Vibrational-rotational levels and Franck-Condon factors of diatomic molecules via Morse potentials in a box

Pekeris' solution for vibrational-rotational levels of the Morse potential includes the unphysical domain − ∞ < r < 0 of the internuclear separation, just like Morse's solution for vibrational levels. In this article, the method of confinement in a box is implemented to obtain the vibrational-rotational J = 1 levels of Morse potentials in the physical domain 0 < r < ∞, as well as the corresponding Franck-Condon factors. Applications to Li₂ and AIO are presented as illustrations. © 1996 John Wiley & Sons, Inc.     

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.     

Franck–Condon factors for the Morse potential

The aim of this study is to establish a new representation for the dynamic algebra of the Morse oscillator and to establish the raising and lowering operators based on the properties of the confluent hypergeometric functions. Using the representation we have obtained a recurrent analytic method for the calculus of the Franck–Condon factors. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 655–660, 1997     

Improving an algebraic approach to calculate approximate Franck–Condon factors of diatomic molecules

Using the BCH theorem, we express the Hamiltonian of a Morse oscillator as a complete series of powers of the creation and annihilation operators for the harmonic oscillator. In this way, we improve the results of a previous work that uses a Bogoliubov–Tyablikov tranformation to calculate the Franck–Condon factors by means of equivalent harmonic oscillators potentials. © 1992 John Wiley & Sons, Inc.     

Simple evaluation of Franck–Condon factors and non‐Condon effects in the Morse potential

The calculation of Franck–Condon factors between different 1‐D Morse potential eigenstates using a formula derived from the Wigner function is discussed. Our numerical calculations using a simple program written in Mathematica are compared with other calculations. We show that our results have a similar accuracy as those calculations performed with more sophisticated methods. We discuss the extension of our method to include non‐Condon effects in the calculation. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem 88: 280–295, 2002     

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.     

Highly accurate vibration–rotation Franck–Condon factors for high levels

Highly accurate vibration–rotation Franck–Condon factors q^ab, for a transition between two diatomic electronic states (a) and (b), are sought. When the potentials of states (a) and (b) are of the RKR type, the computation of q^ab is reduced to that of Franck–Condon integral ?^ab(i) = ∫ ψ^a(r)ψ^b(r) dr in an interval r_i, r_i+1. By using convenient interpolations for the potentials U^a and U^b in the considered interval, this integral becomes ?^ab(i) = ∑ δ (r_i+1 – r_i)ⁿ⁺¹/(n + 1), where the “coupling constants” δ depend uniquely on the eigenvalues E^a and E^b of the considered transition and on the potentials U^a and U^b (the number N of terms depends on the desired accuracy). The method used computes the Franck–Condon factors q^ab without the explicit use of the wave function and by replacing the integrals by simple summations. To test the values of q^ab obtained by this method, the orthogonality rule ∫ ψ_v′ψ_v″ dr = 0 (for v′ ≠ v″) is used for one state or the other. This test, along with other tests, show that the Franck–Condon factors computed by the present method are accurate to nine significant figures for high and low levels.     

Efficient calculation of Franck–Condon factors and vibronic couplings in polyatomics

We present a technique for the calculation of Franck–Condon factors and other integrals between vibronic wave functions belonging to different electronic states. The technique is well suited for the determination of the nonadiabatic or spin‐orbit couplings related to radiationless decays in polyatomics. Rigorous or approximate partitions of the internal coordinate space are exploited to achieve better efficiency and/or to go beyond the harmonic approximation. The technique is tested by computing the Internal Conversion and InterSystem Crossing rates of (CH₃)₃CNO in its ¹(n→π*) state. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 968–975, 2001     

Normally ordered Franck–Condon operator: A new approach for the calculation of Frank–Condon factors

The idea of a Franck–Condon (FC ) operator is introduced, and its normally ordered form is obtained through the newly developed technique of “integration within an ordered product of operators (IWOP ).” It is shown that the FC operator leads to a new approach for the calculation of FC factors. The results of existing theories are viewed, and the connection between the FC operator and the “squeeze-operator” is pointed out.     

The convolution theorem and the Franck–Condon integral

The convolution theorem is used to evaluate the Franck–Condon integral. It is shown that this integral becomes the matrix element between two “squeezed” states. This enables one to evaluate the integral by using boson operators. In addition, a general method is developed to obtain integrals involving Hermite polynomials with a displaced argument. In particular, the two‐center matrix element _g〈m|f(x_e)|n〉_e, is obtained, where f(x_e)=exp(Dx+Fx_e). ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 11–15, 1999     

Chemical bond breaking/forming as a Franck–Condon electronic process

We describe chemical bond changes as Franck–Condon electronic processes within a new theoretical ansatz that we call ‘rigged’ Born–Oppenheimer (R-BO) approach. The notion of the separability of nuclear and electron states implied in the standard Born–Oppenheimer (BO) scheme is retained. However, in the present scheme the electronic wave functions do not depend upon the nuclear coordinate (R-space). The new functions are obtained from an auxiliary Hamiltonian corresponding to the electronic system (r-coordinates) submitted to a Coulomb potential generated by external sources of charges in real space (α-coordinates) instead of massive nuclear objects. A stationary arrangement characterized by the coordinates α_0A, is determined by a particular electronic wave function, ψ(r;α_0A); it is only at this stationary point, where an electronic Schrödinger equation: H_e(r,α_0A)|Ψ(r;α_0A)=E(α_0A)|Ψ(r;α_0A) must hold. This equation permits us to use modern electronic methods based upon analytic first and second derivatives to construct model electronic wave functions and stationary geometry for external sources. If the set of wave functions {Ψ(r;α_0A)} is made orthogonal, the energy functional in α-space, E(α;α_0A)=Ψ(r;α_0A)|H_e(r,α_0A)|Ψ(r;α_0A) is isomorphic to a potential energy function in R-space: E(R;α_0A)=Ψ(r;α_0A)|H_e(r,R)|Ψ(r;α_0A). This functional defines, by hypothesis, a trapping convex potential in R-space and the nuclear quantum states are determined by a particular Schrödinger equation. The total wave function for the chemical species A reads as a product of our electronic wave function with the nuclear wave function (Ξ_ik(R;α_0A)): Φ_ik(r,R)=Ψ_i(r,α_0A)Ξ_ik(R;α_0A). This approach facilitates the introduction of molecular frame without restrictions in the R-space. Two molecules (characterized with different electronic spectra) that are decomposable into the same number of particles (isomers) have the same Coulomb Hamiltonian and they are then characterized by different electronic wave functions for which no R-coordinate ‘deformation’ can possibly change its electronic structure. A bond breaking/forming process must be formally described as a spectroscopic-like electronic process. The theory provides an alternative to the adiabatic as well as the diabatic scheme for understanding molecular processes. As an illustration of the present ideas, the reaction of H₂+CO leading to formaldehyde is examined in some detail.     

Recurrence relations and closed formulas connecting Franck—Condon factors and squeezed states

Recurrence relations and closed formulas which are common to Franck-Condon factors and squeezed states are presented in a unified formulation. Both concepts present a certain parallelism that is explained by formally showing their becoming particular cases of more general concepts. This treatment sets forth the mathematical structure that unifies the two concepts. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 229–232, 1997     

Unified analytical treatment for calculation of the two‐dimensional Franck–Condon factors using the duschinsky transformation

Including binomial expansion theorems, we present an analytical formula for calculating Franck–Condon (FC) factors of two‐dimensional (2D) harmonic oscillators including the Duschinsky effect. The FC principle has various practical applications in quantum modeling of electronic spectra of polyatomic molecules. The 2D FC factors are expressed through the binomial coefficients. Use of the memory of the computer for the calculation of binomial coefficients may extend the limits to large arguments for users and result in speeder calculation, should such limits be required in practice. Accurate numerical results are provided to validate the proposed algorithm. © 2012 Wiley Periodicals, Inc.     

Franck–Condon simulation of photoelectron spectroscopy of HOO: Including Duschinsky effects

Geometry optimization and harmonic vibrational frequency calculations were performed on the and states of HOO and state of HOO⁻. The electron affinity and the term energy () of HOO were calculated at various theory levels. Franck–Condon analyses and spectral simulations were carried out on the and photodetachment processes. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectra. In addition, the equilibrium geometrical parameters of the state of HOO⁻ and state of HOO were obtained in the spectral simulations.     

Exact calculations of Franck–Condon overlaps and of matrix elements of some potentials by means of the coherent state representation

Franck–Condon overlaps are described as the matrix elements of unitary operators related to the spatial displacement and the frequency shift. They are calculated exactly by means of the coherent state representation. Furthermore, the generalized matrix elements of x^j, e^?αx, and e^?βx₂ between two states with different equilibrium coordinates and frequency are evaluated in the same way.     

Laser-induced fluorescence of Cs2 and role of kinetic energy in the Franck–Condon principle

Quantum-mechanical calculation of spectral intensity is done for the transitions from a particular vibrational level of the excited state E ¹∏_u to the discrete vibrational levels and dissociative continuum of the ground state X¹Σ of the Cs₂ molecule. The results are compared with the observed spectra by referring to the conservation of nuclear kinetic energy during the electronic transition.     

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     

Vibrational relaxation of diatomic molecules in condensed monatomic media

A model based on binary-like collisions is presented to describe the vibrational relaxation of diatomic molecules in a monatomic medium. For N₂Ar and COAr, it is shown that the oscillatory motion of host atoms undergoing multiquantrum processes is most important in removing vibrational energy. The translational motion of molecules confined to the cell space also contributes to the energy relaxation by removing the energy mismatch.     

Vibrational energy storage in high pressure mixtures of diatomic molecules

CO/N₂, CO/Ar/O₂, and CO/N₂/O₂ gas mixtures are optically pumped using a continuous wave CO laser. Carbon monoxide molecules absorb the laser radiation and transfer energy to nitrogen and oxygen by vibration–vibration energy exchange. Infrared emission and spontaneous Raman spectroscopy are used for diagnostics of optically pumped gases. The experiments demonstrate that strong vibrational disequilibrium can be sustained in diatomic gas mixtures at pressures up to 1 atm, with only a few Watts laser power available. At these conditions, measured first level vibrational temperatures of diatomic species are in the range T_V=1900–2300 K for N₂, T_V=2600–3800 K for CO, and T_V=2200–2800 K for O₂. The translational–rotational temperature of the gases does not exceed T=700 K. Line-of-sight averaged CO vibrational level populations up to v=40 are inferred from infrared emission spectra. Vibrational level populations of CO (v=0–8), N₂ (v=0–4), and O₂ (v=0–8) near the axis of the focused CO laser beam are inferred from the Raman spectra of these species. The results demonstrate a possibility of sustaining stable nonequilibrium plasmas in atmospheric pressure air seeded with a few percent of carbon monoxide. The obtained experimental data are compared with modeling calculations that incorporate both major processes of molecular energy transfer and diffusion of vibrationally excited species across the spatially nonuniform excitation region, showing reasonably good agreement.