The use of multidimensional Franck-Condon simulations to assess model chemistries: a case study on phenol

Multidimensional Franck-Condon simulations of the dispersed fluorescence spectra of phenol generated with geometries obtained from the highly correlated post-Hartree-Fock methods CASSCF, MRCI, and SACCI are presented. While the simulations based on CASSCF and MRCI optimized geometries are very similar to each other and fail to reproduce the experimentally measured intensities faithfully, the simulations obtained from SACCI optimized geometries are very close to the experimental spectra. The code developed for the multidimensional Franck-Condon simulations is described. It is shown that the integral storage problem common to the evaluation of multidimensional Franck-Condon integrals can be overcome by saving all quantities needed to disk. This strategy allows the code to run on computers with limited resources and is very well suited for application to molecules with a very large number of vibrational modes.     

Franck-Condon factors within damped displacement harmonic oscillators: Solvent-enhanced absorption and fluorescence spectra

Damped harmonic oscillators that take into account local-mode nuclear vibrations interacting with solvent molecules are developed into Franck-Condon factors within displaced harmonic oscillator approximation. This is practically done by scaling an unperturbed Hessian matrix that represents local modes of force constants for molecules in a gaseous phase, and then by diagonalizing the perturbed Hessian matrix it results in direct modification of Huang–Rhys factors which represent normal modes of solute molecule perturbed by solvent environment. For highly symmetric polycyclic aromatic hydrocarbon molecules in which hydrogen atom vibrations in a solution can be scaled equally, one-set scaling parameters constructed into damped Franck-Condon factors can reproduce solvent-enhanced absorption and fluorescence spectra in solution. However, for low symmetry molecules with atoms other than hydrogen and carbon atoms, multi-set scaling parameters constructed into damped Franck-Condon factors can also reproduce solvent-enhanced absorption and fluorescence spectra in solution. Examples for high symmetry perylene in benzene solution with one-set scaling parameters and for low symmetry carbazole in n-hexane solution with multi-set scaling parameters are given, in both cases, the present damped Franck-Condon simulation can reproduce solvent-enhanced absorption and fluorescence spectra in solution.     

Franck-Condon theory of reactive scattering

A Franck-Condon theory of reactive scattering is introduced which generalizes previous works in a number of respects. We consider the collinear AB + C → A + BC reaction where A, B, and C may be polyatomic species and show how the multi-dimensional continuum-continuum Franck-Condon integral can exactly be reduced to a two-dimensional one involving the nonorthogonal reaction coordinates on the reactant and product diabatic surfaces. These integrals are then written in a rapidly convergent series of products of one-dimensional bound-continuum integrals of a form closely related to those studied in recent theories of photodissociation where accurate analytical and numerical methods are available for treating them. The theory is applied to the case of the D + Hl isotope exchange for a model surface to enable a comparison of exact quantum calculations with those of the Franck-Condon theory for the identical surface in both cases. The calculations are all performed at energies where the reaction is classically forbidden. The relative Dl vibrational distributions (as a function of initial Hl state) are accurately reproduced in a fashion that is fairly insensitive to the choice of the reactant and product diabatic surfaces, but the absolute probabilities are shown to be sensitively dependent on this choice.     

Semiclassical bound-continuum Franck-Condon factors uniformly valid at 4 coinciding critical points: 2 Crossings and 2 turning points

A uniform semiclassical approximation to bound-continuum Franck-Condon factors is derived by applying differential topological mapping techniques on a suitable three dimensional integralrepresentation. This approach even holds uniformly if two potentialcurve intersections (real or complex conjugate) and two turning points come close or coincide. The resulting Franck-Condon matrix element is expressed in terms of two generic swallowtail (A₄) integrals whose unfolding parameters are obtained from a single algebraic equation amenable to fast standard routines. Transitional approximations to this result are shown to cover all previously known approaches and to lead to a generalization of a formula of K. Sando and F. H. Mies [1]. A simple and fast trapezoidal method to evaluate the generic swallowtail integrals and other generic integrals of odd determinacy is presented, which even permits the derivation of numerical error bounds.     

The Huggins band of ozone: assignment of hot bands

The "hot bands" of the Huggins band of ozone are assigned, in both the 218 K and the 295 K spectrum. The assignment is based on intensities calculated with three-dimensional vibrational wave functions for the electronic ground state (X) and the excited state (B). The hot-band structures in the 218 K spectrum all can be assigned to transitions starting from vibrational states with one quantum of stretching excitation in the ground electronic state. The 295 K spectrum shows new structures, which are due to transitions originating from vibrational states in the X state with two quanta of excitation of the stretching modes--despite very small Boltzmann factors. All structures in the low-energy range of the 295 K spectrum, even the very weak ones, thus can be uniquely interpreted. The significance of hot bands results from the strong increase of Franck-Condon factors with excitation of the stretching modes in both the lower and/or the upper electronic states, whose equilibrium bond lengths differ significantly.     

A method to reduce the size of the vibronic basis employed in the simulation of spectra using the multimode vibronic coupling approximation

In the time-independent multimode approach for the determination of vibronic spectra involving strongly coupled electronic states, the equilibrium geometry and normal modes of the reference or precursor state are usually employed as the basis for the multimode expansion. This basis, while easily constructed, is generally ill-suited for determining the eigenstates of the observed species. Employing a more computationally effective basis requires the evaluation of Franck-Condon overlap integrals. Using established generalized Hermite polynomial generating function formalisms, an algorithm is developed that can efficiently determine the enormous requisite number of these overlap integrals. It is found that this flexibility in the choice of multimode basis can significantly reduce the size of the basis needed to obtain converged spectral simulations. The previously reported spectrum of the ethoxy (C(2)H(5)O) radical serves as an example of the efficacy of the new technique.     

Theoretical investigation of doubly resonant IR-UV sum-frequency vibrational spectroscopy of binaphthol chiral solution

The doubly resonant IR-UV sum-frequency vibrational spectroscopy (SFVS) of 1,1'-bi-2-naphthol (BN) solution and its dispersion spectra are analyzed and computed using the ZINDO//AM1 calculation and the direct approach of Raman scattering tensor calculation, which is based on calculations of Franck-Condon factors and on differentiation of the electronic transition moments with respect to the vibrational normal modes. The calculated results indicate that, for the most intense vibrational bands observed in the SFVS experiment, the calculated frequencies, symmetry, order, intensities, and pattern of the enhanced vibrational modes agree with experiment qualitatively, and due to the Franck-Condon progression, there are the doublet peaks in the corresponding resonant sum-frequency dispersion spectra. The polarization resonance Raman spectra of BN for the vibrational modes appearing in SFVS are also computed and associated with the experiment SFVS of BN. This direct evaluation approach of Raman tensors may provide a way of assigning the doubly resonant IR-UV SFVS.     

An efficient approach for the calculation of Franck-Condon integrals of large molecules

A general and efficient approach for the calculation of Franck-Condon integrals (FCIs) of large molecules is presented. In a first step, by exploiting the diagonally dominant and sparse structure of the Duschinsky matrix, a model system is constructed for which the Duschinsky matrix takes a block-diagonal form. For each of these blocks separately, the FCIs are calculated discarding all below a certain threshold. From those integrals retained the FCIs of the model system are obtained by simple multiplication. These serve as an estimate for the FCIs of the exact system which are calculated for those integrals which lie above a certain threshold. By systematically decreasing the threshold, the simulation can be reliably converged to the exact result with an arbitrary accuracy. Using this scheme, a considerable reduction of the number of FCIs which have to be calculated is achieved which leads to an improved scaling behavior of the computational effort with system size. The approach has been tested thoroughly for a set of molecules including difficult cases. For the larger systems a speedup of up to three orders of magnitude compared to an exact calculation is observed while the errors can be kept negligible. With this approach accurate calculations of FCIs are feasible also for large molecules encountered in "real-life" chemistry, especially biochemistry and material science.     

Matrix elements for internal conversion

The usual separation of internal conversion matrix elements into a momentum integral of the promoting modes and an overlap integral of the accepting mode is shown to introduce large errors. The proper procedure leads to the replacement of Franck-Condon factors by integrals involving the square root of an energy operator.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. II. Four-electron integrals

An alternative procedure to the classical method for evaluating the four-electron Hylleraas-CI integrals is given. The method consists of direct integration over the r ₁₂ coordinate and integration over the coordinates of one of the electrons, reducing the integrals to lower order. The method based on the earlier work of Calais and L?wdin and of Perkins is extended to the general angular case. In this way it is possible to solve all of the four-electron integrals appearing in the Hylleraas-CI method. The four-electron integrals are expanded in three-electron ones which are in turn expanded in two-electron integrals. Finally the two-electron integrals are expanded into two-electron auxiliary integrals which usually have one negative power. The use of three- and four-electron electron auxiliary integrals is avoided. Some special cases lead to one- and two-electron auxiliary integrals with negative powers which do not converge individually but do converge in combination with others. These relations and their solutions are presented, together with results of various kinds of integrals.     

Multimode approach to hydrogen tunneling

A procedure is developed to improve the quantitative evaluation of hydrogen transfer rates in polyatomic molecules and solids. The aim is to introduce a dynamical model that includes explicitly all vibrations participating in the transfer. This aim favors adoption of the golden-rule approach, since it treats all vibrational modes equally. To simplify the resulting multidimensional transfer integrals, two basic assumptions are introduced: (i) adiabatic separability of the hydrogenic modes directly involved in the tunneling from the other modes, and (ii) negligible anharmonicity for the latter. The number of effectively participating modes can then be reduced drastically by transformation to an appropriate local representation which allows analytical integration over most of these other modes. Those that remain involve vibrations of the atoms between which the hydrogen is transferred. Their frequency, reduced mass and displacement are expressed in terms of the harmonic force field of the system before and after transfer and can be unambiguously evaluated if these force fields are available. These modes replace the empirical effective modes used previously. The theory is applied successfully to single hydrogen transfer in dimethyl-glyoxime and double hydrogen transfer in porphine.     

Quantum chemistry study on internal conversion of diphenyldibenzofulvene in solid phase

We investigate the nonradiative decay process of diphenyldibenzofulvene (DPDBF) in solid phase by using the quantum chemistry methods. To carry out the nonradiative rate constant calculation, we construct a solid phase model based on the ONIOM method. The geometry of the DPDBF molecule is optimized for the ground state by DFT and the first excited state by TD-DFT, and the corresponding vibrational frequencies and normal coordinates are computed. Under displaced-distorted harmonic oscillator potential approximation, Huang-Rhys factors are obtained. Vibronic coupling constants are calculated as a function of the normal mode based on Domcke's scheme. We find that vibronic coupling constants of 12 modes with large reorganization energies are of similar order, and if this result is still valid for other modes, the internal conversion rate would be determined by high frequency modes because they have a significant nuclear factor that is related to Franck-Condon overlap intergrals. We also find that geometrical changes are suppressed due to the stacking effect, which yields small Huang-Rhys values in the solid phase.     

The vibrational structures of furan, pyrrole, and thiophene cations studied by zero kinetic energy photoelectron spectroscopy

The vibrational structures of the electronic ground states ((approximately)X (2)A(2)) of furan, pyrrole, and thiophene cations have been studied by zero kinetic energy (ZEKE) photoelectron spectroscopic method. In addition to the strong excitations of the symmetric a(1) vibrational modes, other three symmetric vibrational modes (a(2), b(1), and b(2)) have been observed unambiguously. These results which cannot be explained by the Franck-Condon principle illustrate that the vibronic coupling and the Coriolis coupling may play important roles in understanding the vibrational structures of the five-membered heterocycle cations. The vibrationally resolved ZEKE spectra are assigned with the assistance of the density function theory calculations, and the fundamental frequencies for many vibrational modes have been determined for the first time. The first adiabatic ionization energies for furan, pyrrole, and thiophene were determined as 8.8863, 8.2099, and 8.8742 eV, respectively, with uncertainties of 0.0002 eV.     

Calculation of Franck-Condon factors including anharmonicity: simulation of the C2H4+X2B3u<--C2H4X1A(g) band in the photoelectron spectrum of ethylene

Our new simple method for calculating accurate Franck-Condon factors including nondiagonal (i.e., mode-mode) anharmonic coupling is used to simulate the C2H4+X2B3u<--C2H4X1A(g) band in the photoelectron spectrum. An improved vibrational basis set truncation algorithm, which permits very efficient computations, is employed. Because the torsional mode is highly anharmonic it is separated from the other modes and treated exactly. All other modes are treated through the second-order perturbation theory. The perturbation-theory corrections are significant and lead to a good agreement with experiment, although the separability assumption for torsion causes the C2D4 results to be not as good as those for C2H4. A variational formulation to overcome this circumstance, and deal with large anharmonicities in general, is suggested.     

The temperature dependence of radiationless transition rates from ab initio computations

The calculation of radiationless transition rates and of their temperature dependence from first principles is addressed by combining reliable electronic computations of the normal modes of the two electronic states with Kubo's generating function approach for the evaluation of the Franck-Condon weighted density of states. The whole sets of normal modes of the involved cofactors have been employed, taking into account the effects of nuclear equilibrium position displacements, of vibrational frequency changes, and of mixing of the normal modes. Application to the case of the elementary electron transfer step between bacteriopheophytin and ubiquinone cofactors of bacterial photosynthetic reaction centers yields a temperature dependence of the electron transfer rates in very good agreement with the experimental data.     

Spectroscopy and thermochemistry of a jet-cooled open-shell polyene: 1,4-pentadienyl radical

The 1,4-pentadienyl (vinylallyl) radical has been observed for the first time by optical spectroscopy. An excitation spectrum is recorded on m/z 67 by resonant two-color two-photon ionization spectroscopy. Several bands are observed with the origin transition identified at 19 449 cm(-1). The spectrum is assigned by a comparison with ab initio frequencies calculated at the CASPT2/cc-pVTZ level of theory, with an accompanying Franck-Condon calculation of the excitation spectrum, including Dushinsky mixing. The b(1) and a(2) outer C-C bond torsional modes are calculated to halve in frequency upon electronic excitation, bringing about their appearance in the excitation spectrum. This can be readily understood by considering the torsional sensitivity of the frontier molecular orbital energies. High-level quantum chemical calculations of the radical stabilization energy, resulting in a value of nearly 120 kJ mol(-1), provide quantitative confirmation that this radical is highly stabilized.     

The role of C-H and C-C stretching modes in the intrinsic non-radiative decay of triplet states in a Pt-containing conjugated phenylene ethynylene

The intrinsic non-radiative decay (internal conversion) from the triplet excited state in phosphorescent dyes can be described by a multi-phonon emission process. Since non-radiative decay of triplet excitons can be a significant process in organic light-emitting diodes, a detailed understanding of this decay mechanism is important if the overall device efficiency is to be controlled. We compare a deuterated Pt(II)-containing phenylene ethynylene with its non-deuterated counterpart in order to investigate which phonon modes control to the non-radiative decay path. We observe that deuteration does not decrease the non-radiative decay rate. A Franck-Condon analysis of the phosphorescence spectra shows that the electronic excitation is coupled strongly to the breathing mode of the phenyl ring and the C≡C carbon stretching modes, while high-energy C-H or C-D stretching modes play an insignificant role. We, therefore, associate the internal conversion process with the carbon-carbon stretching vibrations.     

Ab initio based potential energy surfaces and Franck-Condon analysis of ionization thresholds of cyclic-C3H and linear-C3H

We report a Franck-Condon analysis in reduced dimensionality of the ionization thresholds of linear(l)-C3H and cyclic(c)-C3H using MP2-based potential energy surfaces and CCSD(T)/aug-cc-pVTZ calculations of electronic energies at selected geometries. The potential energy surfaces are fits to tens of thousands of MP2/aug-cc-pVTZ energies for the neutral and cation systems. These fits properly describe the invariance of the potential with respect to all permutations of the three C atoms. The realism of the potential surfaces is assessed by comparing stationary-point structures, energies, and normal-mode frequencies with previous high-level ab initio calculations. Several key vibrational modes in this ionization process are located at saddle points and so a numerical approach to obtain the Franck-Condon factors for those modes is done. On the basis of this analysis combined with a simple harmonic treatment of the energies of the remaining modes and key electronic energy differences obtained with CCSD(T)/aug-cc-pVTZ calculations, we find the threshold ionization energy of l-C3H to be 9.06 eV and for c-C3H we estimate the threshold to be in the range 9.70-9.76 eV. We estimate these values are accurate to within +/-0.05 eV.