Non-nuclear attractor

The other problem in the AIM approach is the presence of non-nuclear attractors in certain metallic systems, such as lithium and sodium clusters. While these are of interest by themselves, they spoil the picture of electrons associated with nuclei forming atoms within molecules. 

There are indications that the appearance of non-nuclear attractors in silicon is basis-set dependent. 

The value of the electron density at the bond critical point correlates with the strength-of the bond, the bond order. As mentioned above there are certain systems like metal clusters which have non-nuclear centred attractors. The corresponding bond critical points have electron densities at least an order of magnitude smaller than normal single bonds, and the value of the density at the local maximum is only slightly larger than at the bond critical point. The non-nuclear attractors are thus only weakly defined. 

The other problem in the AIM approach is the presence of non-nuclear attractors in... 

Fig. 11.2. How does the electronic density change in space Panel (a) illustrates the non-nuclear attractor maxiuium of p). Note that we can tell the signs of some second derivatives (curvatures) computed at the intersection of black lines (slope), the radial curvature is positive, while the two lateral ones (only one of them is shown) are negative. If for the function shown,...

In order to tell whether a particular critical point represents a maximum (non-nuclear attractor), a minimum or a saddle point we have to calculate at this point the Hessian i.e., the matrix of the second derivatives , where = x, 2 = y, h = Now, the stationary point is... 

All three eigenvalues are negative-we have a maximum of p (non-nuclear attractor Fig. 11.2a). 

Levy minimization (p. 679) local density approximation. LDA (p. 687) non-nuclear attractor (p. 670) one-particle density matrix (p. 698) Perdew-Wang functional (p. 688) self-interaction energy (p. 708) spin polarization (p. 687)... 

These are maxima, minima and saddle points. If we start from an arbitrary point and follow the direction of Vp, we end up at a maximum of p. The compact set of starting points which converge in this way to the same maximum is called the basin of attraction of this maximum, and the position of the maximum is known as attractor. The position may correspond to any of the nuclei or to a non-nuclear electronic distribution non-nuclear attractors, Fig. 11.2.a). The largest maxima correspond to the positions of the nuclei. Formally, positions of the nuclei are not the stationary points, because Vp has a discontinuity here connected to the cusp condition (see Chapter 10, p. 504). A basin has its neighbour-basins and the border between the basins (a surface) satisfies Vp - n = 0, where n is a unit vector perpendicular to the surface (Fig. 11.2.b,c). 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

Fig. 15.19 The molecular graphs of the C2H2...HF complex, the MP2/aug-cc-pVTZ (left) and MP2/aug-cc-pVQZ (right) levels, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of Laplacian of electron density are also presented there is non-nuclear attractor (small red circle) between carbon atoms (left) for the MP2/aug-cc-pVTZ level...

Figure 2 shows the B3LYP/6-3H-G optimised geometry of dimer 1. Each lithium nucleus is connected via a bond path (not shown) to the bond critical point located half way between the two boron nuclei. This situation is reminiscent of a so-called conflict structure. Thus even in these circumstances two lithium nuclei are not directly connected by an uninterrupted bond path. Typically a non-nuclear attractor would be found between two lithium nuclei, as in the Li2 molecule for example. Lastly, the atomic charges of dimer 1 are worth listing. Because of symmetry the following list is complete 0.13 for Li ... 

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