Two approaches to describe the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E S is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z S is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E S = −d(ln Z S )/dβ is used to obtain the mean system energy. Employing both classical and quantum mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.

Quantum electron-vibrational dynamics in molecular systems at finite temperature is described using an approach based on the thermo field dynamics theory. This formulation treats temperature effects in the Hilbert space without introducing the Liouville space. A comparison with the theoretically equivalent density matrix formulation shows the key numerical advantages of the present approach. The solution of thermo field dynamics equations with a novel technique for the propagation of tensor trains (matrix product states) is discussed. Numerical applications to model spin-boson systems show that the present approach is a promising tool for the description of quantum dynamics of complex molecular systems at finite temperature.

"Four-wave-mixing" is the generic name for a family of nonlinear electronic and vibrational spectroscopies. These techniques are widely used to explore dissipation, dephasing, solvation, and interstate coupling mechanisms in various material systems. Four-wave-mixing spectroscopy needs a firm theoretical support, because it delivers information on material systems indirectly, through certain transients, which are measured as functions of carrier frequencies, durations, and relative time delays of the laser pulses. The observed transients are uniquely determined by the three-pulse-induced third-order polarization. There exist two conceptually different approaches to the calculation of the nonlinear polarization. In the standard perturbative approach to nonlinear spectroscopy, the third-order polarization is expressed in terms of the nonlinear response functions. As the material systems become more complex, the evaluation of the response functions becomes cumbersome and the calculation of the signals necessitates a number of approximations. Herein, we review alternative methods for the calculation of four-wave-mixing signals, in which the relevant laser pulses are incorporated into the system Hamiltonian and the driven system dynamics is simulated numerically exactly. The emphasis is on the recently developed equation-of-motion phase-matching approach (EOM-PMA), which allows us to calculate the three-pulse-induced third-order polarization in any phase-matching direction by performing three (with the rotating wave approximation) or seven (without the rotating wave approximation) independent propagations of the density matrix. The EOM-PMA is limited to weak laser fields (its domain of validity is equivalent to the approach based on the third-order response functions) but allows for arbitrary pulse durations and automatically accounts for pulse-overlap effects. As an illustration, we apply the EOM-PMA to the calculation of optical three-pulse photon-echo two-dimensional (2D) signals for a generic model system, which represents a characteristic photophysical dynamics of large molecules or chromophores in condensed phases. The EOM-PMA is easy to implement and can straightforwardly be incorporated into any computational scheme, which provides the time-dependent density matrix or wave function of the material system of interest. In particular, EOM-PMA-based computer codes can efficiently be implemented on parallel computers. The EOM-PMA facilitates considerably the computation of four-wave-mixing signals and 2D spectra, in both vibrational and electronic spectroscopy. The EOM-PMA can be extended to higher order optical responses, e.g., heterodyned 3D IR, transient 2D IR, and other six-wave-mixing techniques.

An efficient method has been developed for the calculation of third-order time- and frequency-resolved optical signals. To obtain the general four-wave mixing signal, seven auxiliary density matrices have to be propagated in time. For the special cases of two-pulse photon-echo and transient-grating signals, two or three density matrices, respectively, are required. The method is limited to weak laser fields (it is thus valid within the third-order perturbation theory) but allows for any pulse durations and automatically accounts for pulse-overlap effects. To illustrate the method, we present the explicit derivation of the three-pulse photon-echo signal. Any other third-order optical signal can be calculated in the same manner. As an example, two- and three-pulse photon-echo and transient-grating signals for a weakly damped displaced harmonic oscillator have been calculated.

The recently developed efficient method for the calculation of four-wave mixing signals [M. F. Gelin et al., J. Chem. Phys. 123, 164112 (2005)] is employed for the calculation of two-dimensional electronic photon-echo spectra. The effect of the explicit treatment of vibrations coupled to the electronic transitions is systematically analyzed. The impact of pulse durations, optical dephasing, and temperature on the spectra is investigated. The study aims at an understanding of the mechanisms which may give rise to cross peaks in the two-dimensional electronic spectra and at clarifying the conditions of their detection.

We have developed a variational approach to the description of four-wave-mixing signals of molecular aggregates, in which the third-order response functions are evaluated in terms of the Davydov Ansätze. Our theory treats both singly and doubly excited excitonic states, handling the contributions due to stimulated emission, ground state bleach, and excited state absorption. As an illustration, we simulate a series of optical two-dimensional spectra of model J-aggregates. Our approach may become suitable for the computation of femtosecond optical four-wave-mixing signals of molecular aggregates with intermediate-to-strong exciton-phonon and exciton-exciton coupling strengths.

We have developed a new approach to the computation of third-order spectroscopic signals of molecular rings, by incorporating the Davydov soliton theory into the nonlinear response function formalism. The Davydov D1 and D Ansätze have been employed to treat the interactions between the excitons and the primary phonons, allowing for a full description of arbitrary exciton-phonon coupling strengths. As an illustration, we have simulated a series of optical 2D spectra for two models of molecular rings.

Quantum electron-vibrational dynamics in molecular systems at finite temperature is described using an approach based on Thermo Field Dynamics theory. This formulation treats temperature effects in the Hilbert space without introducing the Liouville space. The solution of Thermo Field Dynamics equations with a novel technique for the propagation of Tensor Trains (Matrix Product States) is implemented and discussed. The methodology is applied to the study of the exciton dynamics in the Fenna-Mathews-Olsen complex using a realistic structured spectral density to model the electron-phonon interaction. The results of the simulations highlight the effect of specific vibrational modes on the exciton dynamics and energy transfer process, as well as call for careful modeling of electron-phonon couplings.