Function, universal

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. The most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibiHty factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

Using these assumptions and a classical partitioning function, integrated over the coordinates and momenta of aH molecules, a universal function was defined ... 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

The relative coverages are in this case universal functions of the parameter y and thus of the distance from the maximum, expressed in (em — e), proportional to (1/Tm) — (1 /T). 

A direct consequence of this feature of the hyperbolic heating schedule is that the relative desorption rates n = Nt/Nm are also universal functions of y, and their shape depends on the desorption order only. They read ... 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

This, at first glance innocuous-looking functional FHK[p] is the holy grail of density functional theory. If it were known exactly we would have solved the Schrodinger equation, not approximately, but exactly. And, since it is a universal functional completely independent of the system at hand, it applies equally well to the hydrogen atom as to gigantic molecules such as, say, DNA FHK[p] contains the functional for the kinetic energy T[p] and that for the electron-electron interaction, Eee[p], The explicit form of both these functionals lies unfortunately completely in the dark. However, from the latter we can extract at least the classical Coulomb part J[p], since that is already well known (recall Section 2.3),... 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

Finally for Cfe(c) 1 the unperturbed (not self-entangled) single-chain re-laxationjust known from good solvent conditions, takes place. S(Q, t)/S(Q, 0) is a universal function of (Q(Q,t)2/3 with Q(Q) = QZ(Q) In Fig 58b a schematic plot of the crossover behavior of the segmental dynamics under 0-conditions is shown. 

The universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

Along with T the kinetic and interaction energies must also be functionals of p(r), such that a universal functional, valid for any number of particles may be defined as... 

The universal functional G[p] is written as the sum of kinetic and exchange energy functionals... 

V. The second term is the classical Coulomb energy of a density distribution p. The quantity F p] is a universal functional of the density, which means that it is uniquely specified by the density p of the interacting electrons and does not depend on the particular external potential V acting on the electrons. The functional F contains whatever is necessary to make the energy in Eq. (6) equal to the expected value in Eq. (2). 

Two approaches to the excited-state problem have been the focus of this chapter. The nonvariational one, based on Kato s theorem, is pleasing in that it does not require a bifunctional, but it presumes that the excited-state density is known. On the other hand, the bifunctional approach is appealing in that it actually generates the desired excited-state density, which results in the generation of more known constraints on the universal functional for approximation purposes. 

If the external magnetic field B(r), and m(r) have only a nonvanishing Z-component, B(r) = (0,0, B(r)) and m(r) = (0,0, m(r)), the universal functional F[p, m] may then be considered as a functional of the spin densities ps(r) and p(r), F[ps(r), p(r)], because the spin density is proportional to the z-component of the magnetization m(r) = p-bPsW P-b is the electron Bohr magneton. It is of worth mentioning that it is possible to define two spin densities that are the diagonal elements of the density matrix introduced by von Barth and Hedin [3]. These correspond to the spin-up (alpha) electrons density pT(r), and the spin-down (beta) electrons density p (r). In terms of these quantities, the electron and spin densities can be written as... 

---