Spatial Distribution Functions

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

In a case when either the second or the third reason takes place, the spatial distribution function will be irreversibly changed. The F, Vk pair can be considered as an example in KCI-SO4 crystals (F is F centre with a trapped electron), which reveals the tunnelling luminescence peak at 3.9 eV [96]. 

These results immediately yield all the internal correlations among chain segments. The spatial distribution function for the pair of segments k-i < k2 is defined as... 

The results summarized above were obtained through the analysis of spatial free energy density profiles using the following methods. Spatial distribution functions for ion occupancies around the macroion are computed on a cubic grid. These distribution functions can be converted to spatial free energy density profiles using the relation ... 

Fig. 11.5. Experimental setup (left) and spatial distribution function (right) of the radiation observed for a laser intensity ao = 5.6...

Probably the most critical question one needs to address in understanding the structure in a molecular liquid is where, in the space defined by the local frame of the central molecule, are we likely to find a neighboring particle. Only after having localized this neighboring particle can we begin to worry about its orientation. The function that provides a direct answer to this question is what we have termed the spatial distribution function (SDF)... 

While not reproducing the full pair distribution function, the SDF does completely describe the local packing structure. As it is only a three-dimensional function, the spatial distribution function can still be visualized and its accumulation is computationally tractable with current computer technology (as we will demonstrate by example below). 

In this chapter we will discuss some of the important issues in the calculation and visualization of SDF data. The principal point of this chapter will be to demonstrate the rich and varied structures present in a variety of molecular systems that can be revealed through spatial distribution function analysis and how these structures can be used to elicidate a more complete understanding of their properties. This chapter is not, however, meant as a complete review of the structure in molecular liquids or solutions discussions on this topic can be found elsewhere (for example see [1-3,19-23]). While the results we will present will come only from our own work, we will also discuss other examples of SDF (or related) analysis carried out on systems... 

As three-dimensional functions, some special considerations are important in the representation, accumulation, and visualization of spatial distribution functions. Here we will discuss these various issues, indicating advantages and disadvantages of different approaches, with functionality and ease of use as principal concerns. 

Systems of molecules with linear symmetry can be viewed as a special application of spherical-polar coordinates. Placing the polar axis along the molecular axis of symmetry allows one to average over the angle 0. The spatial distribution function then becomes a two-dimensional function, g(r,6), which can be much more readily calculated and visualized (for an example, see Figure 1). 

Cartesian coordinates are a convenient alternative representation for a spatial distribution function. Being uniform over the local space, the data structure obtained is easy to represent (access), to normalize, and to visualize. Use of a Cartesian representation becomes a necessity for complex or very flexible molecules. The principal drawbacks of this coordinate system are the size of the data structure it generates (typically about 1,000,000 elements), its inherent inefficiency (since the grid size is determined by the shortest dimension of the smallest feature one hopes to capture), and the fact that its sampling pattern is usually not commensurate with the structures one wants to represent (which can cause artificial surface features or textures when visualized). Obtaining sufficiently well-averaged results in more distant volume elements can be a problem if the examination of more subtle secondary features is desired. See Figures 7, 8 and 9 for examples of SDFs that have utilized Cartesian coordinates. 

In this section we will examine how spatial distribution function analysis can be used to bring new insights and a better understanding of the local structure in pure liquid systems. Implications to other properties, either determined by or related to the pair distribution function, will be discussed. 

Figure 1 (a) Radial distribution functions for soft sphere (solid line) and dipolar soft-sphere fluids at the same temperature and density, (b) The spatial distribution function g(r,0), where 0 is the angle between the dipole and separation vectors, for the dipolar soft-sphere in (a). 

Figure 2 Spatial distribution functions displayed as three-dimensional maps showing the local oxygen density in liquid water. Above TIP4P water at ambient conditions Below PPC water along the co-existence line at 2U0 C. The iso-surfaces shown are for — 1.3 where the surfaces have been cf)k>ied according to their separation from the central molecule, as discu.sse

Figure 5 Oxygen-oxygen spatial distribution functions g r) for a 3 1 water-methanol solution at 25 °C. Above Water-water correlations for g f) = 2.0. Below Methanol-oxygen density around water for an iso-surface threshold of 1.75.

Probably the most notable work on the structure in liquid water based upon experimental data has been that of Soper and co-workers [6,8,10,30,46,55]. He has considered water under both ambient and high temperature and pressure conditions. He has employed both the spherical harmonic reconstruction technique [8,46] and empirical potential structure refinement [6,10] to extract estimates for the pair distribution function for water from site-site radial distribution functions. Both approaches must deal with the fact that the three g p(r) available from neutron scattering experiments provide an incomplete set of information for determining the six-dimensional pair distribution function. Noise in the experimental data introduces further complications, particularly in the former technique. Nonetheless, Soper has been able to extract the principal features in the pair (spatial) distribution function. Of most significance here is the fact that his findings are in qualitative agreement with those discussed above. 

Several other non-aqueous systems have been studied using spatial distribution functions where the data has originated both from experiment (with the subsequent use of RMC or EPSR) or directly from computer simulation. Here we will attempt to summarize some of the liquids examined. 

Figure 7 Abox>e Thr -dinw nsional map of the local oxygen density around luethylamine in an 18 1 mixture of water and metliylamine at 25 The surfaces show n correspond to 2.5 times the bulk value. Below Spatial distribution functions for water oxj gens (red), acetonitrile nitrogens (blue) and methyl groups (green) around a w ater molecule in an equimolar mixture of water and acetonitrile.

Fig. 10 Spatial distribution functions of atoms of the ionic liquid ( 2C iml[tf2N around a central aromatic molecule. Blue contour surfaces enclose regions with a probability of finding C2 atom of imidazolium ring which is 2.4 times the average density. Red contour surfaces enclose regions where the probability of finding O atoms of tf2N anion are 1.8 times the average value in the system. The results were obtained in a liquid-state simulation of 192 ion pairs and 64 aromatic molecules for 1 ns...

Fig. 11 Spatial distribution functions around the imidazolium rings of the cations of O atoms from tf2N anions (red isosurfaces) and C4 (ipso) atoms from toluene (white isosurfaces). Top row [C im [tf N], bottom row ( C C1im][tf,N]. Probability density around the butyl side chain is not shown, since it is not easy to plot it in the same referential due to conformational flexibility of the chain...

Fig. 3 Left Single energy path for the side change of the CI. Right Spatial distribution function around the imidazolium. D IQmim CI conformers I C Clmim][Cl conformers DA chloride in front of the H2 7) chloride anion in front of the additional methyl group. X]t structures with the chloride above respectively below the imidazolium ring. Dc and Tc hydrogen bonded via the rear protons. Reproduced after [92]...

Fig. 4 Spatial distribution functions from FPMD and classical MD reproduced after [70]. Left Traditional MD simulations Right Results of the CPMD simulations. It can be recognized that while the CPMD indicate hydrogen bonding at H2, the traditional MD lacks certain conformers with H2 hydrogen bonding. Figure adapted from [70]. Image Copyright Elsevier...

We recall that p gis(R)dR is the average number of i particles in the element of volume dR, relative to an s molecule at the center of our coordinate system. It is assumed that we have already integrated over all orientations of both i and s to obtain the radial (or the spatial) distribution function, i.e., a function depending on the scalar R only. Hence, p, gis(R)4nR2 dR is the average number of i particles in a spherical shell of radius R and width dR centered at s. We can now write the local composition for each i as... 

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