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     

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.     

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 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.     

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     

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.     

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.     

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     

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.     

Franck–Condon Blockade and Aggregation‐Modulated Conductance in Molecular Devices Using Aggregation‐Induced Emission‐Active Molecules

We report an effective modulation of the quantum transport in molecular junctions consisting of aggregation‐induced‐emission(AIE)‐active molecules. Theoretical simulations based on combined density functional theory and rate‐equation method calculations show that the low‐bias conductance of the junction with a single tetraphenylethylene (TPE) molecule can be completely suppressed by strong electron–vibration couplings, that is, the Franck‐Condon blockade effect. It is mainly associated with the low‐energy vibration modes, which is also the origin of the fluorescence quenching of the AIE molecule in solution. We further found that the conductance of the junction can be lifted by restraining the internal motion of the TPE molecule by either methyl substitution on the phenyl group or by aggregation, a mechanism similar to the AIE process. The present work demonstrates the correlation between optical processes of molecules and quantum transport in their junction, and thus opens up a new avenue for the application of AIE‐type molecules in molecular electronics and functional devices.     

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.     

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.     

Sequence-controlled oligomers fold into nanosolenoids and impart unusual optical properties

Controlled syntheses give unique block oligomers with alternating flexible ethylene glycol and rigid perylenetetracarboxylic diimide (PDI) units. The number of rigid units vary from n=1 to 10. PDI units were stitched together by using efficient phosphoramidite chemistry. The resulting oligomers undergo folding in most solvents, including chloroform. In their ground state, these folded oligomers were characterized by using Fourier transform ion cyclotron resonance mass spectrometry (FTICR‐MS), NMR spectroscopy, and electronic absorption spectroscopy. FTICR‐MS revealed the exact masses of these sequence‐controlled oligomers, which confirmed the chemical composition and validated the synthetic strategy. The NMR neighboring ring‐current effect (NRE) indicates the formation of cofacial π stacks; the stacked aromatic rings have nearly coaxial alignment akin to a nanosoleniod. Nanosolenoidal shielding in π stacks causes all aromatic protons to shift upfield, whereas NOE in a cyclic hetero‐chromophoric dimer supports a rotated, cofacial π‐stacking orientation separated by about 3.5 Å. Electron–phonon coupling is much stronger than excitonic coupling in these self‐folded PDI oligomers; thus, Franck–Condon factors dictate the observed spectral features in visible spectra. The absorbance spectrum exhibits weak hypochromism due to π stacking with increasing stacking units n. Finally, ab initio calculations support the experimental observations, indicating 3.5 Å cofacial spacing in which one molecule is rotated 30° from the eclipsed orientation and higher oligomers can adopt, without a compensating energy penalty, either the right/left‐handed helices or the 1,3‐eclipsed structures. Both theory and experiments validate the nano‐π‐solenoids and their novel photophysical properties.     

Derivation of closed analytical expressions for Rosen–Morse Franck–Condon factors

The expressions for Rosen–Morse Franck–Condon factors derived previously yield a double sum with alternating terms. For higher values of the quantum number the numerical calculation of the Franck–Condon factors by electronic computers using these expressions leads to numerical overflow inspite of the use of double-precision (32 digits) arithmetic. High values for the quantum number in the final ground state of the Rosen–Morse potential occur in molecular nonradiative rate calculations. Furthermore, the expressions show a lack of clearness with respect to the parameters of the potential. For out-of-plane modes exact closed form expressions and exact recurrence relations are derived. Asymptotic expressions for the matrix elements are calculated. Exact closed form expressions for matrix elements with quantum numbers which correspond to regions close to the dissociation barrier are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

GRECP/5e‐MRD‐CI calculation of the electronic structure of PbH

The correlation calculation of the electronic structure of PbH is carried out with the generalized relativistic effective core potential (GRECP) and multireference single‐ and double‐excitation configuration interaction (MRD‐CI) methods. The 22‐electron GRECP for Pb is used and the outer core 5s, 5p, and 5d pseudospinors are frozen using the level‐shift technique, so only five external electrons of PbH are correlated. A new configuration selection scheme with respect to the relativistic multireference states is employed in the framework of the MRD‐CI method. The [6, 4, 3, 2] correlation spin–orbit basis set is optimized in the coupled cluster calculations on the Pb atom using a recently proposed procedure, in which functions in the spin–orbital basis set are generated from calculations of different ionic states of the Pb atom and those functions are considered optimal that provide the stationary point for some energy functional. Spectroscopic constants for the two lowest‐lying electronic states of PbH (²Π_1/2, ²Π_3/2) are found to be in good agreement with the experimental data. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many‐electron dynamics of molecular systems, where an explicitly time‐dependent electronic wave packet is represented as a linear combination of N‐electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one‐electron operator by reducing the multideterminantal, many‐electron wave packet using the Slater‐Condon rules. Here, we introduce a general tool for post‐processing multideterminant configuration‐interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which shares and builds on the modular design of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N‐electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 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.     

The generalized Slater–Condon rules

By relating the blocking structure of the relevant matrix of overlap-integrals to its cofactors, the Slater–Condon rules for the evaluation of an element of a matrix representation of an electronic Hamiltonian in a Slater determinant basis are generalized to the case where not all orbitals are orthogonal. This yields a set of 33 rules, which allows for an efficient implementation of the valence bond theory.     

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.