The total density

According to Eq. (1.22), the structure factor F(H) is the Fourier transform of the electron density p(r) in the crystallographic unit cell. The electron density p(r) is then obtained by the inverse Fourier transformation, or 

Since F(H) is defined at the discrete set of reciprocal lattice points H, the integral in Eq. (5.1) can be replaced by a summation  

The electron density is a real function. The right-hand side of Eq. (5.3) must therefore be real also. This can be shown as follows. Writing 

The period of the plane wave with amplitude F(H), in the direction of the wavevector H, equals l/H. The period is therefore shorter for higher-order reflections, which thus add resolution to the image. As more higher-order reflections are included in the summation, the resolution of the image improves, as illustrated in Fig. 5.1. The improvement is analogous to the increase in resolution in an optical image obtained with shorter-wavelength radiation. 

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One current limitation of orbital-free DFT is that since only the total density is calculated, there is no way to identify contributions from electronic states of a certain angular momentum character /. This identification is exploited in non-local pseudopotentials so that electrons of different / character see different potentials, considerably improving the quality of these pseudopotentials. The orbital-free metliods thus are limited to local pseudopotentials, connecting the quality of their results to the quality of tlie available local potentials. Good local pseudopotentials are available for the alkali metals, the alkaline earth metals and aluminium [100. 101] and methods exist for obtaining them for other atoms (see section VI.2 of [97]). 

Aeeording to Wertheim [8-11], the fluid density at a point 1 of the bulk assoeiating fluid is split into two terms, namely the density of unbonded speeies, po(l)> and the density of (singly) bonded speeies, Pi(l), sueh that the total density is p(l) = po(l) + Pi(l)- We set the matrix of densities for a dimerizing fluid... 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

The strength of DFT is tiiat only the total density needs to be considered. In order to calculate the kinetic energy with sufficient accuracy, however, orbitals have to be reintroduced. Nevertheless, as discussed in Section 6.5, DFT has a computational cost which is similar to HF theory, with the possibility of providing more accurate (exact, in principle) results. 

The total density is the sum of die a and /3 contributions, p = Pa + Pp, and for a closed-shell singlet these are identical (p, = pp). Functionals for the exchange and correlation energies may be formulated in terms of separate spin-densities however, they are often given instead as functions of the spin polarization C, (normalized difference between p and pp), and the radius of the effective volume containing one electron, rs-... 

LSDA may also be written in terms of the total density and the spin polarization. 

This functional contains derivatives of the orbitals, not just the gradient of the total density, and is computationally slightly more expensive. Despite the apparent difference in functional form, exchange expressions (6.24) and (6.25) have been found to provide results of similar quality. 

Ed is the center of gravity of the d band, 9 E) the total density of states and Ep the Fermi energy. 

It is well known that in bulk crystals there are inversions of relative stability between the HCP and the FCC structure as a fxmction of the d band filling which follow from the equality of the first four moments (po - ps) of the total density of states in both structures. A similar behaviour is also expected in the present problem since the total densities of states of two adislands with the same shape and number of atoms, but adsorbed in different geometries, have again the same po, pi, P2/ P3 when the renormalization of atomic levels and the relaxation are neglected. This behaviour is still found when the latter effects are taken into account as shown in Fig. 5 where our results are summarized. 

Figure 12-11. Thickness dependence of the electron only j(V) characteristics at L=0.22, 0.31, and 0.37 pm. Solid lines have been calculated for an exponential distribution of electron traps of the total density 101 cnTJ and a characteristic temperature T,.= 1500 K (Ref. [41[).

Here, is the total density of traps and 7o is a characteristic temperature that accounts for the slope of the distribution. Results are gathered in Table 14-4. L both materials, a comparable characteristic temperature was found, while the total density of traps was ten times higher in 6T than in DH6T. 

However, although it allowed a correct description of the current-voltage characteristics, this model presents several inconsistencies. The main one concerns the mechanism of trap-free transport. As noted by Wu and Conwell [1191, the MTR model assumes a transport in delocalized levels, which is at variance with the low trap-free mobility found in 6T and DH6T (0.04 cm2 V-1 s l). Next, the estimated concentrations of traps are rather high as compared to the total density of molecules in the materials (see Table 14-4). Finally, recent measurements on single ciystals [15, 80, 81] show that the trap-free mobility of 6T could be at least ten times higher than that given in Table 14-4. 

The next step is the decomposition of the total density into single particle densities which are related to single particle wave functions by... 

Owing to their strong bond on Ru(OOOl), mixed COa 0.55 V, the shift of the equilibrium between water and adsorbed OHad/Oad towards the latter increases the density of the respective species in the intermixed adlayer, which increases the repulsions between the adsorbed species and hence leads to more weakly bound OHad/Oad and COad species. These latter species are less stable against COOHad or CO2 formation, because of the reduced reaction barrier ( Brpnsted-Polanyi-Evans relation [Bronstedt, 1928]), and can support a reaction via (14.9) or (14.12), respectively, at low rates. (Note that the total density of the adlayer does not need to remain constant, although also this is possible.)... 

The dynamic structure factor is S(q, t) = (nq(r) q(0)), where nq(t) = Sam e q r is the Fourier transform of the total density of the polymer beads. The Zimm model predicts that this function should scale as S(q, t) = S(q, 0)J-(qat), where IF is a scaling function. The data in Fig. 12b confirm that this scaling form is satisfied. These results show that hydrodynamic effects for polymeric systems can be investigated using MPC dynamics. 

Top DOS contributions of the different bands of a PtX - chain and their superposition to give the total density of states. Bottom COOP contributions of the different bands and their superposition to give the crystal orbital overlap population... 

Given the total density from Eq. (4.17), the temperature follows from the equation of state which depends in turn on what particles are present. For any one species i, with temperature T,. we have from the Fermi-Dirac or Bose-Einstein distribution, Eq. (2.41),... 

Electron-density effects on fi-decay lifetimes also enable the total density to be placed in the range 2500 to 13 000 gm cm-3 all these parameters are characteristic of helium shell-burning zones as expected. 

Table 4.15 summarizes optimized bond lengths and NBO Lewis-like structures for 15 saturated normal-valent H MMH compounds (M = W-Pt) as well as corresponding hydrides of hypovalent Ta for comparison. The accuracy of the localized Lewis-like description (as measured by %pf) is found to be reasonably high both for normal-valent and for hypovalent species, typically >98% of the valence-shell density and >99.5% of the total density. 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

They are, in fact, two possible perturbative approaches. In the first approach, one considers two (or any number) reagents A and B as a single system where A and B are two fragments in interactions. The total density is p(r ) = p4(r/) I p/ (r/) [24]. Displacement of A relative to B is an internal mechanical deformation of the complete system it is related to response function of A I B. For instance, a small displacement U/f of an atom K of B will modify the force on the other atoms by... 

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