Atomic subspaces

Perhaps the most rigorous way of dividing a molecular volume into atomic subspaces is the Atoms In Molecules (AIM) method of Bader.The electron density is the square of the wave function integrated over N — coordinates (it does not matter which coordinates since all electrons are identical). 

Chemists have long been intrigued by the question, Does an atom in a molecule somehow preserve its identity An answer to this question comes from studies on the topological properties of p(r) and grad p(r). It has been shown that the entire space of a molecule can be partitioned into atomic subspaces by following the trajectories of grad p(r) in 3D space. These subspaces themselves extend to infinity and obey a subspace virial theorem (2 (7) + (V) = 0). The subspaces are bounded by surfaces of zero flux in the gradient vectors of p(r), i.e., for all points on such a surface,... 

The Hellmann-Feynman theorem offers a convenient way to highlight the main features of chemical binding. By choosing the nuclear charges as parameters, it becomes possible to define the binding of each individual atom in a molecule without having recourse to an explicit partitioning of that molecule into atomic subspaces. 

Figure 15 presents a schematic view of how the atomic subspaces Cl, C6 and Cl 1 of 1,6-methanojl Ojannulene (35) change upon an approach of Cl to C6. Bond paths (solid lines between atoms), bond critical points (dots) and the traces of the zero-flux surfaces S (A, B) (perpendicular to bond paths) that separate the atomic subspaces are shown in Figure 15a. Clearly, the subspace C11 extends less and less into the region between C1 and C6 until the surfaces of C1 and C6 coincide and a bond path between C1 and C6 is formed. At the same time, the Laplace concentration between Cl and C6 gradually increases and coverges to the one found for a three-membered ring. As shown in Figure 15b, this change corresponds to the valence tautomerism of the l,6-methano[10]annulene to bisnorcaradiene27,54.

Perhaps the most rigorous way of dividing a molecular volume into atomic subspaces... 

In principle, it is not difficult to define the cr-occupancy of the atomic subspace (subspace spanned by AOs of atom A). To this end it is sufficient to calculate... 

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