Centroid of charge distribution

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. 

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

The reason for F is not too far to seek. In the general context of charge distributions, the role of an atom caimot be described solely by reference to the number of electrons it contributes to the molecular fabric. The centroid of its integrated charge density is equally important. 

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.

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.

Here, the centroids of the charge distributions of phases A and B have been introduced ... 

We can now consider the question of the electrical centroids of the positive and the negative charges apart, that is to say, the electrical centroid of the nuclei alone, and the electrical centroid of the electron cloud alone. It may happen that the two points coincide, just as they always do in individual atoms, where the positive centroid is identical with the nucleus, and where also the centroid of the negative charge cloud, on account of the central symmetry of its charge distribution, always coincides with the nucleus. In general, however, the two centroids will be distinct from each other consequently the external action of the molecule is like that of an electric dipole. In this case we speak of a permanent electric dipole moment, and denote it by the vector... 

The above moments of a charge distribution can be related to the statistical moments of random variables. The orbital centroid vector is namely identical to the central first moment M(Q of a random variable , with density function p(x) ... 

Note that a distinction is made between electrostatic and polarization energies. Thus the electrostatic term, Ue e, here refers to an interaction between monomer charge distributions as if they were infinitely separated (i.e., t/°le). A perturbative method is used to obtain polarization as a separate entity. The electrostatic and polarization contributions are expressed in terms of multipole expansions of the classical coulomb and induction energies. Electrostatic interactions are computed using a distributed multipole expansion up to and including octupoles at atom centers and bond midpoints. The polarization term is calculated from analytic dipole polarizability tensors for each localized molecular orbital (LMO) in the valence shell centered at the LMO charge centroid. These terms are derived from quantum calculations on the... 

Again, if i = j and k = the electron-repulsion integral takes on the simple point-charge asymptotic form of 1/R, where this time R is the distance between the centroids of the two distributions. Since the basis functions are always atom-centred functions, the centroid of the diagonal charge distributions pu) are the relevant atoms and so the distances R are actually inter-atomic distances. 

D distance between the centre-of-mass and the centroid of the nuclear charge distribution,... 

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