6.8 Coupled-Cluster Methods

6.8.6 Cholesky Decomposition with CC (CD-CC)

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:

(μν|λσ)P=1MBμνPBλσP, (6.39)

The rank of Cholesky decomposition, M, is typically 3-10 times larger than the number of basis functions N (Ref. Aquilante:2009); it depends on the decomposition threshold δ and is considerably smaller than the full rank of the matrix, N(N+1)/2 (Refs. Aquilante:2009, Beebe:1977, Wilson:1990). Cholesky decomposition removes linear dependencies in product densities (μν|,Aquilante:2009 allowing one to obtain compact approximation to the original matrix with accuracy, in principle, up to machine precision.

Decomposition threshold δ 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 δ=10-3 gives a good balance between accuracy and compactness of the rank. Tolerance of δ=10-2 can be used for exploratory calculations and δ=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 7.9.9) when appropriate.