THC-CCSD

The tensor hypercontraction method decomposes the two-electron integrals as:

\[ (ij|kl) = \sum_{PQ} X_i^P X_j^P Z^{PQ} X_k^Q X_l^Q \]

This decomposition is typically obtained by introducing a molecular grid and defining the X-factors as

\[ X_i^P = \chi_i(r_P) \]

where \(\chi\) are the atomic basis functions and \(p\) indexes the gridpoints. Then, the Z-factor can be obtained from a least-squares fitting procedure. The construction of Z formally scales as \(O(N^5)\) where \(N\) is the number of atomic basis functions. However, one can introduce an auxiliary basis set (as in density fitting) and reduce the computational complexity of forming the Z-factors to \(O(N^4)\):

To reduce the computational complexity of the CCSD method from \(O(N^6)\) to \(O(N^4)\), it is necessary to introduce a THC facorization of the doubles amplitudes as well:

\[ t_{ij}^{ab} = \sum_{WV} y_{i}^{W} y_{a}^{W} T^{WV} y_{j}^{V} y_{b}^{V} \]

While the scaling of THC-CCSD is reduced to quartic, there are complications in attempting to define the THC-CCSD Lagrangian and therefore its extension to the calculation of properties or excited states is difficult. To address this issue, the Rank-Reduced (RR) formulation of CCSD was introduced. In RR-CCSD, we pursue a factorization of the doubles amplitudes of the form:

\[ t_{ij}^{ab} = \sum_{RS} U_{ia}^{R} T^{RS} U_{jb}^{S}\]

Where \(U_{ia}^{R}\) is a projector onto the low-rank subspace in which the doubles amplitudes are solved. In practice, the projector is formed by a truncated eigendecomposition of an approximate set of doubles: the first-order MP2 doubles or second-order MP3 doubles amplitudes. The MP2 amplitudes can be obtained either through the canonical perturbation theory expression, with an approximate Laplace transform for the energy denominator, or through a CD-like approach. The full amplitude option is generally only useful for debugging purposes, and the Laplace transform or CD algorithm should be used for production-level calculations.

In the THC-RR-CCSD method, which combines the THC factorization with the RR formalism, we have obtain a \(O(N^4)\) scaling CCSD by performing a CP factorization of the projectors:

\[ U_{ia}^{R} = \sum_{W} y_{i}^{W} y_{a}^{W} \tau^{WR}\]

And inserting this expression into the RR-CCSD working equations.

To run a THC-CCSD calculation in TeraChem, one needs to include

ccbox yes
ccbox_thcccsd yes

in the input file.

Summary of relevant keywords

Keyword	Type	Default	Description
ccbox_thcccsd_maxiter	integer	50	Maximum number of iterations in the CCSD equations
ccbox_thcccsd_diisvecs	integer	5	Maximum number of DIIS vectors when solving the CCSD equations
ccbox_thcccsd_r_convthre	float	1.0e-8	Convergence threshold for amplitudes
ccbox_thcccsd_e_convthre	float	1.0e-8	Convergence threshold for CCSD energy
ccbox_thcccsd_n4	boolean	no	Force \(O(N^4)\) algorithm? Usually, the \(O(N^5)\) algorithm is faster because of a smaller prefactor
ccbox_thcccsd_projector	string	mp2_lt	Which projector is used for the amplitudes?

Need a lot about projectors Choices are mp2_full, mp3_full, mp2_cd, and mp2_lt

References

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1. B. S. Fales, E. R. Curtis, K. G. Johnson, D. Lahana, S. Seritan, Y. Wang, H. Weir, T. J. Martinez and E. G. Hohenstein, Performance of Coupled-Cluster Singles and Doubles on Modern Stream Processing Architectures, J. Chem. Theory Comput. 16 4021 (2020). ↩