Shape function electron density, variations

Notice that how the shape function naturally enters this discussion. Because the number of electrons is fixed, the variational procedure for the electron density is actually a variational procedure for the shape function. So it is simpler to restate the equations associated with the variational principle in terms of the shape function. Parr and Bartolotti have done this, and note that because the normalization of the shape function is fixed,... 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

The density per particle of shape function cr(r) is seen to be a simpler alternative for the density as the fundamental carrier of information in the study of the electronic structure of atoms and molecules, up to the limit that the shape function holographic theorems can be formulated. Just as in DFT, a variational theorem can be written down and an alternative set of reactivity descriptors can be conceived. The long-range behaviour of the function yields information which reconciles density and shape function theory. 

Hence, Eq. (8) to (11) demonstrate that the immersion of the particles in a medium having the electron density p j leads to three different components of the measured intensity which depend differently on the contrast p-Pm- The first term I lq) presents the Fourier-transform of the shape function T(r) and its careful determination therefore allows us to deduce all the information solely due to the shape of the particle. The third term, on the other hand, which dominates the measured scattering function at low contrast is related to the interference due to the internal variation of the electron density. 

In the difference term one recognizes a fluctuation term, f(r) the deviation of the Fukui function from the average electron density per electron, multiplied by the functional derivative of E with respect to a(r) at constant number of electrons. This quantity (5E/5o(i))ig is a response function of the type (5E/5v(r))j. mentioned in the Introduction and measures the sensitivity of the system s energy to variations in the shape factor. Its evaluation seems to be far from trivial but it is possible to get already an idea of what might be factors of importance in this response function. Adopting an orbital formalism and using a Koopmans type approximation one arrives at an approximate expression... 

Most neutron scattering work has been performed with LDL [295,296,314,315, 319]. This work confirmed the quasi-spherical shape of LDL and the coincidence of the centres of gravity of the hydrocarbon and polar regions by the use of H20- H20 contrast variation, although a slightly smaller Rq of 7.7-8.0 nm is now determined (Table 12) [295,296]. The radial nuclear scattering density distribution function exhibits less pronounced fluctuations than the corresponding electron function. 

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