Two-body correlation matrix

A. The Pure Two-Body Correlation Matrix within the 2-RDM Formalism... 

That is, all the information about the three important matrices 2-RDM, 2-HRDM, and 2-G is contained and available in the pure two-body correlation matrix. Moreover, the spin properties of both the pure two-body correlation matrix and the 2-G matrices play a central role in this purification procedure. 

That is, both the 2-CM and the 2-G matrix have common elements, but a given element occupies different positions in each matrix. In other words, while the labels of the row/column of the 2-CM refer, as in the 2-RDM, to two particlesitwo holes, the labels of the row/column of the 2-G matrix refer to particle-hole/hole-particle. Thus, although both the 2-CM and the 2-G matrices describe similar types of correlation effects, only the 2-CM describes pure two-body correlation effects. This is because the 2-CM natural tensorial contractions vanish, and thus there is no contribution to the natural contraction of the 2-RDM into the one-body space whereas the 2-G natural tensorial contractions are functionals of the 1-RDM. 

Two-body correlation contributions were included in the NRDD calculations as explained in sections 3.7 and 3.8. For the NR calculations based on the Paris-Hamburg g-matrix the correlation terms in section 3.7 were heuristically added to the first-ordo potentials. The relativistic optical potential correlation contributions were not included in the IA2 or the RIA calculations since these effects are small [Ka 90, Lu 87]. 

Note that a quasi-atomic energy is different from a free-atom energy, since it depends on the molecular density matrices. The two-body density matrix can be written in terms of the density and the pair-correlation function (h) ... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

A very important property of the 2-CM is that [15, 83] the decomposition of the 2-HRDM yields a two-body hole correlation matrix that coincides with the 2-CM. Thus... 

Strutinsky procedure. Because asymmetric shapes is so shallow, it is worthwhile to deal with octupole correlation effects using a microscopic, two-body interaction treatment of the octupole-octupole residual interaction [CHA80]. The pairing force matrix elements, G. come from a density dependent delta interaction. This set of matrix elements [CHA77] explains many features of the actinides at low and... 

Another recommendation was to examine the correlation matrix of the covariates prior to the analysis and determine whether any two covariates were correlated. If any two correlated covariates were found to be important predictor variables, one could possibly transform the variables into a composite variable, such as the transformation of height and weight into body surface area or body mass index, or to use only the covariate with the greatest predictive value in the model and not to include the other covariate. An untested approach would be to use principal component analysis and then use one or more of the principal components as the covariate in a model. 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

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