chloropropane) = 10^{(13.98±0.08)} exp(?26, 225 ± 130) K/T) s^?1; 590–1020 K; 1–3 bar
• k(chlorocylopentane) = 10^{(13.65 ± 0.10)} exp(?24,570 ± 160) K/T) s^?1; 590–1020 K; 1–3 bar
• k(chlorocylohexane) = 10^{(14.33 ± 0.10)} exp(?25,950 ± 180) K/T) s^?1; 590–1020 K; 1–3 bar

Including systematic uncertainties, expanded standard uncertainties are estimated to be about 15% near 600 K rising to about 25% at 1000 K. At 2 bar and 1000 K, the reactions are only slightly under their high‐pressure limits, but falloff effects rapidly become significant at higher temperatures. On the basis of computational studies and Rice–Ramsperger–Kassel–Marcus (RRKM)/Master Equation modeling of these and reference dehydrochlorination reactions, reported in more detail in an accompanying article, the following high‐pressure limits have been derived:

• k_∞ (2‐chloropropane) = 5.74 × 10⁹T^1.37 exp(?25,680/T) s^?1; 600–1600 K
• k_∞ (chlorocylopentane) = 7.65 × 10⁷T^1.75 exp(?23,320/T) s^?1; 600–1600 K
• k_∞ (chlorocylohexane) = 8.25 × 10⁹T^1.34 exp(?25,010/T) s^?1; 600–1600 K

© 2011 Wiley Periodicals, Inc.

1 This article is a U.S. Government work and, as such, is in the public domain of the United States of America.

Int J Chem Kinet 44: 351–368, 2012     

A laser flash photolysis/IR diode laser absorption study of the reaction of chlorine atoms with selected alkanes

A high‐resolution IR diode laser in conjunction with a Herriot multiple reflection flow‐cell has been used to directly determine the rate coefficients for simple alkanes with Cl atoms at room temperature (298 K). The following results were obtained: k(Cl + n‐butane) = (1.91 ± 0.10) × 10^?10 cm³ molecule^?1 s^?1, k(Cl + n‐pentane) = (2.46 ± 0.12) × 10^?10 cm³ molecule^?1 s^?1, k(Cl + iso‐pentane) = (1.94 ± 0.10) × 10^?10 cm³ molecule^?1 s^?1, k(Cl + neopentane) = (1.01 ± 0.05) × 10^?10 cm³ molecule^?1 s^?1, k(Cl + n‐hexane) = (3.44 ± 0.17) × 10^?10 cm³ molecule^?1 s^?1 where the error limits are ±1σ. These values have been used in conjunction with our own previous measurements on Cl + ethane and literature values on Cl + propane and Cl + iso‐butane to generate a structure activity relationship (SAR) for Cl atom abstraction reactions based on direct measurements. The resulting best fit parameters are k_p = (2.61 ± 0.12) × 10^?11 cm³ molecule^?1 s^?1, k_s = (8.40 ± 0.60) × 10^?11 cm³ molecule^?1 s^?1, k_t = (5.90 ± 0.30) × 10^?11 cm³ molecule^?1 s^?1, with f( ? CH₂^? ) = f (? CH₂^? ) = f (?C?) = f = 0.85 ± 0.06. Tests were carried out to investigate the potential interference from production of excited state HCl(v = 1) in the Cl + alkane reactions. There is some evidence for HCl(v = 1) production in the reaction of Cl with shape n‐hexane. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 34: 86–94, 2002     

An ab initio chemical kinetic study on the reactions of H,OH, and Cl with HOClO3

The mechanisms for reactions of H, HO, and Cl with HOClO₃, important elementary processes in the early stages of the ammonium perchlorate (AP) combustion reaction, have been investigated at the CCSD(T)/6‐311+G(3df,2p)//PW91PW91/6‐311+G(3df) level of theory. The rate constants for the low‐energy channels have been calculated by statistical theory. For the reaction of H and HOClO₃, the main channels are the production of H₂ + ClO₄ (k_1a) and HO + HOClO₂ (k_1b); k_1a and k_1b can be represented as 1.07 × 10^?17 T^1.97 exp(?7484/T) and 6.08 × 10^?17T^1.96 exp(?7729/T) cm³ molecule^?1 s^?1, respectively. For the HO + HOClO₃ reaction, the main pathway is the H₂O + ClO₄ (k_2a) production process, with the predicted rate constant k_2a = 1.24 × 10 ^?8 T^?2.99 exp(1664/T) for 300–500 K and k_2a = 1.27×10^?19 T^2.12 exp(?1474/T) for 500–3000 K. For the Cl + HOClO₃ reaction, the formation of HOCl + ClO₃ (k_3a) and HCl + ClO₄ (k_3b) is dominant, with k_3a = 1.33×10^?12 T^0.67 exp(?9658/T) and k_3b = 1.75×10¹⁶ T^1.63 exp(?11156/T) cm³ molecules^?1 in the range of 300–3000 K. In addition, the heats of formation of ClO₃ and HOClO₃ have been predicted based on several isodesmic and/or isogyric reactions with Δ_fH^o₀ (ClO₃) = 47.0 ± 1.0 and Δ_fH^o₀ (HOClO₃) = 5.5 ± 1.5 kcal/mol, respectively. These data may be used for kinetic simulation of the AP decomposition and combustion reaction. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 42: 253–261, 2010     

Kinetic study of the reactions of the hydroxyl radical with ethane and propane

Absolute rate coefficients for the reactions of the hydroxyl radical with ethane (k₁, 297–300 K) and propane (k₂, 297–690 K) were measured using the flash photolysis–resonance fluorescence technique. The rate coefficient data were fit by the following temperature-dependent expressions, in units of cm³/molecule·s: k₁(T) = 1.43 × 10^?14T^1.05 exp (?911/T) and k₂(T) = 1.59 × 10^?15T^1.40 exp (-428/T). Semiquantitative separation of OH-propane reactivity into primary and secondary H-atom abstraction channels was obtained.     

Bond dissociation energies and radical heats of formation in CH3Cl,CH2Cl2, CH3Br,CH2Br2, CH2FCl,and CHFCl2

An analysis of thermochemical and kinetic data on the bromination of the halomethanes CH_4–nX_n (X = F, Cl, Br; n = 1–3), the two chlorofluoromethanes, CH₂FCl and CHFCl₂, and CH₄, shows that the recently reported heats of formation of the radicals CH₂Cl, CHCl₂, CHBr₂, and CFCl₂, and the C? H bond dissociation energies in the matching halomethanes are not compatible with the activation energies for the corresponding reverse reactions. From the observed trends in CH₄ and the other halomethanes, the following revised ΔH°_f,298 (R) values have been derived: ΔH°_f(CH₂Cl) = 29.1 ± 1.0, ΔH°_f(CHCl₂) = 23.5 ± 1.2, ΔH_f(CH₂Br) = 40.4 ± 1.0, ΔH°_f(CHBr₂) = 45.0 ± 2.2, and ΔH°_f(CFCl₂) = ?21.3 ± 2.4 kcal mol^?1. The previously unavailable radical heat of formation, ΔH°_f(CHFCl) = ?14.5 ± 2.4 kcal mol^?1 has also been deduced. These values are used with the heats of formation of the parent compounds from the literature to evaluate C? H and C? X bond dissociation energies in CH₃Cl, CH₂Cl₂, CH₃Br, CH₂Br₂, CH₂FCl, and CHFCl₂.     

Darstellung,Kristallstrukturen, Schwingungsspektren und Normalkoordinatenanalysen der mer‐Trihalogeno‐tris‐Pyridin‐Osmium(III)Komplexe mer‐[OsX3Py3], X = Cl,Br, I

Synthesis, Crystal Structures, Vibrational Spectra, and Normal Coordinate Analyses of the mer ‐Trihalogeno‐tris‐Pyridine‐Osmium(III) Complexes mer‐[OsX₃Py₃], X = Cl, Br, I By reaction of the hexahalogenoosmates(IV) with pyridine and iso‐amylalcohol mer‐trihalogeno‐tris‐pyridine‐osmium(III) complexes are formed and purified by chromatography. X‐ray structure determinations on single crystals have been performed of mer‐[OsBr₃Py₃] (monoclinic, space group P2₁/n, a = 9.098(5), b = 12.864(5), c = 15.632(5) Å, β = 90.216(5)°, Z = 4) and mer‐[OsI₃Py₃] (monoclinic, space group P2₁/n, a = 9.0952(17), b = 13.461(4), c = 15.891(10), β = 91.569(5)°, Z = 4). The pyridine rings are twisted propeller‐like against the N₃ meridional plane with mean angles of 49° (Cl), 46° (Br), 44° (I). Based on the molecular parameters of the X‐ray structure determinations and assuming C₂ point symmetry, the IR and Raman spectra are assigned by normal coordinate analysis. Due to the stronger trans influence of pyridine as compared with the halide ligands for N'–Os–X ^· axes significantly different valence force constants are observed in comparison with symmetrically coordinated octahedron axes: f_d(OsCl) = 1.74, f_d(OsCl^·) = 1.49, f_d(OsBr) = 1.43, f_d(OsBr ^· ) = 1.18, f_d(OsI) = 0.99, f_d(OsI ^· ) = 0.96, f_d(OsN) between 1.96 and 2.07 and f_d(OsN') between 2.13 and 2.32 mdyn/Å.     

Theoretical study and rate constants calculation for the reactions X + CF3CH2OCF3 (X = F,Cl, Br)

The multiple‐channel reactions X + CF₃CH₂OCF₃ (X = F, Cl, Br) are theoretically investigated. The minimum energy paths (MEP) are calculated at the MP2/6‐31+G(d,p) level, and energetic information is further refined by the MC‐QCISD (single‐point) method. The rate constants for major reaction channels are calculated by canonical variational transition state theory (CVT) with small‐curvature tunneling (SCT) correction over the temperature range 200–2000 K. The theoretical three‐parameter expressions for the three channels k_1a(T) = 1.24 × 10^?15T^1.24exp(?304.81/T), k_2a(T) = 7.27 × 10^?15T^0.37exp(?630.69/T), and k_3a(T) = 2.84 × 10^?19T^2.51 exp(?2725.17/T) cm³ molecule^?1 s^?1 are given. Our calculations indicate that hydrogen abstraction channel is only feasible channel due to the smaller barrier height among five channels considered. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2012     

Hydroxyl radical reactions with halogenated ethanols in aqueous solution: Kinetics and thermochemistry

Laser flash photolysis combined with competition kinetics with SCN^? as the reference substance has been used to determine the rate constants of OH radicals with three fluorinated and three chlorinated ethanols in water as a function of temperature. The following Arrhenius expressions have been obtained for the reactions of OH radicals with (1) 2‐fluoroethanol, k₁(T) = (5.7 ± 0.8) × 10¹¹ exp((?2047 ± 1202)/T) M^?1 s^?1, (2) 2,2‐difluoroethanol, k₂(T) = (4.5 ± 0.5) × 10⁹ exp((?855 ± 796)/T) M^?1 s^?1, (3) 2,2,2‐trifluoroethanol, k₃(T) = (2.0 ± 0.1) × 10¹¹ exp((?2400 ± 790)/T) M^?1 s^?1, (4) 2‐chloroethanol, k₄(T) = (3.0 ± 0.2) × 10¹⁰ exp((?1067 ± 440)/T) M^?1 s^?1, (5) 2, 2‐dichloroethanol, k₅(T) = (2.1 ± 0.2) × 10¹⁰ exp((?1179 ± 517)/T) M^?1 s^?1, and (6) 2,2,2‐trichloroethanol, k₆(T) = (1.6 ± 0.1) × 10¹⁰ exp((?1237 ± 550)/T) M^?1 s^?1. All experiments were carried out at temperatures between 288 and 328 K and at pH = 5.5–6.5. This set of compounds has been chosen for a detailed study because of their possible environmental impact as alternatives to chlorofluorocarbon and hydrogen‐containing chlorofluorocarbon compounds in the case of the fluorinated alcohols and due to the demonstrated toxicity when chlorinated alcohols are considered. The observed rate constants and derived activation energies of the reactions are correlated with the corresponding bond dissociation energy (BDE) and ionization potential (IP), where the BDEs and IPs of the chlorinated ethanols have been calculated using quantum mechanical calculations. The errors stated in this study are statistical errors for a confidence interval of 95%. © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 40: 174–188, 2008     

The use of single exponential orbitals in one-center molecular orbital calculations

Single-exponential Slater type orbitals of the form ψ₁ = (1 + L₁(r, θ) + L₂(r, θ) +…?+ L_n(r, θ)) exp (? αr) are examined for their potential use as one-center molecular orbitals. These are then to be used as molecular fragments in a LCMO study. The system examined is HeH^{+ +} with calculated energies and dipole moments being compared to the exact values. These functions behave best in the region of chemical interest (the bonding region) and thus demonstrate a possible usefulness in LCMO calculations and in the field of one-electron diatomics.     

Laser magnetic resonance spectroscopy of ClO and kinetic studies of the reactions of ClO with NO and NO2

Far-infrared rotational transitions in ClO(X²∏_3/2, υ = 0) have been observed using laser magnetic resonance (LMR) with an optically pumped spectrometer. Five observed transitions at wavelengths between 444 and 713 µm have been compared with values predicted with spectroscopic constants from the literature. LMR detection of ClO has been used to study its reactions with NO and NO₂ in a discharge flow system under pseudo-first-order conditions for ClO. The measured rate constants are k(ClO + NO) = (7.1 ± 1.4) × 10^?12 exp[(270 ± 50)/T] cm³/molec·s for the temperature range of 202 < T < 393 K; k(ClO + NO₂ + M) = (2.8 ± 0.6) × 10^?33 exp[(1090 ± 80)/T] cm⁶/molec²·s (M = He, 250 < T < 387 K), (3.5 ± 0.6) × 10^?33 exp[(1180 ± 80)/T] (M = O₂, 250 < T < 416 K), and (2.09 ± 0.3) × 10^?31 (M = N₂, T = 297 K). All measurements were made at low pressures, between 0.6 and 6.6 torr. These results are compared with those from other studies.     

Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3ⁿ redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θ_pqr are given by introducing Cartesian tensor rotation matrix elements X( R ). A polynomial expression and a recursion relation for X( R ) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D( R ). The expressions for X( R ) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.     

Effect of counterions on photoprocesses of thiacarbocyanine in a binary solvent blend

A model describing the effect of counterion X⁻ (X = Cl, I) on the deactivation kinetics of the S ₁ state of thiacarbocyanine Cy⁺X⁻ is presented. According to the model, the ion pair Cy⁺X⁻ in a binary solution is characterized by a distribution function f(r) over interatomic distances r, which depends on the composition of the mixture. The assumption of kinetically independent local states of the ion pair, which decay with the rate constants k _i(r)(i = 1–4 is the index of the decay channel), is made. The statistic analysis of the experimental data in terms of the model permitted us to find the functions f(r) and to estimate the parameters of the constants k _i(r).     

Pyrolysis of vinylacetylene between 300 and 450°C

Vinylacetylene was pyrolyzed at 300–450°C in a packed and an unpacked static reactor with a pinhole bleed to a quadrupole mass spectrometer. The reactant and C₈H₈ products were monitored continuously during a reaction by mass spectrometry. In some runs, the products were also analyzed by gas chromatography after the run. In these runs CH₄, C₂H₆, C₃H₆, and C₂H₄ were also detected. The reaction for vinylacetylene removal and C₈H₈ formation is homogeneous, second order in reactant, and independent of the presence of a large excess of N₂ or He. However, C₈H₈ formation is about half-suppressed by the addition of the free-radical scavengers NO or O₂. The rate coefficient for total vinylacetylene removal is 1.7 × 10⁶ exp(?79 ± 13 kJ/mol RT) L/mol · s. The major reaction for C₄H₄ removal is polymerization. In addition four C₈H₈ isomers, carbon, and small hydrocarbons are formed. The three major C₈H₈ isomers are styrene, cyclooctatetraene (COT), and 1,5? dihydropentalene (DHP). The C₈H₈ compounds are formed by both molecular and free-radical processes in a second-order process with an overall k ? 3 × 10⁸ exp(?122 kJ/mol RT) L/mol · s (average of packed and unpacked cell results). The molecular process occurs with an overall k = 8.5 × 10⁷ exp (?118 kJ/mol RT) L/mol · s. The COT, DHP, and an unidentified isomer (d), are formed exclusively in molecular processes with respective rate coefficients of 4.4 × 10⁴ exp(?77 kJ/mol RT), 1.7 × 10⁵ exp(?89 kJ/mol RT), and 3.1 × 10⁹ exp(? 148 kJ/mol RT) L/mol · s. The styrene is formed both by a direct free-radical process and by isomerization of COT.