Atomic basins

Atoms defined in this way can be treated as quantum-mechanically distinct systems, and their properties may be computed by integrating over these atomic basins. The resulting properties are well-defined and are based on physical observables. This approach also contrasts with traditional methods for population analysis in that it is independent of calculation method and basis set. 

The border between two three-dimensional atomic basins is a two-dimensional surface. Points on such dividing surfaces have the property that the gradient of the electron density is perpendicular to the normal vector of the surface, i.e. the radial part of the derivative of the electron density (the electronic flux ) is zero. 

Once the molecular volume has been divided, the electron density may be integrated within each of the atomic basins to give a net atomic charge. As the dividing surface is... 

Figure 9.2 Dividing surface between to atomic basins...

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

The total energy, for example, may be written as a sum of atomic energies, and these atomic energies are again almost constant for the same structural units in different molecules. The atomic basins are probably the closest quantum mechanical analogy to the chemical concepts of atoms within a molecule. 

At the heart of the AIM theory is the definition of an atom as it exists in a molecule. An atom is defined as the union of a nucleus and the atomic basin that the nucleus dominates as an attractor of gradient paths. An atom in a molecule is thus a portion of space bounded by its interatomic surfaces but extending to infinity on its open side. As we have seen, it is convenient to take the 0.001 au envelope of constant density as a practical representation of the surface of the atom on its open or nonbonded side because this surface corresponds approximately to the surface defined by the van der Waals radius of a gas phase molecule. Figure 6.15 shows the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and the p = 0.001 au envelope. It is clear that atoms in molecules are not spherical. The well-known space-filling models are an approximation to the shape of an atom as defined by AIM. Unlike the space-filling models, however, the interatomic surfaces are generally not flat and the outer surface is not necessarily a part of a spherical surface. 

Having defined an atom in a molecule, we can, at least in principle, determine any of the properties of an atom in a molecule. The simplest to illustrate is the atomic volume, which is simply the sum of all the volume elements that occupy all the space defined by the interatomic surfaces and the p = 0.001 au contour. More exactly, it is the integral of all the volume elements dr over the atomic basin. If we denote the atomic basin by Si, then the volume of the atom is given by. 

Another property that is in principle easily evaluated is the electron population of the atom N(Sl). This is obtained by integrating the density of a volume element over the atomic basin ... 

The atomic charge q(Sl) is then simply obtained as — N(Sl), or the electron population subtracted from the charge of the nucleus inside the atomic basin. 

The atomic dipole moment can be obtained by integrating the moment of a volume element prfidr over the atomic basin. The atomic dipole moment M(fl), where is a vector centered on the nucleus of the atom, is then... 

The interatomic (zero-flux) surfaces partition the molecule into separate nonoverlapping atoms (atomic basins), which... 

The AIM theory provides a clear and rigorous definition of an atom as it exists in a molecule. It is the atomic basin bounded by the interatomic surfaces. The interatomic... 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

Fig. 7.1 The electron density p(t) is displayed in the and

In order to frame the QCT method within the general expression (3), we need to give an expression for the weight operator u ci. From the above discussion, it is clear that this operator will depend on r and that it is a binary operator, so its value is either 0 or 1. For a given atom A, the operator vanishes at every point in space, except within the basin of A, where it is equal to one. This way all atomic basins are indeed mutually exclusive. 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

The disadvantage of the physical space analysis is that the calculation of atomic basins and the subsequent integration is not always straightforward, and definitely requires much more time than the Hilbert space analysis (recall the latter is instantaneous). Our experience shows that the latter analysis does provide satisfactory information so that it is not necessary to perform the physical space analysis. 

Typically, gradient paths are directed to a point in space called an attractor. It is obvious that gradient paths should be characterized by an endpoint and a starting point, which can be infinity or a special point in the molecule. All nuclei represent attractors, and the set of gradient paths is called an atomic basin, This is one of the cornerstones of AIM theory becanse the atomic basin corresponds to the portion of space allocated to an atom, where properties can be integrated to give atomic properties. For example, integration of the p function yields the atom s population. 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

It is important that L(r) vanishes when the integration is performed over the zero-flux surface atomic basin. This is because the integral over (r) can be replaced by the surface integral over the flux at the surface (Bader 1990) ... 

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