Hirshfeld

Hirshfeld, D. S. and Kolb, J., Minimize the Cost of Producing Retoi mulated Gasoline, Fuel Reformulation, Vol. 4, No. 2 March/April 1994... 

Figure 6.9 (a) Standard deformation density of tetrafluoroterephthalonitrile in the molecular plane. Contour interval is 0.1 e A-3, terminated at 1.5 e A 3. (b) Molecular diagram with a box around the fragment shown in the deformation map (a). (Reproduced with permission from F. L. Hirshfeld, Acta Crystallogr., B40, 613, 1984.)... 

Tab. 1.3 Comparison between different sets of atomic point charges for a zwitterionic Gly-Ala dipeptide in aqueous solution. D-RESP electrostatic potential derived charges [12] fitted to all 36 configurations. Hirshfeld average value of the Hirshfeld charges [89c] along the full trajectory, Amber AMBER 1995 force field [86], Gromos GROMOS96 force field [85], The charges of equivalent atoms are imposed to be equal.

All the different AIM methods that will be discussed below basically use this same approach but quite different in the nature of n (. Chronologically, we will discuss the Mulliken AIM, the Hirshfeld AIM, and the Bader AIM. This last approach will henceforth be called quantum chemical topology (QCT). There are more AIM methods, but most of them can be easily understood by the three selected emblematic approaches. 

This approach, also often called the stockholder scheme, was introduced in 1977 by Hirshfeld [22]. The central idea of the Hirshfeld method originates in x-ray crystallography. It proposes to divide the electron density among the atoms in a molecule, guided by a promolecular density. More precisely, once a molecular geometry is known, a promolecular density p°(r) is composed by simply summing the density of each atom A (denoted p°A(r)) in an isolated state ... 

This share is used as the Hirshfeld operator. The assumption behind the Hirshfeld AIM is that this same weight operator can be used to divide the electron density of the molecule via... 

This so-called Hirshfeld scheme is particularly popular within the so-called conceptual density functional theory (DFT) [26,27], The weighting function, which identifies the AIM as one that is most similar to the isolated atom [28], has been shown to be directly derivable from information entropy [6,29-33]. Here again, the atoms do not... 

The properties of a quantum mechanical system such as an AIM are readily calculated from any method as long as they involve an operator acting on the electron density, e.g., for the case of the dipole moment. The problem would seem to become harder for other properties, although the introduction of property densities allows us to generally introduce AIM expectation values [45], The expectation value of a property A for atom a in the Hirshfeld and QCT methods can be written as... 

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