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.^{857, 905, 906} Extension of the FNO
approach to ionized states within EOM-CC formalism was recently introduced and
benchmarked;^{514} see Section 7.8.8.

The FNOs are computed as the eigenstates of the virtual-virtual block of the MP2 density matrix [$\mathcal{O}({N}^{5})$ 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. 905, 906. 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.^{514}

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.^{514} 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.^{514} 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. 514).

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.^{514}

CC_FNO_THRESH

Initialize the FNO truncation and sets the threshold to be used for both
cutoffs (OCCT and POVO)

TYPE:

INTEGER

DEFAULT:

None

OPTIONS:

range
0000-10000
$abcd$
Corresponding to $ab.cd$%

RECOMMENDATION:

None

CC_FNO_USEPOP

Selection of the truncation scheme

TYPE:

INTEGER

DEFAULT:

1
OCCT

OPTIONS:

0
POVO

RECOMMENDATION:

None

$molecule 0 1 O H 1 1.0 H 1 1.0 2 100. $end $rem METHOD = CCSD(T) BASIS = 6-311+G(2df,2pd) CC_FNO_THRESH = 6500 65% of the virtual space CC_FNO_USEPOP = 0 $end