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(May 16, 2021)

While conventional coupled-cluster and equation-of-motion methods allow one to
tackle electronic structure ranging from well-behaved closed shell molecules to
various open-shell and electronically excited species,
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Annu. Rev. Phys. Chem.

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meta-stable electronic states, so-called resonances, present a difficult case
for theory. By using complex scaling and complex absorbing potential
techniques, we extended these powerful methods to describe auto-ionizing
states, such as transient anions, highly excited electronic states, and
core-ionized species.
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CC and EOM-CC calculations can also be carried out using complex basis functions (CBFs),
as described in Sections 4.9.5 and 8.7. In addition,
users can employ stabilization techniques using charged sphere and scaled
atomic charges options.
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J. Chem. Phys.

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These methods are only available within
CCMAN2. The complex CC/EOM code is engaged by COMPLEX_CCMAN;
the specific parameters should be specified in the *$complex_ccman* section.

COMPLEX_CCMAN

Requests complex-scaled or CAP-augmented CC/EOM calculations.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Engage complex CC/EOM code.

RECOMMENDATION:

Not available in CCMAN. Need to specify CAP strength or complex-scaling parameter
in *$complex_ccman* section.

The *$complex_ccman* section is used to specify the details of the
complex-scaled/CAP calculations, as illustrated below. If user specifies
CS_THETA, complex scaling calculation is performed.

$complex_ccman CS_THETA 10 Complex-scaling parameter theta=0.01, r->r exp(-i*theta) CS_ALPHA 10 Real part of the scaling parameter alpha=0.01, ! r->alpha r exp(-itheta) $end

Alternatively, for CAP calculations, the CAP parameters need to be specified.

$complex_ccman CAP_ETA 1000 CAP strength in 10-5 a.u. (0.01) CAP_X 2760 CAP onset along X in 10^-3 bohr (2.76 bohr) CAP_Y 2760 CAP onset along Y in 10^-3 bohr (2.76 bohr) CAP_Z 4880 CAP onset along Z in 10^-3 bohr (4.88 bohr) CAP_TYPE 1 Use cuboid cap (CAP_TYPE=0/2 will use spherical/Voronoi CAP) $end

One can also add real absorbing potential by using CAP_RE_ETA; it follows the same format as CAP_ETA. For example, this setup would add purely real absorbing potential with $\eta $=0.01:

$complex_ccman CAP_ETA 0000 CAP strength in 10-5 a.u. (0.00) CAP_RE_ETA 1000 real CAP strength in 10-5 a.u. (0.01) CAP_X 2760 CAP onset along X in 10^-3 bohr (2.76 bohr) CAP_Y 2760 CAP onset along Y in 10^-3 bohr (2.76 bohr) CAP_Z 4880 CAP onset along Z in 10^-3 bohr (4.88 bohr) CAP_TYPE 1 Use cuboid cap (CAP_TYPE=0/2 will use spherical/Voronoi CAP) $end

The CAP_TYPE field specifies the type of the CAP. The current
options are: spherical CAP (CAP_TYPE = 0), cuboid CAP
(CAP_TYPE = 1), and smooth Voronoi
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J. Chem. Theory Comput.

(2015),
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pp. 4627.
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CAP
(CAP_TYPE = 2). In the calculations with a Voronoi CAP, the onset is
specified by the CAP_X variable.

CS_THETA is specified in radian$\times $ 10${}^{-3}$.
CS_ALPHA, CAP_X/Y/Z are specified in a.u.$\times $
10${}^{-3}$, *i.e.*, CS_THETA = 10 means $\theta =0.01$; CAP_ETA
is specified in $\times {10}^{-5}$ ${E}_{h}$. The CAP is calculated by numerical
integration and the default grid is (${N}_{r}=99,{N}_{\mathrm{\Omega}}=590$).
For testing the accuracy of
numerical integration, the numerical overlap matrix is calculated and compared
to the analytical one. If the performance of the default grid is poor, the grid
type can be changed using the keyword XC_GRID (see
Section 5.5 for further details). When CAP calculations are
performed, CC_EOM_PROP = 1 by default; this is necessary for
calculating first-order perturbative correction.

EOM-CCSD with complex basis functions CBFs (see Section 4.9.5) can be enabled by setting COMPLEX_CCMAN = TRUE and enabling complex basis functions with COMPLEX_EXPONENTS = TRUE. As with mean-field calculations the complex basis must be specified as in described in Section 8.7.

Advanced users may find the following options useful. Several ways of
conducing complex calculations are possible, *i.e.*, complex scaling/CAPs can be
either engaged at all levels (HF, CCSD, EOM), or not. When applied at post
Hartree-Fock level, CAP can either be added to all blocks of the Fock matrix or
restricted to the virtual-virtual block only. The latter approach, known as
projected CAP,
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J. Chem. Phys.

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improves the stability of the calculation
results with respect to CAP onset by reducing a CAP-induced perturbation on the
target states through the occupied orbital space.

This type of CAP projection is currently implemented only for EE/EA calculations and is
invoked by setting PROJ_CAP key in the *$complex_ccman* section as
follows. PROJ_CAP = 1 deploys CAP/EOM-CCSD with projected CAP added
at the CCSD and EOM steps. PROJ_CAP = 2 deploys CAP/EOM-CCSD/MP2/MP2T
with projected CAP added at the EOM step. The latter implies that
$T$-amplitudes (Sec. 7.10.2) are obtained from a real-valued
calculation (for zero CAP strength) and can be reused to generate complex
eigenvalue trajectories by specifying ETA_STEP and NSTEPS
parameters in *$complex_ccman*.

PROJ_CAP = 3 deploys another form of CAP projection
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Int. J. Quantum Chem.

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in which the CAP Hamiltonian
is projected onto the subspace spanned by a set of pre-computed EOM eigenvectors. By default,
the excited state eigenvectors are obtained from a single real-valued calculation, and the CAP matrix
represented in the state basis is printed in the output for each irreducible representation.
This functionality is available for all EOM-CC models for which transition properties
between EOM target states are available. To generate eigenvalue trajectories, CAP_ETA
should be set to a non-zero CAP strength, and subsequent points are specified using
the ETA_STEP and NSTEPS parameters in *$complex_ccman*. Trajectories
are written to a separate output file for each irreducible representation. Additionally, first-order
perturbative corrections can be obtained by setting PROJ_PROP = 1. Note that
when PROJ_PROP = 1, the initial set of real eigenvectors are obtained using the
complex valued code at zero CAP strength. As such, first-order perturbative corrections
are only available for complex EOM-CC models.

By default, if COMPLEX_CCMAN is specified, the EOM calculations are conducted using complex code. Other parameters are set up as follows:

$complex_ccman CS_HF = true CS_CCSD = true $end

Alternatively, the user can disable complex HF. These options are experimental and should only be used by advanced users. For CAP-EOM-CC, only CS_HF = TRUE and CS_CCSD = TRUE is implemented.

Non-iterative triples corrections are available for all complex scaled and CAP-augmented CC/EOM-CC models and requested in analogy to regular CC/EOM-CC (see Section 7.10.25 for details).

Molecular properties and transition moments are requested for complex scaled or CAP-augmented CC/EOM-CC calculations in analogy to regular CC/EOM-CC (see Section 7.10.20 for details). Natural orbitals and natural transition orbitals can be computed and the exciton wave-functions can be analyzed, similarly to real-valued EOM-CCSD (same keywords are used to invoke the analysis). Analytic gradients are available for complex CC/EOM-CC only for cuboid CAPs (CAP_TYPE = 1) introduced at the HF level (CS_HF = TRUE), as described in Ref. 85. The frozen core approximation is disabled for CAP-CC/EOM-CC gradient calculations. Geometry optimization can be requested in the same way as in regular CC/EOM-CC (see Section 7.10.20 for details).