Vibronic Eigenstates Research Articles

Thermo-Field Dynamics Approach to Photo-induced Electronic Transitions Driven by Incoherent Thermal Radiation.

The effects of thermal light-matter interaction on the dynamics of photo-induced electronic transitions in molecules are investigated using a novel first principles approach based on the thermo-field dynamics description of both the molecular vibrational modes and of the radiation field. The developed approach permits numerically accurate simulations of quantum dynamics of electronic/excitonic systems coupled to nuclear and photonic baths kept at different temperatures. The baths can be described by arbitrary spectral densities and can have any system-bath coupling strengths. In agreement with the results obtained previously by less rigorous methods, we show that the excitation process obtained by the continuous interaction with the suddenly turned-on thermal radiation field creates a mixed ensemble having a nonnegligible component consisting of a superposition of vibronic eigenstates which can sustain coherent oscillations for relatively long times. The results become especially relevant for the dynamics of electronic transitions upon sunlight excitation. Analytical results based on time-dependent perturbation theory support the numerical simulations and provide a simple interpretation of the time evolution of quantum observables.

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine.

Unraveling the density matrix of a non-stationary quantum state as an explicit function of a few observables provides a complementary view of quantum dynamics. We have recently developed a practical way to identify the minimal set of the dominant observables that govern the quantal dynamics even in the case of strong non-adiabatic effects and large anharmonicity [Komarova et al., J. Chem. Phys. 155, 204110 (2021)]. Fast convergence in the number of the dominant contributions is achieved when instead of the density matrix we describe the time-evolution of the surprisal, the logarithm of the density operator. In the present work, we illustrate the efficiency of the proposed approach using an example of the early time dynamics in pyrazine in a Hilbert space accounting for up to four vibrational normal modes, {Q10a, Q6a, Q1, and Q9a}, and two coupled electronic states, the optically dark and the bright states. Dynamics in four-dimensional (4D) configurational space involve 19,600 vibronic eigenstates. Our results reveal that the rate of the ultrafast population decay as well as the shape of the nuclear wave packets in 2D, accounting only for {Q10a,Q6a} normal modes, are accurately captured with only six dominant time-independent observables in the surprisal. Extension of the dynamics to 3D and 4D vibrational subspace requires only five additional constraints. The time-evolution of a quantum state in 4D vibrational space on two electronic states is thus compacted to only 11 time-dependent coefficients of these observables.

Intramolecular vibrations enhance the\xa0quantum efficiency of excitonic energy transfer

We study the impact of underdamped intramolecular vibrational modes on the efficiency of the excitation energy transfer in a dimer in which each state is coupled to its own underdamped vibrational mode and, in addition, to a continuous background of environmental modes. For this, we use the numerically exact hierarchy equation of motion approach. We determine the quantum yield and the transfer time in dependence of the vibronic coupling strength, and in dependence of the damping of the incoherent background. Moreover, we tune the vibrational frequencies out of resonance with the excitonic energy gap. We show that the quantum yield is enhanced by up to 10% when the vibrational frequency of the donor is larger than at the acceptor. The vibronic energy eigenstates of the acceptor acquire then an increased density of states, which leads to a higher occupation probability of the acceptor in thermal equilibrium. We can conclude that an underdamped vibrational mode which is weakly coupled to the dimer fuels a faster transfer of excitation energy, illustrating that long-lived vibrations can, in principle, enhance energy transfer, without involving long-lived electronic coherence.

Time resolved mechanism of the isotope selectivity in the ultrafast light induced dissociation in N2.

The time evolution of a vacuum ultraviolet excited N2 molecule is followed all the way from an ultrafast excitation to dissociation by a quantum mechanical simulation. The primary aim is to discern the role of the excitation by a pulse short compared to the vibrational period, to discern the different coupling mechanisms between different electronic states, nonadiabatic, spin orbit, and to analyze the origin of any isotopic effect. We compare the picture in the time and energy domains. The initial ultrafast excitation pumps the molecule to a coherent electronic wave packet to which several singlet bound electronic states contribute. The total nonstationary wave function is given as a coherent sum of nuclear wave packets on each electronic state times the stationary electronic wave function. When the wave packets on different electronic states overlap, they are coupled in a mass-dependent manner whether one uses an adiabatic or a diabatic electronic basis. A weak spin-orbit coupling acts as a bottleneck between the bound singlet part of phase space and the triplet manifold of states in which dissociation takes place. To describe the spin-orbit perturbation that is ongoing in time, an energy-resolved eigenstate representation appears to be more intuitive. In the eigenstate basis, the singlet-to-triplet population transfer is large only between those vibronic eigenstates that are quasiresonant in energy. The states in resonance are different for different excitation energy ranges. The resonances are mass dependent, which explains the control of the isotope effect through the profile of the pulse.

Vibronic eigenstates and the geometric phase effect in the 2E″ state of NO3.

The 2E″ state of NO3, a prototype for the Jahn-Teller effect, has been an enigma and a challenge for a long time for both experiment and theory. We present a detailed theoretical study of the vibronic quantum dynamics in this electronic state, uncovering the effects of tunnelling, geometric phase, and symmetry. To this end, 45 vibronic levels of NO3 in the 2E″ state are determined accurately and analyzed thoroughly. The computation is based on a high quality diabatic potential representation of the two-sheeted surface of the 2E″ state developed by us [W. Eisfeld et al., J. Chem. Phys. 140, 224109 (2014)] and on the multi-configuration time dependent Hartree approach. The vibrational eigenstates of the NO3- anion are determined and analyzed as well to gain a deeper understanding of the symmetry properties of such D3h symmetric systems. To this end, 61 eigenstates of the NO3- anion ground state are computed using the single sheeted potential surface of the 1A1 state published in the same reference quoted above. The assignments of both the vibrational and vibronic levels are discussed. A simple model is proposed to rationalize the computed NO3 spectrum strongly influenced by the Jahn-Teller couplings, the associated geometric phase effect, and the tunnelling. Comparison with the available spectroscopic data is also presented.

Simulation of femtosecond two-dimensional electronic spectra of conical intersections.

We have simulated femtosecond two-dimensional (2D) electronic spectra for an excited-state conical intersection using the wave-function version of the equation-of-motion phase-matching approach. We show that 2D spectra at fixed values of the waiting time provide information on the structure of the vibronic eigenstates of the conical intersection, while the evolution of the spectra with the waiting time reveals predominantly ground-state wave-packet dynamics. The results show that 2D spectra of conical intersection systems differ significantly from those obtained for chromophores with well separated excited-state potential-energy surfaces. The spectral signatures which can be attributed to conical intersections are discussed.

Two-dimensional Fourier transform electronic spectroscopy at a conical intersection

We report measurement and modeling of two-dimensional (2D) electronic spectra of a silicon naphthalocyanine (SiNc) in benzonitrile, a system for which the polarization anisotropy reveals passage through a square-symmetric Jahn-Teller conical intersection in ∼100 fs [D. A. Farrow, W. Qian, E. R. Smith, A. A. Ferro, and D. M. Jonas, J. Chem. Phys. 128, 144510 (2008)]. The measured 2D Fourier transform (FT) spectra indicate loss of electronic coherence on a similar timescale. The 2D spectra arising from femtosecond vibronic dynamics through the conical funnel are modeled by full non-adiabatic treatment of the coupled electronic and vibrational dynamics for a pair of un-damped Jahn-Teller active vibrations responsible for both electronic decoherence and population transfer. Additional damped Jahn-Teller active modes that can cause only decoherence or population transfer are treated with analytical response functions that can be incorporated into the numerical non-adiabatic calculation by exploiting symmetry assignment of degenerate vibronic eigenstates to one of two electronic states. Franck-Condon active totally symmetric modes are incorporated analytically. The calculations reveal that these conical intersection dynamics alone are incapable of destroying the coherence of the initially prepared wavepacket on the experimentally observed timescale and predict an unobserved recurrence in the photon echo slice at ∼200 fs. Agreement with the experimental two-dimensional electronic spectra necessitates a role for totally symmetric vibrational dynamics in causing the echo slice to decay on a ∼100 fs timescale. This extended model also reproduces the ∼100 fs ultrafast electronic anisotropy decay in SiNc when an "asymmetric solvation mode" with a small stabilization energy of ∼2 cm(-1) is included. Although calculations show that inhomogeneities in the energy gap between excited states can broaden the anti-diagonal 2D lineshape, the anti-diagonal width is dominated by totally symmetric vibrational motions in SiNc. For this shallow conical intersection, the non-adiabatic dynamics destroy electronic coherence more slowly than they destroy electronic alignment.

Electronic flux densities in vibrating H2+in terms of vibronic eigenstates

A theoretical study of the electronic and nuclear flux densities of a vibrating ${{\mathrm{H}}_{2}}^{+}$ molecular ion is presented. The time-dependent wave function is represented in the basis of vibronic eigenstates which are numerically obtained from the complete nonrelativistic Hamiltonian without the clamped-nuclei approximation. A one-center expansion in terms of $B$-splines and Legendre polynomials is employed to solve the corresponding eigenvalue equation. The electronic and nuclear flux densities are then calculated from the total wave function through their quantum-mechanical definition. Analysis of the flux densities close to the turning points shows that the nuclear wave packet takes longer time (1.4 fs) to change its direction compared to the electronic one (1 fs).

Intermediate vibronic coupling in charge transfer states: Comprehensive calculation of electronic excitations in sexithiophene crystal

A comprehensive theory of linear vibronic coupling in a coupled manifold of Frenkel and charge-transfer states in an infinite molecular crystal is presented and applied for sexithiophene. The approach, valid in the intermediate-coupling regime, includes up to three-particle terms of the Philpott expansion, with the vibronic wavefunctions represented in the Lang-Firsov basis. As a stringent test, the scheme is used to reproduce the complete set of available sexithiophene absorption and electroabsorption spectra within a unified theoretical framework. The input is based primarily on independent calculations and to some extent on independent experiments, with explicit fitting contained within the limits set by the estimated inherent errors of a priori parameter estimates. Reasonably good quantitative agreement with experimental spectra is achieved. The results resolve some existing interpretational ambiguities and expose some peculiarities of electric field effect on vibronic eigenstates of Frenkel parentage, highlighting the role of charge-transfer interactions.

Non-Born–Oppenheimer wavepacket dynamics in polyatomic molecules: vibrations at conical intersections in DABCO

The role of vibrational dynamics in the vicinity of conical intersections is investigated using the first two electronically excited states of 1,4-diazabicyclo[2,2,2]octane (DABCO) by combining time-resolved photoelectron spectroscopy with ab initio computation. Upon resonant excitation of the origin band of the short-lived S2 (1E') state, oscillations in the electronic population between the S2 (1E') and the S1 (1A'1) electronic states are observed with a period of -3 ps. Ab initio computations are employed to characterise these low-lying excited states, which arise from single excitations into the 3s and 3p Rydberg orbitals. Although Rydberg states are generally only weakly coupled, DABCO exhibits rapid nonadiabatic dynamics. This implies that strong coupling occurs only in the immediate vicinity of a conical intersection, enabling unique identification of those vibrations which generate the nonadiabatic transitions. To this end, seams of conical intersection are located at energetically relevant geometries, engendered by differential distortions of the S1 and S2 potentials due to vibronic coupling and a Jahn-Teller-distorted S2 minimum energy point. From an analysis of the conical intersection topography, those vibrations leading to a maximal modulation of the coupling between the electronic states are readily identified. The observed oscillation in the decay of S2 state population is thereby assigned to the beat frequency between two sets of vibronic eigenstates within the S1 manifold, coherently prepared together with another set at the S2 band origin, and whose nominal e' degeneracy is lifted due to differential coupling to the Jahn-Teller-distorted components of S2.

Vibrational dynamics around the conical intersection: a study of methoxy vibrations on the X̃2E surface

The large-amplitude vibrational dynamics of methoxy (CH(3)O), resulting from the conical intersection at the C(3v) geometry, are investigated using variational methods. A two-dimensional model Hamiltonian consisting of two vibrational degrees of freedom and the coupling between them is presented in a diabatic representation of the electronic degrees of freedom. The observed complex dynamics are understood in terms of the multiple timescales that arise as the initial wave packet passes through the conical intersection. This model Hamiltonian is extended to include a full-dimensional treatment of methoxy. A quartic potential is calculated using both single and multiple configuration methods. The vibronic eigenstates are calculated using a series of basis set contraction schemes taking a primitive basis of harmonic oscillator wave functions as the starting point. Our final results, including spin-orbit coupling, are compared to experiment.

Modelling excitonic energy transfer in the photosynthetic unit of purple bacteria

Molecular mechanics and quantum chemical configuration interaction calculations in combination with exciton theory were used to predict vibronic energies and eigenstates of light harvesting antennae and the reaction centre and to evaluate excitation energy transfer rates in the photosynthetic unit of purple bacteria. Excitation energy transfer rates were calculated by using the transition matrix formalism and exciton basis sets of the interacting antenna systems. Energy transfer rates of 600–800 fs from B800 ring to B850 ring in the LH2 antenna, 3–10 ps from LH2 to LH2 antenna, 2–8 ps from LH2 to LH1 antenna and finally 30–70 ps from LH1 to the reaction centre were obtained. Dependencies of energy transfer rates on lateral and vertical inter-complex distances were determined. The results indicate that a fair amount of spatial heterogeneity of antenna complexes in the photosynthetic membrane is tolerated without much loss in excitation energy transfer efficiency.

Coherent Nuclear and Electronic Dynamics in Primary Charge Separation in Photosynthetic Reaction Centers: A Redfield Theory Approach

Primary charge separation dynamics is modeled in the pheophytin-modified Rhodobacter sphaeroides R-26 reaction center (RC). To explain the observed spectral evolution, it is assumed that the process is coupled to coherent nuclear motion. A density matrix equation with the Redfield relaxation superoperator is used for simulation of the electron-vibrational dynamics and its spectral signatures. The model includes two diabatic states, i.e., an excited state P* of the primary donor (i.e., special pair, P), and a charge-transfer state (P+B-, which is the primary photoproduct in the pheophytin-modified RC). The strong coupling of these states with two collective nuclear modes is supposed. The mixing of diabatic states (with different displacements along each of the two nuclear coordinates) results in a complicated potential surface that determines the dynamics of the excited-state wave packet. The coupled nuclear and charge-transfer dynamics is calculated in the basis of vibronic eigenstates obtained by numerical diagonalization of the electron-vibrational Hamiltonian. The third-order nonlinear response associated with excited-state dynamics is calculated, including the P* → P stimulated emission (SE) and the P+B- → P+(B-)* excited-state absorption (ESA). The model allowed us to obtain a quantitative fit of the experimental kinetics of the SE near 900−950 nm and the ESA in the 1020-nm region of the pheophytin-modified Rhodobacter sphaeroides R-26 RC (Yakovlev, A. G.; Shkuropatov, A. Ya.; Shuvalov, V. A. FEBS Lett. 2000, 466, 209). By use of the parameters adjusted from the fit, we have obtained a direct visualization of the electron-vibrational wave packet evolution, including the surface-crossing dynamics superimposed with oscillatory motion along two reaction coordinates in the P* and P+B- states. It is concluded that nonequilibrated vibrational modes involved in electron-transfer play an important role in photoproduct formation in bacterial RC. We found that the specific configuration of two vibrational coordinates (obtained from the modeling) determines high efficiency of charge separation both for coherent and noncoherent excitation.

A multimode vibronic treatment of absorption, resonance Raman, and hyper-Rayleigh scattering of excitonically coupled molecular dimers

The linear absorption spectra, resonance Raman excitation profiles and depolarization dispersion curves, and hyper-Rayleigh scattering profiles are calculated for excitonically coupled homodimers of a model electron donor–acceptor “push–pull” conjugated chromophore as a function of dimer geometry. The vibronic eigenstates of the dimer are calculated by diagonalizing the matrix of transition dipole couplings among the vibronic transitions of the constituent monomers. The absorption spectra show the usual red- or blueshifted transitions for J-type or H-type dimers, respectively. When the electronic coupling is large compared with the vibronic width of the monomer spectrum, the dimer absorption spectra exhibit simple Franck–Condon progressions having reduced vibronic intensities compared with the monomer, and the resonance Raman excitation profiles are shifted but otherwise only weakly perturbed. When the coupling is comparable to the vibronic width, the H-dimer absorption spectra exhibit irregular vibronic frequency spacings and intensity patterns and the effects on the Raman excitation profiles are larger. There is strong dispersion in the Raman depolarization ratios for dimer geometries in which both transitions carry oscillator strength. The first hyperpolarizabilities are somewhat enhanced in J-dimers and considerably reduced in H-dimers. These effects on the molecular β will amplify the effects of dimerization on the ground-state dipole moment in electro-optic materials formed from chromophore-doped polymers that must be electric field poled to obtain the net alignment needed for a macroscopic χ(2).

Laser induced dispersed fluorescence spectroscopy of 107 vibronic levels of NO2 ranging from 12 000 to 17 600 cm−1

We report the laser induced dispersed fluorescence spectra (LIDFS) of NO2 recorded when exciting 107 vibronic levels lying between 11 961 and 17 627 cm−1. These levels result from the vibronic interactions between the highly excited vibrational levels of the electronic ground state, X̃ 2A1, and the isoenergetic levels of the electronic excited state, à 2B2. One observes that the intensity patterns of almost half of the LIDFS can be interpreted in terms of Franck–Condon factors (FCFs) between a single bright parent level belonging to à 2B2 and the low-lying vibrational levels of X̃ 2A1. This means that the vibronic interactions occurring between the vibrational levels of X̃ 2A1 and à 2B2 are not strong enough to systematically mix together several bright parent states of à 2B2. We have also found that vibronic eigenstates scattering over several hundreds of cm−1 can be assigned to the same bright parent state. However, these features, which are characteristic of individual vibronic levels, must be faced with the statistical properties of the NO2 visible spectrum, well-known to exemplify molecular quantum chaos, at least above 16 500 cm−1 [J. Chem. Phys. 103, 1732 (1995)]. Complementarily, we briefly present results of the breakdown of vibrational and rotational selection rules, as observed in LIDFS. This phenomenon reflects the onset of strong vibronic and rovibronic interactions that progressively destroy the above mentioned one-bright-parent-state model and leads, at higher energies, to a regime of rovibronic chaos.

Femtosecond pump-probe spectroscopy of polyatomic molecules in condensed phases.

A theory for ultrafast pump-probe spectroscopy of large polyatomic molecules in condensed phases is developed. A multimode Brownian oscillator model is used to account for high-frequency molecular vibrations and local intermolecular modes as well as collective solvent motions. A semiclassical picture is provided using the density matrix in Liouville space. The pump field creates a doorway state that propagates for a specified time interval, and the spectrum is calculated by finding its overlap with a window state, prepared by the probe pulse. The doorway and the window states are wave packets in phase space. For high-frequency modes and with long pulses they are expanded in the vibronic eigenstates, whereas for low-frequency modes and with impulsive pulses the Wigner (phase-space) representation is more adequate. Conditions for the observation of quantum beats, spectral diffusion, and solvation dynamics (dynamical Stokes shift) are specified.

The Effect of an E-Type Quadratic Jahn-Teller Coupling on the Eigenstates in theT×t2Vibronic States

Numerical studies of the vibronic eigenstates in the case when an orbital triplet states is coupled linearly and quadratically to a t 2 vibrational mode have been made. For a fixed value of the linear Jahn-Teller coupling strength in the intermediate coupling region, the effects of an E -type quadratic Jahn-Teller coupling strength on the vibronic eigenstates of the Jahn-Teller system have been investigated. As the quadratic Jahn-Teller coupling strength increases, the feature of an orthorhombic Jahn-Teller distortion has been clearly shown in the vibronic properties in the sixfold degenerate ground state.

The visible absorption spectrum of NO 2: A three-mode nuclear dynamics investigation

The vibronic band origins of the visible absorption spectrum of NO 2 are calculated theoretically with the aid of a simple model Hamiltonian for the coupled electronic and vibrational motions. Including all three vibrational modes in the calculation and using ab initio values of the relevant parameters, we obtain satisfactory qualitative agreement with experiment. In particular, the observed high density and irregular intensity distribution of the band origins is reproduced correctly by the calculation. The present results confirm unambiguosly that the anomalous vibronic structure of the 2B 2 ← 2A 1 transition is caused by strong nonadiabatic interactions between the 2 B 2 and 2 A 1 electronic states of NO 2. They also show that simple deconvolution procedures, which are often used to deperturb irregular spectra, are not applicable to the 2B 2 ← 2A 1 transition of NO 2. To further explore the strength of the nonadiabatic effects in NO 2, we calculate the mixing of the different electronic species in the vibronic eigenstates and compare it to several relevant experimental quantities.

Vibronic reduction factors of the excited states in the E

Extensive numerical studies of the vibronic eigenstates in the case when an orbital doublet state is coupled linearly and nonlinearly to an e vibrational mode with cubic symmetry have been made. For a fixed value of the linear Jahn-Teller coupling in the strong-coupling region, the effects of the quadratic Jahn-Teller coupling on the vibronic properties, such as Ham's reduction factors or the reduction factors for vibrational angular-momentum operators, have been studied. At a critical value of the quadratic Jahn-Teller coupling coefficient, where the behavior of near-level crossing in the excited E states has been seen, the abrupt change of Ham's reduction factor and the fluctuations in both the admixture coefficient and the reduction factor for the vibrational angular-momentum operator have been found in the excited E state. Similar effects of the lowest-order anharmonic term on the vibronic reduction factors have been found.

The vibronic properties of the T×t2Jahn-Teller system in the strong-coupling region

Numerical studies of the vibronic eigenstates in the case when an orbital triplet state is coupled linearly to a t2 vibrational mode in cubic symmetry have been made. The vibronic energy levels and eigenvectors, as a function of pure linear coupling in the strong-coupling region, have been computed again. A detailed analysis of the vibrational contributions in the vibronic eigenvectors has been made. The possibility of, and conditions under which, the symmetry is lowered from cubic to trigonal have been investigated from the properties of the vibronic eigenvectors. The effect of a local strain on a trigonal Jahn-Teller distortion is discussed from the vibronic model.