3-body correlation effects

An analysis of the values taken by the different elements of the correlation matrices was recently reported [15] for the ground state of the Beryllium atom. This analysis suggested that the contributions of the 1 -, 2- and 3-body correlation effects differed according to the kind of orbitals involved in a given element. In particular, the highest occupied homo) and lowest empty (lumo) orbital of the HF configuration seemed to play an important role. 

In the third group the non-negligible correlation effects are only due to the 3-body correlation effects. Most of these elements are the same type as the second group ones. The new types appearing in this group are h l l2, h2nn (where the order of the two lumo s may differ in both elements) and h h hy, h h h where may also be one or the other of the n orbitals. 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

Three-Body Correlation Effects in Third-Order Reduced... 

Three-Body Correlation Effects in Third-Order Reduced Density Matrices Table 1 Graph representation of elements of RDM s and related quantities... 

Three-body correlation effects in third-order reduced density matrices... 

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. 

Thus, one can think of the R12 wave function as representing many-body correlation effects through two types of terms /(r12)O0 responsible for the short-range two-body Coulomb correlation and y describing conventional many-body correlation. The second term is expanded in terms of Slater determinants composed of orbitals from a finite orbital basis set (OBS) ... 

Eieiat. describes relativistic effects (such as variations in spin couplings - see Chap. A) and 8Econ. other electron-electron (and also electron-vibrational) many-body correlation effects (which are not included in Hartree-Fock calculations). 

There are two results in the data reported in Table 2 which I find both striking and surprising. Unexpectedly it appears that the largest error of 3D which occurs in the elements simultaneously involving the homo and the lumo is not due to the omission of three-body correlation effects, as the developments of Nakatsuji and Yasuda [15,16] seemed to imply, but is rather a result of subtracing instead of adding this three-body contribution to the joint effects of the one- and the two-body contributions. That is, in the FCI calculation and for these elements, the three correlation matrices practically cancel each... 

It is obvious that more sophisticated relativistic many-body methods should be used for correct treating the NEET effect. Really, the nuclear wave functions have the many-body character (usually, the nuclear matrix elements are parameterized according to the empirical data). The correct treating of the electron subsystem processes requires an account of the relativistic, exchange-correlation, and nuclear effects. Really, the nuclear excitation occurs by electron transition from the M shell to the K shell. So, there is the electron-hole interaction, and it is of a great importance a correct account for the many-body correlation effects, including the intershell correlations, the post-act interaction of removing electron and hole. 

The band calculations with potentials generated from local density functional approximations give the most detailed one electron band structures. As was discussed in section 2, one obtains Fermi surface dimensions in good agreement with experiments. We will mention briefly the basic principle of the calculation in order to illustrate that the calculation is surprisingly unsophisticated in terms of many-body correlation effects to be discussed later. 

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