Das Franck‐Condon‐Prinzip

The quantum mechanical model is fully recognized and used for rotational and vibronic transitions in molecules, where according to the model, transitions occur between discrete, the resonance condition matching states, only. It is also accepted that selection rules are implied for rotational and vibronic transitions. Why is it so hard to recognize that this established quantum mechanical model is also valid for vibronic transitions, where the Franck‐Condon principle instead of a selection rule applies to the population of vibronic states in the electronic excitation state? In any case, supposed illustrative and comfortable but wrong explanations, also if they are widely used, must not replace more sophisticated, correct quantum mechanical models. In scientific publications as well as in teaching only the quantum mechanical version of the Franck‐Condon principle should be used. Terms like “Franck‐Condon point” or ”Franck‐Condon region of the photochemical reaction" should be avoided in the future.     

On the hydrogen atom beyond the Born–Oppenheimer approximation

Recently a new formulation of quantum mechanics has been suggested which is based on the concept of signed particles, that is, classical objects provided with a position, a momentum and a sign simultaneously. In this article, we comment on the plausibility of simulating atomic systems beyond the Born–Oppenheimer approximation by means of the signed particle formulation of quantum mechanics. First, to show the new perspective offered by this new formalism, we provide an example studying quantum tunnelling through a simple Gaussian barrier in terms of the signed particle formulation. Then, we perform a direct simulation of the hydrogen atom as a full quantum two‐body system, showing that the formalism can be a very promising tool for first‐principle‐only quantum chemistry.     

Three‐body molecular states of the system in the Born–Oppenheimer approximation

In this work, we present a quantum mechanical treatment of the three‐body molecular system in the Born–Oppenheimer (BO) approximation, were the nuclei dynamics is evaluated over the potential energy surfaces (PES) induced by the electronic states. The PES corresponding to the two lowest electronic levels are the ones described by Martinazzo et al. (Chem. Phys. 2003, 287, 335), and are used to write the three‐body Schrdinger equation of the three atomic system. We use the generalized Sturmian functions method to expand the wave functions in each (distinguishable) pair of relative coordinates or Jacobi pairs, and analyze the convergence differences between the series. A partial‐wave decomposition of the potential is proposed to simplify the Hamiltonian matrix element calculation. Bound states are considered for the ground and first excited electronic PES, the spreading of energies after sudden electronic transitions studied, and the break‐up probability induced by the sudden change of the PES evidenced.     

Vibrational–rotational energies of all H2 isotopomers using Monte Carlo methods

Using variational Monte Carlo techniques, we have computed several of the lowest rotational–vibrational energies of all the hydrogen molecule isotopomers (H₂, HD, HT, D₂, DT, and T₂). These calculations do not require the excited states to be explicitly orthogonalized. We have examined both the usual Gaussian wave function form as well as a rapidly convergent Padé form. The high‐quality potential energy surfaces used in these calculations are taken from our earlier work and include the Born–Oppenheimer energy, the diagonal correction to the Born–Oppenheimer approximation, and the lowest‐order relativistic corrections at 24 internuclear points. Our energies are in good agreement with those determined by other methods. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Organic Microcrystal Vibronic Lasers with Full‐Spectrum Tunable Output beyond the Franck–Condon Principle

The very broad emission bands of organic semiconductor materials are, in theory, suitable for achieving versatile solid‐state lasers; however, most of organic materials only lase at short wavelength corresponding to the 0–1 transition governed by the Franck–Condon (FC) principle. A strategy is developed to overcome the limit of FC principle for tailoring the output of microlasers over a wide range based on the controlled vibronic emission of organic materials at microcrystal state. For the first time, the output wavelength of organic lasers is tailored across all vibronic (0–1, 0–2, 0–3, and even 0–4) bands spanning the entire emission spectrum.     

Effect of molecular vibrations on the MD/QC‐simulated absorption spectra

Effect of molecular vibrations on the absorption spectra simulated via a sequential approach combining molecular dynamics (MD) with quantum‐chemical calculations has been investigated. Simulated spectra have been obtained from the time‐dependent density functional theory results averaged over series of molecular geometries retrieved from Born–Oppenheimer MD trajectories. Distributions of bond lengths have been analyzed and related to the features of calculated spectra. For NVE simulations of small systems, absorption spectra exhibit bimodal bandshape as a result of classical treatment of vibrations. For NVE trajectories of larger systems or simulations in the NVT ensemble calculated absorption bands are symmetric, however, they may not agree with the results of Franck–Condon analysis. These results are practical manifestations of effects predicted theoretically from general principles. Consequences for the modeling of absorption spectra have been discussed. © 2013 Wiley Periodicals, Inc.     

On vibronic coupling in nonradiative transitions of polyatomic molecules

The vibronic coupling between quasidegenerate adiabatic Born—Oppenheimer states has been calculated by going beyond the Condon approximation. A simplified model in which accepting modes are distorted (non-totally symmetric) and of the same frequency, has been considered. The decay rate obtained with this approach is one order of magnitude larger than in the Condon scheme and seems to be practically independent of the symmetry of the accepting modes.     

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     

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.     

Why do molecules echo atomic periodicity?

Franck–Condon factors are investigated for sequences of free main‐group diatomic molecules. Theory‐based Condon loci (parabolas) and Morse‐potential loci are plotted on Deslandres tables to verify if they, indeed, follow the largest Franck–Condon factors. Then, the inclination angles of the Condon loci are determined. Thus, entire band systems are quantified by one variable, the angle. For all available isoelectronic sequences, this angle increases from a central minimum toward magic‐number molecular boundaries. The theory for the Condon locus gives the angle in terms of the ratio of the upper‐state to the lower‐state force constants. It is concluded that the periodicity is caused due to the fact that this ratio becomes larger as rare‐gas molecules are approached, a trend that probably points to the extreme cases of the rare‐gas molecules themselves. Thus, molecular periodicity echoes atomic periodicity in that data plots have extrema at molecules with magic‐number atoms, yet it does not echo the details of atomic periodicity in series between those molecules. © 2013 The Authors. International Journal of Quantum Chemistry Published by Wiley Periodicals, Inc.     

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.     

Non‐born–oppenheimer density functional theory for excited states by using green's function techniques

We have proposed a numerical scheme for the non‐Born–Oppenheimer density functional calculation based upon the Green function techniques within the GW approximation for evaluating quasiparticle excitations of the electronic and nuclear motion in the full quantum mechanical treatment. We calculate the excitation energy and the orbital energy of a hydrogen molecule, a muon molecule, and a positronium–hydrogen complex within the treatment of the dynamical screening. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 354–362, 2001     

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     

Nonadiabatic sudden increase of the cooperative kinetic effect at lattice energy stabilization—Microscopic mechanism of superconducting state transition: Model study of MgB2

The theory of the nonadiabatic electron–vibration interactions has been applied to the study of MgB₂ superconducting state transition. It has been shown that at nonadiabatic conditions in which the Born–Oppenheimer approximation is not valid and electronic motion is dependent not only on the nuclear coordinates but also on the nuclear momenta, the fermionic ground‐state energy of the studied system can be stabilized by nonadiabatic electron–phonon interactions at broken translation symmetry. Moreover, the new arising state is geometrically degenerate; i.e., there are an infinite number of different nuclear configurations with the same fermionic ground‐state energy. The model study of MgB₂ yields results that are in a good agreement with the experimental data. For distorted lattice, with 0.016 Å/atom of in‐plane out‐of‐phase B? B atoms displacements out of the equilibrium (E_2g phonon mode) when the nonadiabatic interactions are most effective, it has been calculated that the new arising state is 87 meV/unit cell more stable than the equilibrium–high symmetry clumped nuclear structure at the level of the Born–Oppenheimer approximation. The calculated T_c is 39.5 K. The resulting density of states exhibits two‐peak character, in full agreement with the tunneling spectra. The peaks are at ±4 meV, corresponding to the change of the π band density of states, and at ±7.6 meV, corresponding to the σ band. The superconducting state transition can be characterized as a nonadiabatic sudden increase of the cooperative kinetic effect at lattice energy stabilization (NASICKELES). © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

The Primary Steps in Excited‐State Hydrogen Transfer: The Phototautomerization of o‐Nitrobenzyl Derivatives

The quantum yield for the release of leaving groups from o‐nitrobenzyl “caged” compounds varies greatly with the nature of these leaving groups, for reasons that have never been well understood. We found that the barriers for the primary hydrogen‐atom transfer step and the efficient nonradiative processes on the excited singlet and triplet surfaces determine the quantum yields. The excited‐state barriers decrease when the exothermicity of the photoreaction increases, in accord with Bell–Evans–Polanyi principle, a tool that has never been applied to a nonadiabatic photoreaction. We further introduce a simple ground‐state predictor, the radical‐stabilization energy, which correlates with the computed excited‐state barriers and reaction energies, and that might be used to design new and more efficient photochemical processes.     

Absorption,resonance, and near‐resonance Raman studies of the tetracyanoquinodimethane neutral and its monoanion in terms of density functional theory and complete active space self‐consistent field methods

The electronic structure of the 1¹B_1u and 1²B_3u excited electronic states of the tetracyanoquinodimethane (TCNQ) neutral and its charged derivative are studied within the framework of complete active space self‐consistent field (CASSCF) and Becke's three‐parameter hybrid method with Lee–Yang–Parr correlation functional (B3LYP) methods applied to the level aug‐cc‐p‐VDZ basis set. Both CASSCF/aug‐cc‐p‐VDZ and B3LYP/aug‐cc‐p‐VDZ treatments provide the ground‐state and the excited state geometries; these are then used to assess the Franck–Condon (FC) parameters in the 1¹B_1u state of the neutral TCNQ and in the 1²B_3u state of the TCNQ monoanion. The quality of numerical results is then tested on the base of available experimental near‐resonance and resonance Raman data. The studies are performed in terms of the vibronic model, which takes both FC and mode‐mixing (Dushinsky) effects into account. This somewhat simplified vibronic model leads to very good agreement between the theory and the Raman experiments concerning both neutral TCNQ and its monoanion. In particular, the calculated excitation profiles of the ν₂ = 2215 cm^?1, ν₄ = 1389 cm^?1, ν₅ = 1195 cm^?1, and ν₉ = 336 cm^?1 fundamentals are shown to be in excellent agreement with those for the TCNQ monoanion. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Measures of stochasticity for nonstationary initial preparations of coupled oscillators

The contributions of different eigenstates of a nonlinearly coupled oscillator system to the expansion of a local wave packet are analyzed from an information theoretical point of view. Such a wave packet can be considered as a nonstationary vibrational state of an electronically excited manifold of a molecule after Franck–Condon type initial preparation. The distributions of these contributions are compared to their individual stochastic ideals using Ruch's concept of “mixing distances.” The stochastic ideals are constructed via a probability diffusion process between neighboring states of the original distributions, representing an initial preparation corresponding to a Hamiltonian with only irregular eigenstates. Gaussian minimum uncertainty wave packets as initial states in a two-dimensional nonlinear oscillator system with classically regular and chaotic energy ranges are studied numerically. It is found that distance measures, partly reflecting the “mixing distance” of a distribution from its stochastic ideal, show a large fluctuation in the classical regular energy range and a small fluctuation in the range where most of the classical trajectories move chaotically. This indicates that for this type of initial preparation process the actual location of the initial state in space plays the dominant role for the dynamics in the low-energy range while for wave packets starting near the dissociation energy of the model system this location becomes unimportant.     

Distinguishing between keto–enol and acid–base forms of firefly oxyluciferin through calculation of excited‐state equilibrium constants

Although recent years have seen much progress in the elucidation of the mechanisms underlying the bioluminescence of fireflies, there is to date no consensus on the precise contributions to the light emission from the different possible forms of the chemiexcited oxyluciferin (OxyLH₂) cofactor. Here, this problem is investigated by the calculation of excited‐state equilibrium constants in aqueous solution for keto–enol and acid–base reactions connecting six neutral, monoanionic and dianionic forms of OxyLH₂. Particularly, rather than relying on the standard Förster equation and the associated assumption that entropic effects are negligible, these equilibrium constants are for the first time calculated in terms of excited‐state free energies of a Born–Haber cycle. Performing quantum chemical calculations with density functional theory methods and using a hybrid cluster‐continuum approach to describe solvent effects, a suitable protocol for the modeling is first defined from benchmark calculations on phenol. Applying this protocol to the various OxyLH₂ species and verifying that available experimental data (absorption shifts and ground‐state equilibrium constants) are accurately reproduced, it is then found that the phenolate‐keto‐OxyLH^– monoanion is intrinsically the preferred form of OxyLH₂ in the excited state, which suggests a potential key role for this species in the bioluminescence of fireflies. © 2014 Wiley Periodicals, Inc.