Zero-differential Overlap Approximations for Molecules

2 Zero-differential Overlap Approximations for Molecules and Crystals 

As was seen in Sec.6.1 in EHT the AO basis functions are only used for the calculation of overlap integrals (molecules) or lattice sums of overlap integrals (crystals). In the MR method kinetic energy integrals or their lattice sums are also included. These integrals in fact can be expressed through overlap integrals. All other contributions to the one-electron Hamiltonian matrix elements are based on empirical parameters. 

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. 

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 

Equations (6.28) and (6.29) are used in CNDO/1 and CNDO/2 methods, respectively. 

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A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

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 ... 

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. 

Quantum-chemical calculations on conjugated hydrocarbons support the spectroscopic estimate, (3 Rq) = -2.40eV, and all-electron descriptions are appealing as soon as they become feasible. There are too many levels of theory to enumerate here, but quantitative ones are not yet applicable to conjugated polymers. Moreover, we are interested in excited states, which remain challenging even in molecules. The rationale for ct-tt separability, for the Coulomb potential V(R), and for the zero differential overlap (ZDO) approximation were discussed [1] in connection with the PPP model. Hubbard [33] considered the same issues for d electrons in transition metals. Quantum cell models [12,13,34] for frontier orbitals of any kind implicitly invoke ZDO to obtain two-center interactions. In many cases, the relevant transfer integrals t, Hubbard repulsion U, and intersite interactions V(R) are small and hence difficult to evaluate in... 

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