Bader method

One further theoretical method that merits consideration at this point is the topological theory of molecular structure exemplified by Bader (1985, 1990). In this method a topological description of the total electron density in the molecule is used. A major advantage of this method is that it allows the total interaction between various centres to be probed. Cremer et al. (1983) used the Bader method to examine the homotropylium cation [12] and concluded that it was indeed homoaromatic. 

Note that the Bader method is similar in philosophy except that there any weight function can be only either exactly one or either exactly zero. In the original Hirshfeld... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

The individual gauges for atoms in molecules (IGAIM) method is based on Bader s atoms in molecules analysis scheme. This method yields results of comparable accuracy to those of the other methods. However, this technique is seldom used due to large CPU time demands. 

Bader, M. A Systematic Approach to Standard Addition Methods in Instrumental Analysis, /. Chem. Educ. 1980, 57, 703-706. 

Perhaps the most rigorous way of dividing a molecular volume into atomic subspaces is the Atoms In Molecules (AIM) method of Bader.The electron density is the square of the wave function integrated over N — coordinates (it does not matter which coordinates since all electrons are identical). 

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]. 

So far we have considered the shape of the electron density of a limited inner region of each atom but not of the complete atom. How do we find the shape of the complete atom In other words, how do we find the interatomic surfaces that separate one atom from another and define the size and shape of each atom The atoms in molecules (AIM) theory developed by Bader and coworkers (4) provides a method for doing this. 

Several methods have been used for analyzing the electron density in more detail than we have done in this paper. These methods are based on different functions of the electron density and also the kinetic energy of the electrons but they are beyond the scope of this article. They include the Laplacian of the electron density ( L = - V2p) (Bader, 1990 Popelier, 2000), the electron localization function ELF (Becke Edgecombe, 1990), and the localized orbital locator LOL (Schinder Becke, 2000). These methods could usefully be presented in advanced undergraduate quantum chemistry courses and at the graduate level. They provide further understanding of the physical basis of the VSEPR model, and give a more quantitative picture of electron pair domains. 

Cherif F. Matta is about to complete his Ph D. in theoretical/quantum chemistry with Professor Richard F. W. Bader at McMaster University (Canada). He has taught general, physical and quantum chemistry for over five years. His main research interest is in developing new theoretical methods for calculating the properties of large complex molecules from smaller fragments. He applied the new method to obtain accurate properties of several opioids molecules. He is also interested in QSAR of the genetically-encoded amino acids. 

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. 

Simon H, Bader J, Gunther H, Neumann S, Thanos I (1985) Angew Chem 97 541 Angew Chem Int Ed Engl 24 539 Thanos I, Bader J, Gunther H, Neumann S, Krauss F, Simon (1987) In Mosbach K (ed) Methods Enzymol 136 302... 

All the different AIM methods that will be discussed below basically use this same approach but quite different in the nature of n (. Chronologically, we will discuss the Mulliken AIM, the Hirshfeld AIM, and the Bader AIM. This last approach will henceforth be called quantum chemical topology (QCT). There are more AIM methods, but most of them can be easily understood by the three selected emblematic approaches. 

Several forms of wf have already been used within the field of MQS. These methods include the Hirshfeld partitioning [30], Bader s partitioning based on the virial theorem within atomic domains in a molecule [64], and the Mulliken approach [65]. For more information on all the three methods, refer to Chapter 15. 

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. 

There are several alternative computational methods for obtaining NMR data such as the GIAO method (Wolinski et al., 1990), the LORG method (Bouman and Hansen, 1989), and IGAIM (Keith and Bader, 1992). 

The microscopic world of atoms is difficult to imagine, let alone visualize in detail. Chemists and chemical engineers employ different molecular modelling tools to study the structure, properties, and reactivity of atoms, and the way they bond to one another. Richard Bader, a chemistry professor at McMaster University, has invented an interpretative theory that is gaining acceptance as an accurate method to describe molecular behaviour and predict molecular properties. According to Dr. Bader, shown below, small molecules are best represented using topological maps, where contour lines (which are commonly used to represent elevation on maps) represent the electron density of molecules. 

Method/Basis Set Mulliken Lowdin Hirshfeld Bader... 

When space is partitioned with discrete boundaries, as in Eq. (6.7) and in the Bader virial partitioning method, the moments can be derived directly from the structure factors by a modified Fourier summation, as described for the net charge in chapter 6. 

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