Harmonic Oscillator Approximation Research Articles

Observation of the Rabi-resonance spectrum.

By use of a double-resonance technique, a spectrum of Rabi resonances has been observed. These resonances are enhancements in the dynamic response of a quantum system's population to rapid changes in the phase of a perturbing electromagnetic field. Theoretically, it is shown that the appearance of a spectrum, as opposed to a single Rabi resonance, is due to the fact that the phase of the field changes discretely by \ensuremath{\pi} radians, rather than sinusoidally. In the presence of \ensuremath{\pi}-radian phase changes, the quantum system's dynamic response is found to be similar to a damped harmonic oscillator driven by a \ensuremath{\delta}-function force term. In this harmonic-oscillator approximation the Rabi-resonance spectrum arises naturally, resulting from a resonant enhancement of the atomic coherence.

Decomposition of methylamino and aminomethyl radicals. The heats of formation of methyleneimine (CH2NH) and hydrazyl (N2H3) radical

AbstractA kinetic analysis of the thermal decomposition of methylamino and aminomethyl radicals into methyleneimine, reactions (1) and (2): equation image leads to ΔH(CH2NH) = 25.0 ± 3 kcal/mol in excellent agreement with ion cyclotron resonance spectroscopy measurements and to a pi bond energy of Eπ = 55.0 kcal/mol in CH2NH which is comparable but smaller than to the corresponding value in CH2CH2 (63.7 kcal/mol). Assuming that Eπ(CH2NH) = 0.5 [Eπ(CH2CH2) + Eπ(NHNH)] then requires that Eπ(NHNH) = 46.8 kcal/mol in diimine and BDE(N2H3‐H) = 87.5 kcal/mol i.e. about 11.5 kcal/mol larger than current data for hydrazine but otherwise consistent with additional evidence.The entropy and heat capacity of methyleneimine, calculated from recent infrared and microwave spectroscopic data using the rigid rotor harmonic oscillator approximation, are also reported.

Structure dependent regularities of zero-point plus heat content energies in organic molecules

A representative collection of heat content plus zero-point energies deduced in the harmonic oscillator approximation suggests a number of simple additivity rules which allow ZPE + (HT − H0) to be related to structural features. Accurate evaluations are obtained for alkanes, alkenes, alkynes, cycloalkanes, chloroalkanes, and amines, as well as for large and small cycloalkanes in which CH2 is replaced by O.

A theoretical analysis of the Jahn–Teller effect in the photoelectron spectrum of methane

An analysis of the Jahn–Teller effect in the photoelectron spectrum of methane was performed by the use of the ab initio self-consistent field and configuration interaction calculations. The molecular structures of the CH+4 ion, which have a local energy minimum, are found to be in the C2v and C3v symmetry. The vertical excitation energy is calculated to be 14.23 eV and the adiabatic ionization potentials of the CH+4 ion including zero-point vibrational energy are estimated to be 12.25 and 12.68 eV for the C2v and C3v symmetry, respectively. We calculated the normal modes of vibration and the Franck–Condon factor in the harmonic oscillator approximation, and assigned the vibrational band below 13.10 eV of the CH+4 ion.

Isotopic Product Rule for Solid State Lithium Oxalate

Abstract The Teller-Redlich type isotopic product rule was applied to solid state anhydrous lithium oxalate, 6Li2C2O4, and 7Li2C2O4 By assuming accidental degeneracies of some observed broad bands, the fundamental wavenumbers satisfactorily reproduced the isotopic product ratios calculated, even for the harmonic oscillator approximation.

Fourier transform photoelectron spectroscopy: The correlation function and the harmonic oscillator approximation

The correlation function describes the time development of the wave packet placed by photoabsorption or photoionization onto the potential surface of an upper electronic state. The function can be obtained as a Fourier transform of the electronic band, and gives information about the features of the final state. The analytical expressions for the correlation function within the harmonic oscillator approximation are presented. Because of some unique properties of the correlation function, the expressions can be used to obtain accurate geometric details of the final state from experimental data. The approach is tested on some photoelectron spectra of diatomics and compared to known data. The method yields the equilibrium internuclear distance with an accuracy of ±0.0025 Å, and resolves the sign uncertainty present in the conventional harmonic Franck–Condon analysis. The comparison of the experimental data with the predictions of the harmonic model gives a deeper insight into the behavior of a wave packet in an anharmonic potential.

Polarisability of finite crystals: one-dimensional model

A simple one-dimensional model is used for electronic polarisability calculations. First-principles results are compared on the one hand with variational many-body molecular-like ones, and on the other with the harmonic oscillator approximation. Limitations of applicability of the latter two methods are evaluated.

Comparative analysis and search for regularities in the behavior of Franck-Condon factors for diatomic molecules

A study has been carried out of the reliability of Franck-Condon factors (FCFs) obtained in the simple harmonic oscillator (SHO) approximation of Nicholls. We have made a quantitative statistical comparison of FCFs for > 300 molecular transitions, obtained in the SHO approximation, with values calculated for the same transitions using the most accurate approximations with Rydberg-Klein-Rees or Morse potentials. The behavior of FCFs has been studied by using the SHO model in all groups of isovalent and isoelectronic diatomic molecules, and variations of the FCFs in these groups are shown to have a smooth character. The results allow reasonable estimation of FCFs for isovalent or isoelectronic molecules if accurate calculations are available for at least one of the molecular systems.

Thermodynamic Properties of Twenty-One Monocyclic Hydrocarbons

The available structural parameters, fundamental frequencies, and relative energies of different stable conformers, if any, for cyclopropane, cyclopropene, cyclobutane, cyclobutene, 1,3-cyclobutadiene, cyclopentane, cyclopentene, 1,3-cyclopentadiene, cyclohexane, cyclohexene, 1,3-cyclohexadiene, 1,4-cyclohexadiene, cycloheptane, cycloheptene, 1,3-cycloheptadiene, 1,3,5-cycloheptatriene, cyclooctane, cyclooctene, 1,3-cyclooctadiene, 1,5-cyclooctadiene, and 1,3,5,7-cyclooctatetraene were critically evaluated and the recommended values selected. Molecular constants for some molecules were estimated as the experimental values for these compounds are not available. This information was utilized to calculate the ideal gas thermodynamic properties C○p, S○, −(G○−H○0)/T, H○−H○0, and log Kf from 100 to 1500 K. The thermal functions were obtained using the rigid-rotor harmonic-oscillator approximation. The contributions derived for the inversion motion of cyclobutane and cyclopentene were obtained from energy levels calculated with the potential functions. For cyclopentane the pseudorotational contributions to thermal functions were calculated by assuming the pseudorotation as the free rotation of the molecule. The calculated values of the thermal functions are compared with those reported in other work. Agreement with experimental data, where such are available, is satisfactory within the experimental uncertainties.

Quantum vibrational chaos in the ArHCl van der Waals molecule

Quantum mechanical ro-vibrational calculations are presented for the ArHCl van der Waals complex using Hutson and Howard's empirical M5 potential. Analysis of the nodal structures, second differences and overlapping avoided crossings suggests that the higher bound states of ArHCl are chaotic. This chaos is made evident by perturbing the angular part of the kinetic energy term, e.g. by isotopic substitution or vibrational excitation of HCl. Calculations with J > 0 show increased level crossings but no significant increase in chaos because of an extra nearly good quantum number. The low-lying states of ArHCl appear to be poorly modelled by either the harmonic oscillator or free rotor approximations. Comparison with other molecules for which vibrational chaos has been predicted are made.

Competitive forms of symmetry breaking in linear antiferromagnetic systems

Two different forms of symmetry breaking are considered for linear antiferromagnetic systems (S = 1/2 ). Their relative stability is examined by considering small fluctuations in the harmonic oscillator approximation. Imaginary frequencies correspond with an unstable phase, and the ground state represents an absolute minimum of the total energy, including contributions from the zero-point fluctuations.

Calculating normal modes of vibration for a molecule modeled in the harmonic oscillator approximation

We propose here a simple procedure for calculating free vibrational modes and vibrational frequencies for any molecule.

Generalized Langevin theory for many-body problems in chemical dynamics: Formulation for molecular solvents, equilibrium aspects

This paper deals with equilibrium statistical mechanical aspects of the extension of the MTGLE approach for chemical reaction dynamics in liquids [S. A. Adelman, Adv. Chem. Phys. 53, 611 (1983)] so that it is applicable to molecular as well as monatomic solvents. This extension is necessary in order to conveniently treat energy exchange between the solute molecules and the solvent vibrational degrees of freedom. The analysis yields a separation of equilibrium solvent density fluctuations into translation-rotational (TR) and vibrational (V) contributions. The TR fluctuations are treated within the rigid solvent molecule model. The V fluctuations are treated within the harmonic oscillator approximation. The analysis is carried out in terms of a set of generalized solvent phase space coordinates S=pvvpww where v and w are, respectively, the V and TR coordinates and where pv and pw are the corresponding conjugate momenta. In this coordinate system, canonical ensemble distribution function for the solvent given that the solute is fixed at configuration point r0, denoted by fCA[S;r0], may be factorized as fCA[S;r0] =fCA[ pvv] fCA[ pw w;v0r0] where fCA[ pww;v0r0] and fCA[pvv] are respectively: the rigid solvent model approximation to fCA[S;r0]; a vibrational phase space probability distribution function which governs equilibrium mean field V fluctuations in the pure solvent. This vibrational probability distribution function may be written as fCA[ pvv] =Z−1 exp[−β(Tv+W(v))] where Z=pure solvent partition function, where Tv =pure solvent kinetic energy, and where W[v] is a vibrational potential of mean force. The above factorization of fCA[S;r0] holds if: (i) Coriolis coupling between the v and w coordinates is ignored; (ii) The solvent vibrational frequencies are sufficiently high. Within the harmonic approximation to W[v], fCA[ pvv] =[2πkβT]−rNs det ω2 exp [−β(Tv+ 1/2 yTω2y)] where y are the mass-weighted displacements of the v coordinates from equilibrium, where rNs =number of solvent normal modes, and where ω2 is an (rNs×rNs) -dimensional dynamical matrix which determines the vibrational frequency spectrum of the pure solvent within the harmonic approximation. This dynamical matrix may be constructed from the canonical ensemble distribution function of the pure rigid molecule solvent. Thus given the harmonic approximation to W[v] the canonical ensemble distribution function of the nonrigid solvent fCA[S;r0] may be constructed from the equilibrium properties of the rigid molecule solvent.

Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation

Dielectric functions of partly amorphized GaAs layers produced by deep ion implantation of different doses of 270-keV As+ ions in crystalline (c-) GaAs have been measured from 1.5 to 6.0 eV by spectroscopic ellipsometry. We show that these dielectric functions cannot be described as physical mixtures of amorphous (a-) and c-GaAs, as such models exhibit strong deviations with respect to the data near 3 eV. We determine representations of these dielectric functions as finite sums of harmonic oscillators, which allows us to describe these spectra as analytic functions of a single parameter related to the amount of damage. In this harmonic oscillator approximation (HOA), we show that damage profiles of ion-implanted material can be nondestructively determined from ellipsometric spectra in terms of multilayer models. In a representative case, good agreement is found between damage profiles determined nondestructively in the HOA and destructively by chemical etching.

Nonadiabatic processes involving quantum-like and classical-like coordinates with applications to nonadiabatic electron transfers

Nonadiabatic processes may involve both classical-like and quantum-like coordinates. A semiclassical analysis is used to treat the contribution of the former to the Franck–Condon factor in the reaction rate expression, thereby avoiding the usual harmonic oscillator approximation. Microcanonical and canonical rate constants are calculated, yielding an expression which includes contributions from both types of coordinates. The results are applied to nonadiabatic electron transfer reactions in solution, and show how ΔG0 enters the final rate expression, even though ΔE0 is present in the initial Golden Rule nonadiabatic formula. This result avoids an approximation which has arisen in the nonadiabatic electron transfer literature.

Hydrogen isotope exchange between fluoroform and water. 5. Equilibrium deuterium distribution. The temperature dependence of .alpha.

A knowledge of the equilibrium deuterium distribution between water and fluoroform is useful for design of the redeuteration stage of a heavy water production process based on IR laser-activated decomposition of CF/sub 3/D in excess CF/sub 3/H. The first measured fractionation factor data are presented for the temperature range 70-130 /sup 0/C. The deuterium is favored in the fluoroform (..cap alpha.. = 1.139 at 130 /sup 0/C, 1.215 at 70 /sup 0/C, and 1.287 at 25 /sup 0/C (extrapolated)). These values are approximately 10-20% higher than various reported theoretical estimates calculated by using isotopic reduced partition function ratios based on molecular vibrational frequencies (with harmonic oscillator approximation). ..delta..H/sup 0/ = -1.23 kJ/mol was calculated from the temperature dependence of ln ..cap alpha...

Calculation of thermodynamic functions of halogen derivatives of silane and germane

Molar heat capacity, absolute entropy, relative enthalpy and relative Gibbs energy were calculated for all halogen derivatives of silane and germane. The calculation was performed for the ideal gas state at atmospheric pressure in the temperature range 100 to 1 500 K in the usual rigid rotator - harmonic oscillator approximation. In the case of molecules where the necessary experimental values of fundamental vibrational frequencies were missing, the calculation was performed using the predicted frequencies. On the basis of the values of molar heat capacities obtained in both these ways, a critical examination of the influence of the inaccuracy of the predicted vibration frequencies on the accuracy of calculated ideal gas thermodynamic functions was made.

Recoupling matrix elements and decay

Recoupling matrix elements are evaluated in the harmonic oscillator approximation for all possible angular and radial excitations in processes where quarks recombine. A diagrammatic representation is given. Their use is demonstrated in calculating the transition potential for ρ→2π in a pair creation model.

Resonance raman studies of models for the reduced states of cytochrome P 450

The catalytic cycle of cytochrome P 450 includes four stable states: a low spin ferric resting state A, a substrate bound high spin ferric state B, a high spin ferrous state C and a low spin ferrous oxy state D. When the system in the state C is exposed to carbon monoxide instead of oxygen a low spin ferrous carboxy state D′ is generated. Several models have been proposed for states B, C, D′ and recently state D [1]. A large variety of spectroscopic techniques have been used to probe the similarities between the model compounds and the actual enzymatic states. Resonance Raman spectroscopy (RR) is a very sensitive technique to investigate specifically the active site of hemoproteins. RR spectra have been obtained for Cyt P 450 from various origins [2–4]: when compared to those of other hemoproteins, the frequencies of the so-called ‘oxidation state’ marker band [5] are unusually low in states C and D′; ‘spin marker bands’ frequencies [5] have been used to monitor the coordination of the iron atom in the ferric states A and B. Recently the FeS stretching mode has been detected at 351 cm −1 in oxidized Cyt P 450 CAM [6]. We report here the results of a RR study of [Fe II(T pivPP)(X −)(L)]18C6Na +* complexes, as models for states C, D′ and D (an investigation of a state A model has already been published [7]). Using the nomenclature propsed in [8], the main RR frequencies are given in Table 1. t001 RR Frequencies (cm −1) of [Fe II(T pivPP)(X −)(L)]Na +18C6 Complexes. Complex Porphyrin vibration FeL vibr. X − L A B C D C 6HF 4S − 1341 a a a 369 Cl − 1343 1355 1494 1545 369 OH − 1344 1355 a a 371 C 6HF 4O − 1343 1354 a 1545 369 C 6HF 4S − CO 1364 a 1567 380 479 C 6HF 4S − O 2 1366 a a 379 a Not observed All the pentacoordinated ferrous species exhibit very similar porphyrinic frequencies. They compare well with the frequencies of the typical high spin ferrous complex Fe(TPP)(2-Me Im) (A = 1345, B = 1361, C = 1500 and D = 1538 [8]). Moreover the A frequency of the carboxy adduct is very close to that of Fe(TPP)(py)(CO) [12], whereas that of the oxy adduct is the same as that of Fe(T pivPP)(1-Me Im)(O 2) [9]. Therefore our RR data do not stress any special π donor properties of the RS − ligand that would induce an extra lowering of the oxidation state marker band frequencies. Soret excitation of the low frequency RR spectrum is readily accessible for the carboxy adduct λ max Soret 448 nm, λ exc 454,5 nm: it reveals a new strong polarized band at 479 cm −1. The intensity of this band decreases with partial photodissociation of the CO ligand. An isotopic substitution experiment, using 13CO, induces a 5 cm −1 lowering of its frequency. This is in good agreement with a calculated shift of −5 cm −1 for the stretching vibration of the FeCO moiety, using the harmonic oscillator approximation. This leads to the assignment of this band to the FeCO stretching vibration. This value is to be compared to those observed for MbCO, HbCO [10] and P 450CO (work in progress).

A quantum-mechanical internal-collision model for state-selected unimolecular decomposition

A quantum-mechanical theory for state-selected unimolecular decomposition is developed in terms of the cumulative effect of a succession of internal collisions. An application to the model decomposition of CO2 to CO(1Σ+)+ O(1D) is developed within the restrictions of a forced harmonic oscillator approximation for the internal scattering matrix. Preliminary results over a restricted energy range yield a distribution of lifetimes over two orders of magnitude, 0.6–60 ps, with evidence of a statistically determined behaviour on average but with wide fluctuations and with significant quantum effects on the branching ratios to different product channels.