Effective method for the computation of optical spectra of large molecules at finite temperature including the Duschinsky and Herzberg-Teller effect: the Qx band of porphyrin as a case study

The authors extend their recent method for the computation of vibrationally resolved optical spectra of large molecules, including both the Duschinsky rotation and the effect of finite temperature in the framework of the Franck-Condon (FC) approximation, to deal with the more general case of the Herzberg-Teller (HT) model, where also the linear dependence of the transition dipole moment on the nuclear coordinates is taken into account. This generalization allows us to investigate weak and vibronically allowed transitions by far extending the range of application of the method. The calculation of the spectra of sizable molecules is computationally demanding because of the huge number of final vibrational states that must be taken into account, and the inclusion of HT terms further increases the computational burden. The method presented here automatically selects the relevant vibronic contributions to the spectrum, independent of their frequency, and it is able to provide fully converged spectra with a modest computational requirement. The effectiveness of the method is illustrated by computing the HT absorption and fluorescence Q(x) spectra of free-base porphyrin both at T=0 K and at room temperature, performing for the first time an exact treatment of vibrations in harmonic approximation. Q(x) spectra are compared to experiments and FC/HT interferences are analyzed in detail.     

Resonantly enhanced multiphoton ionization and zero kinetic energy photoelectron spectroscopy of benzo[g,h,i]perylene

We report zero kinetic energy (ZEKE) photoelectron spectroscopy of benzo[g,h,i]perylene (BghiP) via resonantly enhanced multiphoton ionization (REMPI). Our analysis concentrates on the vibrational modes of both the first electronically excited state and the ground cationic state. Extensive vibronic coupling due to a nearby electronically excited state manifests through strong Franck-Condon (FC) forbidden bands, which are stronger than even the FC allowed bands in the REMPI spectrum. Theoretical calculations using Gaussian are problematic in identifying the electronic configurations of the excited electronic states and predicting the transition energies. However, by setting the keyword for the second excited electronic state, both density functional theory and configuration interaction methods can reproduce the observed spectrum qualitatively. The general agreement significantly helps with the vibrational assignment. The ZEKE spectra demonstrate propensity in preserving the vibrational excitation of the intermediate electronic state. In addition, almost all ZEKE spectra exhibit a similar vibrational distribution, and the distribution can be reproduced by an FC calculation from the vibronic origin of the first excited electronic state to the cationic state using Gaussian 09. These results suggest a remarkable structural stability of BghiP in accommodating the additional charge. All observed vibrational bands of the cation are IR active, establishing the role of ZEKE spectroscopy in mapping out far-infrared bands for astrophysical applications.     

Efficient calculation of short-range Coulomb energies

An efficient algorithm for the calculation of short-range Coulomb energies is examined. The algorithm uses a boxing scheme and a prescreening for negligible integrals to evaluate the short-range Coulomb energy via computational work that scales only linearly with the size of the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 921–927, 1999     

A new method to calculate Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect

Calculations of Franck-Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. We have developed a new method for calculating Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect. Closed-form formulas of two-, three-, and four-dimensional Franck-Condon factors were derived straightforwardly by using the properties of Hermite polynomials and Gaussian integrals. This new method was applied to study the photoelectron spectra of H(2)O(+)(B (2)B(2)) and D(2)O(+)(B (2)B(2)), whose equilibrium geometries and harmonic vibrational frequencies were calculated by using the coupled cluster singles and doubles with perturbative triples [CCSD(T)] method together with the basis sets of 6-311++G(3df,2pd) and aug-cc-pVTZ. The adiabatic ionization energies were computed by using the CCSD(T) method extrapolated to the complete basis set limit with aug-cc-pVXZ (X=D,T,Q,5). It was found that the simulated photoelectron spectra were mainly composed of nu(2) progressions and the combination bands of nu(1) and nu(2), whereas pure nu(1) transitions should be too weak to be observable, contrary to the literature reports. It was also found that the first discernible peak in the experimental photoelectron spectra did not correspond to the adiabatic transition. The adiabatic ionization energies of H(2)O(+)(B (2)B(2)) and D(2)O(+)(B (2)B(2)) are proposed to be 16.78 and 16.83 eV, about 0.40 and 0.58 eV lower than the best experimental values, respectively. Conversely, the calculated ionization energies are in agreement with the proposed values within 0.02 eV.     

Breakdown of the mirror image symmetry in the optical absorption/emission spectra of oligo(para-phenylene)s

The absorption and emission spectra of most luminescent, pi-conjugated, organic molecules are the mirror image of each other. In some cases, however, this symmetry is severely broken. In the present work, the asymmetry between the absorption and fluorescence spectra in molecular systems consisting of para-linked phenyl rings is studied. The vibronic structure of the emission and absorption bands is calculated from ab initio quantum chemical methods and a subsequent, rigorous Franck-Condon treatment. Good agreement with experiment is achieved. A clear relation can be established between the strongly anharmonic double-well potential for the phenylene ring librations around the long molecular axis and the observed deviation from the mirror image symmetry. Consequences for related compounds and temperature dependent optical measurements are also discussed.     

Symmetric double-well potential model and its application to vibronic spectra: studies of inversion modes of ammonia and nitrogen-vacancy defect centers in diamond

In this paper, we have studied the vibronic transitions between two symmetric double-well potentials by proposing a model Hamiltonian consisting of a harmonic oscillator and a parturition described by a Gaussian function that leads to a double minima potential with a barrier between the two energy minima. Making use of the contour integral form of Hermite polynomials, we present a new formula that can calculate Franck-Condon factors of the system rigorously. The simulated vibronic spectra of ammonia and the negatively charged nitrogen-vacancy center in diamond are presented as an application of the formula.     

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.     

Ab initio calculations and Franck-Condon simulation of the absorption spectra of GeCl2 including anharmonicity.

Geometrical parameters, vibrational frequencies and relative electronic energies of the X1A1, ?3B1 and A1B1 states of GeCl2 have been calculated at the CCSD(T) and/or CASSCF/MRCI level with basis sets of up to aug-cc-pV5Z quality. Core electron correlation and relativistic contributions were also investigated. RCCSD(T)/ aug-cc-pVQZ potential energy functions (PEFs) of the X1A1 and ?3B, states, and a CASSCF/MRCl/aug-cc-pVQZ PEF of the A1B1 state of GeCl2 are reported. Anharmonic vibrational wavefunctions of these electronic states of GeCl2, obtained variationally using the computed PEFs, are employed to calculate the Franck-Condon factors (FCFs) of the ?-X and A-X transitions of GeCl2. Simulated absorption spectra of these transitions based on the computed FCFs are compared with the corresponding experimental laser-induced fluorescence (LIF) spectra of Karolczak et al. [J. Chem. Phys. 1993, 98, 60-70]. Excellent agreement is obtained between the simulated absorption spectrum and observed LIF spectrum of the ?-X transition of GeCl2, which confirms the molecular carrier, the electronic states involved and the vibrational assignments of the LIF spectrum. However, comparison between the simulated absorption spectrum and experimental LIF spectrum of the A-X transition of GeCl2 leads to a revision of vibrational assignments of the LIF spectrum and suggests that the X1A1 state of GeCl2 was prepared in the experimental work, with a non-Boltzmann vibrational population distribution. The X(0,0,1) level is populated over 4000 times more than expected from a Boltzmann distribution at 60 K, which is appropriate for the relative population of the other low-lying vibrational levels, such as the X(1,0,0) and X(0,1,0) levels.     

High Resolution Laser Excitation Spectra and Franck-Condon Factors of A2Π-X2Σ+ Electronic Transition of MgF

Magnesium monofluoride (MgF) is proposed as an ideal candidate radical for direct laser cooling. Here, the rotationally resolved laser spectra of MgF for the A²Π-X²Σ⁺ electronic transition system were recorded by using laser induced fluorescence technique. The MgF radicals were produced by discharging SF₆/Ar gas mixtures between the tips of two magnesium needles in a supersonic jet expansion. We recorded a total of 19 vibrational bands belonging to three sequences of Δv=0, ±1 in the region of 348-370 nm. Accurate spectroscopic constants for both X²Σ⁺ and A²Π states are determined from rotational analysis of the experimental spectra. Spectroscopic parameters, including the Franck-Condon factors (FCFs), are determined from the experimental results and the Rydberg-Klein-Rees (RKR) calculations. Significant discrepancies between the experimentally measured and RKR-calculated FCFs are found, indicating that the FCFs are nearly independent of the spin-orbit coupling in the A²Π state. Potential energy curves (PECs) and FCFs determined here provide necessary data for the theoretical simulation of the laser-cooling scheme of MgF.     

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-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals

The authors propose a new algorithm, "local K" (LK), for fast evaluation of the exchange Fock matrix in case the Cholesky decomposition of the electron repulsion integrals is used. The novelty lies in the fact that rigorous upper bounds to the contribution from each occupied orbital to the exchange Fock matrix are employed. By formulating these inequalities in terms of localized orbitals, the scaling of computing the exchange Fock matrix is reduced from quartic to quadratic with only negligible prescreening overhead and strict error control. Compared to the unscreened Cholesky algorithm, the computational saving is substantial for systems of medium and large sizes. By virtue of its general formulation, the LK algorithm can be used also within the class of methods that employ auxiliary basis set expansions for representing the electron repulsion integrals.     

Franck-Condon simulation, including anharmonicity, of the photodetachment spectrum of P2H(-): restricted-spin coupled-cluster single-double plus perturbative triple and unrestricted-spin coupled-cluster single-double plus perturbative triple -F12x potential energy functions of P2H and P2H(-)

Geometry optimization and harmonic vibrational frequency calculations have been carried out on the X?(2)A(') state of P(2)H and the X?(1)A(') state of P(2)H(-) using the restricted-spin coupled-cluster single-double plus perturbative triple excitation [RCCSD(T)] and explicitly correlated unrestricted-spin coupled-cluster single-double plus perturbative triple excitation [UCCSD(T)-F12x] methods. For RCCSD(T) calculations, basis sets of up to the augmented correlation-consistent polarized valence quintuple-zeta (aug-cc-pV5Z) quality were employed, and contributions from extrapolation to the complete basis set limit and from core correlation of the P 2s(2)2p(6) electrons were also included. For UCCSD(T)-F12x calculations, different atomic orbital basis sets of triple-zeta quality with different associated complementary auxiliary basis sets and different geminal Slater exponents were used. When the P 2s(2)2p(6) core electrons were correlated in these F12x calculations, appropriate core-valence basis sets were employed. In addition, potential energy functions (PEFs) of the X?(2)A(') state of P(2)H and the X?(1)A(') state of P(2)H(-) were computed at different RCCSD(T) and UCCSD(T)-F12x levels, and were used in variational calculations of anharmonic vibrational wavefunctions, which were then utilized to calculate Franck-Condon factors (FCFs) between these two states, employing a method which includes allowance for anharmonicity and Duschinsky rotation. The photodetachment spectrum of P(2)H(-) was then simulated using the computed FCFs. Simulated spectra obtained using the RCCSD(T)/aug-cc-pV5Z and UCCSD(T)-F12x(x = a or b)/aug-cc-pCVTZ PEFs are compared and found to be essentially identical. Based on the computed FCFs, a more detailed assignment of the observed vibrational structure than previously reported, which includes "hot bands," has been proposed. Comparison between simulated and available experimental spectra has been made, and the currently most reliable sets of equilibrium geometrical parameters for P(2)H and its anion have been derived. The photodetachment spectrum of P(2)D, yet to be recorded, has also been simulated.     

Cavity ring-down spectroscopy and theoretical calculations of the S1(1B3u)<--S0(1Ag) transition of jet-cooled perylene

As part of our long-term program to test the diffuse interstellar band-polycyclic aromatic hydrocarbon hypothesis, we have investigated the S(1)<--S(0) electronic transition of neutral perylene (C(20)H(12)) in a combined experimental and theoretical study. Jet-cooled perylene was prepared with a pulsed discharge slit nozzle and detected by cavity ring-down spectroscopy. A number of vibronic features were observed in the 24 000-24 900 cm(-1) spectral range. Density functional and ab initio calculations were performed to determine the geometries, harmonic vibrational frequencies, and normal coordinates of both the S(0) and S(1) electronic states. A rotational temperature of 52+/-5 K was derived from a rotational contour analysis of the vibronic band associated with the 0-0 transition. A Franck-Condon treatment was carried out to calculate the vibronic spectrum of the S(1)<--S(0) transition. A good agreement was found between the calculated and the experimental spectra. A vibrational assignment is proposed and six normal modes are identified. The contribution of neutral compact polycyclic aromatic hydrocarbons to the diffuse interstellar bands is briefly discussed.     

Franck-Condon simulation of the single vibronic level emission spectra of HSiF and DSiF including anharmonicity

Potential energy functions (PEFs) of the X (1)A(') and A (1)A(") states of HSiF have been computed using the coupled-cluster single-double plus perturbative triple excitations and complete-active-space self-consistent-field multireference internally contracted configuration interaction methods, respectively, employing augmented correlation-consistent polarized-valence quadruple-zeta basis sets. For both electronic states of HSiF and DSiF, anharmonic vibrational wavefunctions and energies of all three modes have been calculated variationally with the ab initio PEFs and using Watson's Hamiltonian for nonlinear molecules. Franck-Condon factors between the two electronic states, allowing for Duschinsky rotation, were computed using the calculated anharmonic vibrational wavefunctions. These Franck-Condon factors were used to simulate the single vibronic level (SVL) emission spectra recently reported by Hostutler et al. in J. Chem. Phys. 114, 10728 (2001). Excellent agreement between the simulated and observed spectra was obtained for the A (1)A(")(1,0,0)-->X (1)A(') SVL emission of HSiF. Discrepancies between the simulated and observed spectra of the A (1)A(")(0,1,0) and (1,1,0) SVL emissions of HSiF have been found. These are most likely, partly due to experimental deficiencies and, partly to inadequacies in the ab initio levels of theory employed in the calculation of the PEFs. Based on the computed Franck-Condon factors, minor revisions of previous vibrational assignments are suggested. The calculated anharmonic wave functions of higher vibrational levels of the X (1)A(') state show strong mixings between the three vibrational modes of HSi stretching, bending, and SiF stretching.     

Vibrationally resolved circular dichroism spectra of a molecule with isotopically engendered chirality

We present a theoretical study of vibrationally resolved circular dichroism spectra, both in the adiabatic and non-adiabatic frameworks, with a full account of Franck-Condon and Herzberg-Teller vibrational contributions for the former. Model calculations have been performed on 2(R)-deuteriocyclopentanone, whose chirality is due solely to isotopic substitution. This molecule has two distinct, nearly isoenergetic, half-chair conformations in equilibrium, and its demanding nature in terms of computational accuracy makes it a perfect candidate for performing a detailed comparison between different vibronic models. Comparisons are made with experimental spectra, and we also consider temperature effects. In order to reproduce the experimental spectrum, it is necessary to consider the geometry relaxation occurring during the n→π* transition.     

Charge transfer vibronic transitions in uranyl tetrachloride compounds

The electronic and vibronic interactions of uranyl (UO(2))(2+) in three tetrachloride crystals have been investigated with spectroscopic experiments and theoretical modeling. Analysis and simulation of the absorption and photoluminescence spectra have resulted in a quantitative understanding of the charge transfer vibronic transitions of uranyl in the crystals. The spectra obtained at liquid helium temperature consist of extremely narrow zero-phonon lines (ZPL) and vibronic bands. The observed ZPLs are assigned to the first group of the excited states formed by electronic excitation from the 3σ ground state into the f(δ,?) orbitals of uranyl. The Huang-Rhys theory of vibronic coupling is modified successfully for simulating both the absorption and luminescence spectra. It is shown that only vibronic coupling to the axially symmetric stretching mode is Franck-Condon allowed, whereas other modes are involved through coupling with the symmetric stretching mode. The energies of electronic transitions, vibration frequencies of various local modes, and changes in the O═U═O bond length of uranyl in different electronic states and in different coordination geometries are evaluated in empirical simulations of the optical spectra. Multiple uranyl sites derived from the resolution of a superlattice at low temperature are resolved by crystallographic characterization and time- and energy-resolved spectroscopic studies. The present empirical simulation provides insights into fundamental understanding of uranyl electronic interactions and is useful for quantitative characterization of uranyl coordination.     

Calculation of vibronic spectra of polyatomic molecules in the franck—condon and herzberg—teller approximations: Part I. Methods for calculating matrix elements

A number of methods, both accurate and approximate, have been developed to calculate the integrals arising in the theory of vibronic spectra of polyatomic molecules. These methods make it possible to calculate the vibrational structure of the electron spectra of polyatomic molecules in the most general approximations (one-electron, adiabatic) without imposing any particular restrictions (neglect of the Dushinsky effect, temperature distribution of molecules, etc.). A comparison is made of the suggested methods, the approximate methods are analyzed for limits of accuracy, and the optimal fields of application are pointed out.     

Vibronic coupling and intramolecular dynamics of pyrene as revealed by the S0→S2 excitation spectrum in a supersonic jet

The fluorescence excitation spectrum of pyrene obtained in a supersonic jet for the S₀→S₂ transition shows a complicated structure due to the interaction of discrete levels of S₂ with the quasi-continuous levels belonging to S₁. The intensity distribution pattern in this region has been evaluated from quantum-mechanically calculated quantities, such as the vibronic coupling integrals, and Franck-Condon factors deduced from independent experiments. The dynamics in pyrene following excitation into S₂ are discussed.     

Vibronic induced one- and two-photon absorption in a charge-transfer stilbene derivate

Both the electronic and the vibronic contributions to one- and two-photon absorption of a D-pi-D charge-transfer molecule (4-dimethylamino-4'-methyl-trans stilbene) are studied by means of density functional response theory combined with a linear coupling model. Vibronic profiles of the first four excited states are fully explored. The dominating vibrational modes for both Franck-Condon and Herzberg-Teller contributions are identified. The Franck-Condon contribution dominates the spectra of first, second, and fourth excited states. The Herzberg-Teller contribution is on the other hand of comparable size for the third excited state, where its inclusion leads to a blueshift with respect to the vertical transition. A similar vibronic coupling behavior is found for both one- and two-photon absorptions.