Electron density minimum

The shortest cation-anion distance in an ionic compound corresponds to the sum of the ionic radii. This distance can be determined experimentally. However, there is no straightforward way to obtain values for the radii themselves. Data taken from carefully performed X-ray diffraction experiments allow the calculation of the electron density in the crystal the point having the minimum electron density along the connection line between a cation and an adjacent anion can be taken as the contact point of the ions. As shown in the example of sodium fluoride in Fig. 6.1, the ions in the crystal show certain deviations from spherical shape, i.e. the electron shell is polarized. This indicates the presence of some degree of covalent bonding, which can be interpreted as a partial backflow of electron density from the anion to the cation. The electron density minimum therefore does not necessarily represent the ideal place for the limit between cation and anion. 

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

Firstly, the new (Gourary-Adrian) method of the Electron Density Minimum, which has recently been discussed (4, 7 c). We here only point out (cp. Table 1) that, at the distances concerned, this electron density is very low and that such radii may sometimes vary greatly from to case. Thus for F in LiF 1.16 A, in CaF2 1.40 A, have been proposed as structural radii 35),... 

Wasastjema s method and the Relative Size method agree quite well, but disagree with the Electron-Density Minimum values 39). We here therefore use our earher arguments to refine Wasastjerna s method and show that this reinforces the Relative Size argument for which more evidence is then provided. 

Close-packed spheres occupy 74.04% of a total volume, hence the hard-sphere radius of I" in these 2 1 salts in 2.03 A. Correction for the electrostatic attraction alone would give a monovalent iodide radius of about 2.24, an opposite repulsion-correction for the different co-ordination number would reduce this to about 2.10 A for the monovalent sodium-chloride type (see Appendix). Such values are consistent with our earlier estimates, but incompatible with the electron-density minimum value (4) of 1.94 A. 

The essential difference is that aU the other methods give isoelectric anions considerably larger than cations, while the Electron Density Minimum method does not. Yet, for example, the mineralogy of oxides and silicates suggests large oxide anions often in contact (47) and there is evidence for anion-anion contact in sulphides and selenides (2). This evidence for anion-anion contact is the strongest evidence for the older systems of ionic radii. 

The balance of evidence here also seems at present against the Electron-Density-Minimum radii. 

This distinction between electron-cloud radii and structural radii is then used to refine the system of ionic radii due to Pauling and Goldschmidt. Some further examples of anion-anion contact are discussed, and a value deduced for the crystal radius of the hydride anion. These cases of anion-anion contact argue for the Pauling tradition and against the new electron-density-minimum (EDM) radii. 

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

It is also important that sufficiently large basis sets are used. The 6—31G(d) basis set should be considered the absolute minimum for reliable results. Some studies have used locally dense basis sets, which have a larger basis on the atom of interest and a smaller basis on the other atoms. In general, this results in only minimal improvement since the spectra are due to interaction between atoms, rather than the electron density around one atom. 

Fig. 5. Ionospheric electron density vs height above the earth at the extremes (A = minimum, B = maximum) of the 11-yr sunspot cycle during (a) day and (b) night (54). D, E, F, F, and F2 are conventional labels for the indicated regions of the ionosphere.

MO calculations can provide the minimum-energy structure, total energy, and overall electron density of a given molecule. However, this information is in the form of the sum of the individual MOs and cannot be easily dissected into contributions by specific atoms or groups. How can the properties described by the MOs be related to our concept of molecules as a colleetion of atoms or functional groups held together by chemical bonds ... 

This shows that, when we have found the correct electron density matrix and correctly calculated the Hartree-Fock Hamiltonian matrix from it, the two matrices will satisfy the condition given. (When two matrices A and B are such that AB = BA, we say that they commute.) This doesn t help us to actually find the electron density, but it gives us a condition for the minimum. 

A better idea of the real size of an ion in a molecule can now be obtained from a study of electron density distributions, which it has recently become possible to obtain from accurate X-ray crystallographic studies of crystals. Figure 2.2 shows a contour map of the electron density distribution obtained in an X-ray crystallographic study of crystalline sodium chloride. The position of minimum electron density between two adjacent ions seems to be... 

Figure 2.2 A contour plot of the electron density in a plane through the sodium chloride crystal. The contours are in units of 10 6 e pm-3. Pauling shows the radius of the Na+ ion from Table 2.3. Shannon shows the radius of the Na+ ion from Table 2.5. The radius of the Na+ ion given by the position of minimum density is 117 pm. The internuclear distance is 281 pm. (Modified with permission from G. Schoknecht, Z Naiurforsch 12A, 983, 1957 and J. E. Huheey, E. A. Keiter, and R. L. Keiter, Inorganic Chemistry, 4th ed., 1993, HarperCollins, New York.)...

The definition of the radius of an ion in a crystal as the distance along the bond to the point of minimum electron density is identical with the definition of the radius of an atom in a crystal or molecule that we discuss in the analysis of electron density distributions in Chapter 6. The radius defined in this way does not depend on any assumption about whether the bond is ionic or covalent and is therefore applicable to any atom in a molecule or crystal independently of the covalent or ionic nature of the bond, but it is not constant from one molecule or crystal to another. The almost perfectly circular form of the contours in Figure... 

Figure 6.5 Contour plot of the electron density in a plane containing the nuclei of (a) CO and (b) CI2. both drawn at the same scale. Along the internuclear axis, the electron density reaches its minimum value at a point marked by a square. For CI2 this is the midpoint.

---