ZDO

I lc. Ci ond reason why the ZDO approximation is not applied to all pairs of orbitals is that the major contributors to bond formation are the electron-core interactions between pairs of orbila l.s and the nuclear cores (i.e. These interachons are therefore not subjected to the ZDO approximation (and so do not suffer from any transformation problems). 

C. Semi-Empirical Models that Treat Electron-Electron Interactions 1. The ZDO Approximation... 

Most methods of this type are based on the so-called zero-differential overlap (ZDO) approximation. Their development begins by using an approximation to the atomic-orbital-based two-electron integrals introduced by Mulliken ... 

Using the ZDO approximation, the Foek matrix elements over the valenee atomie orbitals (the eores are still treated through an effeetive eleetrostatie potential as above)... 

Contributions to these elements from atoms other than a and b are neglected, again to be consistent with the ZDO approximation. 

They differ among one another in two ways (i) in the degree to whieh they employ the ZDO approximation to eliminate two-eleetron integrals, and (ii) in whether they employ... 

The CNDO and CNDO/S methods apply the ZDO approximation to all integrals, regardless of whether the orbitals are loeated on the same atom or not. In the INDO method, whieh was designed to improve the treatment of spin densities at nuelear eenters and to handle singlet-triplet energy differenees for open-shell speeies, exehange integrals... 

The ZDO approximation is made only for two-center integrals one-center coulomb Za,b = and exchange... 

In a ZDO computation where the overlap matrix is assumed to be the unit matrix, the diagonal elements of the density matrix... 

Brillouin s theorem (Brillouin, 1933) tells us that the singly excited states do not interact with the HF ground state. This theorem is true for all HF wavefunc-tions, and does not depend on the ZDO or LCAO approximations. This means that... 

It is widely accepted that the basis functions used in ZDO 7r-electron tho... 

These new basis functions can easily be shown to be orthonormal. It also turns out that two-electron integrals calculated using these orthogonalized basis functions do indeed satisfy the ZDO approximation much more closely than the ordinart basis functions. 

Although I used the example of ethene, where n =2, the same consideration, apply to ZDO calculations on all conjugated molecules. All overlap matrices are real symmetric, positive definite and so have eigenvalues > 0. 

Aer case, the invariant physical property has to be the same before and after jl sformation. Pople and coworkers proceeded to examine the consequences jtjje context of the all valence electron NDO models. I write NDO rather than ho because the more sophisticated of these models retain many two-electron (ggrals that would be set to zero under the ZDO prescription. 

The most elementary all valence electron NDO model is that known as Ippmplete neglect of differential overlap (CNDO). Segal and Pople introduced (his in 1966. Only valence electrons are explicitly treated, the inner shells being tijicen as part of the atomic core. The ZDO approximation is applied to the WO-electron integrals, so that... 

The treatment follows my discussion of the it n spectra in Chapter 8, I actually performed a CIS calculation on pyridine within the ZDO scheme. If the ground state configuration is... 

The central assumption of semi-empirical methods is the Zero Differential Overlap (ZDO) approximation, which neglects all products of basis functions depending on the same electron coordinates when located on different atoms. Denoting an atomic orbital on centre A as /ja (it is customary to denote basis functions with /j, u, A and cr in semi-empirical theory, while we are using Xn, xs for ab initio methods), the ZDO... 

An ab initio HF calculation with a minimum basis set is rarely able to give more than a qualitative picture of the MOs, it is of very limited value for predicting quantitative features. Introduction of the ZDO approximation decreases the quality of the (already poor) wave function, i.e. a direct employment of the above NDDO/INDO/CNDO schemes is not useful. To repair the deficiencies due to the approximations, parameters are introduced in place of some or all of the integrals. 

Population analysis with semi-empirical methods requires a special comment. These methods normally employ the ZDO approximation, i.e. the overlap S is a unit matrix. The population analysis can therefore be performed directly on the density matrix. In some cases, however, a Mulliken population analysis is performed with DS, which requires an explicit calculation of the S matrix. 

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