Penetration terms

Calculate tlie penetration term, P, associated witli tlie failed bags ... 

The off-diagonal terms (i j) are treated to a similar analysis. Each penetration term involving different basis functions that are on the same atomic centre are given a value of — Vab to maintain invariance. Suppose now that Xi is centred on nucleus A and Xj on nucleus B. We have... 

It mrned out that CNDO/1 calculations gave poor predictions of molecular equilibrium geometries, and this failing was analysed as being due to the treatment of the Uii and the penetration terms Vab-... 

The CNDO/2 model came next (Pople and Segal, 1966). It differs from CNDO/1 in the way it handles Uu and Vab- The penetration term Fab is no longer calculated exactly, but taken as —Z Yab- The atomic terms become... 

For a nucleus at R, the peripheral contribution to the potential 0, due to a spherical density component centered at Rj, consists of a point-charge term and a penetration term. The point-charge term is due to the nuclear charge at Rj and the electronic density within the sphere with radius R, — Ry, centered on Rjt which passes through the nucleus i (Fig. 9.1). The penetration terms are due to the electronic charge outside that sphere. They decay exponentially as the distance Rjj = R, — Rj increases (Hirshfeld and Rzotkiewicz 1974). 

The penetration contribution to the electrostatic potential at R, is evaluated by application of the general expression of Eq. (8.49) for per for the spherical density (lt = ml = 0). The point-charge term, proportional to 1/Rfj-, must subsequently be subtracted. Due to the rapid decrease of the penetration terms with increasing R j, convergence is quickly achieved. For spherically averaged Hartree-Fock atom densities, inclusion of penetration terms for atoms within 10 A of the point under consideration is more than adequate. 

In the CNDO/1 method the penetration terms (Zb-Tab—Vab) were evaluated by approximating all coulomb penetration integrals i>ab as... 

The policy of neglecting penetration terms has been continued by Pople et al. in their INDO method and by Dewar et al. in their MINDO method. 

The Orbital Penetration Term in Aromatic Substitutuent Effects. 

And even when the convergence is rapid, the multipole expansion is not a complete representation of the true electrostatic energy, as it neglects the effects of overlap of the charge distributions of the two molecules in question [12,14], sometimes referred to as a penetration term. Fortunately, the latter complicating effect dies off exponentially with intermolecular separation [15], adding to the validity of the multipole expansion at sufficient spacing. 

The details of the second-order energy depend on the form of exchange perturbation theory used. Most known results are numerical. However, there are some common features that can be described qualitatively. The short-range induction and dispersion energies appear in a non-expanded form and the differences between these and their multipole expansion counterparts are called penetration terms. 

Figure 3.5 Charge penetration in base stacking for the GGiCC base pair step [at 35° Twist and 0.28 A Siide] as a function of Rise, the vertical separation between the base pairs. The difference between the distributed multipole analysis [DMA) value for electrostatics and the quantum mechanical symmetry-adapted perturbation theory [SAPTO/jun-cc-pVDZ) value for electrostatics may be taken as a measure of the charge penetration term. The DMA analysis includes terms up through order 5 [32pole-chai e, hexadecapole-dipole, octopole-quadrupole). Charge penetration rapidly increases in magnitude for smaller intermolecular distances.

The introduction of correction terms to the multipolar expansions of Vg or of ES, acting at short distances and called penetration terms , has been done for formal studies of this problem but it is not used in practical applications. 

Table 13J. The table pertains to two molecules in their eleetronie ground states. For each pair of molecules a short characteristic of their electrostatic, induction and dispersion interactions is given. It consists of the sign of the corresponding interaction type (the minus sign means attraction, the plus sign means repulsion and 0 corresponds to the absence of such an interaction, the penetration terms have been neglected)...

The one-centre expansion of the MEP is not adequate, however, for the large number of chemical applications we mentioned in Sect. 4. In fact the expansion theorem holds for points r lying outside a sphere containing all the elements of the charge distribution. In molecules, this condition is never formally satisfied, because Q r) has an exponential decay. The difference between V (r) and the exact multipole expansion of K (r) is generally called the penetration term ... 

In general, the elementary charge distribution (49) is reduced to the sum of a finite number of Gaussian functions, from which it is immediate to derive an exact multipole expansion composed by a finite number of terms the upper term is the sum of the angular momentum quantum numbers of the two functions. There is still, of course, a penetration term, related to the exponential decay of the functions. Hall has shown, with numerical examples, that this penetration term is reasonably small [75]. 

The terms B and C are due to atomic multipoles (AMs) and penetration of the atom s electron cloud, respectively. The penetration term, C, decays approximately exponentially with increasing /"a and is approximately zero where p r) is negligible. ... 

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