Anharmonic effect on unimolecular reactions with application to the photodissociation of ethylene

The importance of anharmonic effect on dissociation of molecular systems, especially clusters, has been noted. In this paper, we shall present a theoretical approach that can carry out the first principle calculations of anharmonic canonical and microcanonical rate constants of unimolecular reactions within the framework of transition state theory. In the canonical case, it is essential to calculate the partition function of anharmonic oscillators; for convenience, the Morse oscillator potential will be used for demonstration in this paper. In the microcanical case, which involves the calculation of the total number of states for the activated complex and the density of states for the reactant, we make use of the fact that both the total number of states and the density of states can be expressed in the inverse Laplace transformation of the partition functions and that the inverse Laplace transformation can in turn be carried out by using the saddle-point method. We shall also show that using the theoretical approach presented in this paper the total number of states and density of states can be determined from thermodynamic properties and the difference between the method used in this paper and the thermodynamic model used by Krems and Nordholm will be given. To demonstrate the application of our theoretical approach, we chose the photodissociation of ethylene at 157 and 193 nm as an example.     

Algebraic approximation to the Franck–Condon factors for the Morse oscillator

An algebraic approach is proposed to calculate the Franck–Condon factors for the Morse potential of diatomic molecules. The Morse oscillator is approximated by means of a fourth-order anharmonic oscillator. In the second-quantized formalism, this anharmonic Hamiltonian is diagonalized by way of the Bogoliubov–Tyablikov transformation. The Franck–Condon factors are estimated using the harmonic frequency equivalent and the recurrence relations for the Franck–Condon factors of the harmonic oscillator. Overlap integrals are shown for three band systems and compared with values calculated with an RKR potential. Excellent agreement is achieved.     

Franck–Condon factors for diatomic molecules with anharmonic corrections

Some of the band systems of several astrophysically important molecules are calculated and compared with the results obtained by calculations based on realistic Klein–Dunham and Rydberg–Klein–Rees potential functions. The Morse potential is approximated by means of a fourth-order anharmonic oscillator model. In the second-quantized formalism, the anharmonic Hamiltonian is diagonalized by using the Bogoliubov–Tyablikov transformation. The diagonalization process gives a shift in the frequency associated with each normal mode of harmonic vibration of the molecules presented here. The Franck–Condon factors are estimated using this new frequency within the framework of a harmonic oscillator.     

Anharmonic nuclear dynamics in the mixed quantum-classical limit

This study employs mixed quantum-classical dynamics (MQCD) formalism to evaluate the linear electronic dipole moment time correlation function (DMTCF) in which a Morse oscillator serves to model the associated vibrations in a mixed quantum-classical (MQC) environment. While the main purpose of this work is to study the applicability of MQCD formalism to anharmonic systems in condensed phase, approximate schemes to physically evaluate the mathematically divergent integrals have been developed in order to deal with the essential singularities that arise while evaluating the Morse oscillator canonical partition function and the DMTCF in MQC systems in the classical limit. The motivation for numerically and analytically evaluating these divergent integrals is that a partition function of any system should lead to a finite value at any temperature and therefore this divergence is unphysical. Additionally, since a partition function is to signify the number of accessible states to the system at hand, divergent results are not physically acceptable. As such, straightforward approximate analytic expressions, at different levels of rigor, for both the classical Morse oscillator partition function and the DMTCF in MQC systems are derived, for the first time. Calculations of Morse oscillator partition function values using different approaches at various temperatures for CO, HCl, and I(2) molecules, showing good results, are presented to test the expressions derived herein. It is found that this divergence, due to singularity, diminishes upon lowering the temperature and only arises at high temperatures. The gradual diminishing of the singularity upon lowering the temperature is sensible since the Morse potential fits the parabolic potential at low temperatures. Model calculations and discussion of the DMTCF and linear absorption spectra in MQC systems using the molecular constants of CO molecule are provided. The linear absorption lineshape is derived by two methods, one of which is asymptotic expansion.     

Variational method for calculating the anharmonic vibrations of polyatomic molecules using a combined morse-anharmonic basis taking into account the fragmentary structure of molecules

A method of variational solution of anharmonic vibration problems using a mixed Morse—anharmonic basis is proposed. The basis functions are the products of the Morse oscillator eigenfunctions for vibrations of peripheral bonds, the harmonic oscillator eigenfunctions for almost harmonic skeletal and deformation vibrations, and the anharmonic basis functions for essentially anharmonic skeletal and deformation vibrations. The anharmonic basis wave functions are taken as a linear combination of the Morse and harmonic oscillator eigenfunctions. The introduction of the combined Morse—anharmonic functions allows one to factorize the solution of a problem into a series of individual blocks according to the fragmentary structure of molecules. Volgograd Pedagogical University. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 231–238, March–April, 1995. Translated by I. Izvekova     

The supersymmetric quantum mechanics theory and Darboux transformation for the Morse oscillator with an approximate rotational term

Anharmonic potentials with a rotational terms are widely used in quantum chemistry of diatomic systems, since they include the influence of centrifugal force on motions of atomic nuclei. For the first time the Taylor-expanded renormalized Morse oscillator is studied within the framework of supersymmetric quantum mechanics theory. The mathematical formalism of supersymmetric quantum mechanics and the Darboux transformation are used to determine the bound states for the Morse anharmonic oscillator with an approximate rotational term. The factorization method has been applied in order to obtain analytical forms of creation and annihilation operators as well as Witten superpotential and isospectral potentials. Moreover, the radial Schrödinger equation with the Darboux potential has been converted into an exactly solvable form of second-order Sturm–Liouville differential equation. To this aim the Darboux transformation has been used. The efficient algebraic approach proposed can be used to solve the Schrödinger equation for other anharmonic exponential potentials with rotational terms.     

Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multiconfiguration time-dependent Hartree method

In this paper, we present benchmark results for dissipative dynamics of a harmonic oscillator coupled to an anharmonic bath of Morse oscillators. The microscopic Hamiltonian has been chosen so that the anharmonicity can be adjusted as a free parameter, and its effect can be isolated. This leads to a temperature dependent spectral density of the bath, which is studied for ohmic and lorentzian cases. Also, we compare numerically exact multiconfiguration time-dependent Hartree results with approximate solutions using continuous configuration time-dependent self-consistent field and local coherent state approximation.     

Time-dependent wave packet propagation using quantum hydrodynamics

A new approach for propagating time-dependent quantum wave packets is presented based on the direct numerical solution of the quantum hydrodynamic equations of motion associated with the de Broglie–Bohm formulation of quantum mechanics. A generalized iterative finite difference method (IFDM) is used to solve the resulting set of non-linear coupled equations. The IFDM is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a “smart” Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is generalized to treat higher-dimensional problems and anharmonic potentials. The method is applied to a one-dimensional Gaussian wave packet scattering from an Eckart barrier, a one-dimensional Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. The 2D scattering results represent the first successful application of an accurate direct numerical solution of the quantum hydrodynamic equations to an anharmonic potential energy surface.     

Anharmonic Effect in CH<Subscript>3</Subscript>CH<Subscript>2</Subscript>C(=O)OCH<Subscript>2</Subscript>CH<Subscript>3</Subscript> Decomposition

In this paper, using the B3LYP functional and CCSD(T) method with 6-311++G** basis set, the harmonic and anharmonic rate constants in the unimolecular dissociation of ethyl propanoate have been calculated using Rice–Ramsperger–Kassel–Marcus theory. The anharmonic rate constants of the title reaction have also been examined, the comparison shows that, the anharmonic effect especially in the case of high total energies and temperature for channels 3 to 6 is significant, so that the anharmonic effect cannot be neglected for unimolecular dissociation reaction of CH₃CH₂C(=O)OCH₂CH₃ both in microcanonical and canonical systems.     

Semiclassical initial value series representation in the continuum limit: application to vibrational relaxation

A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.     

丁酸甲酯单分子解离的非谐振效应

In this paper, we have used the MP2/6-311++G(2d, 2p) method to conduct a detailed investigation of the potential energy surface for the unimolecular dissociation reaction of methyl butanoate (MB). We have also used the Rice-Ramsperger-Kassel-Marcus (RRKM) theory to calculate the rate constants of the canonical and microcanonical systems at temperatures and total energies ranging from 1000 to 5000 K and 451.92 to 1519.52 kJ·mol^-1, respectively. The results indicated that there was an obvious anharmonic effect for the TS2, TS4 and TS5 pathways, and that this effect was too pronounced to be neglected for the unimolecular dissociation reactions of MB.     

Polarizability and chemical hardness—A combined study of wave function and density functional theory approach

A flexible model for generating the molecular wave function and electron density for diatomic molecules is developed employing the quadratic anharmonic oscillator and Morse potential. The chemical hardness, Fukui function, and polarizability were calculated using the electron density of the molecules, and the values are found to be reasonably good. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Effects of anharmonicity on nonadiabatic electron transfer: a model

The effect of anharmonicity in the intramolecular modes of a model system for exothermic intramolecular nonadiabatic electron transfer is probed by examining the dependence of the transition probability on the exoergicity. The Franck-Condon factor for the Morse potential is written in terms of the Gauss hypergeometric function both for a ground initial state and for the general case, and comparisons are made between the first-order perturbation theory results for transition probability for harmonic and Morse oscillators. These results are verified with quantum dynamical simulations using wave-packet propagations on a numerical grid. The transition-probability expression incorporating a high-frequency quantum mode and low-frequency medium mode is compared for Morse and harmonic oscillators in different temperature ranges and with various coarse-graining treatments of the delta function from the Fermi golden rule expression. We find that significant deviations from the harmonic approximation are expected for even moderately anharmonic quantum modes at large values of exoergicity. The addition of a second quantum mode of opposite displacement negates the anharmonic effect at small energy change, but in the inverted regime a significantly flatter dependence on exoergicity is predicted for anharmonic modes.     

Anharmonic Effect of the Unimolecular Dissociation of Ethyl Radical

The anharmonic and harmonic rate constants have been calculated for the unimolecular dissociation of ethyl radical using the method proposed by Yao and Lin (YL method) at both B3LYP/6‐311++G** and MP2/6‐311++G** levels. The different rate constants indicate that the results obtained from B3LYP and MP2 method are very close. The anharmonic and tunneling effect of the title reaction has also been examined. The comparison shows that, both in microcanonical and canonical systems, the anharmonic rate constants are higher than those for harmonic cases, especially in the case of high total energies and temperatures, which indicates that anharmonic effect of the unimolecular dissociation of ethyl into C₂H₄ and H is so significant that cannot be neglected. The tunneling effect is very small for the decomposition of C₂H₅ radical.     

Effects of anharmonicity on non-radiative transitions in polyatomic molecules

The effect of anharmonicity on the non-radiative transition in large molecules is examined within the Morse potential surface model. The vibrational wavefunctions are assumed to be the product of the harmonic and Morse oscillator wavefunctions. The method of factorization introduced by Gelbart et al. is used for the evaluation of a density weighted Franck-Condon factor. As an example, we choose the intersystem crossing ³ B _1u→¹ A _1g in benzene. The numerical calculation shows that the anharmonicity causes an increase by a numerical factor ～ 10³ in the non-radiative transition rate. The electronic energy distribution over the vibrational modes in the final state is determined and compared with that obtained using the harmonic potential surface model.     

A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed‐form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two‐ and four‐point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Anharmonic Effect of the Unimolecular Isomerization/Decomposition of Benzyne

The anharmonic and harmonic rate constants were calculated for the unimolecular decomposition of o‐benzyne, the isomerization of o‐benzyne to m‐benzyne, the isomerization of m‐benzyne to p‐benzyne and unimolecular decomposition of p‐benzyne by using the Rice–Ramsperger–Kassel–Marcus (RRKM) theory respectively, in the canonical and microcanonical systems. The geometry and the vibrational frequencies were calculated by MP2 and B3LYP methods with 6‐311G(d,p) basis set and the barrier energies were corrected using CBS‐QB3 theory. The anharmonic effect on the reactions was also examined. Comparison of results for the decompositions of benzyne indicate that both in microcanonical and canonical cases, the anharmonic effect on the decomposition of the o‐C₆H₄ and p‐C₆H₄ are significant, while the anharmonic effect on the two isomerizations are not pronounced.     

Parametrization of an anharmonic Kirkwood-Keating potential for AlxGa1-xAs alloys

We introduce a simple semiempirical anharmonic Kirkwood-Keating potential to model A(x)B(1-x)C-type semiconductors. The potential consists of the Morse strain energy and Coulomb interaction terms. The optical constants of pure components, AB and BC, were employed to fit the potential parameters such as bond-stretching and -bending force constants, dimensionless anharmonicity parameter, and charges. We applied the potential to finite temperature molecular-dynamics simulations on Al(x)Ga(1-x)As for which there is no lattice mismatch. The results were compared with experimental data and those of harmonic Kirkwood-Keating model and of equation-of-motion molecular-dynamics technique. Since the Morse strain potential effectively describes finite temperature damping, we have been able to numerically reproduce experimentally obtained optical properties such as dielectric functions and reflectance. This potential model can be readily generalized for strained alloys.     

Coherent control of wavepacket dynamics by locally designed external field

A new coherent control theory for manipulating quantum mechanical dynamics is proposed. The control field is designed locally (in time domain) so as to realize monotonous increase of the overlap between currently evolving wavefunction and the time-dependent target state, which will eventually reach to a desired quantum state under field-free condition. The present theory is applied to one-dimensional harmonic oscillator and Morse potential systems.     

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.