Electron population defined

Classical complexes are identified [1112] as those species in which the central metal ion possesses a well-defined oxidation number and a set of ligands with a discrete electron population. Non-classical complexes , in contrast, involve highly covalent and/or multiple metal-ligand bonding resulting in indistinct oxidation numbers for both participants. 

An important advantage of the finite atoms defined by AIM is that they do not overlap, which is not generally true for orbital-defined atoms. Each atom has a sharp and well-defined boundary inside the molecule, given by its interatomic surfaces. The atoms fit exactly into each other, leaving no gaps. In other words, the shape and the volume of the atoms are additive. This is true also for other physical properties of an atom, such as the electron population and the charge, as seen in Table 6.2 and as indeed has been shown to be true for all other properties. (Bader 1990, Popelier 1999). 

FIGURE 21.2 Profiles of (a) dipole moment (in D), (b) chemical hardness (in kcal/mol), and (c) CO and CS bond electronic populations for the reaction shown in Equation 21.9. Vertical dashed lines indicate the limits of the reaction regions defined in the text. 

In addition to the equations of motion, one needs to specify a procedure to evaluate the observables of interest. Within a quasi-classical trajectory approach, the expectation value of an observable A is given by Eq. (16). For example, the expression for the diabatic electronic population probability, which is defined as the expectation value of the electronic occupation operator, reads... 

Let us investigate to what extent this simple classical approximation is able to describe the nonadiabatic dynamics exhibited by our model. To this end, we consider the diabatic electronic population probability defined in... 

Carefully established correlations between nuclear magnetic resonance (NMR) shifts and atomic electron populations in weU-defined series of closely related compounds can prove valuable for the evaluation of atomic charges in similar systems that are at, or beyond, the limits of practical computational feasibility. We certainly could make good use of them. [Also remember the insight gained with the help of Fig. 5.2 it led to Eq. (5.10).]... 

Strictly speaking, there should be no electron population between pairs of atoms in SHMO since orbitals are assumed not to overlap. However, it is conventional to set all overlap integrals to unity for the purpose of defining a bond order. The bond order, BAb, between centers A and is defined as... 

For the general case of several atoms we define P i as the one-electron population on atom / for molecular orbital j. Pjt is given by the equation... 

Self-consistent analysis of the 1-D fluid and kinetic models presented here can be developed through numerical simulations, where (17.30) and (17.32) or (17.36) and (17.37) can be used to define the initial conditions for the electric field at time 7 = 0, when the ion acceleration process begins. Results from these studies that use the fluid and the kinetic descriptions can be found in the literature, where either a single electron population [88,96-99]... 

Consider a given molecular system consisting of m atoms. In what follows we adopt the AIM resolution to define the canonical AIM chemical potentials (electron population gradient), p = dE/dN = (fiu fi2,..., fim), and the corresponding AIM hardness matrix (electron population hessian) tj = d2E/dN dN = dp/dN = here all differentiations are carried out for the fixed external potential v. This canonical charge-sensitivity information will be used to generate a variety of system charge sensitivities (CS) that probe the responses of the system to various populational perturbations at constant v. 

Let us first define the external MEC in M, consisting of m atoms. Consider the global equilibrium of M in contact with a hypothetical electron reservoir (r) fi0 = fj1 where fi= fi, the chemical potential of r. Let z = N — N° = d/V denotes the vector of a hypothetical AIM electron-population displacements from their equilibrium values N°. Since d/V = - d/Vr, the assumed equilibrium removes the first-order contribution to the associated change due to z in the energy, = M + , of the combined (closed) system (Mir) moreover, taking into account the infinitely soft character of a macroscopic reservoir, the only contribution to the energy change in the quadratic approximation is ... 

We would like to emphasize that, due to the closure constraint, there are only (m — 1) linearly independent internal MEC. Thus, the m vectors defined by Eq. (93) in reality span the (m — 1 )-dimensional space of internal MEC. In order to remove this linear dependence one could adopt the relative internal approach of Sect. 2.1.3. Namely, one then selects the electron population of one atom in the system as dependent upon populations of all remaining atoms, and discards the MEC associated with that atom. All remaining MEC can also be constructed directly from the corresponding internal relative softness matrix. Although the sets of independent internal MEC for alternative choices of the dependent atom will differ from one another, they must span the same (m — 1 )-dimensional linear space of independent internal MEC. For example, in the two-AIM system of Fig. 4 there is only one independent internal MEC direction along the P-line. 

These local mapping transformations, relating AIM electron populations (charges) to bond lengths, can be easily generalized into relations involving collective electron-population- and/or nuclear-position-displacements, e.g., PNM and nuclear normal modes. For example, the bond-stretching normal vibrations, 2b, defined by the fb principal directions, O = d lb/8Rb,... 

The coefficients cjand c2 in Equation 1.1 are a measure of the contribution which each atomic orbital is making to the molecular orbital. When there are electrons in the orbital, the squares of the c-values are a measure of the electron population in the neighbourhood of the atom in question. Thus in each orbital the sum of the squares of all the c-values must equal one, since only one electron in each spin state can be in the orbital. Since Icjl must equal c2 in a homonuclear diatomic like H2, we have defined what the values of c and c2 in the bonding... 

Following a similar approach but using a smaller data set of 369 compounds, Ivanciuc et al. correlated their liquid viscosity (10 Pa s) at 298 K with a mixed set of descriptors to obtain Eq. [48]. This involves three QM descriptors, one topological, and one constitutional descriptor. The QM descriptors were calculated with the AMI Hamiltonian in AMPAC, and CODESSA was used to calculate the descriptors and perform the statistical analyses. The HDCA2 parameter is the same HBD charged surface area used in Eq. [46]. The maximum electrophilic reactivity index, Ep, for a carbon atom is defined by X/ lumo,/A lumo+ 10), with the summation over the valence AOs on a carbon atom in the LUMO. The maximum AO electronic population, Y, models the molecular nucleophilicity and is defined by... 

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