Zero-differential overlap approximation

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. 

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... 

The zero differential overlap approximation can be applied in the localized representation. This was demonstrated by calculating for C H, CioTfio and C14//14, respectively the total energy corrections and the pair correlation energies through second and third order in different approximations. When the strongly local contributions were only... 

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... 

The bottleneck in all self-consistent field calculations is the difficulty of calculating the integrals [Eq. (16)] over the atomic orbitals. Thus, reducing their number has been an imperative requirement and has led to the fundamental zero-differential overlap approximation (ZDO) which assumes... 

Note that the wave functions in the above R and R states share a common determinant. If we now crudely normalize the wave functions using 2 1/2 as a normalization factor (this means we are using a zero differential overlap approximation), using the same approximation the overlap will be the following ... 

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. 

Fischer-Hjalmars, I., Deduction of the zero differential overlap approximation from an orthogonal atomic orbital basis, J. Chem. Phys., 42, 1962-1972, 1965. 

As a consequence of the zero-differential overlap approximation, the normalization condition becomes... 

The 71-electronic densities obtained from an MO calculation which includes the zero differential overlap approximation (Equations 1 and 2) indicate the general disposition of charge in the molecule rather than the exact populations of the AOs. These electronic densities represent the populations of modified AOs which are not well localized. Thus, the relative values of the AO densities, rather than their precise magnitudes, reflect the general electronic behavior of the molecule. 

The transition dipoles and dipole differences between states a) and g) can be expressed in terms of matrix elements between the MO and applying the ZDO (zero differential overlap) approximation as in (68) and (69),... 

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. 

In their treatment of N3 and HN3 Yonezawa and co-workers abandoned the zero differential overlap approximation and considered all valence electrons. The Mulliken approximation... 

In the calculations of Pock matrix elements, the zero differential overlap approximation is adopted. Then we get for a closed shell structure ... 

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... 

Many interesting problems in physical organic chemistry have been clarified by numerical calculations based on next-neighbor interaction and zero-differential-overlap approximations, especially in the field of... 

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... 

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... 

In order to preserve the invariance of charge distributions under rotation of the local coordinate axes of each atom, the integrals K% eSPq and (pp qq) are assumed to be independent of the azimuthal quantum number of atomic orbitals, i.e. the same value is used for any 2 s and 2 p orbitals. Finally, it should be noted that in the case of a electrons the zero-differential-overlap approximation cannot be justified as completely as for n electrons by arguing about orthogonalized Lowdin orbitals, because the expression of the S la matrix cannot be limited to first-order terms 70,71,72). 

Charge densities have been calculated for thiazolylium, 1,3-dithiolylium, and imidazolium ion (by TT-electron SCF-MO) using the zero differential overlap approximation (Table 1) <7iJHC55i> it was found that the highest charge densities are at the heteroatoms of the ions. The difference in energy for an ion and its corresponding carbene predicts the rates of ionization of the H—C(2) bond and was found to be imidazolium < thiazolylium < 1,3-dithiolylium. 

The second part of Pariser and Parr s theory is concerned with the matrix elements between the terms 1Xo, These depend ultimately upon the matrix elements of unity, of the core Hamiltonian, and of the electron repulsion in the system of atomic orbitals. Pariser and Parr, like Pople, introduced the zero differential overlap approximation which abolishes overlap between different atomic orbitals and includes all electron repulsion integrals except those of the type... 

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