Anharmonic Morse Oscillator Research Articles

Infrared Spectral Assignment of Pyrimidine and Pyrazine in the C[sbnd]H Stretching Region by an Effective Spectroscopic Hamiltonian

Mid-Infrared spectra of pyrimidine (PM) and pyrazine (PZ) were recorded in the gas phase using a multi-pass long path gas cell. The IR band structure of these compounds above and below 3000 cm−1 is very broad and contains many humps and shoulders. These humps and shoulders are due to various higher quantum excitation of low-frequency vibrational modes, which participate in Fermi resonance with the nearby CH stretch fundamentals and appears in this region. We constructed an Effective Spectroscopic Hamiltonian (ESH) in a mixed local mode (LM) normal mode (NM) basis to assign the various overtone and combination bands in the CH stretching region of these compounds. The CH stretching vibrations of both PZ and PM were treated as symmetrized anharmonic Morse oscillators in local coordinates and the in-plane deformations down to 1000 cm−1 were treated as normal coordinates. The ESHs were diagonalized and the resulting eigenvalues were subsequently fitted in a given parameter space with the experimentally observed bands. The eigenvalues of the converged Hamiltonian are the anharmonic frequencies and the transition intensities were obtained by summing the squared eigenvector components. The overtone and combination transitions near 3000 cm−1 of both PM and PZ were identified and assigned from the eigenvector coefficients of the ESH matrix. The wavefunctions of a pure CH stretch, overtone of the HCC in-plane bend and due to Type 1 Fermi coupling (resonance between a fundamental with an overtones of a low frequency mode, in this case resonance between the CH strech and the overtone of HCC in-plane bending modes) has been demonstrated pictorially.

Fully and Partially Coherent Pathways in Multiply Enhanced Odd-Order Wave-Mixing Spectroscopy

Nuclear magnetic resonance spectroscopy relies on using multiple excitation pulses to create multiple quantum coherences that provide great specificity for chemical measurements. Coherent multidimensional spectroscopy (CMDS) is the optical analogue of NMR. Current CMDS methods use three excitation pulses and phase matching to create zero, single, and double quantum coherences. In order to create higher order multiple quantum coherences, the number of interactions must be increased by raising the excitation intensities high enough to create Rabi frequencies that are comparable to the dephasing rates of vibrational coherences. The higher Rabi frequencies create multiple, odd-order coherence pathways. The coherence pathways that involve intermediate populations are partially coherent and are sensitive to population relaxation effects. Pathways that are fully coherent involve only coherences and measure the direct coupling between excited quantum states. The fully coherent pathways are related to the multiple quantum coherences created in multiple pulse NMR methods such as heteronuclear multiple quantum coherence (HMQC) spectroscopy with the important difference that HMQC NMR methods have a defined number of interactions and avoid dynamic Stark effects whereas the multiply enhanced odd-order wave-mixing pathways do not. The difference arises because CMDS methods use phase matching to define the interactions and at high intensities, multiple pathways obey the same phase matching conditions. The multiple pathways correspond to the pathways created by dynamic Stark effects. This paper uses rhodium dicarbonyl chelate (RDC) as a model to demonstrate the characteristics of multiply enhanced odd-order wave-mixing (MEOW) methods. Dynamic Stark effects excite vibrational ladders on the symmetric and asymmetric CO stretch modes and create a series of multiple quantum coherences and populations using partially and fully coherent pathways. Vibrational quantum states up to v = 6 are excited. A series of spectra provides different two-dimensional cross sections through the multidimensional parameter space involving two excitation frequencies, the frequency of the output coherence, and the excitation pulse time delays. The spectra allow the identification of 18 different overtone and combination band states. Comparison with a local mode model with two anharmonic Morse oscillators with interbond coupling shows excellent agreement.

Vibrational overtones quenching of near infrared emission in Er3+ complexes

Erbium organic complexes are receiving increasing attention in view of their application in polymeric telecommunication devices, but their use is limited by small emission quantum yield. Non-radiative deactivation of near IR (NIR) transition in Er3+ organic complexes is discussed on the basis of the electronic–vibrational energy transfer. Relevant transition matrix elements necessary to predict quenching effects exerted by various bonds located in the Er3+ coordination sphere are evaluated on the basis of an anharmonic Morse oscillator model and expressed as analytical functions of the fundamental vibrational intensities. The latter are calculated on the basis of various ‘ab initio’ quantum chemical methods, which yield intensity values close to the experimental measurements. Quenching effects in complexes containing a number of common ligands are evaluated and discussed, and a strategy to design highly efficient NIR emitters is proposed.

Quantum and electromagnetic propagation with the conjugate symmetric Lanczos method

The conjugate symmetric Lanczos (CSL) method is introduced for the solution of the time-dependent Schrodinger equation. This remarkably simple and efficient time-domain algorithm is a low-order polynomial expansion of the quantum propagator for time-independent Hamiltonians and derives from the time-reversal symmetry of the Schrodinger equation. The CSL algorithm gives forward solutions by simply complex conjugating backward polynomial expansion coefficients. Interestingly, the expansion coefficients are the same for each uniform time step, a fact that is only spoiled by basis incompleteness and finite precision. This is true for the Krylov basis and, with further investigation, is also found to be true for the Lanczos basis, important for efficient orthogonal projection-based algorithms. The CSL method errors roughly track those of the short iterative Lanczos method while requiring fewer matrix-vector products than the Chebyshev method. With the CSL method, only a few vectors need to be stored at a time, there is no need to estimate the Hamiltonian spectral range, and only matrix-vector and vector-vector products are required. Applications using localized wavelet bases are made to harmonic oscillator and anharmonic Morse oscillator systems as well as electrodynamic pulse propagation using the Hamiltonian form of Maxwell's equations. For gold with a Drude dielectric function, the latter is non-Hermitian, requiring consideration of corrections to the CSL algorithm.

Analysis of Nonequilibrium Dissociation Kinetics in a Boundary Layer Using Various Reaction Models

Values of the nonequilibrium macroscopic reaction rate for a nonisothermal boundary layer of a monatomic diluent gas are calculated using a number of models for thermal dissociation of diatomic molecules — anharmonic Morse oscillators. Analysis is performed for conditions where the diffusive transfer of excited molecules has a significant effect on the population of their upper vibrational levels, which does not only amount to change in vibrational temperature. Under the joint influence of diffusive transfer of molecules, vibrational exchanges, and reactions involving vibrationally excited particles, the local vibrational distribution functions are substantially nonequilibrium. The kinetic models considered take into account the possible contribution of the energy of molecular translational and rotational degrees of freedom to the energy required to overcome the reaction threshold. The effect of multiquantum vibrational—translational exchanges on the distribution of dissociating molecules in their upper vibrational levels is taken into account approximately.

Anharmonic stretching vibrations expressed as local modes

Various models used to interpret X-H stretch (or X-D or other weakly coupled bond modes) overtone spectra in terms of local modes are considered. For general systems of coupled X-H stretching modes we demonstrate systematically the origin of local mode anharmonic coupling terms (in a block-diagonal effective Hamiltonian) via the application of perturbation theory to vibrational Hamiltonians expressed in internal (local) coordinates and which include anharmonic effects as well as quadratic interbond coupling. Two approaches are presented: extension of the approximate anharmonically coupled anharmonic (Morse) oscillator (AACAO) model and a perturbation theory treatment of a generalized local mode model Hamiltonian. These results are tested by application of the derived formulae to model potentials. X-H stretching vibrational energy levels are calculated for ammonia and silane and in the latter case compared with experimental data. Local mode anharmonicity constants are calculated for water and methyl bromid...

Anharmonic stretching vibrations expressed as local modes

Various models used to interpret X-H stretch (or X-D or other weakly coupled bond modes) overtone spectra in terms of local modes are considered. For general systems of coupled X-H stretching modes we demonstrate systematically the origin of local mode anharmonic coupling terms (in a block-diagonal effective Hamiltonian) via the application of perturbation theory to vibrational Hamiltonians expressed in internal (local) coordinates and which include anharmonic effects as well as quadratic interbond coupling. Two approaches are presented: extension of the approximate anharmonically coupled anharmonic (Morse) oscillator (AACAO) model and a perturbation theory treatment of a generalized local mode model Hamiltonian. These results are tested by application of the derived formulae to model potentials. X-H stretching vibrational energy levels are calculated for ammonia and silane and in the latter case compared with experimental data. Local mode anharmonicity constants are calculated for water and methyl bromide and two of its deuterated isotopomers. Comparisons with the harmonically coupled anharmonic oscillator (HCAO) model show that serious systematic errors in the latter may be corrected whilst retaining a relatively simple effective Hamiltonian. The anomalously low magnitude of the HD harmonic interbond (near-resonant) coupling constant determined in an earlier study of CHD2Cl is rationalized by analysis of previously neglected anharmonic effects. This last application demonstrates the relevance of these local mode anharmonic couplings to systems in which Fermi resonances (with non-stretching modes) are also important. Applications to CH3F and CH2D2 are also briefly discussed.

The relationship between the models of anharmonically and harmonically coupled anharmonic oscillators

The relationship between the model of anharmonically coupled anharmonic (Morse) oscillators (ACAO) and the model of harmonically coupled anharmonic oscillators (HCAO) used to interpret X-H stretch overtone spectra in terms of local modes is examined by means of a non-linear co-ordinate transformation described recently by Cooper (1987, Chem. Phys., 112, 67). The transformation allows the ACAO model hamiltonian to be rewritten in an harmonic oscillator basis. Van Vleck perturbation theory may then be applied to further transform the hamiltonian to block diagonal form. With the hamiltonian treated to second order in the perturbation parameter β [=(-2xm /ω m )1/2, where ωm is the harmonic wavenumber of the bond oscillation and xm is the Morse anharmonicity], the eigenvalues of the original ACAO hamiltonian can be reproduced to within a few tenths of a cm-1 at low vibrational levels and a few cm-1 at high vibrational levels, as indicated by calculations on model H2O. The perturbation treatment reveals importa...

Representation of one-dimensional motion in a morse potential by a quadratic hamiltonian

A representation of the algebraic hamiltonian for the anharmonic Morse oscillator as a quadratic form, H = T1ω( 1 2 P 2 + 1 2 Q 2), where P and Q are operators is derived. The commutator of P and Q is an operator that tends to i (times the identity operator) in the harmonic limit. Coherent states and anharmonic normal modes for a linear triatomic molecule are discussed as potential applications.

Uncertainty products for the anharmonic morse oscillator

Uncertainty products for the anharmonic morse oscillator

Temperature dependence of the elastic shear moduli of the cubic metals

A theoretical model for the temperature dependence of the elastic shear moduli of the cubic metals is presented which accounts for the quantitative relationship of these elastic properties to the thermal expansion. The particular phenomenon which the model is concerned with is the extremely rapid temperature variation of the shear moduli, which typically is about 25 times as great as the rate of thermal expansion. The model has the following salient features: (i) The interatomic pair pseudopotential is approximated over the important thermal range by means of a Morse potential; (ii) the thermal expansion is assumed to arise from anharmonic Morse oscillators which are further assumed not to be significantly coupled to each other; (iii) the exact quantum-mechanical solutions for the Morse oscillators are then combined with the Debye model to give an analytic formula for the thermal expansion; (iv) the elastic shear moduli are calculated from this, assuming nearest-neighbor interactions only. The calculated thermal expansions are found to agree quite well with the experimental values, particularly at low temperatures. The rapid temperature variation of the shear moduli is explained by the model in terms of the thermal movement of the nearest neighbors with respect to the interatomic-potential minima. Complete agreement between calculated and experimental shear moduli has been limited, however, by the contributions beyond the nearest neighbors which are not zero, and which have not been included in the formulas.

Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator

The response of a one-dimensional anharmonic Morse oscillator to an intense electromagnetic field has been investigated using both a quasiclassical and quantum mechanical description of the oscillator. The anharmonic nature of the Morse potential reduces the coherence of the quantum excitation process after only a few quanta have been absorbed. The classical and quantum behavior of averaged quantities such as the energy absorbed and the oscillator displacement as a function of time are in good agreement; however, the classical description cannot reproduce the multiphoton resonances. We are led to the conclusion that classical mechanics provides an adequate description of the response of a molecule in an intense laser field provided that multiphoton resonances do not individually play a fundamental role in the process.

Relaxation Rates for Vibrational De-Excitation of Anharmonic Morse Oscillators

To explain an increase of the vibration-translation (V-T) relaxation rate in an expansion flow experiment, the vibrational relaxation of a system of anharmonic Morse oscillators in an isothermal heat bath is studied through an examination of the relaxation manner of the vibration-vibration quasi-steady distribution. It is concluded that an increase of the V-T rate less than a factor of about 2 can be explained as resulting from effects of anharmonicity.