Electron-density difference function

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

A density functional is then used to obtain the energy for the electron density. A functional is a function of a function, in this case, the electron density. The exact density functional is not known. Therefore, there is a whole list of different functionals that may have advantages or disadvantages. Some of these... 

Fig. 15. Integrated electron density differences in the system H2CO-Li+ as a function of the 0-Li+ distance (cf. Ref. 109>. The spacing of the lines is 0.01 (eo) for continuous lines and 0.0025 ( o) for broken lines...

Fig. 20. Integrated electron density differences in the systems H2O-IJ+ and H2CO-Li+ calculated in the CNDO/2 approximation. (Wave functions taken from Ref. 224))...

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

Fig. 15a-c. Scheme of the side chains arrangement of macromolecule (a), function of distribution of electron density Aq along a normal to the smectic plane (b) and one-dimensional correlation function Yi(x) for polymers V (1) and VI (2) (c),08) a) 1 — main chain 2 — mesogenic groups 3 — alkyl group b) Aq — electron density difference between ordered I, and disordered 12 regions E — width of the transitional region... 

The term f. is called the scattering factor of atom j, and it is a mathematical function (called a 8 function) that amounts to treating the atom as a simple sphere of electron density. The function is slightly different for each element, because each element has a different number of electrons (a different value of Z) to diffract the X rays. The exponential term should be familiar to you by... 

Equilibrium geometries calculated at the split valence level almost always show shorter bond distances than the minimum-basis-set results and are generally in better agreement with experiment. Split valence wave functions also give considerably better electron distributions, as evidenced by dipole moments and molecule-atom electron-density differences that compare better with experiment (Hehre et al., 1986 Szabo and Ostlund, 1989). 

The Fukui function is approximated by the electron density difference using the finite difference method,... 

Several approximate, semi-quantitative relations linking the above information-distance densities with the density difference function Ap(r) have been derived and numerically tested for selected linear molecules [29,30]. Since the molecular density is in general only slightly changed relative to the promolecular reference density, as a result of the mainly valence shell, minor reconstruction of the electron distribution,... 

Fig. 16 The development of the SAXS invariant Q = f I q)q dq = n ) 4>- 4>2 function of temperature. The absence of change means that either the change in electron density difference between the two phases is exactly compensated by the change in relative volume fractions of the two different domains or that the electron density difference between the two phases is constant and only a single phase crystallises. 

Figure 6 Valence bond-pair electronic density difference between the GVB-t-MPJl (SCF) and the GVB wave-functions, for the LiH molecule to R=3.015 a.u..(In a.u.).

The PEO/EMAA and PEO/SHS blends exhibit volume-filling spherulites for all compositions examined in this part of the study (i.e., 20%). These observations, in concert with SAXS results, indicate at least partial exclusion of the diluent into regions between lamellar stacks at these compositions. The distribution of the second polymer between the interlamellar and interfibrillar regions can be estimated using measured bulk crystallinities and correlation function parameters as follows. The volume fraction of lamellar stacks is determined from the bulk and linear crystallinities (v, average electron density difference between crystalline and interlamellar amorphous regions can be determined using eqn. 2. 

In general, joint analyses of the spin and electron density source functions provide interesting insights, as the different way the two scalar fields dilute and concentrate in the space lead to reconstractions of these fields which may be totally different. For example, this is the case of the points associated to the lone pair electrons in water triplet [94] or the just mentioned case of the spin density at the terminal C-H bcp in n-alkyl radicals. 

The periodic structure shows an electron density distribution Pe z) as indicated on the left of part (a). It can be described by specifying the long period dao the crystallite thickness dc and the electron density difference Pe,c — Pe,a- The Crystallinity 0c = dc/dac in this example lies below 50%. We first calculate a special correlation function, denoted Kg, z), defined as... 

Q is often called the invariant , for obvious reasons The total integral, as obtained by an integration over all the reciprocal space, depends only on the volume fractions of the two phases and the electron density difference and is invariant with regard to the detailed structure. Equation (A.154) is not a specific property of layered systems, but generally valid. The proof is simple. One has to formulate the Fourier-transformation reverse to Eq. (A.59), expressing the three dimensional electron density correlation function as a function of B q)... 

As can be seen ftom Eq. (4), the invariance Q is proportional to the square average of the electron density fluctuation of the scattering media. When the electron density difference Ap is constant, it is proportional to the volume of the scattering media. In this case, the correlation function can be rewritten using the function yo(0 that depends on the shape of the scattering media ... 

---