Electron correlation 2-particle density matrix

The correlation entropy is a good measure of electron correlation in molecular systems [5, 7]. It is defined using the eigenvalues of the one-particle density matrix IPDM,... 

In ab initio methods the HER approximation is used for build-up of initial estimate for and which have to be further improved by methods of configurational interaction in the complete active space (CAS) [39], or by Mpller-Plesset perturbation theory (MPn) of order n, or by the coupled clusters [40,41] methods. In fact, any reasonable result within the ab initio QC requires at least minimal involvement of electron correlation. All the technical tricks invented to go beyond the HFR calculation scheme in terms of different forms of the trial wave function or various perturbative procedures represent in fact attempts to estimate somehow the second term of Eq. (5) - the cumulant % of the two-particle density matrix. 

The exchange-correlation potential is the source of both the strengths and the weaknesses of the DF approach. In HF theory, the analytical form of the term equivalent to Vxc, the exchange potential, arises directly during the derivation of the equations, but it depends upon the one-particle density matrix, making it expensive to calculate. In DF theory the analytical form of Vxc must be put into the calculations because it does not come from the derivation of the Kohn-Sham equations. Thus, it is possible to choose forms for Vxc that depend only upon the density and its derivatives and which are cheap to calculate (the so-called local and non-local density approximations). The Vxc factor can also be chosen to account for some of the correlation between the electrons, in contrast to HF methods for which additional calculations must be made. The drawback is that there does not appear to be any systematic way of improving the potential. Indeed, many such terms have been proposed. 

In the normal (probability theory) use of the term, two probability distributions are not correlated if their joint (combined) probability distribution is just the simple product of the individual probability distributions. In the case of the Hartree-Fock model of electron distributions the probability distribution for pairs of electrons is a product corrected by an exchange term. The two-particle density function cannot be obtained from the one-particle density function the one-particle density matrix is needed which depends on two sets of spatial variables. In a word, the two-particle density matrix is a (2 x 2) determinant of one-particle density matrices for each electron ... 

Assuming that no correlation among particles is present, the meanings of the structure related parameters of Guinier s law are readily established by application of the mathematical tools of scattering (cf. Chap. 2). Further assumptions state that each particle k is considered to be immersed in a matrix of constant electron density and that its correlation function is monotonously decaying. Thus, the particle is discriminated from the matrix by its number nik of electrons that it has more (respectively, less) than the homogeneous matrix, nik is called the number of excess electrons of the particle k. 

In the following a special class of functions, termed electron localizability indicators [3-8], based on simultaneous evaluation of electron density and electron pair density will be described. This combination is utilized with the aim to analyze the correlation of electronic motion [9]. Apart from the density function point of view, the energy of a molecule can be thought as stemming from two parts - a one-particle terms in wide sense derived from the electron density and a two-particle terms derived from the electron pair density (of course, the full 2-matrix is stiU necessary today). The interplay between the electron density on the one hand and the electron pair density on the other hand could thus elucidate the situation in the molecular system. 

---