Atom-in-molecule model

The third term in equation (127) is the deviation from a rigid atoms in molecules model, that is, it represents a non-perfectly-following part of the density as the nuclei move. 

In the next section, the principal ingredients involved in the ELF will be explained, and their relation with chemical concepts will be clarified. Then, a brief comparison of the ELF with other theoretical related tools, like the atoms in molecules model of Bader, will be done. Next, some elementary concepts from the mathematical theory of topological analysis will be in a rather crude way presented. After that, some applications, extensions and results will be discussed, focusing in particular on applications developed at our group. 

One cannot mention the a priori use of localized orbitals, without emphasising the importance of the hybrid orbitals. It is well-known that hybridization is an extremely fruitful concept in chemistry and in the rationalization of molecular structure and bonding [7]. As it was shown by Maksic, hybridization often takes into account the most essential modifications of the atomic properties, and it can be a very good starting point for a MAM (modified atoms in molecules) model [50]. The SLMOs... 

AIM (atoms in molecules) a population analysis technique AMI (Austin model 1) a semiempirical method... 

The n orbitals on the two CO molecules interact with the same lobe of a vacant 3p orbital on a metal atom in the model for the acute angle coordination, and with different lobes for the obtuse angle coordination (Scheme 29b). Cychc orbital interaction occurs between the occupied 3s orbital and the vacant 3p orbitals on M and the n orbitals, n, and n, of the CO molecules (Scheme 29c). The phase is continuous for the same lobe interaction and discontinuous for the different lobe interaction (Scheme 29d, cf. Scheme 4). The acute-angle coordination is favored. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

In this chapter we give a brief review of some of the basic concepts of quantum mechanics with emphasis on salient points of this theory relevant to the central theme of the book. We focus particularly on the electron density because it is the basis of the theory of atoms in molecules (AIM), which is discussed in Chapter 6. The Pauli exclusion principle is also given special attention in view of its role in the VSEPR and LCP models (Chapters 4 and 5). We first revisit the perhaps most characteristic feature of quantum mechanics, which differentiates it from classical mechanics its probabilistic character. For that purpose we go back to the origins of quantum mechanics, a theory that has its roots in attempts to explain the nature of light and its interactions with atoms and molecules. References to more complete and more advanced treatments of quantum mechanics are given at the end of the chapter. 

To a first approximation, atoms in molecules may be regarded as hard spheres with a segment cut off in the bonding direction, as in the familiar space-filling models. The radius of the atom in a nonbonding direction is called the van der Waals radius. Half the distance between two atoms of the same kind in adjacent molecules at equilibrium is taken as the van der Waals radius (Figure 5.1). In assigning a fixed radius in this way, we assume that atoms... 

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. 

This chapter is based on the VSEPR and LCP models described in Chapters 4 and 5 and on the analysis of electron density distributions by the AIM theory discussed in Chapters 6 and 7. As we have seen, AIM gives us a method for obtaining the properties of atoms in molecules. Throughout the history of chemistry, as we have discussed in earlier chapters, most attention has been focused on the bonds rather than on the atoms in a molecule. In this chapter we will see how we can relate the properties of bonds, such as length and strength, to the quantities we can obtain from AIM. 

MO or VB models. Occasionally atom-in-molecule orbital exponents are used (principally for H atoms, using 1.2) but it is unusual to see any interpretation of this fact. 

M. Alcami, O. Mo and M. Yanez, Modeling Intrinsic Basicities The Use of Electrostatic Potentials and the Atoms-in-Molecules Theory, in Molecular Electrostatic Potentials Concepts and Applications (J. S. Murray and K. Sen, Eds.). Elsevier, Amsterdam 1996. Page 407. 

Sidgwick s discussion raises an important question What are the effective sizes and shapes of atoms in molecules From the viewpoint of the electride ion model of electronic structure, Sigdwick s circles for the fluoride ions in the first column of Fig. 15 are the wrong shape, if nearly the right overall size. In the electride-ion model a fluoride ion is composed of (approximately) spherical domains, but is not itself spherical, in the field of a cation, Fig. 16. Fig. 17 illustrates, correspondingly, the implied suggestion that, on the assumption that non-bonded interactions are not limiting, the covalency limits of an atom will be determined by the radius of the atom s core and by the effective radii, not of the overall van der Waals envelopes of the coordinated ions but, rather, by the radii of the individual, shared electron-pairs. 

Different practical procedures for computing r) have been proposed, that range from the computation of the difference between the energy values of the highest occupied (HOMO) and the lowest unoccupied (LUMO) molecular orbitals [8,9], to the atom in molecules based models [10, 11], to the charge sensitivity analysis [12, 13], to the use of Slater transition state theory [14], to the Janak s extension of DFT for fractional occupancies [15, 16]. Recently, Neshev and Mineva have proposed a scheme for the construction of the internally resolved hardness tensor in... 

Several important developments in the straightforward VB theory have occurred in the past 10 or 15 years. These include the atoms-in-molecules method of Moffitt as modified by Hurley and others, the pair function model of Hurley, Lennard-Jones, and Pople, and the general group function model of McWeeny. These theories can all be usefully discussed within the framework developed in Section 2, and this is done in Section 4. 

---