Multidimensional vibrational spectroscopy for tunneling processes in a dissipative environment

Simulating tunneling processes as well as their observation are challenging problems for many areas. In this study, we consider a double-well potential system coupled to a heat bath with a linear-linear (LL) and square-linear (SL) system-bath interactions. The LL interaction leads to longitudinal (T1) and transversal (T2) homogeneous relaxations, whereas the SL interaction leads to the inhomogeneous dephasing (T2*) relaxation in the white noise limit with a rotating wave approximation. We discuss the dynamics of the double-well system under infrared (IR) laser excitations from a Gaussian-Markovian quantum Fokker-Planck equation approach, which was developed by generalizing Kubo's stochastic Liouville equation. Analytical expression of the Green function is obtained for a case of two-state-jump modulation by performing the Fourier-Laplace transformation. We then calculate a two-dimensional infrared signal, which is defined by the four-body correlation function of optical dipole, for various noise correlation time, system-bath coupling parameters, and temperatures. It is shown that the bath-induced vibrational excitation and relaxation dynamics between the tunneling splitting levels can be detected as the isolated off-diagonal peaks in the third-order two-dimensional infrared (2D-IR) spectroscopy for a specific phase matching condition. Furthermore, this spectroscopy also allows us to directly evaluate the rate constants for tunneling reactions, which relates to the coherence between the splitting levels; it can be regarded as a novel technique for measuring chemical reaction rates. We depict the change of reaction rates as a function of system-bath coupling strength and a temperature through the 2D-IR signal.     

Modeling vibrational dephasing and energy relaxation of intramolecular anharmonic modes for multidimensional infrared spectroscopies

Starting from a system-bath Hamiltonian in a molecular coordinate representation, we examine an applicability of a stochastic multilevel model for vibrational dephasing and energy relaxation in multidimensional infrared spectroscopy. We consider an intramolecular anharmonic mode nonlinearly coupled to a colored noise bath at finite temperature. The system-bath interaction is assumed linear plus square in the system coordinate, but linear in the bath coordinates. The square-linear system-bath interaction leads to dephasing due to the frequency fluctuation of system vibration, while the linear-linear interaction contributes to energy relaxation and a part of dephasing arises from anharmonicity. To clarify the role and origin of vibrational dephasing and energy relaxation in the stochastic model, the system part is then transformed into an energy eigenstate representation without using the rotating wave approximation. Two-dimensional (2D) infrared spectra are then calculated by solving a low-temperature corrected quantum Fokker-Planck (LTC-QFP) equation for a colored noise bath and by the stochastic theory. In motional narrowing regime, the spectra from the stochastic model are quite different from those from the LTC-QFP. In spectral diffusion regime, however, the 2D line shapes from the stochastic model resemble those from the LTC-QFP besides the blueshifts caused by the dissipation from the colored noise bath. The preconditions for validity of the stochastic theory for molecular vibrational motion are also discussed.     

Dissipative quantum dynamics with the surrogate Hamiltonian approach. A comparison between spin and harmonic baths

The dissipative quantum dynamics of an anharmonic oscillator coupled to a bath is studied with the purpose of elucidating the differences between the relaxation to a spin bath and to a harmonic bath. Converged results are obtained for the spin bath by the surrogate Hamiltonian approach. This method is based on constructing a system-bath Hamiltonian, with a finite but large number of spin bath modes, that mimics exactly a bath with an infinite number of modes for a finite time interval. Convergence with respect to the number of simultaneous excitations of bath modes can be checked. The results are compared to calculations that include a finite number of harmonic modes carried out by using the multiconfiguration time-dependent Hartree method of Nest and Meyer [J. Chem. Phys. 119, 24 (2003)]. In the weak coupling regime, at zero temperature and for small excitations of the primary system, both methods converge to the Markovian limit. When initially the primary system is significantly excited, the spin bath can saturate restricting the energy acceptance. An interaction term between bath modes that spreads the excitation eliminates the saturation. The loss of phase between two cat states has been analyzed and the results for the spin and harmonic baths are almost identical. For stronger couplings, the dynamics induced by the two types of baths deviate. The accumulation and degree of entanglement between the bath modes have been characterized. Only in the spin bath the dynamics generate entanglement between the bath modes.     

Condensed-phase relaxation of multilevel quantum systems. I. An exactly solvable model

An analytically solvable model of multilevel condensed-phase quantum dynamics relevant to vibrational relaxation and electron transfer is presented. Exact solutions are derived for the reduced system density matrix dynamics of a degenerate N-level quantum system characterized by nearest-neighbor hopping and off-diagonal coupling (which is linear in the bath coordinates) to a harmonic oscillator bath. We demonstrate that for N> 2 the long-time steady-state system site occupation probabilities are not the same for all sites; that is, they are distributed in a non-Boltzmann manner, which depends on the initial conditions and the number of levels in the system. Although the system-bath Hamiltonian considered here is restricted in form, the availability of an exact solution enables us to study the model in all regions of an extensive parameter space.     

Does ℏ play a role in multidimensional spectroscopy? Reduced hierarchy equations of motion approach to molecular vibrations

To investigate the role of quantum effects in vibrational spectroscopies, we have carried out numerically exact calculations of linear and nonlinear response functions for an anharmonic potential system nonlinearly coupled to a harmonic oscillator bath. Although one cannot carry out the quantum calculations of the response functions with full molecular dynamics (MD) simulations for a realistic system which consists of many molecules, it is possible to grasp the essence of the quantum effects on the vibrational spectra by employing a model Hamiltonian that describes an intra- or intermolecular vibrational motion in a condensed phase. The present model fully includes vibrational relaxation, while the stochastic model often used to simulate infrared spectra does not. We have employed the reduced quantum hierarchy equations of motion approach in the Wigner space representation to deal with nonperturbative, non-Markovian, and nonsecular system-bath interactions. Taking the classical limit of the hierarchy equations of motion, we have obtained the classical equations of motion that describe the classical dynamics under the same physical conditions as in the quantum case. By comparing the classical and quantum mechanically calculated linear and multidimensional spectra, we found that the profiles of spectra for a fast modulation case were similar, but different for a slow modulation case. In both the classical and quantum cases, we identified the resonant oscillation peak in the spectra, but the quantum peak shifted to the red compared with the classical one if the potential is anharmonic. The prominent quantum effect is the 1-2 transition peak, which appears only in the quantum mechanically calculated spectra as a result of anharmonicity in the potential or nonlinearity of the system-bath coupling. While the contribution of the 1-2 transition is negligible in the fast modulation case, it becomes important in the slow modulation case as long as the amplitude of the frequency fluctuation is small. Thus, we observed a distinct difference between the classical and quantum mechanically calculated multidimensional spectra in the slow modulation case where spectral diffusion plays a role. This fact indicates that one may not reproduce the experimentally obtained multidimensional spectrum for high-frequency vibrational modes based on classical molecular dynamics simulations if the modulation that arises from surrounding molecules is weak and slow. A practical way to overcome the difference between the classical and quantum simulations was discussed.     

Vibrational relaxation in simulated two-dimensional infrared spectra of two amide modes in solution

Two-dimensional infrared spectroscopy is capable of following the transfer of vibrational energy between modes in real time. We develop a method to include vibrational relaxation in simulations of two-dimensional infrared spectra at finite temperature. The method takes into account the correlated fluctuations that occur in the frequencies of the vibrational states and in the coupling between them as a result of interaction with the environment. The fluctuations influence the two-dimensional infrared line shape and cause vibrational relaxation during the waiting time, which is included using second-order perturbation theory. The method is demonstrated by applying it to the amide-I and amide-II modes in N-methylacetamide in heavy water. Stochastic information on the fluctuations is obtained from a molecular dynamics trajectory, which is converted to time dependent frequencies and couplings with a map from a density functional calculation. Solvent dynamics with the same frequency as the energy gap between the two amide modes lead to efficient relaxation between amide-I and amide-II on a 560 fs time scale. We show that the cross peak intensity in the two-dimensional infrared spectrum provides a good measure for the vibrational relaxation.     

Transition rates between selected vibrational states in methyl halides; influence of anharmonicities

A semiclassical model of collision induced vibrational relaxation is discussed in terms of an effective collision mass for different values of vibrational energy release. Selected one, two and three quantum transitions of the methyl halides upon collision with rare gases are evaluated in the presence of resonant and nonresonant anharmonic couplings. It is found, that due to the anharmonic coupling the rates between CH stretching modes and the overtones of the CH bending modes become as large as the transfer rates between two CH stretching modes. This is in qualitative agreement with experiments. Without the anharmonic coupling they differ by two orders of magnitude.     

Second-order quantized Hamilton dynamics coupled to classical heat bath

Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.     

Phonon-mediated path-interference in electronic energy transfer

We present a formalism to quantify the contribution of path-interference in phonon-mediated electronic energy transfer. The transfer rate between two molecules is computed by considering the quantum mechanical amplitudes associated with pathways connecting the initial and final sites. This includes contributions from classical pathways, but also terms arising from interference of different pathways. We treat the vibrational modes coupled to the molecules as a non-Markovian harmonic oscillator bath, and investigate the correction to transfer rates due to the lowest-order interference contribution. We show that depending on the structure of the harmonic bath, the correction due to path-interference may have a dominant vibrational or electronic character, and can make a notable contribution to the transfer rate in the steady state.     

Proton transfer reactions in model condensed-phase environments: Accurate quantum dynamics using the multilayer multiconfiguration time-dependent Hartree approach

The recently proposed multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) approach to evaluating reactive quantum dynamics is applied to two model condensed-phase proton transfer reactions. The models consist of a one-dimensional double-well "system" that is bilinearly coupled to a "bath" of harmonic oscillators parameterized to represent a condensed-phase environment. Numerically exact quantum-mechanical flux correlation functions and thermal rate constants are obtained for a broad range of temperatures and system-bath coupling strengths, thus demonstrating the efficacy of the ML-MCTDH approach. Particular attention is focused on the regime where low temperatures are combined with weak system-bath coupling. Under such conditions it is found that long propagation times are often required and that quantum coherence effects may prevent a rigorous determination of the rate constant.     

Generalized Gibbs state with modified Redfield solution: exact agreement up to second order

A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.     

Vibrational spectroscopy and relaxation of an anharmonic oscillator coupled to harmonic bath

The vibrational spectroscopy and relaxation of an anharmonic oscillator coupled to a harmonic bath are examined to assess the applicability of the time correlation function (TCF), the response function, and the semiclassical frequency modulation (SFM) model to the calculation of infrared (IR) spectra. These three approaches are often used in connection with the molecular dynamics simulations but have not been compared in detail. We also analyze the vibrational energy relaxation (VER), which determines the line shape and is itself a pivotal process in energy transport. The IR spectra and VER are calculated using the generalized Langevin equation (GLE), the Gaussian wavepacket (GWP) method, and the quantum master equation (QME). By calculating the vibrational frequency TCF, a detailed analysis of the frequency fluctuation and correlation time of the model is provided. The peak amplitude and width in the IR spectra calculated by the GLE with the harmonic quantum correction are shown to agree well with those by the QME though the vibrational frequency is generally overestimated. The GWP method improves the peak position by considering the zero-point energy and the anharmonicity although the red-shift slightly overshoots the QME reference. The GWP also yields an extra peak in the higher-frequency region than the fundamental transition arising from the difference frequency of the center and width oscillations of a wavepacket. The SFM approach underestimates the peak amplitude of the IR spectra but well reproduces the peak width. Further, the dependence of the VER rate on the strength of an excitation pulse is discussed.     

Interpreting nonlinear vibrational spectroscopy with the classical mechanical analogs of double-sided Feynman diagrams

Observables in coherent, multiple-pulse infrared spectroscopy may be computed from a vibrational nonlinear response function. This response function is conventionally calculated quantum-mechanically, but the challenges in applying quantum mechanics to large, anharmonic systems motivate the examination of classical mechanical vibrational nonlinear response functions. We present an approximate formulation of the classical mechanical third-order vibrational response function for an anharmonic solute oscillator interacting with a harmonic solvent, which establishes a clear connection between classical and quantum mechanical treatments. This formalism permits the identification of the classical mechanical analog of the pure dephasing of a quantum mechanical degree of freedom, and suggests the construction of classical mechanical analogs of the double-sided Feynman diagrams of quantum mechanics, which are widely applied to nonlinear spectroscopy. Application of a rotating wave approximation permits the analytic extraction of signals obeying particular spatial phase matching conditions from a classical-mechanical response function. Calculations of the third-order response function for an anharmonic oscillator coupled to a harmonic solvent are compared to numerically correct classical mechanical results.     

Nonequilibrium molecular dynamics simulations with a backward-forward trajectories sampling for multidimensional infrared spectroscopy of molecular vibrational modes

A full molecular dynamics (MD) simulation approach to calculate multidimensional third-order infrared (IR) signals of molecular vibrational modes is proposed. Third-order IR spectroscopy involves three-time intervals between three excitation and one probe pulses. The nonequilibrium MD (NEMD) simulation allows us to calculate molecular dipoles from nonequilibrium MD trajectories for different pulse configurations and sequences. While the conventional NEMD approach utilizes MD trajectories started from the initial equilibrium state, our approach does from the intermediate state of the third-order optical process, which leads to the doorway-window decomposition of nonlinear response functions. The decomposition is made before the second pump excitation for a two-dimensional case of IR photon echo measurement, while it is made after the second pump excitation for a three-dimensional case of three-pulse IR photon echo measurement. We show that the three-dimensional IR signals are efficiently calculated by using the MD trajectories backward and forward in time for the doorway and window functions, respectively. We examined the capability of the present approach by evaluating the signals of two- and three-dimensional IR vibrational spectroscopies for liquid hydrogen fluoride. The calculated signals might be explained by anharmonic Brownian model with the linear-linear and square-linear system-bath couplings which was used to discuss the inhomogeneous broadening and dephasing mechanism of vibrational motions. The predicted intermolecular librational spectra clearly reveal the unusually narrow inhomogeneous linewidth due to the one-dimensional character of HF molecule and the strong hydrogen bond network.     

The AsO radical detected by infrared laser magnetic resonance

The Pauli master equation for intramolecular vibrational relaxation and the heat bath feedback Bloch equations for radiative pumping of polyatomic molecules can be derived by replacing the standard assumption of random matrix element coupling between zero-order vibrational states by an assumption that relaxation is governed by restricted quantum exchange.     

Absolute intensities of CH stretching overtones in alkenes

We have measured the CH stretching vibrational spectrum of ethene gas in the regions corresponding to 1-5 quanta in the CH stretching vibration with Fourier transform infrared and conventional absorption spectroscopy and have determined the corresponding oscillator strengths. We have calculated the CH stretching vibrational oscillator strengths for a series of alkenes: ethene, propene, 1,3-butadiene, cis-2-butene, and trans-2-butene. The CH stretching intensities are calculated with a simple Morse oscillator local mode model for CH groups and with the harmonically coupled anharmonic oscillator local mode model for CH2 and CH3 groups. The local mode parameters, frequencies, and anharmonicities are obtained from experiments. The harmonic coupling coefficients and the dipole moment functions are calculated with a range of ab initio methods. These include self-consistent-field Hartree-Fock, density functional, correlated, and multireference theories, combined with basis sets ranging from double- to quadruple-zeta quality augmented with polarization and diffuse functions. Variation in calculated oscillator strengths with the choice of ab initio method is systematically studied and compared with observed intensities. From this comparison between the calculated and observed values, we can quantitatively understand the relative usefulness of various ab initio dipole moment functions in calculations of vibrational oscillator strength for alkenes.     

Anharmonic Vibrational Motions In C<Subscript>60</Subscript>: A Potential Energy Surface Derived From Vibrational Self-Consistent Field Calculations

A potential energy surface (PES) is developed for C₆₀ designed to describe vibrational motions valid in the anharmonic limit. The PES is based on a previously existing one that is fit to the harmonic fundamentals and is then modified to generate anharmonicity of all orders and in all terms, but without additional fitted parameters. The resulting Cartesian vibrational motions are decomposed into normal modes, and the anharmonic expansion coefficients are calculated including 2-mode couplings and up to 4th order. The resulting PES is used in a vibrational self-consistent field (VSCF) algorithm to calculate the anharmonically corrected fundamental frequencies. The parameters are then fit to fundamental infrared and Raman frequencies. While it is not possible to assign combination and overtone transitions with sufficient experimental accuracy, conclusions about the effects of anharmonic vibrational coupling in C₆₀ are described.