Autocorrelator

Goldfisher, Autocorrelation function and power spectral density of laser-produced speckle pattern . J. Opt. Soc. Am., vol.55, p.247(1965). 

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

Day P N and Truhlar D G 1991 Benchmark calculations of thermal reaction rates. II. Direct calculation of the flux autocorrelation function for a canonical ensemble J. Chem. Phys. 94 2045-56... 

A typical noisy light based CRS experiment involves the splitting of a noisy beam (short autocorrelation time, broadband) into identical twin beams, B and B, tlnough the use of a Michelson interferometer. One ami of the interferometer is computer controlled to introduce a relative delay, x, between B and B. The twin beams exit the interferometer and are joined by a narrowband field, M, to produce the CRS-type third order polarization in the sample ([Pg.1209]

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

Figure B2.1.2 Modified Michelson interferometer for non-collinear intensity autocorrelation. Symbols used rl, r2, retroreflecting mirror pair mounted on a translation stage bs, beamsplitter x, nonlinear crystal pint, photomultiplier Pibe.

Figure B2.1.3 Output of a self-mode-locked titanium-sapphire oscillator (a) non-collinear intensity autocorrelation signal, obtained with a 100 pm p-barium borate nonlinear crystal (b) intensity spectrum.

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

The intensity autocorrelation measurement is comparable to all of the spectroscopic experunents discussed in the sections that follow because it exploits the use of a variably delayed, gating pulse in the measurement. In the autocorrelation experiment, the gating pulse is just a replica of the time-fixed pulse. In the spectroscopic experiments, the gating pulse is used to mterrogate the populations and coherences established by the time-fixed pulse. 

An interferometric method was first used by Porter and Topp [1, 92] to perfonn a time-resolved absorption experiment with a -switched ruby laser in the 1960s. The nonlinear crystal in the autocorrelation apparatus shown in figure B2.T2 is replaced by an absorbing sample, and then tlie transmission of the variably delayed pulse of light is measured as a fiinction of the delay This approach is known today as a pump-probe experiment the first pulse to arrive at the sample transfers (pumps) molecules to an excited energy level and the delayed pulse probes the population (and, possibly, the coherence) so prepared as a fiinction of time. 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Here, the force F is a linear combination of the components of R it also has a Gaussian distribution and autocorrelation matrix that satisfies the same properties of R t) as shown in eq. (3), with I (the nxn unit matrix) replacing M [71] ... 

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

Fig. 1. Comparison between the CID-CSP, CSP, TDSCF, and the numerically exact autocorrelation functions.

Let us start with a classic example. We had a dataset of 31 steroids. The spatial autocorrelation vector (more about autocorrelation vectors can be found in Chapter 8) stood as the set of molecular descriptors. The task was to model the Corticosteroid Ringing Globulin (CBG) affinity of the steroids. A feed-forward multilayer neural network trained with the back-propagation learning rule was employed as the learning method. The dataset itself was available in electronic form. More details can be found in Ref. [2]. 

Another approach employing the autocorrelation coefficients as descriptors was suggested by Gasteiger et al, [22]. They used the neural networks as a working tool for solving a similarity problem. 

Z eb index, Wiener index. Balaban J index, connectivity indices chi (x), kappa (k) shape indices, molecular walk counts, BCUT descriptors, 2D autocorrelation vector... 

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

A range of physicochemical properties such as partial atomic charges [9] or measures of the polarizabihty [10] can be calculated, for example with the program package PETRA [11]. The topological autocorrelation vector is invariant with respect to translation, rotation, and the conformer of the molecule considered. An alignment of molecules is not necessary for the calculation of their autocorrelation vectors. 

In the calculation of a 3D autocorrelation vector the spatial distance is used as given by Eq. (20). 

Here, the component of the autocorrelation vector a for the distance interval between the boundaries dj (lower) and (upper) is the sum of the products of property p for atoms i and j, respectively, having a Euclidian distance d within this interval. 

In contrast to the topological autocorrelation vector, it is possible to distinguish between different conformations of a molecule using 3D autocorrelation vectors. 

The calculation of autocorrelation vectors of surface properties [25] is similar (Eq. (21), with the distance d XiXj) between two points and Xj on the molecular surface within the interval between d[ and d a certain property p, e.g., the electrostatic potential (ESP) at a point on the molecular surface and the number of distance intervals 1). 

The component of the autocorrelation vector for a certain distance interval between the boundaries 4 and du is the sum of the products of the property p x,) at a point Xi on the molecular surface with the same property p Xj) at a point Xj within a certain distance d Xj,Xj) normalized by the number of distance intervals 1. All pairs of points on the surface are considered only once. 

The 3D autocorrelation vector of the three xylene isomers in Figure 8-4 differ only with respect to the component relating to the two methyl groups. For o-xylene it is... 

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