Zero Differential Overlap Methods

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 

In 1952 Dewar developed Perturbational Molecular Orbital (PMO) theory, a 7t-electron method calibrated directly on the energies of model organic compounds. The accuracy of this simple method is remarkable for 20 conjugated hydrocarbons the average error in the heat of atomization was 6.5 kcal/mol, and, if the worst case, biphenylene, were left out, the average error dropped to 3.33 kcal/mol.  

In 1965, Pople and his co-workers introduced a series of ZDO approximations that generalized the 7t-electron PPP scheme to all valence electrons. 

The important observation here was that approximations such as ZDO implied certain consistency requirements. Foremost among these was the requirement for rotational invariance that is, the same answer should result regardless of the molecule s orientation. As obvious a requirement as this may seem, the way in which to achieve this invariance is, perhaps, not so obvious within a semiempiri-cal scheme. 

Without yet referring to any particular parameterization scheme, we develop the hierarchy of integral approximations suggested by Pople and coworkers and the effect such approximations have on the formation of the Fock matrix. 

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The range of zero differential overlap methods such as CNDO, INDO, and NDDO originally suggested by Pople et al.7z continues to have wide use in spite of the fact that they are less accurate than the methods just discussed. Of these NDDO might be expected to be the most accurate but there has been surprisingly little use of this approach. NDDO neglects integrals by the use of... 

The second group of semiempirical MO LCAO methods is constituted by zero-differential overlap methods (9). Two subgroups can be specified here which somewhat conventionally may be called physical and chemical. The former involves CNDO/2, INDO, and some of their modifications, for example, CNDO/S. These methods are directed to the calculations of the electron characteristics charge distributions, dipole moments, polarizabili-... 

Ihc complete neglect of differential overlap (CNDO) approach of Pople, Santry and Segal u as the first method to implement the zero-differential overlap approximation in a practical fashion [Pople et al. 1965]. To overcome the problems of rotational invariance, the two-clectron integrals (/c/c AA), where fi and A are on different atoms A and B, were set equal to. 1 parameter which depends only on the nature of the atoms A and B and the ii ilcniuclear distance, and not on the type of orbital. The parameter can be considered 1.0 be the average electrostatic repulsion between an electron on atom A and an electron on atom B. When both atomic orbitals are on the same atom the parameter is written , A tiiid represents the average electron-electron repulsion between two electrons on an aiom A. 

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... 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

In the unrestricted Hartree-Fock method, a single-determinant wave function is used with different molecular orbitals for a and jS spins, and the eigenvalue problem is solved with separate F and F matrices. With the zero differential overlap approximation, the F matrix elements (25) become... 

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... 

Sustmann and Binsch132 described a method which started from the same point, but invoked the zero-differential-overlap approximation on the other hand, it was not confined to jr-electrons and the perturbation energy was refined iteratively. Using a MINDO parameterization they then applied the method to Diels-Alder reactions, and were able to account for effects which cannot be explained in terms of simple jr-electron theory, such as the preference for endo addition of cyclopropene to cyclo-pentadiene. 

These methods appear rather simple, yet they were the starting point of a long evolution. Gilles Klopman, whose research interests at Case-Western Reserve University later turned to modeling bioactive molecules, was the first to use Sandorfy s methods. Kenichi Fukui made extensive use of them in his well-known work on the structures and reactions of saturated hydrocarbons and their derivatives. Fukui added his frontier orbital considerations. Around 1959 the milieu of developments in quantum chemistry contributed to inspire William N. Lipscomb to conceive the extended Hiickel method, which was subsequently implemented by Lawrence L. Lohr and Roald Hoffmann.83 Soon thereafter, John Pople and his coworkers introduced self-consistent field methods based on the zero-differential overlap approximation.815... 

Semiempirical methods differ amongst themselves in, amongst other ways, the criteria for setting dS = 0, i.e. for applying zero differential overlap, ZDO. 

In the PPP method, due to Pariser63, Parr64 and Pople65, the assumption of zero differential overlap (ZDO) consists of setting the AO product to zero unless they involve the same orbital on the same center (atom) ... 

The Pariser-Parr-Poplc (PPP) method is based on three assumptions (a) Whenever the factor, (n), (n) (where i n is any electron index) appears in the integrand the integi al vanishes zero differential overlap approximation) (b) The resonance integrals between nearest neighbours are treated as empirical parameters those between non-neighbours are neglected (c) The one-centre Coulomb integrals ii, it) are taken to be equal to /, — E where I, and E, are the ionization potential and electron affinity, respectively, of the atomic orbital tpi, when the atom is in the appropriate valence state. 

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. 

Now, it is just this term which determines to a large extent the importance of the interaction of the n electrons between the atoms P and Q. The integral fiPq may be identified with the parameter f pg of the semi-empirical theories based on the zero-differential-overlap approximation 35,38). in our opinion, there is no general calculation method leading to values for the fipq s which are in good numerical agreement with the pvq parameters fitted on experimental data (electronic spectra, dipole... 

Pure atomic orbitals Extended Htickel theory 40,50) Iterative extended Hiickel theories80 81 82) Quasi SCF diagonal element method85) Kinetic-energy Hiickel theory 58) Zero-differential overlap approximation 61,69) Standard SCF method73 74 75 76,77,78) SCF group function method 83) Random-phase... 

Nowadays, the success of the methods proposed by Hoffmann 50> and by Pople and Segal 51> among the chemists tends to promote the use of pure atomic orbital bases for all-valence treatments. The first method is a straightforward application of the Wolfsberg-Helmholz treatment of complexes to organic compounds and is called the Extended Hiickel Theory (EHT), because its matrix elements are parametrized in the same way as the Hiickel method with overlap for n electrons. The other method, known under the abbreviation Complete Neglect of Differential Overlap (CNDO), includes electron repulsion terms by extending to a orbitals the successful approximation of zero-differential overlap postulated for n electrons. 

A further step in the way of improvements is to consider all the parameters, the a s as well as the /3 s, as approximate expressions of the SCF effective Hamiltonian. This was done using various zero-differential-overlap approximations 51>69>e). The diagonal elements Fpp of the CNDO method are given by an expression completely equivalent to Eq. (6.14) and the off-diagonal elements are of the form... 

The second most popular all-valence electrons procedure is the CNDO/2 method 3>, generalizing the zero-differential overlap Pariser-Parr hypothesis to an otherwise rationalized SCF scheme. A parallel between EHT, iterative EHT and CNDO/2 has been drawn elsewhere 4 5C... 

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