Atom-in-molecule

This results in a population analysis scheme that is less basis set dependent than the Mulliken scheme. Flowever, basis set effects are still readily apparent. This is also a popular technique because it is available in many software packages and researchers find it convenient to use a method that classifies the type of orbital. 

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. 

The AIM scheme is popular due to its reliability with large basis sets for which some other schemes fail. Unfortunately, the numerical surface finding and integration involved in this scheme are not completely robust. For example, nonnuclear attractor compounds like Li2 and Na clusters have maxima in the middle 

Atoms in Molecules.—In this approach, which was first proposed by Moffitt,105 a wavefunction for a particular electronic state of a molecule is constructed from products of atomic wavefunctions, these, moreover, being taken to be exact eigenfunctions of their respective atomic hamiltonians. We confine our attention to the case of diatomic molecules AB so that, according to this procedure, the wavefunction is written as 

The functions on the right-hand side of this equation are termed composite functions and consist of products of atomic eigenfunctions properly coupled and antisymmetrized as follows  

According to the original method of Moffitt, the necessary matrix elements are evaluated according to the following prescription 106-107 

The matrix elements Hab.ca, Aat,ca are calculated using spin valence functions constructed from atomic orbitals. The energies E%, Ef, Ef, Ef are the exact values for the particular states of the participating atoms and may be taken from spectroscopic tables. The corresponding quantities E%, etc., with tildes are the values for these same spectroscopic states obtained from calculations using the orbital wavefunctions. In essence, equation (116) prescribes how the matrix elements Hab.ca, which are obtained from an ab initio spin valence calculation, are to be corrected in order to eliminate known atomic errors. 

According to Hurley,103 the matrix elements Hab.ca, Aab.cd are to be calculated with all orbital exponents optimized so as to achieve the best possible molecular energy. This takes into account to some extent the deformations of the atoms when the molecule is formed, so that the expansion 

4 The Lewis Model as Displayed in the Laplacian of the Electron Density 

The initial attempts to relate this language to quantum mechanics were understandably done through the orbital model that underlies the valence bond and molecular orbital methods employed to obtain the approximate solutions to Schrodinger s equation. The one-electron model, as embodied in the molecular orbital method or its extension to solids, is the method for classifying and predicting the electronic structure of any system (save a superconductor whose properties are a result of collective behavior). The orbital classification of electronic states, in conjunction with perturbation theory, is a powerful tool in relating a system s chemical reactivity and its response to external fields to its electronic structure and to the symmetry of this abstract structure. The conceptual basis of chemistry is, however, a consequence of structure that is evident in real space. 

The spatial distribution of electronic charge in the field of the nuclei and its flow in the presence of external fields are the physical manifestations of the forces operative in matter 

The essential chemical concepts are those that can be recovered in their entirety from physics. Their recovery yields unique definitions and the resulting physical understanding greatly increases their conceptual usefulness. An atom in a molecule, molecular structure, and electron localization are such concepts and they are the topics discussed in this article. 

The topological definition of an atom and the associated ideas of structure and structural stability are introduced in Section 2. These ideas can be presented in a pictorial and qualitative manner. The quantum definition of an open system and its identification with the topological atom are presented in Section 3. The consequences of this identification are explored without presenting its derivation, the approach that must necessarily be followed in this general introduction to the theory. 

The main requirement in the determination of bond orders is to derive rules on how to measure the number of electrons shared between two atoms. For this purpose, a definition of an atom in a molecule is required, which, however, cannot be formulated in a unique and unambiguous way [169]. Quantum chemical calculations are typically performed in the Hilhert-space analysis, where atoms are defined by their basis orbitals. Such an analysis, however, strongly depends on both the atomic basis set chosen and the type of wave function used. The position-space representation, on the other hand, where atoms are defined as basins in three-dimensional physical space does not suffer from these insufficiencies. In this chapter, we present one option for a three-dimensional atomic decomposition scheme and the reader is referred to Refs. [170-173] for further examples. 

In a physical-space analysis, a weight function introduced for every 

In the first case, iv (r) is equal to one in the domain 2, while all other weight functions vanish 

If such a partition scheme is to be employed, the atomic domains 2 need to be defined exphdtly. The most frequently apphed decomposition is Bader s Atoms in Molecules (AIM) approach [174,175] where the atomic domains 2 are divided by surfaces that are determined on the basis of the topology of the electron density. As the atomic volumes (or domains) are determined from the properties of the electron density, the AIM scheme can be apphed to any electron density calculated from a general wave function. In AIM theory, the boundary condition for an atom in a molecule requires the gradient vector field of the electron density Vp(r) to have zero flux [174,176], 

Perhaps the most rigorous way of dividing a molecular volume into atomic subspaces is the Atoms In Molecules (AIM) method of R. Bader. ° The electron density is a 

Hydrogen and carbon AIM basins for cyclopropane dots indicate bond and ring critical 

A more fundamental problem in the AIM approach is the presence of non-nuclear attractors in certain met lic systems, such as lithium and sodium clusters. While these are of interest by themselves, they spoil the picture of electrons associated with nuclei, forming atoms within molecules. It should be noted that non-nuclear attractors can also be found for other systems such as ethyne when a low level of theory is used for calculating the electron density. 

The second derivative of the electron density, the Laplacian V p, provides information on where electron density is depleted or increased. At a bond critical point the sign of the Laplacian has been used for characterizing the nature of the bond, i.e. a negative value indicates a covalent bond, while a positive value indicates an ionic bond or a van der Waals interaction. 

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 given 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, i.e. functional groups react similarly in different molecules. It may be shown that any observable molecular property may be written as a sum of corresponding atomic contributions. 

The concept that substances are composed of molecules, and molecules are composed of atoms, can be traced back to chemical antiquity. Nevertheless, in modem molecular electronic stmcture theory, the atomic constituents differ appreciably from the immutable, indivisible particles envisioned by the ancients. Of course, the signature properties of an atom are only indirectly linked to the positively charged nucleus, which carries virtually the entire atomic mass but occupies only an infinitesimally small portion of the apparent atomic volume. We now understand the atom to be composed of the surrounding quantum mechanical distribution of electrons that occupy the characteristic set of orbitals associated with the nucleus in question. Finding the atom in a molecular wavefunction therefore reduces (as in Chapter 2) to the problem of finding the atomic orbitals and the associated electronic configuration (number of electrons occupying each available atomic orbital) around each nuclear center. 

As before, the Natural Atomic Orbitals (NAOs) serve as the optimal effective atom-like orbitals for describing the overall electron density distribution of the molecular wavefunction, so that finding the atomic electrons in NBO output is not more difficult than in Chapter 2. We shall first examine how the NAOs within the molecular environment differ from the free-space forms encountered in Chapter 2. We use the experience gained there to anticipate the breathing  

Discovering Chemistry With Natural Bond Orbitals, First Edition. Frank Weinhold and Clark R. Landis. 2012 John Wiley Sons, Inc. Published 2012 by John Wiley Sons, Inc. 

Crystallographic electron-density functions exhibit local maxima at the positions of atomic nuclei and, not surprisingly, several efforts to partition the density function into regions that represent individual atoms have been made. 

In addition, deformation densities provide further guidance on how to recognize the presence of chemical bonds and lone pairs as special features of the total density. Significant early suggestions on how to achieve a chemically meaningful partitioning were proposed by Berlin [183] and by Daudel [184]. 

The atoms-in-molecules partitioning indicates that the total density in a molecular crystal can be obtained as the sum over individual atom densities. It is argued [173] that other molecular properties would partition in a related manner and that any physical property of a molecule is given by the sum of its atomic properties. 

Apart from serving to partition crystallographically measured electron densities, the atoms-in-molecules procedure is also used to derive molecular properties from isolated-molecule densities, calculated by ab initio methods. Discrepancies between sum-over-atoms and whole-molecule calculations, 

Bader and associates at Canada s McMaster University have derived a means of describing the electron distribution associated with specific atoms in a molecule, called the atoms in molecules (AIM) method. The foundation of this approach is derived from quantum mechanics and principles of physics. It uses the methods of topology to identify atoms within molecules. The electron density of a molecule is depicted by a series of contours. Bond paths are the paths of maximum electron density between any two atoms. The critical point is a point on the bond path where the electron density is a maximum or a minimum with respect to dislocation in any direction. The bond critical point is defined by the equation 

The critical point is the point at which the gradient vector field for the charge density is zero, that is, either a maximum or minimum along N. The condition Vp(r) N(r) = 0 applied to other paths between two atoms defines a unique surface that can represent the boundary of the atoms within the molecule. The electron density within these boundaries then gives the atomic charge. The combination of electron density contours, bond paths, and critical points defines the molecular graph. This analysis can be applied to electron density calculated by either MO or DFT methods. For a very simple molecule such as Hj, the bond path is a straight line between the nuclei. The 

Atoms in Molecules A Quantum Theory, Oxford University Press, Oxford, 1990. For an introductory discussion of the AIM method for describing electron density, see C. F. Matta and R. J. Gillespie, J. Chem. Ed., 79, 1141 (2002). 

The electron density can also be characterized by its ellipticity, the extent to which it deviates from cylindrical symmetry, reflecting the contribution of rr orbitals. While the C=C bond in ethyne is cylindrically symmetrical, the C-C bonds in ethene and benzene have greater extension in the direction of the rr component. Ellipticity is defined by 

The ellipticity of the electron density in benzene is 0.23, while that in ethene is 

Molecular orbital theory assumes a set of nuclei held together by a collection of electrons. This is a rather different view of matter from the intuitive view that molecules are made of atomic centers held together by electron pair bonds. There is a theoretical method that allows one to recover the concepts of atoms and bonds in chemical structures, and it does so in a rigorous, mathematical way. The approach was developed by Bader and co-workers and is known as atoms in molecules (AIM) or the quantum theory of atoms in 

Weinhold, F. Angew. Chem. Int. Ed. Engl. 2003, 42, 4188. See also reference 187. 

001 au density envelopes of methane, ethane, propane, butane, pentane, and hexane. (Adapted from reference 216.) 

Atoms in Molecules A Quantum Theory Clarendon Press Oxford, 1990. Bader, R. F. W. Hemandez-Trujillo, J. Cortes-Guzman, F. /. Comput. Chem. 2007,28,4. Malta, C. F. Boyd, R. J. The Quantum Theory of Atoms in Molecules From Solid State to DNA and Drug Design Wiley-VCH Weinheim, 2007. 

000 au equals 6.748 e/A A 0,002 electron density surface is a better match for van der Waals radii of molecules in the condensed phase. 

MO calculations can provide the minimum-energy structure, total energy, and overall electron density of a given molecule. However, this information is in the form of the sum of the individual MOs and cannot be easily dissected into contributions by specific atoms or groups. How can the properties described by the MOs be related to our concept of molecules as a colleetion of atoms or functional groups held together by chemical bonds  

The subunits which can be defined can include atoms or collections of atoms corresponding to functional groups. The subunits can be represented as regions of space defined by electon density. These representations correspond well with the qualitative concepts that arise from valence bond stmctures. The mathematical evaluation can assign shape and charge density to atoms. Table 1.17 gives the C and H charge densities in some 

Atoms in Molecules—A Quantum Theory, Oxford University Press, Oxford, U.K., 1990 R. F. W. Bader, Chem. Rev. 91 893 (1991). 

The lesson in these figures is that the qualitative concepts of chemical structures can be given a pictorial representation based on the quantitative application of the principles of quantum chemistry. Various, indeed all, molecular properties can, in principle, be calculated from the electronic distribution these pictures represent. 

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The atom type tlefin es the chemical eii viroiini eii t of an atom. The basic idea is that not all carbon atoms in molecules are the same and can be distinguished by the following ... 

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

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

VViberg and Rablen found that the charges obtained with the atoms in molecules method were relatively invariant to the basis set. The charges from this method were also consistent v it i the experimentally determined C-H bond dipoles in methane (in which the carbon is p isitive) and ethyne (in which the carbon is negative), unlike most of the other methods they examined. 

B.iilcr R F W 1985. Atoms in Molecules. Accounts of Chemistry Research 18 9-15. 

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. 

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

Condensation is the process of reduction of matter into a denser form, as in the liquefaction of vapor or steam. Condensation is the result of the reduction of temperature by the removal of the latent heat of evaporation. The removal of heat shrinks the volume of the vapor and decreases the velocity of, and the distance between, molecules. The process can also be thought of as a reaction involving the union of atoms in molecules. The process often leads to the elimination of a simple molecule to form a new and more complex compound. 

During electrochemical fluorination retention of important functional groups or atoms in molecules is essential. Acyl fluorides and chlorides, but not carboxylic acids and anhydrides (which decarboxylate), survive perfluorination to the perfluorinated acid fluorides, albeit with some cyclization in longer chain (>C4) species [73]. Electrochemical fluorination of acetyl fluoride produces perfluoro-acetyl fluoride in 36-45% yields [85]. Electrochemical fluorination of octanoyl chloride results in perfluorinated cyclic ethers as well as perfluorinated octanoyl fluonde. Cyclization decreases as initial substrate concentration increases and has been linked to hydrogen-bonded onium polycations [73]. Cyclization is a common phenomenon involving longer (>C4) and branched chains. a-Alkyl-substituted carboxylic acid chlorides, fluorides, and methyl esters produce both the perfluorinated cyclic five- and six-membered ring ethers as well as the perfluorinated acid... 

Camparing Charge Methods Atoms in Molecules Analysis... 

Advanced Exercise 8.6 Atoms in Molecules Charges end Bond Orders... 

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. 

Exercise 8.6 Allyl cation Atoms-in-Molecules analysis 0 55 44.7... 

A. Werner (Zurich) work on the linkage of atoms in molecules which has thrown new light on earlier investigations and opened up new fields of research especially in inorganic chemistry. 

The values are called the net atomic populations and the overlap population. Chemists speak of the charges on atoms in molecules, and Mulliken s second contribution was to propose a method of partitioning the overlap population between contributing atoms. He proposed that the overlap populations be divided equally between participating atoms, so giving the gross atomic populations of... 

In Chapter 3, we studied the topic of population analysis. In population analysis, we attempt a rough-and-ready numerical division of the electron density into atom and bond regions. In Mulliken theory, the bond contributions are divided up equally between the contributing atoms, giving the net charges. The aim of the present section is to answer the questions Are there atoms in Molecules , and if so, How can they be defined . According to Bader and coworkers (Bader, 1990) the answers to both questions are affirmative, and the boundaries of these atoms are determined by a particular property of the electron density. 

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