Atomic -electron densities

There is a second point to note in dementi s paper above where he speaks of 3d and 4f functions. These atomic orbitals play no part in the description of atomic electronic ground states for first- and second-row atoms, but on molecule formation the atomic electron density distorts and such polarization functions are needed to accurately describe the distortion. 

If population analysis is not synonymous with the concept of an AIM, it becomes necessary to introduce a proper set of requirements before one can speak of an AIM. An AIM is a quantum object and as such has an electron density of its own. This atomic electron density must obviously be positive definite and the sum of these atomic densities must equal the molecular density. Each atomic density pA(r) can be obtained from the molecular density p(r) in the following way ... 

While the embedded atom method has been formally derived by Daw and Baskes the functions used in computer simulations are t3pically empirically determined. The description presented here will therefore treat this approach as an empirical method. The first step in determining the potential is to define a local electron density at each atomic site in the solid. A simple sum of atomic electron densities has proven to be adequate, and so in most cases a sum of free atom densities is used . The second step is to determine an embedding... 

Table 1 Values of the atomic electron density at the nucleus, p(0) evaluated with the present modified TFD method compared to HF values by means of the percent deviation (%). Also, the values of 2 Z tq are displayed where tq is the switching point among the quantum mechanical and the semiclassical description (see text).

Whenever the atoms under consideration in a given molecule are in the same valence states as in the reference molecule, the relaxation process is such that the potential created by the other atoms at the kth nucleus is the same as would be predicted by leaving the pertinent intemuclear distances and the shapes of atomic electron densities as they are in the reference molecule, with the electron populations changed as required by the new situation. 

In the 1970s, among many other approaches, the method of multipolar expansion of atomic electron density was recognized as the most applicable and accurate. 

Fig. 3.3 The electron density of the homonuclear molecule (upper panel) can be regarded as the sum of the non-interacting or frozen free-atom electron densities (lower panel) and the quantum mechanically induced bond density (middle panel). The dashed curve represents the first-order approximation, eqn (3.26), for the bond density, the deviation from the exact result (full curve) being due to the sizeable value of the overlap integral namely S = 0.59 at / = 2au.

Here I0 is the intensity of the x-ray beam, r0 = e2/mc2 is the classical electron radius (2.82 x 10 15 m)., P(9,) is the polarization of the x-rays it depends on the angle between the polarization and the scattering vector. For horizontally polarized x-rays, it takes the form P(0, < >) = 1 - sin220 sin2t)>, where 20 is the scattering angle and < > the azimuthal angle with respect to the vertical direction. The formfactor ) is the Fourier transform of the atomic electron density ... 

X-ray crystallography seeks to obtain the best model to describe the periodic electron density in a crystal by a least squares fit of the parameters of the model (used to calculate structure factors) against the observed structure factors derived from the diffraction experiment. All models used are atomic in nature, but vary in the complexity of the description of the atomic electron density. 

As we are dealing with spherical harmonics, and as we are trying to model the aspherical atomic electron density, the orientation of the local atom centered coordinate system is, in principle, arbitrary, appropriate linear combinations always giving the same result. However, in practice it is helpful to choose a local coordinate system such that the multipoles are oriented in rational directions, and thus the most important multipole populations will lie in directions that would be expected to represent chemical bonds or lone pairs [2,20], e.g. for an sp2 hybridized atom, defining one bond as the x direction, the trigonal plane as the xy plane, and z perpendicular defines three lobes of the 33+... 

The electronic configuration of free atoms is an important factor in the interpretation of atomic spectra, but less so for the understanding of chemical behaviour. Chemistry happens in crowded environments, which means that atomic electron densities fades to zero far from infinity. SCF wave functions are therefore not appropriate for atoms in a chemical environment. More suitable wave functions are obtained by terminating the SCF calculations at some fixed distance p from the nucleus, rather than infinity. The effect of such a new boundary condition is like applying hydrostatic pressure to the atom. 

When we are dealing with simple atoms as substituents, these effects are straightforward and more or less additive. If we go on adding electronegative chlorine atoms to a carbon atom, electron density is progressively removed from it and the carbon nucleus and the hydrogen atoms attached to it are progressively deshielded. 

In condensed heteroaromatic systems with a ring-junction pyrrolic nitrogen atom, -electron density is always shifted from the electron-rich six-membered ring (formally contains seven -electrons) toward the five-membered ring (formally has six -electrons). As a result, electrophiles are directed to carbon atoms of the latter. Thus, imidazo[l,2-tf]pyridines 160 unsubstituted at C(3) almost always react with electrophiles at that position. 

In the X-N method, the experimental electron density is determined from the X-ray diffraction data. The free atom electron density is determined by placing theoretical free atom electron densities at the atomic nuclear positions determined by a neutron diffraction experiment on the same crystal structure at the same temperature, albeit with a crystal of larger size. In the X-N method, it is frequently found that the atomic thermal parameters determined by neutron diffraction do not agree with those determined by X-ray diffraction, even though the experiments were carried out under identical conditions. This requires the introduction of an empirical scaling factor between the two sets of data, which is effective but disconcerting. 

The self-consistent field calculation requires that as a first approximation V j be calculated for each atom j with its associated atomic electron density. The total potential V is then obtained as a superposition of the atomic potentials ... 

The aspherical atomic electron density is divided into three components ... 

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). 

To improve upon a double- basis, one generally adds polarization functions whose / values correspond to orbitals unoccupied in the free atoms. For example, to expand upon a double- basis for HjO, one would add 2p functions on H and 3d functions on O. Such functions are called polarization functions since they describe the polarization of atomic electron density arising from molecule formation. For example, if a H atom is placed in the electrostatic field of an O atom, its electron density will be polarized along the 0-H bond direction, a change that can be described by the mixing of H 2p character into the His wave function. 

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