6.13.1 Overview

Large computational savings are possible if the virtual space is truncated using the frozen natural orbital (FNO) approach. For example, using a fraction $f$ of the full virtual space results in a $1/{(1-f)}^4$-fold speed up for each CCSD iteration (CCSD scales with the forth power of the virtual space size). FNO-based truncation for ground-states CC methods was introduced by Bartlett and coworkers. ¹¹²⁵ Sosa C. et al.
Chem. Phys. Lett.
(1989), 159, pp. 148. Link ^{, 1182} Taube A. G., Bartlett R. J.
Collect. Czech. Chem. Commun.
(2005), 70, pp. 837. Link ^{, 1183} Taube A. G., Bartlett R. J.
J. Chem. Phys.
(2008), 128, pp. 164101. Link Extension of the FNO approach to ionized states within EOM-CC formalism was recently introduced and benchmarked; ⁶⁵⁶ Landau A. et al.
J. Chem. Phys.
(2010), 132, pp. 014109. Link see Section 7.10.12.

The FNOs are computed as the eigenstates of the virtual-virtual block of the MP2 density matrix [$𝒪(N^5)$ scaling], and the eigenvalues are the occupation numbers associated with the respective FNOs. By using a user-specified threshold, the FNOs with the smallest occupations are frozen in CC calculations. This could be done in CCSD, CCSD(T), CCSD(2), CCSD(dT), CCSD(fT) as well as CCD, OD, QCCD, VQCCD, and all possible triples corrections for these wave functions.

The truncation can be performed using two different schemes. The first approach is to simply specify the total number of virtual orbitals to retain, e.g., as the percentage of total virtual orbitals, as was done in Refs.  1182 Taube A. G., Bartlett R. J.
Collect. Czech. Chem. Commun.
(2005), 70, pp. 837. Link , 1183 Taube A. G., Bartlett R. J.
J. Chem. Phys.
(2008), 128, pp. 164101. Link . The second approach is to specify the percentage of total natural occupation (in the virtual space) that needs to be recovered in the truncated space. These two criteria are referred to as the POVO (percentage of virtual orbitals) and OCCT (occupation threshold) cutoffs, respectively. ⁶⁵⁶ Landau A. et al.
J. Chem. Phys.
(2010), 132, pp. 014109. Link

Since the OCCT criterion is based on the correlation in a specific molecule, it yields more consistent results than POVO. For ionization energy calculations employing 99–99.5% natural occupation threshold should yields errors (relative to the full virtual space values) below 1 kcal/mol. ⁶⁵⁶ Landau A. et al.
J. Chem. Phys.
(2010), 132, pp. 014109. Link The errors decrease linearly as a function of the total natural occupation recovered, which can be exploited by extrapolating truncated calculations to the full virtual space values. This extrapolation scheme is called the extrapolated FNO (XFNO) procedure. ⁶⁵⁶ Landau A. et al.
J. Chem. Phys.
(2010), 132, pp. 014109. Link The linear behavior is exhibited by the total energies of the ground and the ionized states as a function of OCCT. Therefore, the XFNO scheme can be employed even when the two states are not calculated on the same level, e.g., in adiabatic energy differences and EOM-IP-CC(2,3) calculations (more on this in Ref.  656 Landau A. et al.
J. Chem. Phys.
(2010), 132, pp. 014109. Link ).

The FNO truncation often causes slower convergence of the CCSD and EOM procedures. Nevertheless, despite larger number of iterations, the FNO-based truncation of orbital space reduces computational cost considerably, with a negligible decline in accuracy. ⁶⁵⁶ Landau A. et al.
J. Chem. Phys.
(2010), 132, pp. 014109. Link

Because of the limitation of the implementation, point-group symmetry cannot be used with FNO/OSFNO and will be disabled. Please, adjust your input consistently with CC_SYMMETRY = false.

For open-shell species an open-shell FNO (OSFNO) has been developed for the use in EOM-SF-CC calculations. The benchmarks show negligible errors in singlet–triplet gaps for a variety of molecules. For more details, see Section 7.10.12 and the Ref.  ⁹⁶³ Pokhilko P., Izmodenov D., Krylov A. I.
J. Chem. Phys.
(2020), 152, pp. 034105. Link .