Electrons correlation correction factor

In our previous report, however, the calculated multiplet energies tend to be overestimated especially for the doublets. This is due to the underestimation of the effect of electron correlations. Recently, we have developed a simple method to take into account the remaining effect of electron correlations. In this method, the electron-electron repulsion integrals are multiplied by a certain reduction factor (correlation correction factor), c, and the value of c is determined by the consistency between the spin-unrestricted one-electron calculations and the multiplet calculations. The details of this method will be described in another paper (5). In the present paper, the effect of electron correlations on the multiplet structure of ruby is investigated by the comparison between the results with and without the correlation corrections. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

The next step to include electron-electron correlation more precisely historically was the introduction of the (somewhat misleading) so-called local- field correction factor g(q), accounting for statically screening of the Coulomb interaction by modifying the polarizability [4] ... 

The first observation one can make is that the correlation effects for the rotational g tensor in HF, H2O, and NH3 are in general small, 1.5-3.5%, and negative, i.e., correlation reduces the values of the rotational g tensor and therefore the amount of coupling between the electronic and rotational motion. Methane is an exception in that respect, because correlation increases the value of the g factor and because some methods, MPn and CCSD, predict much larger correlation corrections. 

With this correction all but one computed electron correlation energy fell within 10% of the exact solution. As Table 11 shows, the very simple scaling correction yields huge improvements on modest initial electron correlations. The use of the correction factor implies the loss of the variational principle and does not account for the use of different basis sets for example, the Dunning double-zeta basis set for Be is different for LiH [24, 25]. 

The size distributions of the particles in cloud samples from three coral surface bursts and one silicate surface burst were determined by optical and electron microscopy. These distributions were approximately lognormal below about 3/x, but followed an inverse power law between 3 and ca. 60 or 70p. The exponent was not determined unequivocally, but it has a value between 3 and 4.5. Above 70fi the size frequency curve drops off rather sharply as a result of particles having been lost from the cloud by sedimentation. The effect of sedimentation was investigated theoretically. Correction factors to the size distribution were calculated as a function of particle size, and theoretical cutoff sizes were determined. The correction to the size frequency curve is less than 5% below about 70but it rises rather rapidly above this size. The corrections allow the correlation of the experimentally determined size distributions of the samples with those of the clouds, assuming cloud homogeneity. 

The inappropriate scaling of the RLDA with Z, and thus also with j6, becomes particularly obvious for fixed electron number. In Fig. 5.5 the percentage deviations of the RLDA for and Ej are shown for the Ne isoelectronic series. The error for the correlation energy in the RLDA shows little tendency to approach zero with increasing Z, indicating that the relativistic correction factor plotted in Fig. 4.4 is inadequate for electronic structure calculations. 

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