Centroids of charge

Table 15.7. The centroids of charge implied by the second moment of the charge distribution of the nuclear and a framework. The C and H nuclear positions are those of the 6-3IG SCF equilibrium geometry.

Schneider proceeds with a detailed treatment based on point charges placed at the centroids of charge of hybridized nonbonding orbitals. 

The Boys internal criterion consists essentially in separating as much as possible the so-called centroids of charge of the various molecular orbitals. We have systematically used the Boys procedure for describing the electronic structure of chemical species and for determining the electronic mechanism of chemical reactions. The obtained results will be presented in Sections II and IV. 

Moreover, the Boys procedure may also be used for localizing the singly occupied molecular orbitals obtained by the UHF approach (Pee-ters and Leroy, 1977). Thus, as shown in Sections II and IV, the electronic structure of open-shell systems and the mechanism of free-radical reactions can also be described in terms of the corresponding centroids of charge. 

One knows that, for closed-shell systems, RHF and UHF methods give identical results all the molecular orbitals are doubly occupied and then each centroid corresponds to a charge -2. The spatial distributions of the centroids of charge in some diatomic and polyatomic systems are, respectively, shown in Figs. 9 and 10. It is seen that they depend not only on the number of these centroids of charge but also on the number and... 

Fig. 9. Spatial distributions of centroids of charge (o) in some diatomic systems.

Fig. 10. Spatial distributions of the centroids of charge in some polyatomic systems, (a) Number of atomic cores, (b) Number of centroids of charge, (c) Spatial distribution general case, (d) Typical examples.

It is seen that, every time a compound may be described by a unique Lewis formula, the latter corresponds quite well to the picture given by the centroid of charge distribution. Each dot plays the same role as a line in the Lewis formula, but we must emphasize that a dot is the centroid of charge of a doubly occupied localized orbital and not a localized shared electron pair. In other words, the presence of a centroid of charge in a given molecular region does not necessarily imply that two electrons are confined in the same region. This remark can be illustrated by two typical... 

Centroid of Charge Distributions and Lewis Formulas of Various Chemical Species... 

Using a simplified representation of centroid of charge distributions, one obtains the pictures collected in Table V, for a series of model free radicals. When two centroids of different types (a and /3) are near one to the other, we adopt the Linnett notation, i.e., a line, but we do not distinguish between spatial and nonspatial pairs. Moreover, centroids of the CH, NH, or OH bond are not always indicated. 

In HCO, the centroid of charge which appears alone near the carbon atom is that of a cr-localized orbital (of a type ). All the other centroids... 

Centroid of Charge Distributions of Various Types of Free Radicals, C(N,0)X(Y,Z) ... 

In HOO the centroid of charge appearing alone, close to the terminal oxygen atom, corresponds to a 7r-localized orbital (of a type Here... 

In conclusion, the Boys localization procedure may be considered as the mathematification of the qualitative Lewis and Linnett theories. As pointed out, in this perspective a formal signification of the symbols used by these authors can be proposed these represent not localized electrons but centroids of charge of localized orbitals. Within this interpretation, the former theories agree much better with the ideas of quantum mechanics such as the wave character of electrons that prevents their localization. Moreover, the results obtained by the Boys procedure allow us to predict the Lewis and (or) Linnett structures of many chemical species without performing explicit calculations. For example, linear and bent 1,3-dipoles are described by the following general formulas ... 

Another way to define this dispersion is to consider the expectation value of the quadratic moment operator with its origin at the centroid of charge ... 

The center of charge of a pseudologe is never very far from the centroid of charge of the corresponding localized molecular orbital (see Table XI). 

Thus, in first approximation, in closed-shell systems, the LMO s centroids of charge can be identified with centers of charge of(pseudo)loges. So, we find a physical interpretation of Lewis symbols. They no longer... 

LMO values in parentheses distances atom center (centroid) of charge in atomic units. 

Considering then the electronic point of view, we have found that the charge transfer, at the transition state, is always rather small and takes place, in most cases, from the dipole toward the dipolarophile. Moreover, the analysis of the evolution of the centroids of charge along the reaction path demonstrates that all concerted 1,3-dipolar cycloadditions proceed from a cyclic movement of three 7r-electron pairs as shown in Fig. 15 for two typical reactions involving a linear and a bent dipole, respectively. 

Fig. 15. Evolution of the centroids of charge in two typical 1,3-dipolar cycloadditions involving a linear and a bent dipole, respectively.

The elaboration of the so-called exclusive orbitals was proposed by Foster and Boys [6]. This idea was modified by Boys whose method is often used when producing LMOs. In the procedure of Boys the criterion for finding localized molecular orbitals using a convenient unitary transformation is as follows the rj = r. - iv, i.e. the distances of the centroids of charge vectors of the orbitals... 

The charge displacement can be described by the displacement of the centroids of the LMOs. The calculations have been performed on the water dimer, using 6-31G/d basis set. The displacements of the centroids of charges for each bond and... 

We now express the time-doivative of the dipole moment as a cuirent dD(t)/dt = e v (t) where e = 2e, is the effective pair charge and v (t) is the velocity of the centroid of charge. The cuirent density is then... 

The dipole moment of neutral molecule is defined as the difference in the positive and negative centroids of charge ... 

Fig. 4-36. Centroids of charges of the localized valence shell orbitals ( ) and the overlap populations in HB(NH2)2- The direction of the dipole moment vector (+ -) is denoted by the arrow [1],...

Fig. 4-46, p. 220, depicts the centroids of charges and the overlap populations. For a tabulation of canonical (CMC) and localized (LMO) orbitals, quadrupole moments, second moments, and electric field gradients, see the original paper [1]. 

Calculation of the Edmiston-Ruedenberg energy-localized MOs is very time consuming. Boys (and Foster) proposed a method to find localized MOs that is computationally much faster than the Edmiston-Ruedenberg method and that gives similar results in most cases see D. A. Kleier, J. Chem. Phys., 61, 3905 (1974). The Boys method defines the LMOs as those that maximize the sum of the squares of the distances between the centroids of charge of all pairs of occupied LMOs. The centroid of charge of orbital is defined as the point at (xc,yc>Zc), where Xc = ff>i x tf>i), yc = (I y I ). Zc = 4>i z (t>i) Tjj is, the distance between the centroids of LMOs i and j, the Boys LMOs maximize 2j>, 2, rfj. 

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