non-Condon Effects Research Articles

A phenomenological approach to modeling chemical dynamics in nonlinear and two-dimensional spectroscopy

We present an approach for calculating nonlinear spectroscopic observables, which overcomes the approximations inherent to current phenomenological models without requiring the computational cost of performing molecular dynamics simulations. The trajectory mapping method uses the semi-classical approximation to linear and nonlinear response functions, and calculates spectra from trajectories of the system's transition frequencies and transition dipole moments. It rests on identifying dynamical variables important to the problem, treating the dynamics of these variables stochastically, and then generating correlated trajectories of spectroscopic quantities by mapping from the dynamical variables. This approach allows one to describe non-Gaussian dynamics, correlated dynamics between variables of the system, and nonlinear relationships between spectroscopic variables of the system and the bath such as non-Condon effects. We illustrate the approach by applying it to three examples that are often not adequately treated by existing analytical models--the non-Condon effect in the nonlinear infrared spectra of water, non-Gaussian dynamics inherent to strongly hydrogen bonded systems, and chemical exchange processes in barrier crossing reactions. The methods described are generally applicable to nonlinear spectroscopy throughout the optical, infrared and terahertz regions.

Vibronic coupling simulations for linear and nonlinear optical processes: Theory

A comprehensive vibronic coupling model based on the time-dependent wavepacket approach is derived to simulate linear optical processes, such as one-photon absorbance and resonance Raman scattering, and nonlinear optical processes, such as two-photon absorbance and resonance hyper-Raman scattering. This approach is particularly well suited for combination with first-principles calculations. Expressions for the Franck-Condon terms, and non-Condon effects via the Herzberg-Teller coupling approach in the independent-mode displaced harmonic oscillator model are presented. The significance of each contribution to the different spectral types is discussed briefly.

Vibronic coupling simulations for linear and nonlinear optical processes: Simulation results

A vibronic coupling model based on time-dependent wavepacket approach is applied to simulate linear optical processes, such as one-photon absorbance and resonance Raman scattering, and nonlinear optical processes, such as two-photon absorbance and resonance hyper-Raman scattering, on a series of small molecules. Simulations employing both the long-range corrected approach in density functional theory and coupled cluster are compared and also examined based on available experimental data. Although many of the small molecules are prone to anharmonicity in their potential energy surfaces, the harmonic approach performs adequately. A detailed discussion of the non-Condon effects is illustrated by the molecules presented in this work. Linear and nonlinear Raman scattering simulations allow for the quantification of interference between the Franck-Condon and Herzberg-Teller terms for different molecules.

Vibrational manifestations of strong non-Condon effects in the H3O+·X3 (X = Ar, N2, CH4, H2O) complexes: A possible explanation for the intensity in the “association band” in the vibrational spectrum of water

The harmonic approximation provides a powerful approach for interpreting vibrational spectra. In this treatment, the energy and intensity of the 3N- 6 normal modes are calculated using a quadratic expansion of the potential energy and a linear expansion of the dipole moment surfaces, respectively. In reality, transitions are often observed that are not accounted for by this approach (e.g. combination bands, overtones, etc.), and these transitions arise from inherent anharmonicities present in the system. One interesting example occurs in the vibrational spectrum of H(2)O((l)), where a band is observed near 2000 cm(-1) that is commonly referred to as the "association band". This band lies far from the expected bend and stretching modes of the water molecule, and is not recovered at the harmonic level. In a recent study, we identified a band in this spectral region in gas-phase clusters involving atomic and molecular adducts to the H(3)O(+) ion. In the current study we probe the origins of this band through a systematic analysis of the argon-predissociation spectra of H(3)O(+)·X(3) where X = Ar, CH(4), N(2) or H(2)O, with particular attention to the contributions from the non-linearities in the dipole surfaces, often referred to as non-Condon effects. The spectra of the H(3)O(+) clusters all display strong transitions between 1900-2100 cm(-1), and theoretical modeling indicates that they can be assigned to a combination band involving the HOH bend and frustrated rotation of H(3)O(+) in the solvent cage. This transition derives its oscillator strength entirely from strong non-Condon effects, and we discuss its possible relationship to the association band in the spectrum of liquid water.

Comparison of vertical and adiabatic harmonic approaches for the calculation of the vibrational structure of electronic spectra

The calculation of the vibrational structure associated to electronic spectra in large molecules requires a Taylor expansion of the initial and final state potential energy surface (PES) around some reference nuclear structure. Vertical (V) and adiabatic (A) approaches expand the final state PES around the initial-state (V) or final-state (A) equilibrium structure. Simplest models only take into account displacements of initial- and final-state minima, intermediate ones also allow for difference in frequencies and more accurate models introduce the Dushinsky effect through the computation of the Hessians of both the initial and final state. In this contribution we summarize and compare the mathematical expressions of the complete hierarchy of V and A harmonic models and we implement them in a numerical code, presenting a detailed comparison of their performance on a number of prototypical systems. We also address non-Condon effects through linear expansions of the transition dipole as a function of nuclear coordinates (Herzberg-Teller effect) and compare the results of expansions around initial and final state equilibrium geometries. By a throughout analysis of our results we highlight a number of general trends in the relative performance of the models that can provide hints for their proper choice. Moreover we show that A and V models including final state PES Hessian outperform the simpler ones and that discrepancies in their predictions are diagnostic for failure of harmonic approximation and/or of Born-Oppenheimer approximation (existence of remarkable geometry-dependent mixing of electronic states).

Quadratic non-Condon effect in optical spectra of impurity paramagnetic centers in dielectric crystals

Analytical expressions for the absorption and luminescence form functions of impurity paramagnetic centers in dielectric crystals at zero temperature are derived in the adiabatic approximation taking into account the quadratic non-Condon effect. It is proved that, if the optical transition is forbidden due to symmetry selection rules, the non-Condon absorption and luminescence spectra are not mirror symmetric and can contain a zero-phonon line, contrary to the case of linear non-Condon effect. Conditions under which the zero-phonon line is contained in the optical spectra of a symmetry-forbidden transition are determined.

Quantum Chemistry Behind Bioimaging: Insights from Ab Initio Studies of Fluorescent Proteins and Their Chromophores

The unique properties of green fluorescent protein (GFP) have been harnessed in a variety of bioimaging techniques, revolutionizing many areas of the life sciences. Molecular-level understanding of the underlying photophysics provides an advantage in the design of new fluorescent proteins (FPs) with improved properties; however, because of its complexity, many aspects of the GFP photocycle remain unknown. In this Account, we discuss computational studies of FPs and their chromophores that provide qualitative insights into mechanistic details of their photocycle and the structural basis for their optical properties. In a reductionist framework, studies of well-defined model systems (such as isolated chromophores) help to understand their intrinsic properties, while calculations including protein matrix and/or solvent demonstrate, on the atomic level, how these properties are modulated by the environment. An interesting feature of several anionic FP chromophores in the gas phase is their low electron detachment energy. For example, the bright excited ππ* state of the model GFP chromophore (2.6 eV) lies above the electron detachment continuum (2.5 eV). Thus, the excited state is metastable with respect to electron detachment. This autoionizing character needs to be taken into account in interpreting gas-phase measurements and is very difficult to describe computationally. Solvation (and even microsolvation by a single water molecule) stabilizes the anionic states enough such that the resonance excited state becomes bound. However, even in stabilizing environments (such as protein or solution), the anionic chromophores have relatively low oxidation potentials and can act as light-induced electron donors. Protein appears to affect excitation energies very little (<0.1 eV), but alters ionization or electron detachment energies by several electron volts. Solvents (especially polar ones) have a pronounced effect on the chromophore's electronic states; for example, the absorption wavelength changes considerably, the ground-state barrier for cis-trans isomerization is reduced, and fluorescence quantum yield drops dramatically. Calculations reveal that these effects can be explained in terms of electrostatic interactions and polarization, as well as specific interactions such as hydrogen bonding. The availability of efficient computer implementations of predictive electronic structure methods is essential. Important challenges include developing faster codes (to enable better equilibrium sampling and excited-state dynamics modeling), creating algorithms for properties calculations (such as nonlinear optical properties), extending standard excited-state methods to autoionizing (resonance) states, and developing accurate QM/MM schemes. The results of sophisticated first-principle calculations can be interpreted in terms of simpler, qualitative molecular orbital models to explain general trends. In particular, an essential feature of the anionic GFP chromophore is an almost perfect resonance (mesomeric) interaction between two Lewis structures, giving rise to charge delocalization, bond-order scrambling, and, most importantly, allylic frontier molecular orbitals spanning the methine bridge. We demonstrate that a three-center Hückel-like model provides a useful framework for understanding properties of FPs. It can explain changes in absorption wavelength upon protonation or other structural modifications of the chromophore, the magnitude of transition dipole moment, barriers to isomerization, and even non-Condon effects in one- and two-photon absorption.

Probing Two-Photon Properties of Molecules: Large Non-Condon Effects Dominate the Resonance Hyper-Raman Scattering of Rhodamine 6G

Experimentally measured resonance hyper-Raman (RHR) spectra spanning the S(1) ← S(0), S(2) ← S(0), and S(3) ← S(0) transitions in rhodamine 6G (R6G) have been recorded. These spectra are compared to the results of first-principles calculations of the RHR intensity that include both Franck-Condon (A-term) and non-Condon (B-term) scattering effects. Good agreement between the experimental and theoretical results is observed, demonstrating that first-principles calculations of hyper-Raman intensities are now possible for large molecules such as R6G. Such agreement indicates that RHR spectroscopy will now be a routine aid for probing multiphoton processes. This work further shows that optimization of molecular properties to enhance either A- or B-term scattering might yield molecules with significantly enhanced two-photon properties.

Vibrational spectroscopy of water in hydrated lipid multi-bilayers. I. Infrared spectra and ultrafast pump-probe observables.

The vibrational spectroscopy of hydration water in dilauroylphosphatidylcholine lipid multi-bilayers is investigated using molecular dynamics simulations and a mixed quantum/classical model for the OD stretch spectroscopy of dilute HDO in H(2)O. FTIR absorption spectra, and isotropic and anisotropic pump-probe decay curves have been measured experimentally as a function of the hydration level of the lipid multi-bilayer, and our goal is to make connection with these experiments. To this end, we use third-order response functions, which allow us to include non-Gaussian frequency fluctuations, non-Condon effects, molecular rotations, and a fluctuating vibrational lifetime, all of which we believe are important for this system. We calculate the response functions using existing transition frequency and dipole maps. From the experiments it appears that there are two distinct vibrational lifetimes corresponding to HDO molecules in different molecular environments. In order to obtain these lifetimes, we consider a simple two-population model for hydration water hydrogen bonds. Assuming a different lifetime for each population, we then calculate the isotropic pump-probe decay, fitting to experiment to obtain the two lifetimes for each hydration level. With these lifetimes in hand, we then calculate FTIR spectra and pump-probe anisotropy decay as a function of hydration. This approach, therefore, permits a consistent calculation of all observables within a unified computational scheme. Our theoretical results are all in qualitative agreement with experiment. The vibrational lifetime of lipid-associated OD groups is found to be systematically shorter than that of the water-associated population, and the lifetimes of each population increase with decreasing hydration, in agreement with previous analysis. Our theoretical FTIR absorption spectra successfully reproduce the experimentally observed red-shift with decreasing lipid hydration, and we confirm a previous interpretation that this shift results from the hydrogen bonding of water to the lipid phosphate group. From the pump-probe anisotropy decay, we confirm that the reorientational motions of water molecules slow significantly as hydration decreases, with water bound in the lipid carbonyl region undergoing the slowest rotations.

Violation of the Condon Approximation in Semiconducting Carbon Nanotubes

The Condon approximation is widely applied in molecular and condensed matter spectroscopy and states that electronic transition dipoles are independent of nuclear positions. This approximation is related to the Franck-Condon principle, which in its simplest form holds that electronic transitions are instantaneous on the time scale of nuclear motion. The Condon approximation leads to a long-held assumption in Raman spectroscopy of carbon nanotubes: intensities arising from resonance with incident and scattered photons are equal. Direct testing of this assumption has not been possible due to the lack of homogeneous populations of specific carbon nanotube chiralities. Here, we present the first complete Raman excitation profiles (REPs) for the nanotube G band for 10 pure semiconducting chiralities. In contrast to expectations, a strong asymmetry is observed in the REPs for all chiralities, with the scattered resonance always appearing weaker than the incident resonance. The observed behavior results from violation of the Condon approximation and originates in changes in the electronic transition dipole due to nuclear motion (non-Condon effect), as confirmed by our quantum chemical calculations. The agreement of our calculations with the experimental REP asymmetries and observed trends in family dependence indicates the behavior is intrinsic.

Theoretical investigation of the non-Condon effect on electron transfer: Application to organic semiconductor

The non-Condon effect plays an important role in the process of electron transfer (ET). Several theoretical models have been proposed to investigate its effect on ET rates. In this paper, we overview a theoretical method for the calculations of the non-Condon ET rate constants proposed by us, and its applications to organic semiconductors. First, full quantum expressions of the non-Condon ET rates are presented with the electronic couplings having exponential, Gaussian and linear dependences in terms of the nuclear coordinates, respectively. The proposed formulas have closed forms in time domain and they thus can be easily applied in multi-mode systems. Then, the driving force dependences of the ET rates involving the non-Condon effect are calculated with the use of full quantum mechanical formulas. It is found that these dependences show very different properties from the Marcus one. As an example of applications, the approaches are used to investigate the non-Condon effect on the mobility of the organic semiconductor dithiophene-tetrathiafulvalene (DT-TTF). The results manifest that the non-Condon effect enhances ET rates compared with the Condon approximation, and static fluctuations of electronic coupling dominate the ET rate in the DT-TTF, which has been confirmed by the molecular dynamics simulation.

Reduced Density Matrix Theory for Vibrational Absorption Line Shape in Energy Transfer Systems: Non-Condon Effects in Water

Using the projection operator technique, a reduced density matrix theory for linear absorption spectrum of energy transfer systems is developed for the theoretical absorption line shape of the systems with non-Condon transitions. As an application, we considered a model system of OH vibrations of water. In the present model calculation, the OH vibration modes are coupled to each other via intra-molecular coupling mechanism while their intermolecular couplings are turned off. The time-correlation functions appearing in the formulation are calculated from a mixed quantum/classical mechanics method. The present theory is successful in reproducing the exact absorption line shape. Also the present theory was improved from an existing approximate theory, time-averaged approximation approach.

Non-Condon Effects in the One- and Two-Photon Absorption Spectra of the Green Fluorescent Protein

We report calculations of one and two photon absorption (OPA and TPA, respectively) spectra of the green fluorescent protein (GFP) chromophore using the double-harmonic parallel-mode approximation and including explicit dependence of the electronic transition moments on nuclear geometry. The non-Condon effects are found to be more significant for TPA resulting in a different shape of the spectra and a blue shift of 500 cm−1 of the TPA peak absorption relative to OPA. The computed shift is in excellent agreement with the experimentally observed 700 cm−1.

Signature of Nonadiabatic Transitions on the Pump−Probe Infrared Spectra of a Hydrogen-Bonded Complex Dissolved in a Polar Solvent: A Computational Study

The mixed quantum-classical Liouville equation provides a unified and self-consistent platform for modeling the spectral signatures of nonequilibrium solvation dynamics, non-Condon effects, and nonadiabatic transitions. These features are demonstrated in this paper in the context of the pump-probe infrared spectra of the hydrogen stretch in a moderately strong hydrogen-bonded complex dissolved in a polar solvent. Particular emphasis is put on incorporating nonadiabatic transitions and accounting for their unique spectral signature.

Non-Condon effect on charge transport in dithiophene-tetrathiafulvalene crystal

Combining first-principles calculations and non-Condon charge transfer rates proposed by us recently [Y. Zhao and W. Z. Liang, J. Chem. Phys. 130, 034111 (2009)], we investigated non-Condon effect on charge carrier mobility of organic semiconductor dithiophene-tetrathiafulvalene (DT-TTF) crystal. The first-principles results reveal that only several high-frequency intramolecular vibrational modes dominate the reorganization energy, and the nuclear-coordinate dependence of electronic coupling prefers to perform an exponential or Gaussian property for most intermolecular modes rather than a linear one as assumed in conventional models. Furthermore, the electronic coupling of an isolated DT-TTF dimer is indeed affected by the surrounding molecules. The predicted non-Condon mobilities with use of the obtained structure parameters are always greater than those from Condon approximation, and the non-Condon dynamic disorder is not important for DT-TTF, which is also confirmed by molecular dynamics simulation. More interestingly, the bandlike property can be predicted under the hopping mechanism when the nuclear tunneling is incorporated.

On the calculation of rotational anisotropy decay, as measured by ultrafast polarization-resolved vibrational pump-probe experiments

Polarization-resolved vibrational pump-probe experiments are useful for measuring the dynamics of molecular reorientation. The rotational anisotropy observable is usually modeled by the second-Legendre-polynomial time-correlation function of the appropriate molecule-fixed unit vector. On the other hand, more elaborate calculations that include non-Condon effects, excited-state absorption, and spectral diffusion, can be performed using the infrastructure of the nonlinear response formalism. In this paper we present "exact" (within the impulsive limit) results from the nonlinear response formalism, and also a series of approximations that ultimately recover the traditional result mentioned above. To ascertain the importance of these effects not included in the traditional approach, we consider the specific case of dilute HOD in H(2)O. We find that for the frequency-integrated anisotropy decay, it is important to include non-Condon effects. For the frequency-resolved anisotropy decay, non-Condon effects, excited-state absorption, and spectral diffusion are all important. We compare our results with recent experiments.

Computational study of the signature of hydrogen-bond strength on the infrared spectra of a hydrogen-bonded complex dissolved in a polar liquid

The signature of hydrogen-bond strength on the one- and two-dimensional infrared spectra of the hydrogen-stretch in a hydrogen-bonded complex dissolved in a polar liquid was investigated via mixed quantum–classical molecular dynamics simulations. Non-Condon effects were found to intensify with increasing hydrogen-bond strength and to shift oscillator strength from the stable configurations that correspond to the ionic and covalent tautomers into unstable configurations that correspond to the transition-state between them. The transition-state peak is observed to blue shift and increase in intensity with increasing hydrogen-bond strength, and to dominate the spectra in the case of a strong hydrogen-bond. It is argued that the application of multidimensional infrared spectroscopy in the region of the transition-state peak can provide a uniquely direct probe of the molecular events underlying breaking and forming of hydrogen-bonds in the condensed phase.

10.1007/s11175-008-2004-7

The question concerning the applicability of the Born-Oppenheimer approximation (BOA) for the calculation of the transition probability for a nonadiabatic process of charge transfer in a polar environment with allowance made for temperature effects is investigated theoretically. Considered is the transfer of a quantum particle (proton) that interacts with a local vibration mode in a model of bound harmonic oscillators. The model admits an exact solution for wavefunctions of the initial and final states. A calculation shows that BOA is applicable even for very large distances of the proton transfer. At the same time, the exact result and the BOA result severely differ from a probability calculated in a crude Condon approximation. It is demonstrated that the non-Condon effects are in a general case temperature-dependent and may substantially influence the calculated values of the transition probability.

Effectiveness of Perturbation Theory Approaches for Computing Non-Condon Electron Transfer Dynamics in Condensed Phases

A description of electron transfer in condensed-phase media requires models that adequately describe the coupling of the electronic degrees of freedom to the surrounding nuclear coordinates. The spin-boson model has been the canonical model used to understand quantum dynamic processes in condensed-phase media over the last 25 years. Inherent in the standard model of a two-state quantum system coupled to a bosonic bath is the assumption that the Condon approximation is valid. In this context, the Condon approximation assumes that the bath configurations (coordinates) have no effect on the nonadiabatic coupling matrix element. While this is a useful model for electron transfer in small molecular systems, the validity of this approximation is less likely when large-scale motions of solvent molecules are strongly coupled to the electron transfer event, e.g., in molecular clamps and long-range electron transfer in biopolymers. In the present paper a general model for two-state electron transfer which allows for system-bath coupling in both the diagonal and off-diagonal (nonadiabatic) terms is studied. Time-dependent perturbation theory for this Hamiltonian is developed using a small polaron transformation. As noted in several recent studies, in a certain regime of parameter space, the relevant Hamiltonian admits an exact solution, termed the exactly solvable non-Condon Hamiltonian (or NCE). This limit, for which exact solutions are available, is used to benchmark the short- and long-time accuracy of various perturbative approaches. The validated perturbation equations are subsequently used to explore the role of non-Condon effects on electron transfer by systematically increasing the strength of the non-Condon coupling term from zero (i.e., the canonical spin-boson model) to the value that pertains to the exactly solvable non-Condon model (where non-Condon effects are significant).

Waiting Time Dynamics in Two-Dimensional Infrared Spectroscopy

We review recent work on the waiting time dynamics of coherent two-dimensional infrared (2DIR) spectroscopy. This dynamics can reveal chemical and physical processes that take place on the femto- and picosecond time scale, which is faster than the time scale that may be probed by, for example, nuclear magnetic resonance spectroscopy. A large number of chemically relevant processes take place on this time scale. Such processes range from forming and breaking hydrogen bonds and proton transfer to solvent exchange and vibrational population transfer. In typical 2DIR spectra, multiple processes contribute to the waiting time dynamics and the spectra are often congested. This makes the spectra challenging to interpret, and the aid of theoretical models and simulations is often needed. To be useful, such models need to account for all dynamical processes in the sample simultaneously. The numerical integration of the Schrodinger equation (NISE) method has proven to allow for a very general treatment of the dynamical processes. It accounts for both the motional narrowing resulting from solvent-induced frequency fluctuations and population transfer between coupled vibrations. At the same time, frequency shifts arising from chemical-exchange reactions and changes of the transition dipoles because of either non-Condon effects or molecular reorientation are included in the treatment. This method therefore allows for the disentanglement of all of these processes. The NISE method has thus far been successfully applied to study chemical-exchange processes. It was demonstrated that 2DIR is not only sensitive to reaction kinetics but also to the more detailed reaction dynamics. NISE has also been applied to the study of population transfer within the amide I band (CO stretch) and between the amide I and amide II bands (CN stretch and NH bend) in polypeptides. From the amide I studies, it was found that the population transfer can be used to enhance cross-peaks that act as structural markers for beta-sheet structure in proteins. From the amide I/II investigation, it was found that the amide II band and the hydrogen-bond stretch vibration are important parts of the relaxation pathway for the amide I vibration. With the development of simple approximations, it becomes possible to apply the NISE method even to very big systems, such as the OH stretch of bulk water, which can only be described well when large numbers of coupled vibrations are taken into account.