Exponentially correlated Gaussian

Exponentially correlated gaussian type wave function... 

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. 

Variational calculations on using exponentially correlated Gaussian wave functions. 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

The above function is a one-center correlated Gaussian with exponential coefficients forming the symmetric matrix A]. <1) are rotationally invariant functions as required by the symmetry of the problem—that is, invariant with respect to any orthogonal transformation. To show the invariance, let U be any 3x3 orthogonal matrix (any proper or improper rotation in 3-space) that is applied to rotate the r vector in the 3-D space. Prove the invariance ... 

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 ... 

In this Table we also find a second form of exponentially correlated function s]. Such functions have been referred to as explicitly correlated Gaussians (ECG). They take the form,... 

The study of a Brownian particle suspended in a fluid lead also to the introduction of the exponentially correlated Ornstein-Uhlenbeck process [48], the only Markovian Gaussian non-white stochastic process [19, 22]. We present here the Langevin approach to this problem, hence we analyze the forces that act on a single Brownian particle. We suppose the particle having a mass m equal to unity, and we assume the force due to the hits with thermal activated molecules of the fluid to be a stochastic variable. Moreover, due to the viscosity of the fluid, a friction force proportional to the velocity of the particle has to be considered. All this yields the following equation... 

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. 

S. Bubin, L. Adamowicz. Matrix elements of N-partide expUdtly correlated Gaussian basis functions with complex exponential parameters. J. Chem. Phys., 124 (2006) 224317. 

For this so-called Uhlenbeck-Ornstein process, one therefore obtains correlation functions whose decay can range from simple exponential to Gaussian. 

The total peak profile is the sum of these two components, with a relative weight that depends on 0 and I. The Bragg component is mathematically a 5-function, but is in reality broadened by instrumental and sample imperfections. The shape of the diffuse component depends on the form of the correlation function. Many forms are possible, but two common ones are an exponential correlation function, leading to a Lorentzian line shape, and a Gaussian correlation function, leading to a Gaussian Hne shape. One can also use correlation functions that describe a preferred distance (e.g., island-island correlations) or with an in-plane anisotropy [42]. 

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