7.11.8 EOM-CC Calculations of Core-Level States

RIXS is a coherent two-photon process involving core-level states. ⁸⁸⁰ Nanda K. D. et al.
Phys. Chem. Chem. Phys.
(2019), 22, pp. 2629. Link The calculations of RIXS cross-sections require solving response equations, in the same fashion as in calculations of 2PA cross-sections (see Section 7.11.21.6). Currently, only calculations of RIXS cross-sections between the CCSD reference and the EOM-CCSD target states are possible. Because of the resonant nature of RIXS, damped response theory is used to handle singularities in the resolvent. In addition, CVS is enforced on the response vectors to eliminate their coupling with the ionization continuum. To set up RIXS calculation, one needs to set METHOD = EOM-CCSD and to specify the number of excited states to be included by using EE_STATES (note that only 2PA bright states need to be included) and to activate CVS by using CVS_EE_STATES asking for zero states. RIXS calculations is deployed by activating CC_EOM_RIXS. The option for performing wave-function analysis (e.g., NTO analysis) of the complex RIXS one-particle transition density matrices is also available. Conceptual details can be found in Ref.  878 Nanda K. D., Krylov A. I.
J. Chem. Phys.
(2020), 152, pp. 244118. Link . Finally, one can request to compute REXS (cross-section for the “elastic” peak) by CC_REF_PROP = 1 (by default, REXS is not calculated) This feature is only available within fc-CVS-EOM-EE-CCSD.

Note:  Since the RIXS response solutions within EOM-EE-CCSD often do not converge, fc-CVS-EOM-EE-CCSD RIXS calculations with CC_EOM_RIXS = 1 are recommended for smooth convergence. All other options are experimental.

To specify pumping frequencies and damping factor $ϵ$, use $rixs section:

$rixs
damped_epsilon 0.005       Damping factor in hartree.
omega_1 2306503 500 10 0   Pumping frequencies: starting w (cm-1), increment (cm-1),
                           number of frequency points, algo (always 0 for now)
omega_2 2200000 600 20 0   Emitted frequencies:  starting w (cm-1), increment (cm-1),
                           number of frequency points, algo (always 0 for now)
$end

Damping factor (DAMPED_EPSILON) is specified in atomic units (0.005 is a good choice). OMEGA_1 specifies the starting pumping frequency (in cm${}^{-1}$), the increment in frequency scan (in cm${}^{-1}$), number of frequency points, and the algorithm for response equation (use zero until further notice). OMEGA_2 should be specified only for generating 2D RIXS scans. Example 7.11.8.2 illustrates the setup of RIXS calculations.

By default, the damped RIXS response wave functions are computed using the (new) iterative Davidson-like subspace procedure (called Dalton solver) ²¹⁸ Christiansen O. et al.
J. Chem. Phys.
(1998), 108, pp. 2801. Link ^{, 605} Kauczor J. et al.
J. Chem. Phys.
(2013), 139, pp. 211102. Link , which is controlled by the DALTON_XCONV, DALTON_PRECOND_START, DALTON_MAXITER, and DALTON_MAXSPACE keywords.

Damped RIXS response wave functions can also be computed using the (old) iterative DIIS procedure by setting DAMPED_DALTON_SOLVER = FALSE. The DIIS solver is controlled by the CC_MAX_ITER, CC_DIIS_START, CC_DIIS_SIZE, CC_EOM_2PA_ECONV, and CC_EOM_2PA_XCONV.

Note:  For better convergence of the response equations with DIIS solver, increase CC_DIIS_SIZE (25 is recommended) and consider using a larger damping factor.

Note:  Dalton solver uses square of the norm of the residual in estimating convergence (as in the Davidson solver), whereas the DIIS solver uses the norm of the difference of the response vector between iterations.

In example 7.11.8.2, the $1s$ orbital on the oxygen atom is frozen in the CCSD calculation (N_FROZEN_CORE = FC). In the EOM calculation, the CVS approximation is invoked (CVS_EE_SINGLETS), so that the core-excitation energies are obtained as the lowest excitations. The calculation of the oscillator strengths and rotatory strengths is activated by selecting CC_TRANS_PROP = 1 and the libwfa analysis is invoked by STATE_ANALYSIS = TRUE (see Section 10.2.9). Note that rotatory strengths will be zero for non-chiral systems.

Example 7.11.8.2 illustrates CVS-EOM-EE-CCSD calculations in a two-edge molecule (CO). In the present implementation, the calculation should be done separately for each edge. The first job computes carbon-edge states. Since the carbon $1s$ orbital is the highest in energy (among the core $1s$ orbitals of the molecule), the input for the C-edge is similar to example 7.11.8.2. Both the oxygen’s and the carbon’s $1s$ orbitals are frozen in the reference CCSD calculation. In the EOM part, the carbon core-excited states are automatically selected. In this case, using default frozen core settings (N_FROZEN_CORE = FC) is equivalent to specifying N_FROZEN_CORE = 2. In the second input, the oxygen edge is computed. As the core-orbitals of oxygen lie deeper, the frozen core and CVS selection specifically targets the oxygen edge by using a smaller core. The 1$s$ orbital of the oxygen atom is selected by N_FROZEN_CORE = 1. If the molecule has other edges, the deepest lying core orbitals, up to and including those of the edge of interest, should be selected by an appropriate value of N_FROZEN_CORE.

Example 7.11.8.2 illustrates calculation of the XAS spectrum of ammonia using additional set of Rydberg functions placed on the nitrogen; the exponents of the Rydberg functions were generated using Kaufmann’s recipe ⁶⁰⁶ Kaufmann K., Baumeister W., Jungen M.
J. Phys. B: At. Mol. Opt. Phys.
(1989), 22, pp. 2223. Link .

Note:  When ghost atoms are used, the symmetry is turned off. It can be forced on by using FORCE_SYMMETRY_ON keyword.

Note:  Alternatively, this calculation can be set up by adding additional diffuse functions to all (or selected) atoms using even-tempered exponents.

Example 7.11.8.2 illustrates calculation of the XES spectrum of benzene. Examples 7.11.8.2 and 7.11.8.2 illustrate calculations of Dyson orbitals between core-excited and core-ionized states and between core-excited and valence-ionized states.

Example 7.11.8.2 illustrates the setup of RIXS calculations.

Calculations of L-edge spectra using state-interaction approach ¹²⁶³ Vidal M. L. et al.
J. Phys. Chem. Lett.
(2020), 11, pp. 8314. Link entails a two-step procedure. First, Q-Chem computes necessary CVS-EOM states and SOCs. In the second step, the spin-orbit perturbed spectrum is computed by a post-processing script. Scripts, detailed instructions, and examples can be found here .