Zero-differential overlap

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Zero-differential overlap is an approximation that is used to ignore certain integrals, usually two-electron repulsion integrals, in Semi-empirical quantum chemistry methods quantum chemistry molecular orbital methods.

If the molecular orbitals $\mathbf{\Phi}_i \$ are expanded in terms of N basis functions, $\mathbf{\chi}_\mu^A \$ as:-

$\mathbf{\Phi}_i \ = \sum_{\mu=1}^N \mathbf{C}_{i\mu} \ \mathbf{\chi}_\mu^A \$

where A is the atom the basis function is centred on, and $\mathbf{C}_{i\mu} \$ are coefficients, the two-electron repulsion integrals are then defined as:-

$<\mu\nu|\lambda\sigma> = \iint \mathbf{\chi}_\mu^A (1) \mathbf{\chi}_\nu^B (1) \frac{1}{r_{12}} \mathbf{\chi}_\lambda^C (2) \mathbf{\chi}_\sigma^D (2) d\tau\,d\tau \$

The zero-differential overlap approximation ignores integrals that contain the product $\mathbf{\chi}_\mu^A (1) \mathbf{\chi}_\nu^B (1)$ where $μ$ is not equal to $ν$ . This leads to:-

$<\mu \ \nu |\lambda \ \sigma> = \delta_{\mu\nu} \delta_{\lambda\sigma} <\mu\mu|\lambda\lambda>$

where $\delta_{\mu\nu} = \begin{cases}0 & \mu \ne \nu \\ 1 & \mu = \nu \ \end{cases}$

The total number of such integrals is reduced to N(N+1)/2 (approximately N²/2) from [N(N+1)/2][N(N+1)/2 + 1]/2 (approximately N⁴/8), all of which are included in ab initio Hartree-Fock and post Hartree-Fock calculations.

Methods such as the Pariser-Parr-Pople method (PPP) and CNDO/2 use the zero-differential overlap approximation completely. Methods based on the intermediate neglect of differential overlap, such as INDO, MINDO, ZINDO and SINDO do not apply it when A = B = C = D, i.e. when all four basis functions are on the same atom. Methods that use the neglect of diatomic differential overlap, such as MNDO, PM3 and AM1, also do not apply it when A = B and C = D, i.e. when the basis functions for the first electron are on the same atom and the basis functions for the second electron are the same atom.

It is possible to partly justify this approximation, but generally it is used because it works reasonably well when the integrals that remain - $< μμ | λλ >$ - are parameterised.