Harmonic Potential Surfaces Research Articles

Topological aspects of system-bath Hamiltonians and a vector model for multisite systems coupled to local, correlated, or common baths.

Some topological features of multisite Hamiltonians consisting of harmonic potential surfaces with constant site-to-site couplings are discussed. Even in the absence of Duschinsky rotation, such a Hamiltonian assumes the system-bath form only if severe constraints exist. The simplest case of a common bath that couples to all sites is realized when the potential minima are collinear. The bath reorganization energy increases quadratically with site distance in this case. Another frequently encountered situation involves exciton-vibration coupling in molecular aggregates, where the intramolecular normal modes of the monomers give rise to local harmonic potentials. In this case, the reorganization energy accompanying excitation transfer is independent of site-to-site separation, thus this situation cannot be described by the usual system-bath Hamiltonian. A vector system-bath representation is introduced, which brings the exciton-vibration Hamiltonian in system-bath form. In this, the system vectors specify the locations of the potential minima, which in the case of identical monomers lie on the vertices of a regular polyhedron. By properly choosing the system vectors, it is possible to couple each bath to one or more sites and to specify the desired initial density. With a collinear choice of system vectors, the coupling reverts to the simple form of a common bath. The compact form of the vector system-bath coupling generalizes the dissipative tight-binding model to account for local, correlated, and common baths. The influence functional for the vector system-bath Hamiltonian is obtained in a compact and simple form.

Franck-Condon and Herzberg-Teller Signatures in Molecular Absorption and Emission Spectra.

Some molecules of chemical and biological significance possess vibrations with significant Herzberg-Teller (HT) couplings, which render the Franck-Condon (FC) approximation inadequate and cause the breakdown of the well-known mirror-image symmetry between linear absorption and emission spectra. Using a model two-state system with displaced harmonic potential surfaces, we show analytically that the FC-HT interference gives rise to asymmetric intensity modification, which has the same sign for all transitions on one side of the 0-0 absorption line and the opposite sign in the equivalent fluorescence transitions, while the trend is exactly reversed for all transitions on the other side the 0-0 line. We examine the dependence of the absorption-emission asymmetry on the mode frequency, Huang-Rhys factor, and dipole moment parameters to show the recovery of symmetry with particular combinations of parameters and a crossover from fluorescence to absorption dominance. We illustrate the analytical predictions through numerically exact calculations in models of one and two discrete vibrational modes and in the presence of a harmonic dissipative bath. In addition to homogeneous broadening effects, we identify large asymmetric shifts of absorption and emission band maxima, which can produce the illusion of unequal frequencies in the ground and excited potential surfaces as well as a nontrivial modulation of spectral asymmetry by temperature, which results from the enhancement of transitions on one side of the 0-0 line. These findings will aid the interpretation of experimental spectra in HT-active molecular systems.

Two-dimensional mobile breather scattering in a hexagonal crystal lattice.

We describe the full two-dimensional scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated with the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two-dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60^{∘} scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

Theoretical analysis of vibronic structure in absorption and fluorescence spectra of polyatomic molecules beyond the Condon approximation: Application to 11Ag ↔ 21Ag and 11Ag ↔ 11Bu electronic transitions in all-trans-1,3,5,7-octatetraene

Calculations of the vibronic structure in electronic absorption and fluorescence spectra of a large polyatomic molecule based on time- and frequency-domain approaches beyond the Condon approximation are developed. In time-domain approach, the molecular spectra are expressed in terms of Fourier transforms of appropriate time correlation functions, which are evaluated in closed form for displaced–distorted–rotated harmonic potential surfaces at finite temperature. In frequency-domain approach, use is made of the Franck–Condon factors which were evaluated in closed form for the same potential surfaces in our previous work (J. Molec. Spectrosc. 194 (1999) 179). The vibronic structure in dipole-forbidden (but vibronic-allowed) 11Ag↔21Ag and dipole-allowed 11Ag↔11Bu electronic transitions of all-trans-1,3,5,7-octatetraene have been calculated and compared with experimental data as illustrations of the approaches. Our vibronic structure calculations support the hypothesis that a sudden opening of a non-radiative decay channel occurs at 21Ag vibronic levels with energies larger than about 2100cm−1.

A Benchmark Study of Different Methods for Calculating One- And Two-Dimensional Optical Spectra

The accuracy and robustness of several approximate methods for computing linear and nonlinear optical spectra are considered. The analysis is performed in the context of a benchmark model that consists of a two-state chromophore with shifted harmonic potential surfaces that differ in frequency. The exact one- and two-dimensional spectra for this system are calculated and compared to spectra calculated via the following approximate methods: (1) The semiclassical forward-backward initial-value representation (FB-IVR) method; (2) the linearized semiclassical (LSC) method; (3) the standard second-order cumulant approximation which is based on the ground-state equilibrium frequency-frequency correlation function (2OC); (4) an alternative second-order cumulant approximation which is able to account for nonequilibrium dynamics on the excited-state potential surface (2OCa). All four approximate methods can be shown to reproduce the exact results when the frequencies of the ground and excited harmonic surfaces are identical. However, allowing for the ground and excited surfaces to differ in frequency leads to a more meaningful benchmark model for which none of the four approximate methods is exact. We present a comparison of one- and two-dimensional spectra calculated via the above-mentioned approximate methods to the corresponding exact spectra, as a function of the following parameters: (1) The ratio of excited state to ground-state frequencies; (2) Temperature; (3) The horizontal displacement of the excited-state potential relative to the ground-state potential; (4) The waiting time between the coherence periods in the case of two-dimensional spectra. The FB-IVR method is found to predict spectra which are practically indistinguishable from the exact ones over a wide region of parameter space. The LSC method is found to predict spectra which are in good agreement with the exact ones over the same region of parameter space. The 2OC and 2OCa are found to be highly inaccurate unless the frequencies of the ground and excited states are very similar. These observations give credence to the use of the LSC method for modeling spectra in complex systems, where exact or even FB-IVR-based calculations are prohibitively expensive.

Femtosecond stimulated Raman scattering for polyatomics with harmonic potentials: Application to rhodamine 6G

The perturbation theory of stimulated Raman scattering (SRS), with Raman pump on minus pump off and heterodyne detection along the probe direction, is reviewed. It has four third-order polarization terms, labeled as SRS or inverse Raman scattering (IRS): SRS(I), SRS(II), IRS(I), and IRS(II). These four polarizations have a wave packet interpretation. The polarizations, with homogenous and inhomogeneous broadening included, can be written as integrals over four-time correlation functions, and analytic formulas are derived for the latter for multidimensional harmonic potential surfaces with Franck-Condon displacements in the modes which facilitates the calculation of the SRS cross sections. The theory is applied to understand recent experimental results on the femtosecond SRS (FSRS) of a fluorescent dye, rhodamine 6G (R6G), where the Raman pump pulse is about 1 ps long, and the probe pulse is about 10 fs. The calculations compared very well with the R6G experimental results for off-resonance and resonance FSRS spectra spanning both Stokes and anti-Stokes bands, and for negative and positive pump-probe delay times on resonance.

Signatures of Nonequilibrium Solvation Dynamics on Multidimensional Spectra

Multidimensional electronic and vibrational spectroscopies have established themselves over the last decade as uniquely detailed probes of intramolecular structure and dynamics. However, these spectroscopies can also provide powerful tools for probing solute-solvent interactions and the solvation dynamics that they give rise to. To this end, it should be noted that multidimensional spectra can be expressed in terms of optical response functions that differ with respect to the chromophore's quantum state during the various time intervals separating light-matter interactions. The dynamics of the photoinactive degrees of freedom during those time intervals (that is, between pulses) is dictated by potential energy surfaces that depend on the corresponding state of the chromophore. One therefore expects the system to hop between potential surfaces in a manner dictated by the optical response functions. Thus, the corresponding spectra should reflect the system's dynamics during the resulting sequence of nonequilibrium solvation processes. However, the interpretation of multidimensional spectra is often based on the assumption that they reflect the equilibrium dynamics of the photoinactive degrees of freedom on the potential surface that corresponds to the chromophore's ground state. In this Account, we present a systematic analysis of the signature of nonequilibrium solvation dynamics on multidimensional spectra and the ability of various computational methods to capture it. The analysis is performed in the context of the following three model systems: (A) a two-state chromophore with shifted harmonic potential surfaces that differ in frequency, (B) a two-state atomic chromophore in an atomic liquid, and (C) the hydrogen stretch of a moderately strong hydrogen-bonded complex in a dipolar liquid. The following computational methods are employed and compared: (1) exact quantum dynamics (model A only), (2) the semiclassical forward-backward initial value representation (FB-IVR) method (models A and B only), (3) the linearized semiclassical (LSC) method (all three models), and (4) the standard ground-state equilibrium dynamics approach (all three models). The results demonstrate how multidimensional spectra can be used to probe nonequilibrium solvation dynamics in real time and with an unprecedented level of detail. We also show that, unlike the standard method, the LSC and FB-IVR methods can accurately capture the signature of solvation dynamics on the spectra. Our results also suggest that LSC and FB-IVR yield similar results in the presence of rapid dephasing, which is typical in complex condensed-phase systems. This observation gives credence to the use of the LSC method for modeling spectra in complex systems for which an exact or even FB-IVR-based calculation is prohibitively expensive.

Theoretical study on S1(1B3u) state electronic structure and absorption spectrum of pyrazine

Making use of a set of quantum chemistry methods, the harmonic potential surfaces of the ground state (S0(1Ag)) and the first (S1(1B3u)) excited state of pyrazine are investigated, and the electronic structures of the two states are characterized. In the present study, the conventional quantum mechanical method, taking account of the Born-Oppenheimer adiabatic approximation, is adopted to simulate the absorption spectrum of S1(1B3u) state of pyrazine. The assignment of main vibronic transitions is made for S1(1B3u) state. It is found that the spectral profile is mainly described by the Franck-Condon progression of totally symmetric mode ν6a. For the five totally symmetric modes, the present calculations show that the frequency differences between the ground and the S1(1B3u) state are small. Therefore the displaced harmonic oscillator approximation along with Franck-Condon transition is used to simulate S1(1B3u) absorption spectra. The distortion effect due to the so-called quadratic coupling is demonstrated to be unimportant for the absorption spectrum, except the coupling mode ν10a. The calculated S1(1B3u) absorption spectrum is in reasonable agreement with the experimental spectra.

Theoretical investigations of spectroscopy and excited state dynamics of adenine

A set of harmonic potential surfaces have been obtained at the CASSCF/6-311+G ∗∗ level for the ground, n–π ∗, and π–π ∗ states of 9H-adenine. They have been used to analyze experimental R2PI and resonance fluorescence spectra and the excited state lifetimes of adenine. A tentative plausible assignment has been proposed, which takes into account a strong coupling between these electronic states. Using the potential surfaces, the excess energy dependence of the excited state lifetimes of adenine has been interpreted, and the lifetime of the n–π ∗ state at the band origin has also been estimated to be in the order of 10 ns.

Femtosecond spectroscopy study of electronically excited states of Chlorophyll a molecules in ethanol

A femtosecond transient absorption technique is used to investigate the vibrational frequencies of the first electronically excited state of Chlorophyll a. From the quantum beats analysis, we obtain vibrational frequencies within 98–791 cm −1. Based on these frequencies, we calculate the absorption and fluorescence spectra using the displaced–distorted harmonic potential surface model. The ratio of the absorption intensities of the first and second excited states yields ∼7.5. We perform various excitations to obtain anisotropy profiles and find that they vary between −0.005 ± 0.005 and 0.41 ± 0.015. To understand the kinetics appearing in the profiles, we propose a microscopic model.

A Theoretical Analysis of Absorption Spectra and Dynamics of Photosynthetic Reaction Centers

AbstractIn this work, we construct a microscopic model with which one can reconstruct all the experimentally observed characteristics of the RCs including temperature effects. For this purpose, we analyze the absorption spectra of Rb. sphaeroides RCs (R26 and R26. Phe‐a mutants). We apply the excitonic‐vibronic model with the displaced harmonic potential surface model for the nuclear freedoms and determine microscopic properties such as the electronic coupling constants, the vibrational frequencies, and the displacements among the potential surfaces of these electronic states. We also take into account inhomogeneity of RCs in our theoretical analysis. We introduce a thermal expansion model to treat temperature dependent peak shifts observed in the P band of the absorption spectra of the RCs. With the resulting information, we also calculate the rate constants of the dynamic processes occurring in RCs, determining the electronic coupling constants among the electronic states involved in electron transfer and electronic energy transfer of the RCs. Based on the excitonic‐vibronic model, we also investigate the mechanism of the direct path PB*H → P+B−H recently observed in YM210W mutant.

Coupled Electronic States in trans-Dioxo Complexes of Rhenium(V) and Osmium(VI) Probed by Near-Infrared and Visible Luminescence Spectroscopy

trans-Dioxo complexes of rhenium(V) and osmium(VI) with ethylenediamine, imidazole, and oxalate ancillary ligands show luminescence maxima varying between 620 and 900 nm. The resolved vibronic progressions of the rhenium complexes have intensity distributions that cannot be rationalized with harmonic potential energy surfaces. The broadband spectra of the osmium(VI) complexes also show deviations from the band shapes expected for a harmonic ground-state potential surface. We use the large variation of luminescence energies and the vibronic features to show the influence of excited electronic states on the ground-state potential energy surface. The vibronic structure and band envelopes of the luminescence spectra are calculated using the lowest energy adiabatic surface obtained from three interacting electronic states.

Ab Initio Calculations of Vibronic Spectra and Dynamics for Small Polyatomic Molecules: Role of Duschinsky Effect

The Duschinsky effect has been shown to be significant in spectroscopy and dynamics of molecules that involve the π−π* transitions. In this paper, we present a derivation of exact expressions for optical absorption and radiationless transitions in polyatomic molecules with displaced−distorted−rotated harmonic potential surfaces. In the formulation, we take into account the temperature effect exactly. The application of this new formulation is demonstrated for ethylene and allene, where the Duschinsky effect in the first singlet excited electronic state is very strong.

Coherent nuclear motion in a condensed-phase environment: Wave-packet approach and pump–probe spectroscopy

A quantum-mechanical Gaussian wave-packet approach to the theoretical description of nuclear motions in a condensed-phase environment is developed. General expressions for the time-dependent reduced density matrix are given for a harmonic potential surface, and the exact quantum dynamics is found for a microscopic system-plus-bath model. Particular attention is devoted to the influence of initial correlations between system and bath for the outcome of a pump–probe experiment. We show that the standard factorized preparation, compared to a more realistic correlated preparation, leads to significantly different stimulated emission spectra at high temperatures. Recent experiments for the reaction center are analyzed using this formalism.

Langevin modes analysis of myoglobin

Langevin modes describe the behavior of atoms moving on a harmonic potential surface subject to viscous damping described by a classical Langevin equation. We present applications to myoglobin. The Langevin modes are obtained from the gas-phase normal modes using a perturbation expansion. The friction matrix in the Langevin description is assumed diagonal thus ignoring hydrodynamic interactions. The diagonal elements, which are the atomic friction constants on each atom, are weighted according to the surface area accessible to the solvent. Time-dependence of the position and velocity correlation functions are calculated for the Fe atom buried inside the protein for varying values of the external solvent viscosity. Even for negligibly small values of the viscosity, the time correlation functions are overdamped and suggest a substantial damping component from the neighboring protein atoms. The effective friction seen by the Fe atom has contributions from both the solvent friction as well as the internal protein friction. The internal protein friction is estimated to be about 14 ps−1 and is comparable to the friction of water. The viscosity dependence of the inverse of the effective friction on the Fe atom is found to agree qualitatively with the viscosity dependence of the experimentally measured rate of conformational changes in myoglobin.

Coherence motion of photoinduced nonadiabatic charge transfer reaction in solution: A numerical study of pump–probe spectroscopy

We study the photoinduced charge separation processes in solution through a pump–probe spectroscopy theory [Dah-Yen Yang and Sheh-Yi Sheu, J. Chem. Phys. 106, 9427 (1997)] numerically. We investigate the detailed mechanism of nonadiabatic transition processes via the transition differential flux analysis. For the harmonic potential surfaces, an electronic coherence motion is observed in the overdamped exothermic activationless and inverted regimes.

Liquid dynamics theory of high-temperature specific heat

The potential-energy surface underlying the liquid dynamics Hamiltonian is supposed to consist of a large number of intersecting, macroscopically similar, nearly harmonic random valleys. The statistical mechanics replaces each valley with an infinitely extended harmonic potential surface and then corrects for (a) the anharmonic distortion of the potential surface and (b) the boundary condition that limits the extent of the potential surface along lines where neighboring valleys intersect. The ion-motional specific heat then consists of the main quasiharmonic contribution ${C}_{H}=3\mathrm{Nk}$ plus the anharmonic correction ${C}_{A}$ and the boundary correction ${C}_{B}.$ Here we analyze available specific-heat data for monatomic liquids, at temperatures $T$ up to three times the melting temperature ${T}_{m},$ and suggest the following interpretation of the data: (a) ${C}_{A}$ is about the same for both the crystal and liquid at the melting point and ${C}_{A}$ tends toward zero as $T$ increases above ${T}_{m}$ and (b) ${C}_{B}$ is a roughly universal function of ${T/T}_{m},$ running from around zero at ${T=T}_{m}$ to around $\ensuremath{-}0.6Nk$ at ${T=3T}_{m}.$ The quintessential liquid property, the primary difference between a liquid and a crystal, is that the liquid ions move through a vast number of random potential valleys and this property is directly responsible for the boundary specific heat ${C}_{B}.$ A physically based model for ${C}_{B}$ agrees with experiment for mercury to high temperatures.

Electron transfer via bridges

In this paper we explore the energetic control of sequential and superexchange electron transfer (ET) mechanisms on the basis of quantum-mechanical simulations and calculations for long-range ET in DBA systems, where the donor (D) and the acceptor (A) are separated by a bridge (B). We studied ET dynamics in a Franck–Condon (FC) system characterized by three multi-dimensional displaced harmonic potential surfaces, where an initial single vibronic doorway state |α〉 (with energy Eα) in the DBA (≡D) electronic state is coupled to the mediating {|β〉} vibronic quasicontinuum of the D+B−A (≡B) electronic state, which in turn is coupled to the final {|γ〉} vibronic quasicontinuum of the D+BA− (≡A) electronic state. The level structure was described by the vibrational frequencies (for a four-mode harmonic system) and the energy gaps ΔGDB and ΔGDA between the origins of the corresponding electronic states (with nα=1–50, nβ=1000–2000, and nγ=1000–2000 states in the {|α〉}, {|β〉}, and {|γ〉} manifolds, respectively), while the couplings were characterized by the spectral densities and by the pair correlations (specified in terms of correlation parameters ηαα′ and ηββ′) between states belonging to the same manifold. The correlation parameters ηαα′ (α,α′=1–40) for the doorway-quasicontinuum coupling and ηββ′ (β,β′=150–190) for the interquasicontinuum coupling are considerably lower than unity (|ηαα′|⩽0.4 and |ηββ′|⩽0.3), obeying propensity rules with the highest values of |ηαα′| and |ηββ′| which correspond to a single vibrational quantum difference, while for multimode changes between α and α′ or between β and β′ very low values of |ηαα′| or |ηββ′| are exhibited. Radiationless transitions theory was applied for quantum-mechanical simulations based on the dynamcis of wave packets of molecular eigenstates for resonance (ΔGDB<Eα) and for off-resonance (ΔGDB>Eα) coupling. Resonance |α〉–{|β〉}–{|γ〉} coupling results in two-step sequential ET kinetics for all doorway states |α〉, manifesting phase erosion due to weakly correlated intercontinuum coupling, without the need of intermediate state phonon induced thermalization. Off-resonance |α〉–{|β〉} coupling in conjunction with {|β〉}–{|γ〉} resonance interactions results in unistep superexchange ET kinetics. The simulated sequential ET rates and the superexchange rate are in good agreement with the calculated quantum-mechanical rates obtained using the electronic couplings and FC densities. The energy-gap (ΔGDB) dependence of the simulated and the calculated ET rates from a single doorway state reveal a “transition” from sequential to superexchange ET with increasing ΔGDB. For a finite-temperature system, characterized by a fixed ΔGDB (>0) small energy gap, the thermally averaged rate from a canonical ensemble of doorway states will result in the superposition of both superexchange and sequential mechanisms.

Vibronic mode couplings in adsorbed molecules analyzed by doubly resonant sum-frequency generation.

We present multimode model calculations of doubly resonant infrared-visible sum-frequency generation from adsorbed molecules with general harmonic potential surfaces that are valid for finite temperatures. Numerical simulations of spectra recorded versus the frequency of the infrared or the visible sources have been performed on a simple two-mode model with equilibrium position displacements, frequency shifts, and Duschinsky rotation upon electronic excitation. It is argued that quadratic vibronic couplings need to be included if the excited state potential well is to be reliably probed by these techniques. \textcopyright{} 1996 The American Physical Society.

Doubly resonant sum-frequency generation from molecules with general harmonic potential surfaces

A many-body approach is used to analyse the doubly resonant infrared-visible sum-frequency generation from molecules with general harmonic potential surfaces. This multi-mode model includes arbitrary linear and quadratic electron-phonon coupling coefficients, and is valid at any temperature. Numerical simulations have been performed on a simple two-mode model with equilibrium position displacements, frequency shifts and Duschinsky rotation under electronic excitation. Our study leads us to propose these types of experiments to discriminate between the effects of linear and quadratic electron-phonon coupling terms.