Two-electron integrals can be decomposed using Cholesky decompositionKrylov:2013b giving rise to the same representation as in RI and substantially reducing the cost of integral transformation, disk storage requirements, and improving parallel performance:
The rank of Cholesky decomposition, , is typically 3-10 times larger than the number of basis functions (Ref. Aquilante:2009); it depends on the decomposition threshold and is considerably smaller than the full rank of the matrix, (Refs. Aquilante:2009, Beebe:1977, Wilson:1990). Cholesky decomposition removes linear dependencies in product densities ,Aquilante:2009 allowing one to obtain compact approximation to the original matrix with accuracy, in principle, up to machine precision.
Decomposition threshold is the only parameter that controls accuracy and the rank of the decomposition. Cholesky decomposition is invoked by specifying CHOLESKY_TOL that defines the accuracy with which decomposition should be performed. For most calculations tolerance of gives a good balance between accuracy and compactness of the rank. Tolerance of can be used for exploratory calculations and for high-accuracy calculations. Similar to RI, Cholesky-decomposed integrals can be transformed back, into the canonical MO form, using CC_DIRECT_RI keyword.