On Feasibility of “Spectroscopic” Calculations of XH Bond Dissociation Energies for Polyatomic Molecules from Fundamental Vibration Frequencies Using the Anharmonic Molecular Model

This paper discusses the applicability of the variational technique using a minimal Morse-harmonic basis set to calculations of the fundamental spectrum and the potential function parameters for polyatomic molecules. The potential function is assumed to be the sum of the Morse function for XH bonds and the harmonic function for the skeletal and deformation vibrations. The initial approximation for the potential function is found by ab initio calculations and refined by solving the inverse mechanical problem (selecting the scaling indices). The thus selected harmonic part of the potential function gives equally good agreement between the experimental and calculated transition frequencies in both harmonic and anharmonic approximations. The anharmonic (Morse) term of the potential function (bond dissociation energy) is selected by solving the inverse mechanical problem until the best agreement between the experimental and calculated CH bond stretching frequencies has been achieved. Problem solving ends with the construction of a transmission curve in the IR spectrum. Variations of the dipole moment of the molecule induced by vibrations are found by ab initio calculations.     

“Spectroscopic” Calculations of CH Bond Dissociation Energies for Ethane, Propane, Butane, Isobutane, Pentane, Hexane, and Neopentane Using Fundamental Vibration Frequencies

The fundamental spectrum and the parameters of the potential function of a number of saturated hydrocarbon molecules are calculated in an anharmonic approximation. The calculation is performed by the variational technique using a minimal Morse-harmonic basis. The potential function is taken as the sum of the Morse function for CH bonds and the harmonic function for the skeletal and deformation vibrations. The initial approximation for the potential function is found by ab initio calculations in a 6-31G basis and refined by solving the inverse problem. The calculated CH bond dissociation energies depend significantly on the molecular structure and on the position of CH bonds in the molecule. These energies correlate well with the experimental cleavage energies of these bonds. The changes in the dipole moment of the molecule induced by vibrations were found by ab initio calculations in a 6-31G basis. The calculated IR transmission curves are in good agreement with the experimental curves.     

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.     

Method of computer-aided determination of the type of normal vibrations of polyatomic molecules

Correct assignment of calculated and experimental frequencies, as well as corservation of the initial assignment in case of a random coincidence of two frequencies during the solution of an inverse spectral problem, are important problems in spectroscopy of polyatomic molecules. Vibrations are classified according to their form, which is thought to be a more reliable basis for assignment than frequency. We offer a PC program to determine the type of vibrations according to their form (obtained by solving a direct vibrational problem) in a given system of vibrational basis set functions. Optimization of systems of basis set functions for molecules with six-membered rings is discussed. L. Ya. Karpov Physicochemical Scientific Research Institute. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 298–303, March–April, 1995. Translated by L. Smolina     

Analytic quantum mechanics of the morse oscillator

The approximate eigenfunctions of the Morse oscillator, expressed in terms of Laguerre polynomials, are shown to form an approximately orthogonal basis. Analytic expressions for the matrix elements of common operators are obtained within this representation. With such matrix elements in closed form, the Morse oscillator becomes, as the harmonic oscillator has been, a practical building block in molecular theory.     

Analysis of local anharmonicity using Gaussian model potentials and cartesian oscillator basis sets: example, HCN

This paper examines local anharmonic vibrations in molecules using an analysis that starts with an ab initio potential energy surface, fits a model potential constructed of Gaussian basis functions, and proceeds to a quantum mechanical analysis of the anharmonic modes using Cartesian harmonic oscillator basis functions in a variational calculation. The objective of this work is to suggest methods, with origins in nuclear and molecular (electronic) quantum mechanics, that should be useful for the accurate analysis of the local anharmonic motions of hydrogen, and perhaps other atoms or small molecular fragments, residing in molecularly complicated but otherwise harmonic environments.     

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.     

Perturbational and variational treatments of the Morse oscillator

The Morse oscillator hamiltonian is expressed as an infinite expansion in powers of a natural perturbation parameter, the square root of the anharmonicity constant, relative to the simple harmonic oscillator as zeroth-order hamiltonian. A transformation of variables leads to a hamiltonian which involves terms no higher than second order in this natural perturbation parameter. In both cases, the exact bound state eigenvalues of the Morse oscillator are given by second-order perturbation theory. The Schrödinger equation corresponding to the transformed Morse hamiltonian is solved variationally, via a complete set expansion in simple harmonic oscillator eigenstates. Accurate approximations to the exact eigenvalues and eigenfunctions of bound states of the Morse oscillator can be obtained for all but the very highest levels.     

Novel methods for calculating the fine vibronic spectra of polyatomic molecules

Novel approximate methods for calculating the vibrational structure of the electronic spectra of polyatomic molecules—a method for the direct calculation of the overlap integrals of vibrational wave functions for the electronic states involved in a transition and a variational method for the solution of the vibrational problem for the excited states—are discussed. The methods are based on the consideration of the displacement and entanglement of normal coordinates, the quasiorthogonality of the Dushinsky transformation, and the classification of the states by total vibrational quantum numbers. Matrix perturbation theory is employed. It is shown that the accuracy of these methods compares well with the accuracy of the available “exact” techniques (the errors are ∼1 cm⁻¹ for frequency and 10% for relative intensity). At the same time, calculations by the new methods are performed more than two orders of magnitude faster than by the previously known methods. K. A. Timiryazev Agricultural Academy, Moscow. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 217–230, March–April, 1995. Translated by I. Izvekova     

A theory of nuclear motion in polyatomic molecules based upon the morse oscillator: I. Application to the stretching overtone spectra of water and benzene molecules

A theory of molecular vibrations is presented which is based upon the Morse, as opposed to the harmonic, oscillator. As a first application of this model, the stretching overtone spectra of the benzene and water molecules have been calculated, these examples being chosen on account of the high degree of anharmonicity characterizing their potentials and the availability of suitable experimental data.     

The subduction of vibrational states of molecules from the momentum space of simple harmonic motion

A single-particle model of molecular vibrational states is proposed in which the normal modes are projected out of the body vibrations of an infinite simple harmonic sphere. This model assigns the spurious change of mass or centre of mass and leads to removal of mass monopoles and dipoles from the system. These conservation conditions impose strict boundary conditions on the potential and basis functions. On incorporation into the model they result in a set of loop equations in which the potential is proportional to the fundamental vibration. The simplest solutions to these equations strongly resemble the Poschl-Teller generalization of the Morse potential. The solutions have been extended to incorporate the repulsive states and generate the set of net attractive states appropriate to the anharmonic potential.The basis functions of this potential display both angular and radial node structures. The degeneracies between radial and angular mode patterns can be studied by transformation into an angular coordinate space. In this way coupling to other phenomena described in similar angular momentum space can be performed directly before subduction to real displacement space.On leave from the Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200F, B-3030 Leuven-Belgium     

On the decay of highly excited oscillator in a crystal

The diagonal density matrix elements are found for a dynamical subsystem which is either anharmonic or harmonic highly excited oscillator interacting with the crystal vibrations. For the extreme case of the infinitely highly excited oscillator the result is exact. Different limiting cases are considered. In particular for the harmonic oscillator the validity criterion for the balance equation is found.     

Anharmonic oscillator model of overtone spectra for D3h symmetry molecules

The bond stretching vibrations of XF₅ molecules with D_3h symmetry are treated computationally on the Morse oscillator model in which the bond oscillators are coupled harmonically. Each calculation involves four parameters for two types of Morse potential and three parameters for the kinetic-energy, potential coupling terms. The eigenvalue formula for overtone and combination states up to three are presented and can be used to predict all the vibrational energy levels from local mode molecules through normal mode molecules. For PF₅, AsF₅ and VF₅, the coupled Morse oscillator model gives a prediction in good agreement with the experimental data.     

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     

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.     

A discrete look at localization

This article introduces a set of localized orthonormal functions to serve as basis functions for quantum calculations. They are defined to be eigenfunctions of the position operator in a function space. Their properties, including their variances, for a one-dimensional system are developed. The application to simple harmonic motion is considered as an example and, in particular, the time evolution of an initially localized function is calculated and shown to be periodic. The theory can be interpreted as producing a discrete quantization of space with Hamiltonian interactions that are predominantly between nearest neighbors. These functions can also be used in approximate calculations. To illustrate their accuracy, the example of a Morse oscillator treated as a perturbation of a harmonic oscillator is reconsidered. It is shown that the localized functions in a variational calculation lead to a result that is a good approximation for the lowest states. Furthermore, the use of a wave function that is defined only at discrete points can be justified as the first approximation to this, so that its accuracy can also be discussed. © 1995 John Wiley & Sons, Inc.     

The anharmonic effect study of coupled Morse oscillators for the unimolecular reaction

The importance of anharmonic effect in dissociation of molecular systems especially clusters has been noted. In this paper, we shall study the effect of coupled anharmonic oscillator of the standard bilinear form (SBF) Morse oscillator (MO) potential on unimolecular reaction. We shall use the systematic theoretical approach, YL method, proposed by Yao and Lin (YAO L, et. al. J Phys Chem A, 2007, 111(29): 6722-6729), which can evaluate anharmonic effects on the rate constants based on the transition state theory. In treating the anharmonic effect with the Morse oscillator potential on unimolecular reactions under collision-free conditions by using the RRKM (Rice-Ramsperger-Kassel-Marcus) theory, the in-verse Laplace transformation of the partition functions was used to obtain the total amount of state and density of state by using the first-order and the second-order approximations of the saddle-point method. To demonstrate the anharmonic effect of the SBF Morse model, we choose some model systems and a real reaction as examples.     

Accurate theoretical IR and Raman spectrum of Al2O2 and Al2O3 molecules

The accurate harmonic vibration frequencies together with the infrared (IR) and Raman intensities of the most stable conformers of Al₂O₂ and Al₂O₃ molecules have been calculated by the density functional theory (DFT) method with B3LYP exchange–correlation potential and using a set of the augmented correlated consistent basis sets up to quintuple order. The anharmonic vibration frequencies of the non-linear Al₂O₂ molecule have also been calculated. The obtained equilibrium geometrical parameters, harmonic and anharmonic vibration frequencies along with the IR and Raman intensities good converge to their limits with increasing the size of the used basis set. A comparison of the calculated harmonic and anharmonic vibrational frequencies with the available experimental ones points out that the small differences between the calculated harmonic and experimental frequencies can be further substantially reduced when calculations of the anharmonic vibrational frequencies will be available for all types of molecular geometries.     

Franck-Condon factors based on anharmonic vibrational wave functions of polyatomic molecules

Franck-Condon (FC) integrals of polyatomic molecules are computed on the basis of vibrational self-consistent-field (VSCF) or configuration-interaction (VCI) calculations capable of including vibrational anharmonicity to any desired extent (within certain molecular size limits). The anharmonic vibrational wave functions of the initial and final states are expanded unambiguously by harmonic oscillator basis functions of normal coordinates of the respective electronic states. The anharmonic FC integrals are then obtained as linear combinations of harmonic counterparts, which can, in turn, be evaluated by established techniques taking account of the Duschinsky rotations, geometry displacements, and frequency changes. Alternatively, anharmonic wave functions of both states are expanded by basis functions of just one electronic state, permitting the FC integral to be evaluated directly by the Gauss-Hermite quadrature used in the VSCF and VCI steps [Bowman et al., Mol. Phys. 104, 33 (2006)]. These methods in conjunction with the VCI and coupled-cluster with singles, doubles, and perturbative triples [CCSD(T)] method have predicted the peak positions and intensities of the vibrational manifold in the X 2B1 photoelectron band of H2O with quantitative accuracy. It has revealed that two weakly visible peaks are the result of intensity borrowing from nearby states through anharmonic couplings, an effect explained qualitatively by VSCF and quantitatively by VCI, but not by the harmonic approximation. The X 2B2 photoelectron band of H2CO is less accurately reproduced by this method, likely because of the inability of CCSD(T)/cc-pVTZ to describe the potential energy surface of open-shell H2CO+ with the same high accuracy as in H2O+.