Bond-atom polarizability

From tabulated values of n-electron densities (t/,), bond orders [pij), atom-atom (Tti i), bond-bond (rcij.ij), and bond-atom polarizabilities (21, . ), X has been calculated. These values (Table I) resemble those obtained by direct solution of the Hiickel matrices. 

Related quantities, derivable in a similar manner, are the bond-atom polarizability,... 

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

An important addition to the model was the inclusion of virtual particles representative of lone pairs on hydrogen bond acceptors [60], Their inclusion was motivated by the inability of the atom-based electrostatic model to treat interactions with water as a function of orientation. By distributing the atomic charges on to lone pairs it was possible to reproduce QM interaction energies as a function of orientation. The addition of lone pairs may be considered analogous to the use of atomic dipoles on such atoms. In the model, the polarizability is still maintained on the parent atom. In addition, anisotropic atomic polarizability, as described in Eq. (9-28), is included on hydrogen bond acceptors [65], Its inclusion allows for reproduction of QM polarization response as a function of orientation around S, O and N atoms and it facilitates reproduction of QM interaction energies with ions as a function of orientation. 

A common feature of the various methods that we have developed for the calculation of electronic effects in organic molecules is that they start from fundamental atomic data such as atomic ionization potentials and electron affinities, or atomic polarizability parameters. These atomic data are combined according to specific physical models, to calculate molecular descriptors which take account of the network of bonds. In other words, the constitution of a molecule (the topology) determines the way the procedures (algorithms) walk through the molecule. Again, as previously mentioned, the calculations are performed on the entire molecule. 

As the formation of a covalent bond between two atoms implies a (dipolar) deformation of the density, polarizability and reactivity must be related. Indeed, Nagle demonstrated an empirical relation between the atomic polarizabilities (response to a field) and the scales of electronegativities (reactivity) [36]. More... 

The atomic polarizability along the bond is increased relative to a the field felt by the atom is enhanced by the field of the other atom in this direction [46]. 

Orientation effects in benzene derivatives operate in two ways. If the substituent is inductive there are large first order charge displacements at the ortho and para positions, and these can be estimated approximately using the atom polarizabilities (which is very small at the meta position). The changes of bond order, however, and consequently of free valence, vanish in first order and hence depend on Sa. The charge g g at position s therefore increases or decreases from the value unity in the... 

The polarity alternation rule (PAR) considers two kinds of substituents. The donors are. those having unshared electronic pairs or -electrons, and +1 groups. These include OH, OR, OCOR, NH2, NRR, N(R)COR, SH, SR, halogens and alkyl groups. The donor properties of the alkyl groups may reflect the existence of hyperconjugation. On the other hand, the acceptors are electron sinks, i.e. polarizable it-bonds, atoms with empty orbitals, and —I groups. Examples of acceptors are C=0 (aldehydes, ketones, carboxylic acid derivatives), CN, S02, N02, SiRj. 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

B. Indices of Chemical Reactivity Table X presents the following data for models of molecules XV-XXV 7r-electron densities (q), atom-atom polarizabilities (77 ), free valences (F, Nmhx = /3), and exact superdelocalizabilities (Se,8r, and Sn). Table XI gives Wheland s atom-localization energies for a few molecules. Bond orders for molecules XV-XXV are compiled in Table XII. 

A linear correlation between 13C chemical shifts and local n electron densities has been reported for monocyclic (4n + 2) n electron systems such as benzene and nonbenzenoid aromatic ions [76] (Section 3.1.3, Fig. 3.2). In contrast to theoretical predictions (86.7 ppm per n electron [75]), the experimental slope is 160 ppm per it electron (Fig. 3.2), so that additional parameters such as o electron density and bond order have to be taken into account [381]. Another semiempirical approach based on perturbational MO theory predicts alkyl-induced 13C chemical shifts in aromatic hydrocarbons by means of a two-parameter equation parameters are the atom-atom polarizability nijt obtained from HMO calculations, and an empirically determined substituent constant [382]. 

Several approaches have been made to calculate 13C chemical shifts of coumarins by MO methods. Good correlations were found between the 13C chemical shift values of coumarin (also protonated) and the n charge densities calculated by the CNDO/2 method [962], and of coumarins with it charge densities calculated by the Hiickel MO method (which, however, fails for methoxylated coumarins) [965]. Chemical shifts of mono- and dimethoxycoumarins have been correlated with parameters determined by refined INDO MO calculations, in which n bond orders, atom-atom polarizabilities, excitation energies and electron-nucleus distances were taken into consideration [966], In 3-substituted 4-hydroxy and 4-hydroxy-7-methoxycoumarins chemical shifts were found to be related to Swain and Lupton s parameters iF and M [388], according to equation 5.4 (SE = Substitution Effect) ... 

Just as a is the linear polarizability, the higher order terms p and y (equation 19) are the first and second hvperpolarizabilities. respectively. If the valence electrons are localized and can be assigned to specific bonds, the second-order coefficient, 6, is referred to as the bond (hyper) polarizability. If the valence electron distribution is delocalized, as in organic aromatic or acetylenic molecules, 6 can be described in terms of molecular (hyper)polarizability. Equation 19 describes polarization at the atomic or molecular level where first-order (a), second-order (6), etc., coefficients are defined in terms of atom, bond, or molecular polarizabilities, p is then the net bond or molecular polarization. 

Various reactivity indices have been derived for benzenoid hydrocarbons from the following purely topological approaches the Huckel model (HMO), first-order perturbation theory (PMO), the free electron MO model (FEMO), and valence-bond structure resonance theory (VBSRT). Since many of the indices that have been known for a long time (index of free valence Fr, self-atom polarizability ir , superdelocalizability Sr, Brown s index Z, cation localization energy Lr+, Dewar reactivity number Nt, Brown s para-localization energy Lp) have been described in detail by Streitwieser in his well-known volume [23] we will refer here only to some more recent developments. 

Basis set effects begin to be important in three-electron bonds involving atoms of the third-row, since polarization functions are important for reproducing atomic polarizabilities, which are important in atom-ion interactions. For this reason, the BOVB result that was calculated in the modest 6-31+G basis set yields a bonding energy which is some 4-5 kcal/mol smaller than more accurate G2 [47] and CPF [48] calculations, using a close to... 

A large number of polarizable models have been developed for water, many of them with one polarizable site (with a = 1.44 A ) on or near the oxygen position. For these models, the polarizable sites do not typically get close enough for polarization catastrophes (4aa) = 1.4 A, see comments after Eq. [16], so screening is not as necessary as it would be if polarization sites were on all atoms. However, some water models with a single polarizable site do screen the dipole field tensor. Another model for water places polarizable sites on bonds. Other polarizable models have been used for monatomic ions and used no screening of T or gO 15,16,27,34 Polarizable models have been developed for proteins as well, by Warshel and co-workers (with screening of T but not and by Wodak and... 

The ETMC is essentially an interatomic distance matrix (Fig. 3.47), with the diagonal elements containing an electronic structural parameter (atomic charge, polarizability, HOMO energy, etc.). Off-diagonal elements for two atoms that are chemically bonded are used to store information regarding the bond (bond order, polarizability, etc.). Matrices for active compounds in a series are then searched for common features that are not shared by inactive compounds. The successful examples cited are predominately for small, relatively rigid structures where the conformational parameter does not confuse the analysis. 

At the same time, the formally independent particle nature of DFT allows the application of standard interpretative tools developed for the HF approach. This is true not only for the standard MuUiken population analysis, but also for more sophisticated schemes, like the Natural Bond Orbital (NBO) analysis [9], the Atomic Polarizable Tensor population [10], or the Atom in Molecule (AIM) approach [11]. These tools allow the use of familiar and well known models to analyze the molecular wave function and to rationalize it in terms of classical chemical concepts. In short, DFT is providing very effective quantum... 

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