Reduced density-functions correlation densities

Wesolowski, T. A., Parisel, O., Ellinger, Y., Weber, J., 1997, Comparative Study of Benzene---X (X = 02, N2, CO) Complexes Using Density Functional Theory The Importance of an Accurate Exchange-Correlation Energy Density at High Reduced Density Gradients , J. Phys. Chem. A, 101, 7818. 

Solutions in hand for the reference pairs, it is useful to write out empirical smoothing expressions for the rectilinear densities, reduced density differences, and reduced vapor pressures as functions of Tr and a, following which prediction of reduced liquid densities and vapor pressures is straightforward for systems where Tex and a (equivalently co) are known. If, in addition, the critical property IE s, ln(Tc /Tc), ln(PcVPc), and ln(pcVPc), are available from experiment, theory, or empirical correlation, one can calculate the molar density and vapor pressure IE s for 0.5 < Tr < 1, provided, for VPIE, that Aa/a is known or can be estimated. Thus to calculate liquid density IE s one uses the observed IE on Tc, ln(Tc /Tc), to find (Tr /Tr) at any temperature of interest, and employs the smoothing relations (or numerically solves Equation 13.1) to obtain (pR /pR). Since (MpIE)R = ln(pR /pR) = ln[(p /pc )/(p/pc)] it follows that ln(p7p)(MpIE)R- -ln(pcVpc). For VPIE s one proceeds similarly, substituting reduced temperatures, critical pressures and Aa/a into the smoothing equations to find ln(P /P)RED and thence ln(P /P), since ln(P /P) = I n( Pr /Pr) + In (Pc /Pc)- The approach outlined for molar density IE cannot be used to rationalize the vapor pressure IE without the introduction of isotope dependent system parameters Aa/a. 

Density Functional Theory, DFT (B3LYP), CASSCF (Complete Active-State Self-Consistent Field) and MRSD-CI (Multi-Reference Single-Double Correlation Interaction) calculations on the diatomic units AuO, AuO", AuO " and AuO " clearly show that stability of Au-0 bond reduces in this order. This trend is consistent with the molecular orbital diagram of AuO molecule presented in Fig. 10. 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

Yasuda [55] obtained a correlation energy functional Ec[ D] from the first-and second-order density equations together with the decoupling approximations for the 3- and 4-reduced density matrices. The Yasuda functional is capable of properly describing a high-density HEG [56] and encouraging results were reported for atoms and molecules [55]. Some shortcomings of this functional were also pointed out [57]. 

The matrix elements of the reduced density matrix needed to calculate the entanglement can be written in terms of the spin-spin correlation functions and the average magnetization per spin. The spin-spin correlation functions for the ground state are dehned as [62]... 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

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. 

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