Generalised density functions

The local density approximation is highly successful and has been used in density functional calculations for many years now. There were several difficulties in implementing better approximations, but in 1991 Perdew et al. successfully parametrised a potential known as the generalised gradient approximation (GGA) which expresses the exchange and correlation potential as a function of both the local density and its gradient ... 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

The first way has been followed in what has become known as Car-Parrinello molecular dynamics (CPMD) (9). A solute and 60-90 solvent molecules are considered to represent the system, and the QM calculations are performed with density functionals, usually of generalised gradient approximation type (GGA), such as the Becke-Lee-Young-Parr (BLYP) (10) or the Perdew-Burke-Enzerhofer (PBE) (11,12) functionals. It is clear that the semiempirical character of concurrent density functional theory (DFT) methods and the use of these simple functionals imply a number of error sources and do not really provide a method-inherent control procedure to test the reliability of results. Recently it has been shown that these functionals even do not enable a correct description of the solvent water itself, as at ambient temperature they will describe water not as liquid but as supercooled system... 

The knowledge of the structure of titanium in its different oxidation states (II to IV) should bring some information on its coordination to H, Cl and CH3. Here, we present optimised structures of titanium hydrides and chlorides using density functional theory with the generalised gradient approximation (DFT-GGA). Next, for the different oxidation states we study the addition of acids, radicals and bases on the chlorides. 

Car-Parrinello (CP) type simulations [32] achieve the necessary compromise between effort and accuracy by reducing the number of particles as well as the simulation time on the one hand and by the employment of simple generalised gradient approximation density functionals such as PBE [33,34] or BLYP [35] on the other hand. In the case of hydrated ions the reduction of the number of solvent molecules is in many cases chosen too low to properly hydrate the ion and to reflect the surrounding bulk liquid. The reduction of the simulation time is sometimes at the expense of a proper equilibration period. 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Perdew, J., Kurth, S., Zupan, A., Blaha, P. (1999). Accurate density functional with correct formal properties a step beyond the generalised gradient approximation. Phys. Rev. Lett. 82,2544-2547. Erratum 5179. 

We remark in passing that co,i(s) will also yield the Laplace transform of the characteristic function of the configuration space probability density function. Equations (241)-(243) then lead to the generalisation of the Gross-Sack result [39,40] for a fixed axis rotator to fractal time relaxation governed by Eq. (235), namely,... 

K.D.Sen and R.Carb6-Dorca, Inward Matrix Products, Generalised Density Functions and Rayleigh-Schrodinger Perturbation Theory , Institute of Computational Chemistry Technical Report IT-IQC-98-42. (1998)... 

The concept of local hardness has earlier been shown [55] to provide a generalisation of the electrostatic-potential approach used in understanding chemical reactivity. Local hardness is in fact the reciprocal of local compressibility [55]. The various relations involving the energy derivatives would be useful in formulating the local force laws [73] within the density-functional framework, and hence in obtaining new reactivity indices based on the electron density and related quantities. 

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930 Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum ... 

In this chapter, after a brief introduction to MBPT and Hedin s GW approximation, we will summarise some peculiar aspects of the Kohn-Sham xc energy functional, showing that some of them can be illuminated using MBPT. Then, we will discuss how to obtain ground-state total energies from GW. Finally, we will present a way to combine techniques from many-body and density functional theories within a generalised version of Kohn-Sham (KS) DFT. 

Ab initio determination of positron states in solids is possible on the basis of a generalisation of the density-functional theory [97,98]. In the two-component density-functional theory, the ground-state energy of a system of electrons and positrons in an external potential Fext is written as a function of the electron (h ) and positron (n+) densities ... 

Two classes of probabilistic approach are possible. One class is the direct probability density functional (pdf) approach, often generalising the multivariate Gaussian (normal) distribution. The other class consists of assiunp>-tions and rules that define a stochastic process. In this second class of method it is not usually possible to state an explicit pdf for the interpolants the process must be studied via its sample realisations and their properties. The derivation of standard geostatistical results often appears to be rule based but, as shown in the next few pages, can be derived from an explicit pdf. More research using explicit pdfs could lead to new results and insights into the methods of spatial statistics. 

To overcome this problem Dubinin, McEnaney, Stoeckli and Kadlec have proposed generalised forms of the DR equation which in principle should be applicable to heterogeneous microporous solids. In practice, the main problem in adopting this approach is to arrive at a unique solution for the probability density function of E and hence the micropore size distribution. These aspects are discussed in Sme detail by McEnaney and Mays. 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

Consequently the momentum density of any mol ule will have inversion symmetry, even if p(r) does not. It can be useful to take advantage of this inversion symmetry when integrating functions of p(p such as the generalised overlaps to be described in Sect. 3. 

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