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     

An analytical approach for computing Franck‐Condon integrals of harmonic oscillators with arbitrary dimensions

We have developed an analytical approach for computing Franck‐Condon integrals (FCIs) of harmonic oscillators (HOs) with arbitrary dimensions in which the mode‐mixing Duschinsky effect is taken into account. A general formula of FCIs of HOs was obtained and was applied to study the photoelectron spectroscopy of vinyl alcohol and ovalene (C₃₂H₁₄). The equilibrium geometries, harmonic vibrational frequencies and normal modes of vinyl alcohol, ovalene, and their cations were computed at the B3LYP/aug‐cc‐pVTZ or the B3LYP/6‐31G(d) level, from which Franck‐Condon factors were calculated and photoelectron spectra were simulated. The adiabatic ionization energies of vinyl alcohol were also computed by extrapolating the CCSD(T) energies to the complete basis set limit with aug‐cc‐pVXZ (X = D, T, Q, 5). The simulated photoelectron spectra of both vinyl alcohol and ovalene are in agreement with the experiments. The computed adiabatic ionization energies of syn‐ and anti‐vinyl alcohol are in consistent with the experiment within 0.008 eV and 0.014 eV, respectively. We show, for the first time, that the analytical approach of computing FCIs is also efficient and promising for the studies of vibronic spectra of macrosystems. © 2012 Wiley Periodicals, Inc.     

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     

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     

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.     

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.     

Potential‐Energy Surfaces,the Born–Oppenheimer Approximations,and the Franck–Condon Principle: Back to the Roots

The concept of a potential‐energy surface (PES) is central to our understanding of spectroscopy, photochemistry, and chemical kinetics. However, the terminology used in connection with the basic approximations is variously, and somewhat confusingly, represented with such phrases as “adiabatic”, “Born–Oppenheimer”, or “Born–Oppenheimer adiabatic” approximation. Concerning the closely relevant and important Franck–Condon principle (FCP), the IUPAC definition differentiates between a classical and quantum mechanical formulation. Consequently, in many publications we find terms such as “Franck–Condon (excited) state”, or a vertical transition to the “Franck–Condon point” with the “Franck–Condon geometry” that relaxes to the excited‐state equilibrium geometry. The Born–Oppenheimer approximation and the “classical” model of the Franck–Condon principle are typical examples of misused terms and lax interpretations of the original theories. In this essay, we revisit the original publications of pioneers of the PES concept and the FCP to help stimulate a lively discussion and clearer thinking around these important concepts.     

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.     

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.     

Relaxation of excited Franck–Condon states of molecular electron donor–acceptor complexes in liquid solution: the role of orientational isomers

The direct consequences of the presence of ground state orientational isomers of molecular complexes are discussed in terms of the adiabatic potential energy surfaces calculated for the ground and excited states of electron donor–acceptor complexes of tetracyanobenzene with toluene and with mesitylene. Some earlier experimental results that confirm the presence of orientational isomers are also recalled and reviewed, together with the recent results for molecular exciplexes under supersonic molecular beam conditions. Exploration of potential energy surfaces shows that the relaxation pathways of excited Franck–Condon states of the ground state isomers may differ considerably and in liquid solution may be sensitive to physical conditions, which in fact is observed in time-resolved fluorescence spectra of the electron donor–acceptor systems under consideration, upon excitation of high-energy Van der Waals orientational isomers. It is concluded that, in weak electron donor–acceptor complexes in liquid solutions, the role of such isomers may be limited, but it may become crucial for the kinetics and dynamics of excited states if the system is simultaneously capable of forming an exciplex.     

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     

Thomas–Fermi approximation for the quasi‐two‐dimensional electron gas

To take into account static correlation effects in the quasi‐two‐dimensional electron gas a screened Coulombic interaction between particles is studied. The Thomas–Fermi approximation is used and the potential screening appears as a function of the Wigner–Seitz density parameter r_s and the effective width t of the system. With the self‐consistent field theory applied to the modified deformable jellium, the ground‐state energy per particle and the conditions for electron localization are obtained in terms of the interparticle distance and the screening parameter μ. A critical minimum characteristic width t_c is obtained; below t_c no long‐range order is obtained. For larger widths a stable localized state is predicted at finite densities. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 269–276, 2001     

Green's function calculation of through‐bond electronic coupling in donor–bridge–acceptor model systems II: Importance factors applied to atomic sites

Importance factors, associated with the Green's function formalism, are introduced. They are applied for the determination of the relative atomic site contribution to the electronic interaction propagation in a molecular system. The calculation is performed at the Hartree–Fock (self‐consistent) level, using ab initio STO‐3G, 4‐31G, and D95 basis sets. The results are compared with those obtained from the charge densities of the appropriate molecular orbitals at the ab initio STO‐3G level. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

An instability condition for the Hartree–Fock solution of the infinite one‐dimensional system with two‐crossing bands. I. Singlet‐instability check of metallic carbon nanotube

An instability condition is derived for the Hartree–Fock solution so that it can be applied to the system in which the highest occupied and the lowest unoccupied bands cross at the in‐between point in the Brillouin zone. The instability check developed here is further applied to a metallic single‐walled carbon nanotube having the two‐crossing bands toward prediction of its instability. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 574–582, 2000     

Complete one‐ and two‐center partitioning scheme for the total energy in the Hartree–Fock theory

We proposed a complete calculation scheme for attributing the total energy by the Hartree–Fock theory to atoms (E_A) and the region between two atoms (E_AB). It was pointed out that the conventional method using the Fock matrix includes a large amount of mutual contamination in both E_A and E_AB. The new scheme was derived from the basic expression of the total energy. Calculated results by the new scheme satisfy the theoretical requirements. The scaling effect on partitioned energies was also examined. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 35–46, 1999     

Multicanonical Monte Carlo calculation of the free‐energy map of the base–amino acid interaction

The specific interactions between base pairs and amino acids were studied by the multicanonical Monte Carlo method. We sampled numerous interaction configurations and side‐chain conformations of the amino acid by the multicanonical algorithm, and calculated the free energies of the interactions between an amino acid at given C_α positions and a fixed base pair. The contour maps of free energy derived from this calculation represent the preferred C_α position of the amino acid around the base, and these maps of various combinations of bases and amino acids can be used to quantify the specificity of intrinsic base–amino acid interactions. Similarly, enthalpy and entropy maps will provide further details of the specific interactions. We have also calculated the free‐energy map of the orientations of the C_α C_β bond vector, which indicates the preferential orientation of the amino acid against the base. We compared the results obtained by the multicanonical method with those of the exhaustive sampling and canonical Monte Carlo methods. The free‐energy map of the base–amino acid interaction obtained by the multicanonical simulation method was nearly identical to the accurate result derived from the exhaustive sampling method. This indicates that a single multicanonical Monte Carlo simulation can produce an accurate free‐energy map. Multicanonical Monte Carlo sampling produced free‐energy maps that were more accurate than those produced by canonical Monte Carlo sampling. Thus, the multicanonical Monte Carlo method can serve as a powerful tool for estimating the free‐energy landscape of base–amino acid interactions and for elucidating the mechanism by which amino acids of proteins recognize particular DNA base pairs. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 954–962, 2000     

Ab initio calculation of the Dzyaloshinskii–Moriya parameters: Spin–orbit GSO‐HF,DFT, and CI approaches

We present ab initio methods to determine the Dzyaloshinskii–Moriya (DM) parameter, which provides the anisotropic effects of noncollinear spin systems. For this purpose, we explore various general spin orbital (GSO) approaches, such as Hartree–Fock (HF), density functional theory (DFT), and configuration interaction (CI), with one‐electron spin–orbit coupling (SOC1). As examples, two simple D_3h‐symmetric models, H₃ and B(CH₂)₃, are examined. Implications of the computational results are discussed in relation to as isotropic and anisotropic interactions of molecular‐based magnets. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Tuning the Adsorption Energy of Methanol Molecules Along Ni‐N‐Doped Carbon Phase Boundaries by the Mott–Schottky Effect for Gas‐Phase Methanol Dehydrogenation

Engineering the adsorption of molecules on active sites is an integral and challenging part for the design of highly efficient transition‐metal‐based catalysts for methanol dehydrogenation. A Mott–Schottky catalyst composed of Ni nanoparticles and tailorable nitrogen‐doped carbon‐foam (Ni/NCF) and thus tunable adsorption energy is presented for highly efficient and selective dehydrogenation of gas‐phase methanol to hydrogen and CO even under relatively high weight hourly space velocities (WHSV). Both theoretical and experimental results reveal the key role of the rectifying contact at the Ni/NCF boundaries in tailoring the electron density of Ni species and enhancing the absorption energies of methanol molecules, which leads to a remarkably high turnover frequency (TOF) value (356 mol methanol mol^?1 Ni h^?1 at 350 °C), outpacing previously reported bench‐marked transition‐metal catalysts 10‐fold.