The theory of atoms in molecules

An important conclusion of an atomic statement of the virial theorem is that the total energy of a molecule is expressible as a sum of atomic energies. In more general terms, each topologically defined atom makes an additive contribution to the average value of every molecular property. 

As expected, the electron density has maxima at the positions of the nuclei. As a matter of fact, the electron density distribution for any molecule contains the information needed to determine the distribution of nuclear charge as well. It is found that the electron density at the nuclei p(0) is related to the atomic number Z through the empirical expression p(0) = 0.4798Z atomic units, with a mean deviation of 2.6%, for Z 55 (ref. 89). 

Returning to Fig. 8.1, special attention is drawn to the midpoint Fc in the CC axis. The corresponding value p is a minimum or a maximum depending on the axis considered. The first derivatives of p vanish in any case, but the second derivatives are positive for the CC internuclear axis (positive curvature) and negative for the two other cartesian axes. This shows the need to know not only the gradient vector field associated with p 

It is noted that, contrary to isolated atoms, the zero-flux surfaces of bound atoms are in general not spherical. 

It is the use of zero-flux surfaces for the topological definition of atoms or functional grouping of atoms in molecules that maximizes the extent of transferability of their properties between systems. 

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Partitioning Electron Density The Theory of Atoms in Molecules... 

The theory of atoms in molecules of R. F. W. Bader and coworkers provides another, more sophisticated approach to atomic charges and related properties. Jerzy CiosJowski has drawn on and extended this theory, and he is responsible for the AIM faciJityin Gaussian. 

The theory of atoms in molecules defines chemical properties such as bonds between atoms and atomic charges on the basis of the topology of the electron density p, characterized in terms of p itself, its gradient Vp, and the Laplacian of the electron density V p. The theory defines an atom as the region of space enclosed by a zero-/lMx surface the surface such that Vp n=0, indicating that there is no component of the gradient of the electron density perpendicular to the surface (n is a normal vector). The nucleus within the atom is a local maximum of the electron density. 

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. 

The final part is devoted to a survey of molecular properties of special interest to the medicinal chemist. The Theory of Atoms in Molecules by R. F.W. Bader et al., presented in Chapter 7, enables the quantitative use of chemical concepts, for example those of the functional group in organic chemistry or molecular similarity in medicinal chemistry, for prediction and understanding of chemical processes. This contribution also discusses possible applications of the theory to QSAR. Another important property that can be derived by use of QC calculations is the molecular electrostatic potential. J.S. Murray and P. Politzer describe the use of this property for description of noncovalent interactions between ligand and receptor, and the design of new compounds with specific features (Chapter 8). In Chapter 9, H.D. and M. Holtje describe the use of QC methods to parameterize force-field parameters, and applications to a pharmacophore search of enzyme inhibitors. The authors also show the use of QC methods for investigation of charge-transfer complexes. 

In the early 1970s, Dr. Bader invented the theory of "Atoms in Molecules," otherwise known as AIM theory. This theory links the mathematics of quantum mechanics to the atoms and bonds in a molecule. AIM theory adopts electron density, which is related to the Schrodinger description of the atom, as a starting point to mapping molecules. 

The X-ray ED formalism, which is closely connected to the theory of atoms in molecules, is still evolving and there are some unsolved problems. Thus, it appears that in simple inorganic structure the multipole model may be nonunique. ... 

A theory is only justified by its ability to account for observed behaviour. It is important, therefore, to note that the theory of atoms in molecules is a result of observations made on the properties of the charge density. These observations give rise to the realization that a quantum mechanical description of the properties of the topological atom is not only possible but is also necessary, for the observations are explicable only if the virial theorem applies to an atom in a molecule. The original observations are among the most important of the properties exhibited by the atoms of theory (Bader and Beddall 1972). For this reason and for the purpose of emphasizing the observational basis of the theory, these original observations are now summarized. They provide an introduction to the consequences of a quantum mechanical description of an atom in a molecule. 

Laplacian of the charge density V p(r), along with p(r) and Vp(r), serve to define the conceptual models of chemistry and they provide the necessary basis for the theoretical description of these models. The Laplacian of the charge density V p(r), as demonstrated in the two preceding chapters, plays a dominant role throughout the theory of atoms in molecules. It is shown here that the Laplacian provides a link between theory and the chemical models of geometry and reactivity that are based upon the Lewis model. 

The Laplacian of the charge density plays a central role in the theory of atoms in molecules where it appears as an energy density, that is. 

The determination of a property density at some point in a molecule by the total distribution of particles in the system is essential to the definition of atomic contributions to the electric and magnetic properties of a system. The densities for properties resulting from the molecule being placed in an external field must describe how the perturbed motion of the electron at r depends upon the field strength everywhere inside the molecule, a point that has been emphasized by others (Maaskant and Oosterhoff 1964). This requirement is met by the definition of an atomic property as determined by the theory of atoms in molecules. Property densities for a molecule in the presence of external electric and magnetic fields have been defined and discussed by Jameson and Buckingham (1980) and the present introduction follows their presentation. 

In this work we explore using only properties derived from the gradient vector field of the charge density, using the theory of atoms in molecules (AIM) In this way we demonstrate the usefulness of the charge density to understand such properties as metallicity when ice is in such extreme conditions (> 100 GPa, since above this pressure the tunneling effect of the proton ceases). The pressure is applied via an external isotropic force of compression. 

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