6.8.6 Cholesky decomposition with CC (CD-CC)

Two-electron integrals can be decomposed using Cholesky decomposition²⁵⁰ 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, $M$, is typically 3-10 times larger than the number of basis functions $N$ (Ref. 38); it depends on the decomposition threshold $\delta$ and is considerably smaller than the full rank of the matrix, $N(N+1)/2$ (Refs. 38, 75, 1005). Cholesky decomposition removes linear dependencies in product densities $(\mu \nu |$,³⁸ allowing one to obtain compact approximation to the original matrix with accuracy, in principle, up to machine precision.

Decomposition threshold $\delta$ 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 $\delta ={10}^{-3}$ gives a good balance between accuracy and compactness of the rank. Tolerance of $\delta ={10}^{-2}$ can be used for exploratory calculations and $\delta ={10}^{-4}$ for high-accuracy calculations. Similar to RI, Cholesky-decomposed integrals can be transformed back, into the canonical MO form, using CC_DIRECT_RI keyword.

Note:  Cholesky decomposition is available for all CCMAN2 methods, including energy, analytic gradients, and properties calculations. For maximum computational efficiency, combine with FNO (see Sections 6.11 and 7.8.9) when appropriate.