Correlation reduced

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32), Flectrons in the antisymmetric orbital r r have a 7ero probability of occurring at the node in u where U] = rj. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge... 

If a random process is described as statistically homogeneous, then its statistical moments are translation invariant, i.e., they do not depend on the positions X and X2 only on their difference. Thus the correlation reduces to ... 

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. 

This correlation reduces to the solution of Kronig and Bruijsten for small Pe and recovers Levich s result for large Pe. As shown in Fig. 41, the best fit of the data corresponds to n = 3, but = 2 fits the data nearly as well. 

PURIFICATION OF CORRELATED REDUCED DENSITY MATRICES REVIEW AND APPLICATIONS... 

Lowdin, who contributed in no small measure to the development of formal many-electron theory through his seminal work on electron correlation, reduced density matrices, perturbation theory, etc. many times expressed his concerns about the theoretical aspects of density functional approaches. This short review of the interconnected features of formal many-electron theory in terms of propagators, reduced pure state density matrices, and density functionals is dedicated to the memory of Per-Olov Lowdin. 

Somewhat later calculations by Kofranek et al. were able to incorporate correlation into the spectral data of the HCN dimer. The SCF data in Table 3.76 - °° confirm the patterns obtained earlier by Somasundram et al. with a different basis set. The correlated results in the next two columns of the table indicate no major changes in patterns. Curiously enough, the inclusion of correlation reduces all of the red shifts while enlarging all the blue... 

From Table 3.77, it appears that correlation exerts only a minor effect upon the CH stretching intensities. The absolute values of the monomers are diminished somewhat, but the dimerization-induced magnification of both the donor and acceptor are increased relative to the SCF results. The internal HCN bends are virtually unaffected. These trends are consistent with the SCF results of Somasundram et al. achieved with a different basis set, and recorded in Table 3.74. But quite dramatic effects are observed in the CN stretches. In the first place, correlation reduces the intensity of the HCN monomer band by an order of magnitude. Regarding the intensification caused by dimerization, SCF calculations from Tables 3.74 and 3.77 predict a factor of perhaps 3-5. But this magnification enlarges when correlation is included the dimer/monomer ratio is calculated to be 10 for the proton acceptor and 64 for the donor. Recent experimental measurements support the correlated data in that enhancements of approximately 30 are seen. 

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