Atomic-Orbital-Basis Algorithms

The last summation may be interpreted as the back-transformation of the two virtual indices on the T2 amplitude into the AO basis, that is. 

If this set of half-AO amplitudes is computed and stored (using two standard C( T ) steps ), the amplitudes may be subsequently contracted with the AO-basis integrals to give 

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Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

Representation of each molecular orbital as a linear combination of atomic orbitals (atomic basis sets). Atomic basis sets are usually represented as Slater type orbitals or as combinations of Gaussian functions. The latter is very popular, due to a very fast algorithm for the computation of bielectronic integrals. 

Another Unear-scaUng MP2 algorithm was proposed in [129] that is based on the atomic-orbitals Laplace-transform (LT) MP2 method [137]. In this method, the energy denominators (5.32) in (5.30) are ehminated by Laplace-transformation, which paves the way to express the MP2 energy directly in the basis. The price to pay is the additional Laplace integration, which is carried out by quadrature over a few (8-10) points. For each of the quadrature points an integral transformation has to... 

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