ZDO approximation

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... 

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. 

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 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. 

As noted above, many integrals describing electron repulsion are very small in magnitude, especially those such as 4> (1)< jv(1) when p v. The simplest semi-empirical approach, termed the zero-differential overlap (ZDO) approximation, is therefore to assume that these integrals can be ignored. Mathematically expressed, this is equivalent to the following ... 

At a first glance we are confronted with a striking contradiction Differential overlap is neglected in the CNDO method due to the ZDO approximation (16). Therefore there is no exchange contribution to the energy of interaction in CNDO or related calculations, z1 ex=0. 

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... 

Although semiempirical theories make some rather drastic approximations (e.g., the ZDO approximation), the use of empirical parameters partially compensates for these approximations, and allows the theories to give useful results for molecules too large to treat accurately by ab initio methods. For details of two-electron semiempirical theories, see Murrell and Harget, Pople and Beveridge. 

Use the zero differential overlap (ZDO) approximation for the two-electron integrals to show that the singly excited configurations and transition dipoles are equal. When they are allowed to interact two states are formed ... 

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