7.10.9 EOM-CC Calculations of Metastable States: Super-Excited Electronic States, Temporary Anions, and More

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,⁶⁰³ 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.^{129, 514, 513} 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.⁶¹² 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.

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 Voronoi¹⁰⁵⁶ 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-CC 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,⁹⁸⁹ 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. 7.10.3) 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 ¹⁰⁵⁷ 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. The complete set of one-particle state and transition OPDMs between each pair of states (which is all that is required for CAP projection) will be exported to checkpoint file for further analysis when GUI = 2 and PROJ_CAP = 3.

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 7.10.26 for details).

To take account of the impact of the environment on electronic resonances CAP-EOM-CCSD and CBF-EOM-CCSD can be combined with projection-based embedding,⁸⁵⁹ as described in Sections 7.10.3.1 and 11.6.

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 7.10.21 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. 87. 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 7.10.21 for details).