Exponentially correlated integral

The recursive four-body formulas increased the importance of having good recursive methods for the three-body exponentially correlated integrals with one index equal to — 1. An additional method for dealing with these integrals was briefly sketched by the present author [5], but the material there presented gave neither a full description of the formula nor its method of derivation. The present communication provides the missing derivation and discusses a class of finite summations that are relevant thereto. 

The solvation dynamics of bulk water have been well studied. Jarzeba et al. [33] obtained a correlation function with 160 fs (33%) and 1.2 ps (67%), and Jimemez et al. [34] reported an initial Gaussian-type component (frequency 38.5 ps-1 25 fs in time width, 48%) and two exponential decays of 126 fs (20%) and 880 fs (35%). Using Eq. (6), the correlation function we obtained for bulk water, as shown in Fig. 4, is best fitted by double exponential decays integrated with an initial Gaussian-type contribution through a stretched mode c t) = + c2e tlZl, where for a pure Gaussian-type decay, / = 2. The... 

The power-exponential correlation function (3), for example, is of this product form. To computere x), the integral on the right-hand side of (22) is evaluated as... 

By using the fitting function (16) and Eq. (15), we numerically reproduce Gg(f feq), and the reproduced curve well approximates the numerical result as shown in Fig. 9b. Note that cjg(f feq) is proportional to f2 in the limit of f 0, since Cp(s feq) in Eq. (15) goes to the constant Cp(0 feq). On the other hand, in the limit of f —> oo, cjg(f feq) is proportional to f, because both Cp(s feq) and sCp(s teq) are almost zeros in the long-time region, and hence their integrals become constants. The crossover from f2 to f is also observed if we assume an exponential correlation function, and hence we conclude that diffusion at equilibrium is normal as expected although a stretched exponential is present. 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

A practical reason that pre-exponential r,y have not been used with exponentially correlated four-body wavefunctions has been the difficulty of managing analytical formulas for the integrals that thereby result that difficulty has now been reduced in importance by the author s recent presentation[10] of a recursive procedure for the integral generation. 

The family of variational methods with explicitly correlated functions includes the Hylleraas method, the Hyller-aas Cl method, the James-Coolidge and the KcAos-Wolniewicz approaches, as well as a method with exponentially correlated Gaussians. The method of explicitly correlated functions is very successful for two-, three-, and four-electron systems. For larger systems, due to the excessive number of complicated integrals, variational calculations are not yet feasible. 

Abstract A new recursive procedure is reported for the evaluation of certain three-body integrals involving exponentially correlated atomic orbitals. The procedure is more rapidly convergent than those reported earlier. The formulas are relevant to ab initio electronic-structure computations on three- and four-body systems. They also illustrate techniques that are useful in the evaluation of summations involving binomial coefficients. 

Keywords Three-body integrals Binomial summations Exponentially correlated orbitals... 

For electronic-structure computations involving exponentially correlated orbitals in atomic systems, it is convenient to generate the necessary integrals using recurrence... 

Recursive methods have also been reported for exponentially correlated four-body atomic systems, where the integrals have the generic form... 

Experimentally, one usually integrates over one direction by having a detector with a sufficiently wide acceptance. Using an exponential correlation function... 

Now the phase shift depends on the in-plane index h, where we have chosen a surface unit cell with the bulk periodicity, meaning that the superstructure reflections correspond to half-order values for h. If we assume that both domains are equally probable (0 = 0.5) and that an exponential correlation function applies, we now find for the two components (after integration over k)... 

From the above discussion, it is apparent that the exponential asymptotic behaviour of KmU) characterizes the correlation between collisions rather than collision itself. Hence the quantity tm defined in Eq. (1.67) cannot be considered as a collision time. To determine the true duration of collision let us transform Eq. (1.63) to the integral-differential equation as was done in [51] ... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

To describe bound stationary states of the system, the cji s have to be square-normalizable functions. The square-integrability of these functions may be achieved using the following general form of an n-particle correlated Gaussian with the negative exponential of a positive definite quadratic form in 3n variables ... 

To model the experimental data we used a global-fit procedure to simulate EPS, integrated TG, heterodyne-detected TG, and the linear absorption spectrum simultaneously. The pulse shape and phase were explicitly taken into account, which is of paramount importance for the adequate description of the experimental data. We applied a stochastic modulation model with a bi-exponential frequency fluctuation correlation function of the following form ... 

Figure 20. Integrated signals Ei and 2 for OHBA (a), ODBA (b), and HAN (c) plotted as a function of the time delay at the indicated excitation wavelength. Note the change in ordinate time scales. Signal Ei always followed the laser cross-correlation, indicating a rapid proton transfer reaction. The decay of signal 2 was fitted via single exponential decay, yielding the time constant for internal conversion of the Si keto state in each molecule. See color insert.

---