Correlation functions reliability

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

Correlation functions are sometimes believed to be more reliable than their spectra... 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

In Ref. 80 we carried out a W1 and W2 investigation for all six cases with X,Y F, Cl, Br, in order to assess the performance of a number of DFT exchange-correlation functionals. W2 is in excellent agreement with experiment where reliable experimental data are available in some other cases, the W1 calculations either suggest revisions or provide the only reliable data available (see Ref. 80 for details). 

During last decades the DFT based methods have received a wide circulation in calculations on TMCs electronic structure [34,85-88]. It is, first of all, due to widespread use of extended basis sets, allowing to improve the quality of the calculated electronic density, and, second, due to development of successful (so called - hybrid) parameterizations for the exchange-correlation functionals vide infra for discussion). It is generally believed, that the DFT-based methods give in case of TMCs more reliable results, than the HER non-empirical methods and that their accuracy is comparable to that which can be achieved after taking into account perturbation theory corrections to the HER at the MP2 or some limited Cl level [88-90]. 

Analysis of the correlation functions demonstrates also impressive general agreement between the superposition approximation and computer simulations. Note, however certain overestimate of the similar particle correlations, X r,t), at small r, especially for d = 1. In its turn the correlation function of dissimilar particles, Y(r,t), demonstrates complete agreement with the statistical simulations. Since the time development of concentrations is defined entirely by Y(r, t), Figs 5.2 and 5.3 serve as an additional evidence for the reliability of the superposition approximation. An estimate of the small distances here at which the function Y (r, t) is no longer zero corresponds quite well to the earlier introduced correlation length o, equation (5.1.47) as one can see in fact that at moment t there are no AB pairs separated by r < o-... 

Quantitative deviations are seen also from the correlation shown in Fig. 5.9. The correlation functions of dissimilar particles Y (r, t) are in good agreement with simulations, which results also in a reliable reproduction of the decay kinetics for nA(t) - unlike behaviour of the correlation functions of the similar particles Xv r,t) which is very well pronounced for XA(r,t). Positive correlations, Xu(r,t) > 1 as r < , argue for the similar particle aggregation, and the superposition approximation tends to overestimate their density. The obtained results permit to conclude that the approximation (2.3.63) of the three-particle correlation function could be in a serious error for the excess of one kind of reactants. 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

The longest value of ( r ) that can be reliably measured is determined by the longest sampling interval in the correlator times the number of channels, the dark count in the photomultiplier tube, the long term stability in the laser, and whether full correlation or clipping is employed. At present 100 s is a practical maximum for measured values of (r). In order to determine a relaxation time of 100 s, it is desirable to measure the correlation function for at least 1000 relaxation times. This means that run times of 105 s are required. This places severe requirements on the long term stability of all parts of the system. Routine measurements of (r) are probably better restricted to 10 s. 

Detailed high-frequency (terahertz) dynamical studies of glasses have been probed by inelastic X-ray scattering (IXS) [139], The advantage of this technique is that with reliable measurements it allows determination of the so-called nonergodicity parameter f(q, T) as a function of wavevector q this quantity is defined by the long time limit of the density-density correlation function F(q, t) divided by the static structure factor [15],... 

Solvation dynamics are measured using the more reliable energy relaxation method after a local perturbation [83-85], typically using a femtosecond-resolved fluorescence technique. Experimentally, the wavelength-resolved transients are obtained using the fluorescence upconversion method [85], The observed fluorescence dynamics, decay at the blue side and rise at the red side (Fig. 3a), reflecting typical solvation processes. The molecular mechanism is schematically shown in Fig. 5. Typically, by following the standard procedures [35], we can construct the femtosecond-resolved emission spectra (FRES, Stokes shifts with time) and then the correlation function (solvent response curve) ... 

---