Time correlation function normalized

FIG. 4 Time-resolved fluorescence Stokes shift of coumarin 343 in Aerosol OT reverse micelles, (a) normalized time-correlation functions, C i) = v(t) — v(oo)/v(0) — v(oo), and (b) unnormalized time-correlation functions, S i) = v i) — v(oo), showing the magnitude of the overall Stokes shift in addition to the dynamic response, wq = 1.1 ( ), 5 ( ), 7.5 ( ), 15 ( ), and 40 (O) and for bulk aqueous Na solution (A)- Points are data and lines that are multiexponential fits to the data. (Reprinted from Ref 38 with permission from the American Chemical Society.)... 

Many dynamic properties can be defined as time-correlation functions of a quantity. For vector X(t), for instance, in the non-normalized form... 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

The power spectrum distribution function, can be decomposed into a continuous and a discrete part, C7c(co) and Gd(co), respectively ... 

The normalized time-correlation function can thus be decomposed in a corresponding way,... 

We shall now briefly review the Kubo-Oxtoby theory of vibrational line-shape. The starting point for most theories of vibrational dephasing is the stochastic theory of lineshape first developed by Kubo [131]. This theory gives a simple expression for the broadened isotropic Raman line shape (/(< )) in terms of the Fourier transform of the normal coordinate time correlation function by... 

The experimental observables are either the lineshape function 7(co), as in the classical experiments, or the normal coordinate time correlation function, (2(O)2(0) as in the time domain experiments of Tominaga and Yoshihara [126]. The normal coordinate time correlation is related to the frequency modulation time correlation function by... 

An important ingredient of Oxtoby s work was the decomposition of the force on the normal coordinate, — (dV/dQ), in terms of the force on the atoms involved. Assuming that the forces on the different atoms of the diatomic are uncorrelated and that the area of contact of each atom with the solvent is a half-sphere, Oxtoby derived the following expression for the frequency-time correlation function ... 

The calculated ln(j2(f)<2(0)) for the first three quantum levels for the CH3I case is shown in Fig. 12. The overtone normal coordinate time correlation function, Q(t)Q(0)), is obtained from the frequency-modulation time corre-... 

C(2)(t,q) can be related to the normalized first-order electric field time correlation function g (t,q) by [9,10]... 

In dynamic LLS [45,46], the intensity-intensity time correlation function G(2 t, q) in the self-beating mode was measured. For a Poisson distribution of the number of photons, G 2)(f, q) can be related to the normalized first-order electric field time correlation function g (f, q) as [46]... 

For monodisperse systems the normalized polarized and depolarized time correlation functions can be expressed as (7)... 

In a dynamic light scattering experiment, the measured intensity-intensity time-correlation function g<2)(tc), where tc is the delay time, is related to the normalized electric field correlation function g(1)frc), representative of the motion of the particles, by the Siegert relation [18] ... 

Each molecular vibration factor in Equation (3) is a type of molecular time correlation function for the internal vibrational dynamics. In the harmonic approximation, i) and f) would reduce to the harmonic vibrational eigenstates and the qj would be the actual molecular normal modes. Then one has the simplification... 

FIG. 1 Normalized intensity-intensity time correlation function of poly (urethane-urea) microcapsule suspension at c = 5 x 10-5 g/cm measured at 60 and at 30 C. r(l/l ) is the characteristic decay time. (Reprinted with permission from the paper entitled An experimental investigation on the structure of microcapsules, by T. Dobashi. F. Yeh, Q. Ying, K. Ichikawa, and B. Chu. Langmuir // 4278. Copyright 1995 American Chemical Society.)... 

It is well known that the fluctuational behavior of the solvent molecules is characterized by the normalized time correlation function of the fluctuation, p (r), which is expressed as a following equation using observed time resolved fluorescence spectra. 

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J — lu, respectively. Note that, in this context I is the moment of inertia tensor The correlation function in equation (6) is calculated over the spherical harmonics. If m = 0, this reduces to time correlation function over Legendre polynomials ... 

Such types of equations can also be formulated for the tangential component of the pressure tensor, pjz), which for simple molecules is a curve with a maximum, as sketched in fig, 2,2. In passing it is noted that the asymmetrical components of this tensor can in principle also be obtained, both in the bulk liquid and near surfaces, and that, from the integral over the time-correlation function of these components, the viscosity is obtainable. Such computations are veiy demanding of computer power. For liquids near hard walls the viscosity appears to be anisotropic, the normal viscosity being higher than the tangential one ). 

Equation (6.14) associates the zero frequency component of the velocity time correlation function with the long-time diffusive dynamics. We will later find (see Section 6.5.4) that the high frequency part of the same Fourier transform, Eq. (6.15), is related to the short-time dynamics of the same system as expressed by its spectrum of instantaneous normal modes. 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

Time dependence of the normalized total moment-moment time correlation function [ J M(t)] for the water molecules, both in the CsPFO micellar solution (solid line) and in neat water. Circles are the simulation data and the continuous line is a multiexponential fit. 

We concentrate on the role of quantum interference in the correlation of photons emitted from a coherently driven V-type atom, recently analyzed by Swain et al. [58]. We calculate the normalized second-order two-time correlation function g (R, t R, t + x) for the fluorescent field emitted from a three-level V-type atom driven by a coherent laser field coupled to both atomic transitions. The fluorescence field is observed by a single detector located at a point R = RR, where R is the unit vector in the direction of the observation. 

Figure 25. (a) Normalized time correlation function for torsional fluctuations of the Tyr-21 ring in the protein, (b) Normalized time correlation function for torsional fluctuations of the tyrosine ring in the isolated tyrosine fragment. 

It should be noted that the imaginary time correlation function in Eq. (2.4) provides a measure of the localization of quantum particles in condensed media [17-19,52]. From this point on, the notation denotes an averaging by integrating some centroid-dependent function over the centroid position q weighted by the normalized centroid density p q ) Z. An alternative method for defining the correlation function... 

Dynamic light scattering measurements were performed with a Malvern photon correlation system eqxiipped with a krypton ion laser KR 165-11 from Spectra Physics (1 =647.1 nm). The intensity time correlation function (TCP) was recorded by a Malvern autocorrelator. The electric field TCP g,(t) normalized to the base line of the intensity TCP, and its first cumulant F = -Slng (t)/3t at time to were calculated as usual ( )by an on-line computer where 80 cheinnels of a total of 96 chemnels were used for the recording of the TCP, and the leist 12 channels, shifted by 164 seusple times, were used for the detection of the beise line. 

Figure 9.3. The simulated normalized spectral density function, J(cd) = J(a>)/ oda> ct) V(ct>), for water with immersed donor and acceptor molecules of radii 3.5 A at different separation (from [41c] with permission. Copyright (1997) by the American Institute of Physics), as compared against the experimental results (circles, data taken from [202a]). The corresponding time correlation function is shown in Figure 9.4.

Substituting Eqs. (3.47) and (3.48) into Eq. (3.58), the time correlation function of the end-to-end vector, after some algebraic derivation, can be expressed in terms of the normal modes as ... 

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