Brownian oscillation

The effective frequencies that characterize solvent response can be characterized more quantitatively from several points of view, including generalized Langevin theory [367-372], Brownian oscillators [373, 374], and instantaneous normal modes [375],... 

Fluctuations of positions along the step edge-in contrast to those of the q-modes — are interdependent. This problem of coupled Brownian oscillators has a rich history (Wax, 1954). 

Fig. 1. Puinp-probe transients for solutions of 6 M NaBr and 1 M NaCl in HD0 D20. The solid curves are calculated with a Brownian oscillator model using a time constant tc of 25 ps for the 0 -H- -Br- hydrogen bond and of 12 ps for the O-H- -Cl- hydrogen bond. The dashed curves are calculated with the same model using rc = oc.

We found that rc of the O H- - -Cl- hydrogen-bond length increases from 14 2 ps at 25 °C, to 24 5 ps at 65 °C, to 30 6 ps at 85 °C. This increase of rc can be explained within the framework of the Brownian oscillator model. The time constant rc is related to the frequency cchb of the hydrogen-bond stretch vibration via rc = t/o hr [11], with 7 the damping of the hydrogen-bond stretch vibration. An increase in temperature leads to a decrease of wI1B, and thus to an increase of rc. 

The frequency (o and gate pulse time-delay td dependent fluorescence up-conversion signal F(cojd) have been theoretically modeled by using an overdamped multi-mode Brownian oscillator model [5], The deconvoluted fluorescence spectrum is given by... 

In the Brownian oscillator overdamped model as an attempt to simulate the solvation dynamics, the explicit forms of the line-shape functions gr(t) are... 

Time-resolved fluorescence of coumarin C522 was determined in water and in host-guest complex with p-cyclodextrin, representing free aqueous and cavity restricted environments, respectively. Experimental fluorescence clearly showed faster dynamics in a case of water. The time parameters of monoexponential fit for water and p-cyclodextrin at 500 nm and 520 nm were determined to be 1.37 ps and 2.02 ps, and 2.97 ps and 7.14 ps, respectively. Multi-mode Brownian oscillator model, as an attempt to simulate the solvation dynamics, supported these fluorescence dynamics results. 

If the phonon distribution of the model Eq. (8) spans a dense spectrum - as is generally the case for the extended systems under consideration, which are effectively infinite-dimensional - the dynamics induced by the Hamiltonian will eventually exhibit a dissipative character. However, the effective-mode construction demonstrates that the shortest time scales are fully determined by few effective modes, and by the coherent dynamics induced by these modes. The overall picture thus corresponds to a Brownian oscillator type dynamics, and is markedly non-Markovian [81,82],... 

Besides, the notation Q(f) for the time-dependent H-bond bridge coordinate interacting with the thermal bath may be viewed as the coordinate of a Brownian oscillator Q(f), which is a time-dependent stochastic variable ... 

In order to compute Eq. (158), write the Langevin equations governing the dynamics of the Brownian oscillator. In the present situation that leads us to consider three time-dependent stochastic variables S(t), Q(f), and v(t), described by the following three equations ... 

Here, (Q2) is the fluctuation of the Brownian oscillator coordinate at equilibrium, the classical and quantum statistical expressions of which are given in Table L.l Now, take the Laplace transform of both sides of Eq. (L.3),... 

Let us consider the two usual Langevin equations (161) and (162), dealing with the Brownian oscillator together with the definition of the stochastic coordinate S... 

DeBoeij WP, Pshenichnikov MS, Wiersma DA. On the relation between the echo-peak shift and Brownian oscillator correlation function. Chem Phys Lett 1996 253 53-60. 

To demonstrate the potential of two-dimensional nonresonant Raman spectroscopy to elucidate microscopic details that are lost in the ensemble averaging inherent in one-dimensional spectroscopy, we will use the Brownian oscillator model and simulate the one- and two-dimensional responses. The Brownian oscillator model provides a qualitative description for vibrational modes coupled to a harmonic bath. With the oscillators ranging continuously from overdamped to underdamped, the model has the flexibility to describe both collective intermolecular motions and well-defined intramolecular vibrations (1). The response function of a single Brownian oscillator is given as,... 

We first consider the intermolecular modes of liquid CS2. One of the details that two-dimensional Raman spectroscopy has the potential to reveal is the coupling between intermolecular motions on different time scales. We start with the one-dimensional Raman spectrum. The best linear spectra are based on time domain third-order Raman data, and these spectra demonstrate the existence of three dynamic time scales in the intermolecular response. In Fig. 3 we have modeled the one-dimensional time domain spectrum of CS2 for 3 cases (A) a single mode represented by the sum of three Brownian oscillators, (B) three Brownian oscillators, and (C) a distribution of 20 arbitrary Brownian oscillators. Case (A) represents the fully coupled, or isotropic case where the liquid is completely homogeneous on the time scales of the simulation. Case (B) deconvolutes the linear response into the three time scales that are directly evident in the measured response and is in the limit that the motions associated with each of the three timescales are uncoupled. Case (C) is an example where the liquid is represented by a large distribution of uncoupled motions. 

Figure 3 Brownian oscillator simulations for the ID (left) and 2D (right) Raman response for liquid CS2 at room temperature. (A) The response is modeled as a single oscillator represented by a sum of the three Brownian oscillators in (B). (B) The response is modeled by three independent Brownian oscillators. (C) The response is modeled by 20 randomly distributed Brownian oscillators. Note that all three cases reproduce the same ID time dependent response but exhibit clear differences in the 2D responses.

To provide an example of the two-dimensional response from a system containing well-defined intramolecular vibrations, we will use simulations based on the polarized one-dimensional Raman spectrum of CCI4. Due to the continuous distribution of frequencies in the intermolecular region of the spectrum, there was no obvious advantage to presenting the simulated responses of the previous section in the frequency domain. However, for well-defined intramolecular vibrations the frequency domain tends to provide a clearer presentation of the responses. Therefore, in this section we will present the simulations as Fourier transformations of the time domain responses. Figure 4 shows the Fourier transformed one-dimensional Raman spectrum of CCI4. The spectrum contains three intramolecular vibrational modes — v2 at 218 cm, v4 at 314 cm, and vi at 460 cm 1 — and a broad contribution from intermolecular motions peaked around 40 cm-1. We have simulated these modes with three underdamped and one overdamped Brownian oscillators, and the simulation is shown over the data in Fig. 4. 

Figure 4 The ID polarized Raman spectrum from liquid CCI4 at room temperature fit with a sum of four Brownian oscillators.

For the two-dimensional response we will first consider the case of nonlinear polarizability coupling and simulate the response using Equation (21) and the Brownian oscillators used to fit the one-dimensional spectrum in Fig. 4. Figure 5 shows the simulations for the limiting cases where the system is (a) fully uncoupled, ay = <5y, and (b) fully coupled,... 

Palese S, Mukamel S, Miller RJD, Lotshaw WT. Interrogation of vibrational structure and line broadening of liquid water by Raman-induced Kerr effect measurements within the multi-mode Brownian oscillator model. I Phys Chem 1996 100 10380-10388. 

Figure 7. Left column. The potential energy functions V — /c, (solid lines) and their curvatures (dotted lines) for different values of c c = 2 (linear oscillator), and c — 4,6,8 (strongly non-linear oscillators). Middle column. Typical sample paths of Brownian oscillators, a = 2, with the potential energy functions shown on the left. Right column Typical sample paths of Levy oscillators, a = 1. On increasing m the potential walls become steeper, and the flights become shorter in this sense, they are confined.

This is the multidimensional generalization of the position autocorrelation function of a Brownian oscillator and provides such a function with a number of unknown parameters. Using this approximate memory function allows replacement of the GLE in Eq. (5.1) by... 

Figure Al.6.25. Modulus squared of the rephasing, (a), and non-rephasing, R-, (b), response functions versus final time t for a near-critically overdamped Brownian oscillator model M f). The time delay between the second and third pulse, T, is varied as follows (a) from top to bottom, 7=0, 20, 40, 60, 80, 100,...

---