Symmetrized autocorrelation function

We note that classical dipole autocorrelation functions are real and symmetric in time, C(—t) = C(t). 

Binary interactions. Dipole autocorrelation functions of binary systems are readily computed. For binary systems, it is convenient to obtain the dipole autocorrelation function, C(t), from the spectral profile, G(co). Figure 5.2 shows the complex correlation function of the quantum profile of He-Ar pairs (295 K) given in Figs. 5.5 and 5.6. The real part is an even function of time, 91 C(—t) = 91 C(t) (solid upper curve). The imaginary part, on the other hand, is negative for positive times it is also an odd function of time, 3 C(—t) — — 3 C(t) (solid lower curve, Fig. 5.2). For comparison, the classical autocorrelation function is also shown. It is real, positive and symmetric in time (dotted curve). In the case considered, the... 

Fig. 5.2. The dipole autocorrelation function of He-Ar at 295 K, according to a quantal (solid lines) and a classical calculation (dotted). The quantum correlation function is complex the real part is symmetric and positive (91) while the imaginary part (3) is anti-symmetric and negative at positive frequencies.

The classical dipole correlation function is symmetric in time, C(—t) = C(f), as may be seen from Eq. 5.59 by replacing x by x — t the classical scalar product in Eq. 5.59 is, of course, commutative. Classical line shapes are, therefore, symmetric, J(—. Furthermore, classical dipole autocorrelation functions are real. 

We will briefly consider several desymmetrization procedures that have been mentioned in the literature. These may simply employ various factors applied to the symmetric, classical profile, G(co), or alternatively attempt to correct the classical dipole autocorrelation functions in the time domain. 

Exercise. For one-step processes W is a tridiagonal matrix. With the aid of (3.8) a similarity transformation can be constructed which makes the matrix symmetric, as in (V.6.15). Prove in this way that any finite one-step process has a complete set of eigenfunctions, and that its autocorrelation function consists of a sum of exponentials.510... 

Equation III is merely stating that a correlation function of a periodic signal is also periodic while Equation IV states that the maximum amount of correlation of a periodic function occurs when the same point of that signal is compared with itself and that the correlation is diminished as one compares two points that are farther and farther apart (up to a difference of T). Equation V states that the autocorrelation function is symmetric about t = 0. 

The autocorrelation function S(t) shown in Figure 7.20 reflects these two types of motion the wide oscillations with period Tss represent the slow symmetric stretch vibration and the rapid oscillations with period Tas represent the fast anti-symmetric stretch vibration. Because the two exit channels are so exceedingly narrow leakage into the product channels is rather weak. Although no barrier hinders the dissociation the system is trapped in the inner region for a long time. 

Fig. 7.20. Autocorrelation function for the dissociation of EHI. The wide oscillation reflects the symmetric stretch motion of the two iodine atoms with respect to the stationary hydrogen atom with period Tss whereas the fast oscillations manifest the anti-symmetric stretch motion of hydrogen between the two iodine atoms with period Tas. Adapted from Engel (1991a).

The wavepacket in the lBi state, 3>b, performs large-amplitude symmetric stretch motion leading to recurrences in the autocorrelation function the recurrences in turn cause vibrational structures in the absorption spectrum. 

Power spectra of the autocorrelation functions of the linear and angular velocities parallel and perpendicular to the C3 symmetrical axes have also been examined by Neusy et al. (32). In the rotator phase, there is good agreement with the Raman data (36). The calculated characteristic time (r4) for reorientation of the C3 axes from one [111] direction to another and also the reorientation time (r3) for rotation of molecules around the C3 axes were similar... 

V(x) is assumed to be the usual symmetrical double-well potential [V(x) - 8x /2 + fix /4] the third term on the right-hand side of Eq. (77) is the coupling between the Brownian particle and the external radiation field, which is characterized through its autocorrelation function... 

Inserting Eq. (138) into the expression for the autocorrelation function (137) and noting that the integral is symmetric in the times, the delta function restricts the integration to the lesser of the two times, so introducing the notation for the lesser time t < and greater time t> we obtain [53]... 

The diffuse structures in the absorption spectrmn of ozone in the Chappuis band, shown in Figs. 12 (b) and (c), has likewise puzzled experimentalists and theorists for a long time. The recmrence structmes of the autocorrelation function provide a reasonable explanation, although not all details are perfectly reproduced by the calculations. The main structures reflect excitation of the symmetric stretch motion, just hke for H2S. 

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