Pair-density functional theory

The importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

The Slater hull constraints are not directly applicable to existing approaches to pair-density functional theory because they are formulated in the orbital representation. Toward the conclusion of this chapter, we will also address A-representability constraints that are applicable when the spatial representation of the pair density is used. 

The preceding approach can be viewed as an orbital representation analogue for a recently proposed Kohn-Sham-based pair-density functional theory [17],... 

P. Ziesche, Pair density-functional theory — a generalized density-functional theory. Phys. Lett. A 195, 213-220 (1994). 

P. Ziesche, Attempts toward a pair density functional theory. Int. J. Quantum Chem. 60, 149-162 (1996). 

F. Furche, Towards a practical pair density-functional theory for many-electron systems. Phys. Rev. A 70, 022514 (2004). 

P. W. Ayers and M. Levy, Using the Kohn—Sham formalism in pair density-functional theories. Chem. Phys. Lett. 416, 211-216 (2005). 

Higuchi, K. Higuchi, M. Computational pair density functional theory a proposal for the kinetic energy functional. Phys. Rev. B 2010, 82, 155135. 

Higuchi, M. Higuchi, K. Pair density-functional theory by means of the correlated wave function. Phys. Rev. A 2007, 75, 042510. 

Nagy, A. Spherically and system-averaged pair density functional theory. J. Chem. Phys. 2006,125, 184104. 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

The basis set is 6-31G(d,p), and electron correlation at the MP2 level is included. A similar structure is obtained with the AMI and PM3 semi-empirical methods. Density functional theory at the B3LYP/6-31G(dp,p) level also produced the same structure for this ion-pair. The only observed differences between the semi-empiri-cal and the ab initio structures were slightly shorter hydrogen bonds (PM3 and AMI) between FI, F2, and F5 and the G2-F1 (H18) on the imidazolium ring. 

Perdew, J. P., Ernzerhof, M., Burke, K., Savin, A., 1997, On-Top Pair-Density Interpretation of Spin Density Functional Theory, With Applications to Magnetism , Int. J. Quant. Chem., 61, 197. 

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