The finite field approach to the third- and fifth-order Raman response of liquids

The thirdand fifth-order time-domain Raman response can be used in order to probe the lowfrequency dynamics of liquids. This Raman response can be calculated using a finite field method. This method will be described and compared to the time correlation function methods that can also be used. The advantages of the finite field method will be addressed and the calculated thirdand fifth-order response will be presented for liquid carbon disulfide. The calculated third-order response is shown to agree very well with experiments. For the fifth-order response the problem of contamination of the experimental results by thirdorder cascaded response will be addressed. Introduction In the beginning of last century Raman scattering was discovered [1, 2] and Raman spectroscopy has since then become a technique applied to solve many scientific questions. Within the last fifteen years a Correspondence/Reprint request: Dr. Thomas l. C. Jansen, Theoretical Chemistry, Materials Science Centre, Rijksuniversiteit Groningen (RuG), Nijenborgh 4, 9747 AG Groningen, The Netherlands. E-mail: ljsn@mail.rochester.edu Thomas l. C. Jansen et al. 152 number of femtosecond techniques have been developed, relying on stimulated light scattering instead of spontaneous light scattering. In these experiments the sample is perturbed by a pair of short optical laser pulses through a Raman event. After a delay, allowing free evolution in the sample, the state of the sample is probed by a third optical pulse. The (heterodyned) optical Kerr effect [3,4] and transient grating scattering [5,6] are examples of such techniques. They allow the observation of the induced motion in real time rather than as resonances, which is an advantage when studying the short time dynamics that is hidden in the wings of the frequency domain spectra. The time and frequency domain spectra are related by Fourier transforms and in principle contains the same information. Two-dimensional Raman spectroscopy was suggested by Tanimura and Mukamel [7] to give a more detailed understanding of liquid dynamics. Two Raman perturbations are applied to the sample in this technique, separated by a time delay allowing free evolution. The sample is then probed with a fifth optical pulse after a second time delay. These experiments can be expected to contain much more information about the dynamics than the simple one-dimensional experiments discussed above. Tanimura and Mukamel [7] showed that the two limiting line broadening mechanisms, homogeneous and inhomogeneous line broadening, gives rise to distinctively different spectra. In this way information on whether the observed relaxation is due to ultrafast fluctuations on a local molecular scale or by density fluctuations on a much larger length scale can be obtained. Furthermore, the two-dimensional response is also sensitive to anharmonicities and it contains information about mode coupling [7], equivalent to the well-known coupling effects between spins in two-dimensional nuclear magnetic resonance (NMR) [10]. Various groups have tried to measure the two-dimensional response experimentally [11-14]. Unfortunately all these measurements were shown to be dominated by thirdorder cascade processes [14] containing no information that is not obtainable from the simpler third-order experiments. More recent experiments utilizing heterodyne detection techniques [15] and multi color optical fields [16-18] report to have overcome this difficulty. Theory is needed in order to interpret both the thirdand fifth-order measurements. Concepts like homogeneous and inhomogeneous line broadening are based on abstract models and have an unclear physical interpretation. Brownian oscillator models [7, 19,21] and quantum Fokker-Planck models [22, 23] are more detailed, but still abstract and difficult to interpret. Furthermore, these models cannot be used to predict intensities of the thirdand fifth-order responses or even intensity ratios between the fifth-order response and the cascaded third-order response. A basic understanding of the high-order Raman experiments and an interpretation of these in terms of physical relevant information are only possible using more microscopic models. The miscroscopic information needed to give spectral predictions and understanding of the underlying physics can be provided from Molecular Dynamics (MD) simulations. In the instantaneous normal modes method (INM) method [24-30] snapshots of the potential surface from MD simulations are used to describe the motion that gives rise to the Raman spectrum. This method is limited to the description of phenomena on a very short time scale because of the use of snapshots and therefore properties like diffusive motion can not be described properly. The finite field approach to Raman response 153 Using full molecular dynamics simulation data the nonlinear Raman spectra can be predicted employing classical time correlation functions (TCF) [27,31-37]. The thirdorder response is easily obtained in this way, but for the fifthand higher-order response this becomes very time consuming [27-36] and the approach has only been realized for the fifth-order response of a small ensemble of liquid xenon [27, 36]. The finite field (FF) method can be employed instead, making the calculation of the fifth-order response numerically much less expensive [39, 40]. This nonequilibrium MD method is based on a simulation of the experiment by applying actual forces in the calculations, caused by the laser fields. Recently also a number mode-coupling schemes were employed [41-45] of which some were shown to reproduce the main features of the time correlation function calculations on xenon [41-43]. Here the theory behind the finite field and time correlation function methods will be described. The third-order response obtained for liquid carbon disulfide with the two methods will be compared and the fifth-order response obtained with the finite field method will be presented. Furthermore, the problem with cascading third-order response in the fifth-order experiments will be adressed and a theoretical estimate of the intensity ratio between the cascaded and true response will be given. Time-resolved Raman scattering In a time-domain one-dimensional Raman experiment an initial laser pulse pair perturbs the sample and after a delay t1 the nuclear dynamics (Eq. (1b)), following the impact of the initial pulse pair, is probed by a third laser pulse. If the delay is zero a pure electronic response (Eq. (1a)) will also arise. This is illustrated in the energy diagrams Figure 1. The signal is governed by the third-order response function