Radial correlation

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

The presented result is different from the radial correlation function y(r) = An js21 (s)(sm 2nrs)/2nrs) ds, which is computed from the isotropic scattering intensity by means of the three-dimensional Fourier transform. 

Figure 8.23. The chord length distributions g (r) and g (—r) found in the 2nd derivative y" (r) of the radial correlation function. The example shows g(r) of a suspension of 10 wt.-% of silica (reproduced from a handout of DENISE TCHOUBAR)...

Figure 8.35 shows for homogeneous identical spheres the radial correlation function (Guinier and Fournet [65] p. 12-19 Letcher and Schmidt [192])... 

Figure 8.35. The homogeneous sphere of radius R. Radial correlation function, ys (r), distance distribution function (DDF) ps (r) and chord length distribution (CLD) gs (r)...

In the alkaline earth atom case, the polarized orbital pairs are formed by mixing the ns and np orbitals (actually, one must mix in equal amounts of pi, p i, and po orbitals to preserve overall S symmetry in this case), and give rise to angular correlation of the electron pair. Use of an (n+l)s2 CSF for the alkaline earth calculation would contribute in-out or radial correlation because, in this case, the polarized orbital pair formed from the ns and (n+l)s orbitals would be radially polarized. 

Therefore, the weakness of the fluctuation effects in the A + B — B reaction as compared to the case of A + B —> 0 (Section 5.1) does not permit us to restrict ourselves to the radial correlations between A s and B s subtle effects leading to equation (5.2.1) arise due to the angular correlations of reactants B in the vicinity of a single A. The demonstrated peculiarity of the A + B — B reaction requires an adequate specified mathematical approach. 

T. Morishita, C.D. Lin, Hyperspherical analysis of radial correlations in four-electron atoms, Phys. Rev. A 71 (2005) 012504. 

For an experimental verification of this exponent see [KSA88, LIM90, HAD91].) This n value results from the influence of radial correlations described by the hyperspherical angle a f because without these correlations a linear power law results (see [Rau71]). 

It follows that the GVB calculation takes into account some radial correlation, not present at the VBSCF level, hence the higher energy of the latter. Of course, this is not a limitation of the VBSCF method, because one could have started with ionic structures that match the ones embedded in the GVB wave function. 

As mentioned previously, parameter a may be viewed as the effective nuclear charge felt by either one of the two electrons. Such an interaction is commonly called the screening effect. Furthermore, as described by this wavefunction, the two electrons move independently of each other, i.e., angular correlation is ignored. Electron correlation may be taken as the tendency of the electrons to avoid each other. For helium, angular correlation describes the two electrons inclination to be on opposite sides of the nucleus. On the other hand, radial correlation, or screening effect, is the tendency for one electron to be closer to the nucleus, while the other one is farther away. A one-parameter trial function that does take angular correlation into account is... 

Another optional improvement concerns the description of the ionic VB structures. At the simplest level, the active ionic orbital is just a unique doubly occupied orbital as in 10 or 11. However this description can be improved by taking care of the radial correlation (also called "in-out" correlation) of the two active electrons, and this can be achieved most simply by splitting the active orbital into a pair of singly occupied orbitals accommodating a spin-pair, much as in GVB theory. This is pictorially represented in 12 and 13 which represent improved descriptions of 10 and 11. 

The radial correlation leads to a dynamical screening of the nuclear charge, imposed by one electron on the other. This allows the electrons to start at the inner boundary of the Coulomb zone with rt r2, but to leave the Coulomb zone with > r2 (or r, < r2). 

The question always arises as to how many configurations should be included. In the helium atom the radial correlation is improved by expanding in 5 orbitals, for example,... 

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