Zero Differential Overlap 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 ... 

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... 

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 ... 

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... 

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 ... 

The two-electron integrals pq kl] are < p(l)0fc(2) e2/ri2 0,(l)0j(2) > and may involve as many as four orbitals. The models of interest are restricted to one and two-center terms. Two electrons in the same orbital, [pp pp], is 7 in Pariser-Parr-Pople (PPP) theory[4] or U in Hubbard models[5], while pp qq are the two-center integrals kept in PPP. The zero-differential-overlap (ZDO) approximation[3] can be invoked to rationalize such simplification. In modern applications, however, and especially in the solid state, models are introduced phenomenologically. Particularly successful models are apt to be derived subsequently and their parameters computed separately. 

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 PPP-theory [4,60a,61] electron-electron interaction is introduced via the two-electron integrals yn applying the zero differential overlap (ZDO) approximation. In ZDO approximation all integrals containing products of basis-functions which are not centered at the same site are assumed to vanish. This... 

The integral J can be simplified to the double sum of two-centre Coulombic repulsion integrals -yM (Equation 4.29) by adopting the zero differential overlap (ZDO) approximation, that is, we assume that atomic orbitals located on different atoms do not overlap, 0 for pffv. 

In the evaluation of the effective PPP Fock matrix elements, T, 1 1 , the zero differential overlap (ZDO) approximation is used. 

The Hubbard model can be derived from eqn.(2) by (i) making a zero differential overlap (ZDO) approximation and (ii) by assuming the range of Coulomb interactions to be truncated to just on-site repulsion by virtue of strong shielding of the interactions by the conduction electrons in metals. The ZDO approximation [25] restricts the nonzero electron repulsion integrals to those of the form... 

When the orthonormal set us(C) is now employed for representations of the hamiltonian and other relevant operators, it is in approximate theories taken as a justification to neglect in a first approximation integrals involving the product density of spin orbitals associated with different atomic centers. This is the so-called Zero Differential Overlap (ZDO) approximation. The charge density operator, for instance, would then become... 

If a zero differential overlap (ZDO) approximation (47) is invoked, as in semiempirical CNDO theories, the gross atomic population reduces to the electron density Paa (24). 

In several semi-empirical methods of quantum chemistry (e.g., in the Huckel method), we assume the Zero Differential Overlap (ZDO) approximation i.e., that feX/ iXk) kl and hence the second terms in /t. as well as in /tyig, are equal to zero. and therefore... 

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,... 

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. 

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 zero differential overlap (ZDO) approximation has been made here so that only two center integrals are considered. Note that... 

The complete neglect of differential overlap (CNDO) method [1] rests on the zero differential overlap (ZDO) approximation, which means that all the products of... 

The simplest NDO scheme is the CNDO method which fully utilizes the Zero Differential Overlap (ZDO) approximation for the two-electron integrals. The ZDO condition means that we neglect all two-electron integrals in which the differential overlap of two different basis functions is present ... 

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... 

How are electron-electron repulsion terms handled (n = not included, c = zero differential overlap (ZDO) approximation applied to all integrals, i = ZDO not applied to one-center integrals, d = ZDO not applied to one-center integrals nor to a two-center integral if both orbitals of an electron are on the same nucleus)... 

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