Interparticle coordinates

For a theoretical calculation of relaxation times one. must write the temporal autocorrelation functions of several functions Fn of the interparticle coordinates riS(t), 0y(O, and interparticle distance and where 0,/O and external magnetic field Ho (here particle refers to magnetic nuclei and atoms). The relaxation rates are proportional to the Fourier intensities of these autocorrelation functions at selected frequencies. For example, Torrey (16) has written for this autocorrelation function the equivalent ensemble average... 

Its form in terms of the interparticle coordinates n, r2 and n2 has been given in equation (3.46). 

By making a series of approximations, Colle and Salvetti then obtained the following expression in terms of interparticle coordinates R = (rj + Vq) and u = rj — r2... 

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. 

In contrast to the four-body problem, real exponential wavefunctions in the interparticle coordinates led to readily evaluated integrals for three-body systems, and that fact was exploited in the context of extensive configuration interaction for adiabatic systems as long ago as 1977 by Thakkar and Smith [12]. Starting in 1987, this method was also applied to nonadiabatic systems by Petelenz and Smith [33,34], and to many adiabatic and nonadiabatic systems by Frolov, both alone [35,36] and in collaboration with Thakkar [37] and with Smith [38]. 

This communication outlines formulas that can be used when the energy is described (for S states) entirely in terms of the interparticle coordinates r,- and extends earlier work [11, 12] that shows how the combinations of integrals that describe the kinetic energy can be related to the overlap and Coulomb interaction integrals that enter the evaluation of the electrostatic potential energy. 

The matrix elements of interest here are of the forms ( (a, n) (/ , m)), (i/f(oe, n) V Vf(/ , m)), and n) T Vf(/ , m)), where V and T are, respectively, the Coulomb and kinetic-energy operators. For a four-body system, with respective charges qi,. ..,q4 and masses mi,m, with all quantities expressed in Hartree atomic units, these matrix elements can be written entirely in terms of the interparticle coordinates [12], with the potential-energy matrix elements given as... 

It should be noted that for the set of interparticle coordinates, if, in the course of distinguishing axes, we fix the positions of two or more particles (e.g. nuclei) in the distinguished subspace, the number of particles N in Eq. (7) needs to be reinterpreted. To avoid confusion, in such cases we shall henceforth exclude the particles fixed in the subspace in our definition of N. The centrifugal factor is then (D— r) — l)(D — r) — 2N — l), which renders this formulation equivalent to treatment using the generalization of the set of radial coordinates and cosines of angles. 

The applications of this model is mostly restricted to hydrogen and helium like systems. Winkler [25] has calculated the detachment energies of H embedded in a variety of Debye plasmas. He used a correlated description of the two-particle wave functions introducing interparticle coordinate in the expansion of the basis set. The linear variational parameters are determined by solving the generalized secular equation... 

Figure 5-1 Interparticle coordinates for a three-particle system consisting of two electrons and a nucleus.

There are several techniques for going beyond the SCF method and thereby including some effects of electron correlation. Some extremely accurate calculations on small atoms and molecules, making explicit use of interparticle coordinates, were described in Section 7-8. There is one general technique, however, that has traditionally been used for including effects of correlation in many-electron systems. This technique is called configuration interaction (Cl). 

This idea is readily extended to the Born-Oppenheimer electronic Hamiltonian by noting that x -r xe implies that interparticle coordinates should be scaled asr re . For 0 0, the operator H 6) is non-Hermitian and therefore admits complex eigenvalues. In its simplest form, the CCR method consists of determining these eigenvalues. 

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