Two body correlations

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

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

Basic Properties of the Pure Two-Body Correlation Matrices... 

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. 

We may also readily calculate corrections to the IBI approximation, obtaining as a bonus a more precise definition of its limits of validity. First-order corrections have two origins, the three-body correlation function and the effect of a third body on the two-body correlation function. 

Fig. 5.22. First-order corrections to the wavefunction (a) two-body correlation p and q are excited to k and fc (b) p is excited to k by interacting with unexcited states n (c) is the exchange interaction from p to k via n (d) is the excitation from ptok through the perturbation. The dashed line is the Coulomb interaction while the cross indicates an interaction via the perturbation. For a Hartree-Fock potential, (b), (c) and (d) sum to zero (after H.P. Kelly [241]).

As mentioned above, in the absence of an external field, e can be expressed in terms of the response of each particle to the field set up by the others. In the model under consideration that field is a sum of pair terms, so the key statistical mechanical quantity involved in the expression of e is the two-body correlation function. Its systematic use unavoidably entails a heavy dose of terminology and notation, which we now introduce in the language of Refs. 4 and 5. 

In writing the final form for qi, the integrations over the angular variables (0 have been carried out. The function h l,2, E) is referred to as a two-body correlation function and carries the full effects of the two-body potential tpiij). The polarization is now given by the following expression ... 

In the preceding paragraphs we have shown that we have two models at hand which show the same static structure on the level of the two-body correlation functions. Do they have the same dynamics The high temperatme behavior of the CRC model (curve at T = 353 K) and the behavior of the FRC model agree. One observes a crossover from short time ballistic and vibrational motion to a subdifiusive Rouse-like regime determined by the connectivity of the chains. For the CRC model at... 

Another strong point of the simulation approach is its ability to selectively change parts of the model Hamiltonian. In this way one can compare a chemically realistic model of PB with a freely rotating chain version of the same polymer and does not have to switch to a completely different polymer with some of the same properties like is unavoidable in experiments [33]. With this approach we could establish that identical structure on the two-body correlation function level (single chain and liquid structure factors) does not imply identical dynamics which raises questions on the applicability of the mode-coupling theory of the glass transition to polymer melts. 

A reasonable way to estimate the size of the correlation correction to the relativistic tp potential is to make use of the semi-relativistic multiple scattering formalism [Ra 85] discussed in section 4.2. In analogy with the Watson theory, the leading correlation correction is quadratic in the relativistic NN invariant scattering operator and is proportional to two-body correlations in the target nucleus. 

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]. 

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