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# 6.10.8 Cholesky Decomposition with CC (CD-CC)

(July 14, 2022)

Two-electron integrals can be decomposed using Cholesky decomposition 317 Epifanovsky E. et al.
J. Chem. Phys.
(2013), 139, pp. 134105.
giving rise to the same representation as in RI and substantially reducing the cost of integral transformation, disk storage requirements, and improving parallel performance:

 $(\mu\nu|\lambda\sigma)\approx\sum_{P=1}^{M}B_{\mu\nu}^{P}B_{\lambda\sigma}^{P},$ (6.50)

The rank of Cholesky decomposition, $M$, is typically 3-10 times larger than the number of basis functions $N$ (Ref.  43 Aquilante F., Pedersen T. B., Lindh R.
Theor. Chem. Acc.
(2009), 124, pp. 1.
); it depends on the decomposition threshold $\delta$ and is considerably smaller than the full rank of the matrix, $N(N+1)/2$ (Refs.  43 Aquilante F., Pedersen T. B., Lindh R.
Theor. Chem. Acc.
(2009), 124, pp. 1.
, 87 Beebe N. H. F., Linderberg J.
Int. J. Quantum Chem.
(1977), 12, pp. 683.
, 1292 Wilson S.
Comput. Phys. Commun.
(1990), 58, pp. 71–81.
). Cholesky decomposition removes linear dependencies in product densities $(\mu\nu|$, 43 Aquilante F., Pedersen T. B., Lindh R.
Theor. Chem. Acc.
(2009), 124, pp. 1.
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 6.13 and 7.10.12) when appropriate.