Atomic basin definition

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. 

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... 

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. 

The atoms-in-molecules partitioning of electron density (6.3.1) can now be seen in different perspective. The total crystallographically measured electron density is essentially that of the promolecule, which by definition must partition into atomic densities. Calculated densities, on the other hand, can only be obtained after assuming a set of nuclear coordinates. Partitioning into a set of atomic basins therefore simply demonstrates a degree of self-consistency between synthesis and analysis of the density function. 

If we want to assign local-spin values to (atomic) subsystems, a definition of an atomic basin is required. Clark and Davidson proposed a framework where local molecular spin operators are obtained from projections of the total molecular spin operator [112, 118] upon atomic basins A. The one- and twoexpectation value of the local spin operator (S ) and of diatomic products of local spin operators (S Sg), respectively. However, Clark and Davidson s decomposition leads to nonvanishing... 

The partitioning of the space into separate non-overlapping atomic basins, exhausting all three-dimensional space, entails the definition of atomic properties that add-up to yield the corresponding molecular counterparts. Such atomic properties are obtained by integrating each corresponding property density over the bounded region of real space occupied by the atomic basin. 

Many of the rigorous definitions of atomic and bond properties that are described in this article invoke the concepts of the quantum-mechanical theory of atoms in molecules (AIMs) (see Atoms in Molecules). The AIMs consist of nuclei and disjoint portions of Cartesian space called atomic basins. Direct integration of property densities over those basins yields the first-order properties of AIMs, whereas the calculations of the second-order properties and quantities such as atomic electronegativities and similarities are somewhat more involved. Topological analysis of the electron density p r) that accompanies the construction of AIMs yields a wealth of other information, including the location of major interactions within molecules. ... 

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian... 

While the individual contributions to the virial tC ( 2) can be given physical interpretations, care must be exercised in this regard. The value of the total virial tC( 2) is independent of the choice of origin used in the definition of the virial operator, but this is not the case for the basin and surface terms treated separately. The origin-dependent terms appearing in the total virial for an atom are... 

The atomic contribution to the magnetic susceptibility is obtained in analogy with the definition of (Q), by a basin contribution obtained by integration of /(r) over the basin of the atom. 

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