Zero differential overlap calculations

Semiempirical methods are widely used, based on zero differential overlap (ZDO) approximations which assume that the products of two different basis functions for the same electron, related to different atoms, are equal to zero [21]. The use of semiempirical methods, like MNDO, ZINDO, etc., reduces the calculations to about integrals. This approach, however, causes certain errors that should be compensated by assigning empirical parameters to the integrals. The limited sets of parameters available, in particular for transition metals, make the semiempirical methods of limited use. Moreover, for TM systems the self-consistent field (SCF) procedures are hardly convergent because atoms with partly filled d shells have many... 

The zero differential overlap approximation can be applied in the localized representation. This was demonstrated by calculating for C H, CioTfio and C14//14, respectively the total energy corrections and the pair correlation energies through second and third order in different approximations. When the strongly local contributions were only... 

The bottleneck in all self-consistent field calculations is the difficulty of calculating the integrals [Eq. (16)] over the atomic orbitals. Thus, reducing their number has been an imperative requirement and has led to the fundamental zero-differential overlap approximation (ZDO) which assumes... 

The second group of semiempirical MO LCAO methods is constituted by zero-differential overlap methods (9). Two subgroups can be specified here which somewhat conventionally may be called physical and chemical. The former involves CNDO/2, INDO, and some of their modifications, for example, CNDO/S. These methods are directed to the calculations of the electron characteristics charge distributions, dipole moments, polarizabili-... 

The 71-electronic densities obtained from an MO calculation which includes the zero differential overlap approximation (Equations 1 and 2) indicate the general disposition of charge in the molecule rather than the exact populations of the AOs. These electronic densities represent the populations of modified AOs which are not well localized. Thus, the relative values of the AO densities, rather than their precise magnitudes, reflect the general electronic behavior of the molecule. 

Finding a way out. Calculations can be considerably simplified by using approximation of the zero differential overlap (ZDO). The essence of this approximation is that the overlap of different AOs yfpi and is assumed to be equal to zero for any element dx of the volume of a molecule ... 

Analogous to the PPP method for planar 7r-systems, semiempirical all-valence methods can be and were extended to include Cl, thus giving rise to a family of procedures based on the CNDO, INDO and NDDO variants of the zero-differential overlap (ZDO) approximation, many of which were applied also to the discussion of Cl effects in radical cations. Due to the parametric incorporation of dynamic correlation effects, such procedures often yield rather accurate predictions of excited-state energies and they continue to be the methods of choice for treating very large chromophores which are not amenable to ab initio calculations. 

In the calculations of Pock matrix elements, the zero differential overlap approximation is adopted. Then we get for a closed shell structure ... 

Figure 3.10. INDO/MRDCI calculated transition densities for 5-8 between the S0 and Si on the left-hand side and between the Si and S2 on the right-hand side. The radii of the circles are proportional to the zero differential overlap (ZDO) transition densities associated with each atom. (From Ref. [284] with permission of the American Institute of Physics.)...

Now, it is just this term which determines to a large extent the importance of the interaction of the n electrons between the atoms P and Q. The integral fiPq may be identified with the parameter f pg of the semi-empirical theories based on the zero-differential-overlap approximation 35,38). in our opinion, there is no general calculation method leading to values for the fipq s which are in good numerical agreement with the pvq parameters fitted on experimental data (electronic spectra, dipole... 

Many interesting problems in physical organic chemistry have been clarified by numerical calculations based on next-neighbor interaction and zero-differential-overlap approximations, especially in the field of... 

Charge densities have been calculated for thiazolylium, 1,3-dithiolylium, and imidazolium ion (by TT-electron SCF-MO) using the zero differential overlap approximation (Table 1) <7iJHC55i> it was found that the highest charge densities are at the heteroatoms of the ions. The difference in energy for an ion and its corresponding carbene predicts the rates of ionization of the H—C(2) bond and was found to be imidazolium < thiazolylium < 1,3-dithiolylium. 

PPP (Pariser-Parr-Pople) [14-16] is an SCF (self-consistent field) Jt-electron theory, assuming o — jt separability. Only a single (2pz) atom orbital is considered on each atom and the Ji-electron Hamiltonian includes electron-electron interactions with ZDO (zero differential overlap) approximation. All integrals are determined by semiempirical parameters. The PPP method can only be used to calculate those physical properties for which jt electrons are mainly responsible. 

If AOs of different atoms did not overlap with one another, overlap integrals would vanish, and so would the integrals p,v a) if and Xv were AOs of different atoms or if x and Xu were AOs of different atoms. The number of the TERIs that had to be calculated would then be far less. This is the basis of the NDDO approximation, " the most rigorous of several so-called zero differential overlap (ZDO) approximations. In the NDDO approximation, the integrals (/uj/ A(t) are neglected unless Xfi and Xv are AOs of the same atom, and xa and x[Pg.470]

The two-electron integrals require the main computational effort in a HF calculation and their number is significantly reduced in semiempirical methods by the zero differential overlap (ZDO) approximation. This basic semiempirical assumption sets products of functions for one electron but located at different atoms equal to zero (i.e. /xa(1)vb(1) = 0, where and vb are two different orbitals loeated on centers A and B, respectively). The overlap matrix, S, is set equal to the unit matrix, S v = and the two-electron integrals (/xv Act) are zero, unless fx = v and k = a, that is,... 

For this study, the CNDO/2 MO method is chosen because the partitioning of the total energy to atoms and atom pairs is possible in this scheme due to the zero-differential overlap approximation. This feature of the CNDO/2 MO method is important for the comparison of the MO results with the energy terms in the MFF method. As a model compound for the calculation, 1,2-dimethoxyethane (1,2-DME) was chosen because the previous statistical mechanical conformational analysis of a series of polyethers indicated this to be... 

As we have seen, EHT is a nonself-consistent method but the self-consistency over charge and configuration is included in the MR approximation. The Ab-initio HE SCF method requires the self-consistent calculation of the density matrix (see Chap. 4). This feature of the HE approach is maintained in the semiempirical methods, based on the zero differential overlap (ZDO) approximation. This approximation is used to reduce the number of multicenter integrals appearing in HE LCAO calculations. 

The spin-Hamiltonian VB theory uses the same approximations as the qualitative theory presented above to calculate the Hamiltonian matrix elements, but with a few simplifications. The theory is restricted to determinants having one electron per AO this restriction excludes ionic structures or molecules bearing lone pairs. As such, the theory has mainly been applied to conjugated polyenes. Another simplification is the zero-differential overlap approximation, which means that all overlaps are neglected in the formulas. 

The effect of the environment (which is denoted by s) surrounding the reacting system (r, referred to as solute) can be included in MO calculations in a fairly straightforward way if the overlap integrals between the solute and solvent are neglected [7]. The effect of the solvent polarisation on the solute electronic states can be estimated within the zero differential overlap approximation by rewriting the diagonal matrix elements of r as... 

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