Duschinsky Matrix Research Articles

Generating Function Approach to Single Vibronic Level Fluorescence Spectra.

An efficient time-dependent generating function method to compute vibronic emission and absorption spectra arising from transitions from a singly excited vibrational initial state is presented. In contrast to existing finite temperature approaches that intrinsically contain these transitions weighted by a Boltzmann factor, the current approach allows one to calculate these transitions individually. Using vibrational frequencies and normal modes computed by the second-order approximate coupled cluster (CC2) method, this formalism is used to compute the single vibronic level (SVL) fluorescence spectra of anthracene. Calculated spectra are in excellent agreement with spectra measured in jet-cooled expansion experiments. Duschinsky mixing is necessary to explain intensities of certain peaks. In a few cases, CC2, however, underestimates Duschinsky mixing, leading to too low peak intensities. An empirical correction of the Duschinsky matrix is presented. The presented method has the potential to facilitate the assignment and interpretation of SVL fluorescence spectra.

Computational Spectroscopic Characterization of a Bistable Binuclear Complex [(CO)2(benzoate)FeII/III(terephthalate)CoIII/II(benzoate)(CO)2

Electric dipole moments, polarizabilities, and IR, Raman, optical rotatory dispersion, and electronic and vibrational circular dichroism spectra of the four cis–trans isomers of the proposed [(CO)2(benzoate)FeII/III(terephthalate)CoIII/II(benzoate)(CO)2]+ binuclear complex, having bistablity due to intramolecular charge transfer (IMCT), is investigated using the time-dependent density functional theory ((TD)DFT) B3LYP/6–31G(d,p)[LanL2DZ] method. Results show that the two FeII–CoIII and FeIII–CoII IMCT states of this binuclear complex have distinctly different spectroscopic, optical, and electric response properties, and are sensitive to the cis–trans arrangement of the ligands around the two metallic centres. Furthermore, intrinsic reaction coordinates inter-connecting the two IMCT states are identified using the Duschinsky matrix method. Only one or two of the normal coordinates remain almost (above 80 %) intact during the IMCT reaction which denotes global changes in the bonding strengths and potential energy hypersurface of this bistable binuclear complex. Analysis of the calculated spin densities characterizes the IMCT transition state structures of the trans–trans, cis–cis, and trans–cis isomers as early, early, and late transition states, respectively.

Algorithm for solving the inverse vibronic problem on the basis of SVL fluorescence spectra

Computer methods whereby the inverse vibronic problem is solved on the basis of resonance fluorescence spectra with the use of modern quantum-mechanical methods for constructing structuraldynamic models of polyatomic molecules are discussed. An algorithm is proposed for solving the inverse vibronic problem according to resonance fluorescence spectra under laser excitation, and the corresponding calculation programs are constructed. The initial program data are acquired by means of an original software package which implements the scaling of quantum-mechanical force fields in two electronic states. The Duschinsky matrix and the initial matrix of shifts in normal coordinates caused by electron excitation are calculated in the Cartesian and natural vibrational coordinates. The program data are taken from quantum-molecular models based on calculations performed via ab initio modern quantum-mechanical methods and density functional theory. The algorithm is tested through the calculation of a model molecular system.

A quantum-mechanical analysis of trans-acrolein vibrational spectra in the ground S0 and excited T1 and S1 electronic states

The optimization of geometrical parameters and calculation of the force fields of trans-acrolein (CH2CHCHO) in the ground S0 and lowest excited triplet (T1) and singlet (S1) electronic states are performed at the CASPT2/def2-TZVPP theoretical level. Co-assignments of the vibrational frequencies calculated from the corresponding force fields are validated by calculations of the analogs of the Duschinsky matrix for the pairs of the (S0, T1) and (S0, S1) electronic states. The results obtained by comparing the vibrational frequencies calculated from the scaled force fields with the experimental vibrational frequencies made it possible to corroborate some details of the molecular structure in these electronic states.

Revisiting Vertical Models To Simulate the Line Shape of Electronic Spectra Adopting Cartesian and Internal Coordinates.

Vertical models for the simulation of spectroscopic line shapes expand the potential energy surface (PES) of the final state around the equilibrium geometry of the initial state. These models provide, in principle, a better approximation of the region of the band maximum. At variance, adiabatic models expand each PES around its own minimum. In the harmonic approximation, when the minimum energy structures of the two electronic states are connected by large structural displacements, adiabatic models can breakdown and are outperformed by vertical models. However, the practical application of vertical models faces the issues related to the necessity to perform a frequency analysis at a nonstationary point. In this contribution we revisit vertical models in harmonic approximation adopting both Cartesian (x) and valence internal curvilinear coordinates (s). We show that when x coordinates are used, the vibrational analysis at nonstationary points leads to a deficient description of low-frequency modes, for which spurious imaginary frequencies may even appear. This issue is solved when s coordinates are adopted. It is however necessary to account for the second derivative of s with respect to x, which here we compute analytically. We compare the performance of the vertical model in the s-frame with respect to adiabatic models and previously proposed vertical models in x- or Q1-frame, where Q1 are the normal coordinates of the initial state computed as combination of Cartesian coordinates. We show that for rigid molecules the vertical approach in the s-frame provides a description of the final state very close to the adiabatic picture. For sizable displacements it is a solid alternative to adiabatic models, and it is not affected by the issues of vertical models in x- and Q1-frames, which mainly arise when temperature effects are included. In principle the G matrix depends on s, and this creates nonorthogonality problems of the Duschinsky matrix connecting the normal modes of initial and final states in adiabatic approaches. We highlight that such a dependence of G on s is also an issue in vertical models, due to the necessity to approximate the kinetic term in the Hamiltonian when setting up the so-called GF problem. When large structural differences exist between the initial and the final-state minima, the changes in the G matrix can become too large to be disregarded.

Calculation of Franck–Condon factors and simulation of photoelectron spectra of the HCCl− anion: Including Duschinsky effects

Halomethylenes (such as HCCl) are particularly critical intermediates in many gas-phase reaction environments. The structures, spectra, and reactivity of environment-related molecules have been of many studies interest. In the present work, geometry optimization and frequency calculations have been carried out on the X˜2A″ state of HCCl− and the X˜1A′ and a˜3A″ states of HCCl at CCSD(T) theory. The single point energy, electron affinity and term energy of HCCl have been computed up to the CCSD(T)/aug-cc-pV5Z level and extrapolated to the complete basis set limit. The Duschinsky matrix and displacement vector have been considered at the CCSD(T) level of theory and the result shows that the normal mode mixing effects play a main role for HCCl(a˜3A″)–HCCl−(X˜2A″) transition, and which can be neglected for the HCCl(X˜1A′)–HCCl−(X˜2A″) process. Franck–Condon analysis and spectral simulations have been performed on the singlet and triplet photo-detachment processes, respectively. We have merged the singlet and triplet transitions together for the first time and the results show that the simulated spectra are very consistent with the previous experimental one.

Temperature Dependence of Radiative and Nonradiative Rates from Time-Dependent Correlation Function Methods.

The temperature dependence of the rate constants in radiative and nonradiative decays from excited electronic states has been studied using a time-dependent correlation function approach in the framework of the adiabatic representation and the harmonic oscillator approximation. The present work analyzes the vibrational aspect of the processes, which gives rise to the temperature dependence, with the inclusion of mode-mixing, as well as of frequency change effects. The temperature dependence of the rate constants shows a contrasting nature, depending on whether the process has been addressed within the Franck-Condon approximation or beyond it. The calculation of the Duschinsky matrix and the shift vector between the normal modes of the two states can be done in Cartesian and/or internal coordinates, depending on the flexibility of the investigated molecule. A new computational code has been developed to calculate the rates of intersystem crossing, internal conversion, and fluorescence for selected molecules as functions of temperature.

A combined ab initio and Franck–Condon simulation study of the photodetachment spectra of the HCBr− anion

Geometry optimization and harmonic vibrational frequency calculations have been performed on the X˜2A″ state of HCBr−, and X˜1A′ and a˜3A″ states of HCBr. The term energy and electron affinity of HCBr were calculated and extrapolated to the complete basis set limit. The Duschinsky matrix and displacement vector were calculated at the CCSD(T)/aug-cc-pVQZ level. The normal mode mixing effects played a minor role and can be neglected for the HCBr(X˜1A′)–HCBr−(X˜2A″) photodetachment. Spectral simulations involved the HCBr(X˜1A′)–HCBr−(X˜2A″) and HCBr(a˜3A″)–HCBr−(X˜2A″) photodetachments were carried out on HCBr− at CCSD(T)/aug-cc-pVQZ level, respectively.

High Resolution Electronic Spectroscopy of Vibrationally Hot Bands of Benzimidazole

Rotationally resolved electronic spectra of seven vibrationally excited bands in the electronic spectrum of benzimidazole have been measured and analyzed. From the vibrational contributions to the rotational constants, an assignment of the hot bands could be made on the basis of anharmonic corrections to the harmonic normal modes and by using the information contained in the Duschinsky matrix calculated by second order coupled cluster (CC2) theory. Fluorescence emission and (hot) absorption spectra of benzimidazole from Jalviste and Treshchalov [Chem. Phys. 1993, 172, 325] have been simulated using Franck-Condon integrals obtained from CC2 optimized geometries and Hessians.

Harmonic Models in Cartesian and Internal Coordinates to Simulate the Absorption Spectra of Carotenoids at Finite Temperatures.

When large structural displacements take place between the ground state (GS) and excited state (ES) minima of polyatomic molecules, the choice of a proper set of coordinates can be crucial for a reliable simulation of the vibrationally resolved absorption spectrum. In this work, we study two carotenoids that undergo structural displacements from GS to ES minima of different magnitude, from small displacements for violaxanthin to rather large ones for β-carotene isomers. Their finite-temperature (77 and 300 K) spectra are simulated at the harmonic level, including Duschinsky effect, by time-dependent (TD) and time-independent (TI) approaches, using (TD)DFT computed potential energy surfaces (PES). We adopted two approaches to construct the harmonic PES, the Adiabatic (AH) and Vertical Hessian (VH) models and, for AH, two reference coordinate frames: Cartesian and valence internal coordinates. Our results show that when large displacements take place, Cartesian coordinates dramatically fail to describe curvilinear displacements and to account for the Duschinsky matrix, preventing a realistic simulation of the spectra within the AH model, where the GS and ES PESs are quadratically expanded around their own equilibrium geometry. In contrast, internal coordinates largely amend such deficiencies and deliver reasonable spectral widths. As expected, both coordinate frames give similar results when small displacements occur. The good agreement between VH and experimental line shapes indicates that VH model, in which GS and ES normal modes are both evaluated at the GS equilibrium geometry, is a good alternative to deal with systems exhibiting large displacements. The use of this model can be, however, problematic when imaginary frequencies arise. The extent of the nonorthogonality of the Dushinsky matrix in internal coordinates and its correlation with the magnitude of the displacement of the GS and ES geometries is analyzed in detail.

Vibrational co-assignment of the calculated frequencies in the ground and two lowest excited electronic states

The feasibility of co-assignments of the vibrational frequencies in the ground and T1 and S1 excited electronic states has been demonstrated for trans-C2O2F2. Matrices analogous to the Duschinsky matrix were used to juxtapose the vibrational frequencies of this molecule calculated at the CASPT2/cc-pVTZ level in the ground S0 and excited triplet T1 and singlet S1 electronic states. The calculations suggest that the calculated CC and CF stretching frequencies of trans-C2O2F2 in these three electronic states should be mutually reassigned in comparison with the previous interpretation.

Isotopic shifts versus potential energy distribution?

Optimizations of geometries and calculations of vibrational wavenumbers and potential energy distributions for trans/ cis oxalyl fluoride were performed at the MP2(FC)/aug-cc-pVQZ theoretical level. The isotopic shifts were calculated for trans- 13C 2O 2F 2, cis- 13C 2O 2F 2, trans- 14C 2O 2F 2 and cis- 14C 2O 2F 2 from the unscaled force fields. The mutual co-assignments of all the calculated wavenumbers were obtained using matrices analogous to the Duschinsky matrix. The assignments of the calculated wavenumbers of trans- and cis- 12C 2O 2F 2 to a specific vibration based on the potential energy distribution and the isotopic shifts were compared, and it is shown that the assignments of the C–F and C–C stretching vibrations ( a g / a 1) depend dramatically on which of these is employed in making the determination. The validity of the latter approach is clearly demonstrated.

Integrated computational approach to vibrationally resolved electronic spectra: Anisole as a test case

A new general and effective procedure to compute Franck-Condon spectra from first principles is exploited to elucidate the subtle features of the vibrationally resolved optical spectra of anisole. Methods based on the density functional theory and its time-dependent extension for electronic excited states [B3LYP6-311+G(d,p) and TD-B3LYP6-311+G(d,p)] have been applied to geometry optimizations and harmonic frequency calculations. Perturbative anharmonic frequencies [J. Chem. Phys. 122, 014108 (2005)] have been calculated for the ground state, and the Duschinsky matrix elements have been used to evaluate the corresponding anharmonic corrections for the first excited electronic state. The relative energetics of both electronic states has been refined by single point calculations at the coupled clusters (CC) level with the aug-cc-pVDZ basis set. Theoretical spectra have been evaluated using a new optimized implementation for the effective computation of Franck-Condon factors. The remarkable agreement between theoretical and experimental spectra allowed for revision of some assignments of fundamental vibrations in the S(1) state of anisole.

Multidimensional Franck-Condon simulations of photodetachment spectra for the formate-water cluster anion: Investigating H atom transfer along the HCOOH+OH reaction coordinate

A new multidimensional Franck-Condon (FC) simulation methodology was applied to an anionic-neutral cluster transition for the first time to investigate the use of photodetachment spectroscopy of the HCOO(-).H(2)O anion as a means to study the HCOO.H(2)O and HCOOH.OH neutral clusters. For the HCOO(-).H(2)O to HCOO.H(2)O transition, vibrationally resolved simulated spectra were obtained across the threshold detachment region, indicating that photodetachment spectroscopy of the respective anionic cluster should provide detailed structural information on the bifurcated HCOO.H(2)O neutral cluster. The simulations predict that the photodetachment spectra should display prominent progressions of both the intermolecular stretch and the in-plane OCO bending mode. In contrast, for the HCOO(-).H(2)O to HCOOH.OH transition, the vibronic FC simulations resulted in transitions with negligible intensities, despite the fact that the geometries of the respective anionic and neutral systems were similar. The low FC intensities were traced to the large off-diagonal elements of the Duschinsky matrix for this transition, which arise due to the considerable differences in the vibrational wave functions following hydrogen transfer.

Low-frequency vibrations specific for conformers of 1-aminoindan studied by UV-UV hole-burning spectroscopy

The UV-UV hole-burning spectra of the jet-cooled 1-aminoindan were measured for the first time. Complicated spectral features observed in the laser-induced fluorescence excitation spectrum due to two conformers, R and B, were firmly separated. On the basis of fluorescence measurements and B3LYP/cc-pVTZ calculations, low-frequency ring twisting and ring puckering modes were assigned. These modes are coupled in the S1 state due to the Duschinsky rotation. The Duschinsky matrix was calculated from the normal modes predicted by quantum chemical calculations. The coupling between the twisting and puckering modes for conformer B is stronger than that for conformer R. The twisting mode was observed at 0+99 cm(-1) in the S1 state for conformer B, while not for conformer R. The Franck-Condon activity of the twisting mode substantially differs between the two conformers. The transition to the twisting level for conformer B would be allowed by the Duschinsky rotation. The fluorescence lifetime of conformer vibronic levels was also measured and differed for each conformer.

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.

CASSCF and CASPT2 studies on the structures, transition energies, and dipole moments of ground and excited states for azulene.

The electronic structure of azulene molecule has been studied. We have obtained the optimized structures of ground and singlet excited states by using the complete active space self-consistent-field (CASSCF) method, and calculated vertical and 0-0 transition energies between the ground and excited states with second-order Møller-Plesset perturbation theory (CASPT2). The CASPT2 calculations indicate that the bond-equalized C(2v) structure is more stable than the bond-alternating C(s) structure in the ground state. For a physical understanding of electronic structure change from C(2v) to C(s), we have performed the CASSCF calculations of Duschinsky matrix describing mixing of the b(2) vibrational mode between the ground (1A(1)) and the first excited (1B(2)) states based on the Kekule-crossing model. The CASPT2 0-0 transition energies are in fairly good agreement with experimental results within 0.1-0.3 eV. The CASSCF oscillator strengths between the ground and excited states are calculated and compared with experimental data. Furthermore, we have calculated the CASPT2 dipole moments of ground and excited states, which show good agreement with experimental values.

Ab Initio Computation of the Duschinsky Mixing of Vibrations and Nonlinear Effects

We present an analysis of the Duschinsky effect and its application to real molecules. We discuss the many subtle aspects of applying the theory to calculations and give examples of a nonlinear normal coordinate transformation. We show how to judge if nonlinear effects are small enough to be neglected through use of the zero-order axis-switching approximation, which allows calculation of Franck−Condon factors (FCF). However, even with the zero-order axis-switching approximation, nonorthogonality can occur in the Duschinsky matrix, and this must be corrected to allow proper FCF calculations. We have calculated the Duschinsky effect for two systems that form the anion in an electron-transfer ion pair, V(CO)6- and Co(CO)4-. The formation of the D3d neutral vanadium species is accompanied by a small geometric distortion and small Duschinsky effect, despite the change in point group from Oh. We discuss how to perform the calculations to properly represent degenerate vibrations and how to test if the linear app...

Transfer of force constants and scaling factors in vibrational problems

This paper discusses the predictive abilities of the methods of force constant and scaling factor transfer to the force fields of related molecules in solving vibrational problems for the latter. It is noted that the method of scaling the quantum mechanical force fields and transferring the scaling factors is preferable. The vibrational spectra of the rotational isomers of butadiene-1,3 are considered as an example. The vibrational frequency assignments are tested using a matrix analogous to the Duschinsky matrix.

The S(1A g)–S1(1B2u) vibronic transition in benzene: An ab initio study

The four e2g false origin bands and the a1g progression of the S0(1Ag)–S1(1B2u) transition in benzene are simulated ab initio with the recently introduced configuration interaction singles (CIS) with 6-31G orbitals. The ground and excited state CC and CH bond lengths are optimized and compared with the experiment; the CC bond elongation upon excitation is found to be slightly underestimated. The vibrational force fields are calculated at the stationary points of S0 and S1. The 1Ag force field is calculated at the Hartree–Fock level while the 1B2u force field is calculated at the CIS level of theory. The two force fields are scaled to fit the experimental frequencies and the normal mode rotation upon excitation, i.e., the Duschinsky matrix, is obtained. In agreement with previous empirical fitting of the S1(1B2u) vibrational frequencies, the Duschinsky matrix is found to be nearly diagonal with the exception of the b2u modes submatrix which shows a large amount of mixing. The mixing of the b2u modes is larger before scaling but is subsequently reduced after scaling. The normal modes and the optimized geometries are used to calculate the amount of displacement, upon excitation, of the equilibrium position of the totally symmetric modes. This displacement causes the Franck–Condon progression and is slightly underestimated by the calculation. The intensity of the four e2g false origins in the absorption spectrum of S1 is calculated and the Herzberg–Teller intensities of the four bands are found to be very close to the experiment. In particular, the relative intensity of the CCC bend (ν6) and CC stretch (ν8) bands is nicely reproduced. This result is discussed in light of similar calculations at the semiempirical level of theory. We conclude that CIS can be of great value for the unravelling of vibronic spectra of conjugated systems.