Pair-density function

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). 

Keywords Local atomic structure Neutron scattering Pair-density function ... 

In addition, many of the ferroelectric solids are mixed ions systems, or alloys, for which local disorder influences the properties. The effect of disorder is most pronounced in the relaxor ferroelectrics, which show glassy ferroelectric behavior with diffuse phase transition [1]. In this chapter we focus on the effect of local disorder on the ferroelectric solids including the relaxor ferroelectrics. As the means of studying the local structure and dynamics we rely mainly on neutron scattering methods coupled with the real-space pair-density function (PDF) analysis. 

Egami (2001) has developed pair-density-function analysis of pulsed neutron-diffraction data to determine the structure in a time interval that is short relative to a bond-length fluctuation. This method provides a direct observation of the pattern of bond-length fluctuations. Several other measurements have been shown to provide indirect signatures of the presence of bond-length fluctuations (Goodenough and Zhou, 2001). For example,... 

Another function often denoted g r) is called the pair distribution function. It is normalized so that, as r -r oo, g(r) 1 and has the property that for r shorter than the distance of closest approach of pairs of atoms g r) becomes zero. It is closely related to the pair density function, p(r) = pog(r). Clearly, p(r) oscillates about, and then asymptotes to, the average number density of the material, po at high-r and becomes zero as r 0. The relationship between these correlation functions is given by ... 

As in the classical kinetic theory of gases, the pair density function is impossible to determine anal3dically and some closure approximation has to be made. In the population balance derivation we normally adopt a common assumption in kinetic theory and makes the coarse approximation /( )(x,r x, r, f) This assumption implies that... 

Atomic pair-density function analysis of X-ray diffraction patterns is used to resolve the local stmcture of the catalysts from 0.15 to 2 nanometers (nm). 

Pair-density function analysis of X-ray diffraction patterns shows that active iron oxide (FeOx) catalysts are disordered at the nanoscale by the platinum and also have micropores, while inactive catalysts do not share these features. 

Although the catalysts are termed amorphous, they can have nanoscale order in their atomic stmctures that can have a significant bearing on their activity and transport properties. These nanoscale structural properties are not easily discerned with conventional materials science tools, but can be elucidated using an emerging technique atomic pair-density function analysis of X-ray diffraction (PDF-XRD). ... 

NaAlH4 Sodium Tetrahydroaluminate PDF Pair-Density Function... 

Another, a similar route that considers interactions between individual elements of a system, is the reference interaction site model, RISM. The theory involves computations of the system structure by means of the probability density function, which describes location of all N particles of the system. The binary interactions define the pair-density function ... 

Soncini and Lazzeritti calculated the one- and two-bonds nuclear spin-spin coupling densities and the Fermi hole densities for hydrogen fluoride, water, ammonia, and methane molecules. The pair density function p2(xi, X2) determines the probability of two electrons being found simultaneously at points Xi = fiT]i and X2 = tit] , where i and fi are coordinates in three-dimensional space, and rii and TI2 are the spin variables of the two electrons. For a system described by a one determinant wavefunction of occupied spin-orbitals < >, (x), that is, a wavefunction in the HF approximation, the pair density function becomes... 

Levy, M. Ziesche, P. The pair density functional of the kinetic energy and its simple scaling property. J. Chem. Phys. 2001,115,9110-9112. 

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. 

---