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# Theoretical Details To Partial AO Algorithms

The basic idea of partial AO algorithms is easily explained for the following term:

{$Z_{ij}^{ab}=\sum_{e,f}\langle ab|ef\rangle t_{ij}^{ef}$}

which is calculated as

a) partial transformation of the amplitudes from MO to AO representation

{$t_{ij}^{\mu \nu}=\sum_{e,f} c_{\mu e} c_{\nu f} t_{ij}^{ef}$}

b) contraction with AO integrals

{$Z_{ij}^{\mu \nu}=\sum_{\sigma \rho}\langle \mu \nu | \sigma \rho \rangle t_{ij}^{\sigma \rho}$}

c) back transformation to the MO representation

{$Z_{ij}^{ab}=\sum_{\mu,\nu}c_{\mu a}c_{\nu b}Z_{ij}^{\mu \nu}$}

In this way, a full transformation of the AO two-electron integrals is avoided and the disk space requirements are drastically reduced.