ZDO methods

This allowed one to approximate the four-orbital integrals of electron-electron interaction by much simpler expressions  

The simplification is achieved due to the fact that for an integral on the left side, which may potentially involve AOs coming from four different centers and cannot be easily calculated at least for the Slater-type AOs, representation on the right is given in terms of no more than two-center quantities, which can be easily calculated for the Slater functions. 

In the above equalities the basis functions are orthogonal (OAOs)  

if the differential overlaps of the original AO basis are approximated by the formula eq. (2.28), it turns out that applying the Lowdin transformation S 2 to the set of the AOs makes the products i.e. the differential overlaps of the symmetrically orthogonal OAOs vanishing  

CNDO methods. The simplest among the methods developed within the ZDO paradigm was the CNDO method, which uses the Complete Neglect of Differential Overlap so that 

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Recently, the MNDO type methods (MNDO [32], AMI [33] and PM3 [34]) have been tested for their ability to produce reliable MEP maps. These semi-empirical methods, just as the CNDO and INDO methods, are ZDO methods, and are based on the more sophisticated NDDO approximation [35]. 

The first of the zero differential overlap (ZDO) methods was the simple tt-electron method due to Htickel. Historically this method was very important in that it showed rather quickly that molecular orbital methods that... 

Table 2 presents an abstract of Table 1 in ref. 12. In general, all three methods MNDO, AMI, and PM3 give rather remarkable results. In many cases, however, PM3 is more accurate than AMI, and both are more accurate than MNDO. MNDO predicts sterically crowded molecules to be too unstable and favors, in contrast, small rings, a shortcoming in most ZDO methods largely corrected in AMI and PM3. Of special interest is the observation that PM3 successfully reproduces the heats of formation of hypervalent compounds without the use of d orbitals. 

In generating a Mulliken population from a ZDO calculation, however, an assumption such as that given in Eq. [14a] must be made that is, a model that is characterized principally through integrals must be related to an actual basis. The symmetrically orthogonalized basis is not an inherent feature in any of the ZDO methods discussed above, with the exception of the SINDOl method. 

J. D. Head and M. C. Zerner, Chem. Phys. Lett., 131,359 (1986). An Approximate Hessian for Molecular Geometry Optimization. (Introduced is an approximate analytical Hessian that decreases the amount of work required by a factor of N in ZDO methods, where N is the size of the basis.)... 

If the MOs are obtained with zero differential overlap (ZDO) methods (see Chapter 2.38), then the overlap integrals, S, between different AOs are neglected, and the contribution of the... 

The KS orbital energies can also be used for qualitative interpretation of the electronic spectra of atoms and molecules " and band gaps in solids.In the HF or semiempirical ZDO methods, the unoccupied MOs are subject to the self-consistent field of all N electrons, whereas the occupied MOs are subject to the self-consistent field of the N - 1) electrons (an electron in an occupied orbital does not interact with itself). So, for unoccupied orbitals of the N-electron system, the MO energy Sa corresponds to the interactions of an extra N + l)th electron. For the excitation of an electron from the occupied MO to the unoccupied MO an electron in electron affinity, namely with , . As a result, the MO energy differences, Ea — Si, obtained from HF or semiempirical INDO/S calculations are not estimates of transition energies, they have to be combined with appropriate Jand Kintegrals (see Chapter 2.38). 

The general analysis of Rudenberg s approximation in the HF LCAO method for molecules [218] and solids [223] has shown that EHT and zero differential overlap (ZDO) approximations can be considered as particular cases of Rudenberg s integral approximation. ZDO methods, considered in the next section, were applied to a wide class of molecules and solids, from purely covalent to purely ionic systems. Therefore, they are more flexible compared to the MR approximation, which is more appropriate for ionic systems. 

Depending on the level of the approximations used for other integrals ZDO methods differ. In the CNDO (complete neglect of differential overlap) method [205,236] all two-electron integrals are approximated by Coulomb integrals according to... 

The first apphcations of ZDO methods for extended systems refer to the electronic-structure calculations of regular polymer chains when the one-dimensional (ID) periodicity is taken into account. The corresponding modifications of molecular ZDO equations can be found in the literature for the CNDO method in [263-266], for the INDO and MINDO methods in [267,268], for the MNDO, PM3 and AMI methods in [269-272]. 

The considered CNDO method for periodic systems formally corresponds to the model of an infinite crystal or its main region consisting of L primitive cells. This semiempirical scheme was also apphed for the cychc-cluster model of a crystal allowing the BZ summation to be removed from the two-electron part of matrix elements. In the next section we consider ZDO methods for the model of a cyclic cluster. 

The practical cyclic-cluster calculations for crystals were made by EHT, MR and different ZDO methods. In these calculations one has to take into account the mul-tipbcation law of the cyclic-cluster symmetry group. This requires the modifications (compared with molecules) of Fock matrix elements in the LCAO approximation. 

The EHT method is noniterative so that the results of COM apphcation depend only on the overlap interaction radius. The more complicated situation takes place in iterative Mulliken-Riidenberg and self-consistent ZDO methods. In these methods for crystals, the atomic charges or the whole of the density matrix are calculated by summation over k points in the BZ and recalculated at each iteration step. The direct lattice summations have to be made in the surviving integrals calculation before the iteration procedure. However, when the nonlocal exchange is taken into account (as is done in the ZDO methods) the balance between direct lattice and BZ summations has to be ensured. This balance is automatically ensured in cychc-cluster calculations as was shown in Chap. 4. Therefore, in iterative MR and self-consistent ZDO methods the increase of the cyclic cluster ensures increasing accuracy in the direct lattice and BZ summation simultaneously. This advantage of COM is in many cases underestimated. 

CCM has been implemented in various ZDO methods CNDO-INDO [276,277,280, 303-310], MINDO/3 [311], MNDO [271], NDDO(AMl, PM3), [312]. AU the semiem-pirical methods stand or faU by their parameter set. The ZDO methods parameter... 

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