The core-valence separation (CVS) scheme^{Cederbaum:1980} allows one
to extend standard methods for excited and ionized states to the core-level
states. In this approach, the excitations involving core electrons are
decoupled from the rest of the configurational space. This allows one to reduce
computational costs and decouple the highly excited core states from the
continuum. Currently, CVS is implemented within EOM-EE/IP-CCSD for energies
and transition properties (oscillator strengths, NTOs, Dyson orbitals, exciton
descriptors). CVS-EOM-EE-CCSD can be used to model NEXAFS, while CVS-EOM-IP-CCSD
can be used to model XPS and XES. These methods can also be used to compute transient
absorption spectra, e.g., valence pump/X-ray probe experiments.^{Vidal:2020}
The calculation of non-linear spectra, such as RIXS, is also possible.

In Q-Chem, a slightly different version of CVS-EOM-EE-CCSD than the original
theory by Coriani and Koch^{Coriani:2015} is implemented: the reference
coupled-cluster amplitudes do not include core electrons.^{Vidal:2018}
To distinguish this method from the original,^{Coriani:2015} we refer in what follows to
the Q-Chem implementation as frozen-core-ground-state/core-valence-separated
EOM (FC-CVS-EOM) approach.^{Vidal:2018}

In the FC-CVS-EOM approach the ground-state parameters (amplitudes and
Lagrangian multipliers) are computed within the frozen-core approximation,
whereas the core-excitation energies and transition strengths
are obtained imposing that
at least one index in the EOM excitation (and ionization) operators refer to a
core occupied orbital.
Likewise, CVS is enforced in the calculation of the response
states in RIXS.^{Nanda:2020}

To ensure the best convergence of EOM equations, the calculation is
*edge-specific* with respect to the highest lying edges (or deepest lying
core orbitals): the frozen-core and CVS spaces are selected for each edge such
that the core orbitals we are addressing in the excited state calculations are
explicitly frozen in the ground state calculation and specifically included in
the EOM calculation. Examples 7.9.6.3 and 7.9.6.3 below
illustrate this point.

Although the convergence of FC-CVS-EOM is much more robust that that of regular EOM-CCSD, sometimes calculations would collapse to low-lying artificial states. If this happens, rerun the calculation using CVS_EOM_SHIFT to specify an approximate onset of the edge.

To invoke the CVS approximation, use METHOD = CCSD and CVS_EE_STATES instead of EE_STATES to specify the desired target states (likewise, CVS_EE_SINGLETS and CVS_EE_TRIPLETS can be used in exactly the same way as in regular EOM calculations). For ionized states, use CVS_IP_STATES or CVS_IP_ALPHA/CVS_IP_BETA. Preconverging single amplitudes can be invoked by CVS_EOM_PRECONV_SINGLES. Transition properties and Dyson orbitals can be computed either within CVS manifold or between CVS and valence manifolds (see Section 7.9.25 for definition of Dyson orbitals). CVS-EOM-CCSD is only available with CCMAN2.

Note: Core electrons must be frozen in CVS-EOM calculations. The exact definition of the core depends on the edge, so using default values may be not appropriate.

CVS_EOM_SHIFT

Specifies energy shift in CVS-EOM calculations.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$
corresponds to $n\cdot {10}^{-3}$
hartree shift (i.e., 11000 = 11 hartree); solve for
eigenstates around this value.

RECOMMENDATION:

Improves the stability of the calculations.

CVS_EOM_PRECONV_SINGLES

When not zero, singly excited vectors are converged prior to a full excited
states calculation (CVS states only). Sets the maximum number of iterations for pre-converging
procedure.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
do not pre-converge
1
pre-converge singles

RECOMMENDATION:

Sometimes helps with problematic convergence.

RIXS is a coherent two-photon process involving core-level states.^{Nanda:2020}
The calculations of RIXS cross-sections require solving response equations, in the same
fashion as in calculations of 2PA cross-sections (see section 7.9.18.1).
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. Nanda:2020a.
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.

CC_EOM_RIXS

Whether or not the RIXS scattering moments and cross-sections will be calculated.

TYPE:

INTEGER

DEFAULT:

0
do not compute RIXS cross-sections

OPTIONS:

1
Perform RIXS within fc-CVS-EOM-EE-CCSD using the response wave functions of the CCSD reference state only
2
Perform RIXS within fc-CVS-EOM-EE-CCSD response theory along with the wave-function analysis of RIXS transition density matrices
11
Perform RIXS within the standard EOM-EE-CCSD using the response wave functions of the CCSD reference state only
12
Use $\sigma $-intermediates for RIXS response calculations within the standard EOM-EE-CCSD

RECOMMENDATION:

Use 1 to deploy fc-CVS-EOM-EE-CCSD with robust convergence

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 $\u03f5$, 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 7.9.6.2 illustrates the setup of RIXS calculations.

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

In example 7.9.6.3, 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 is activated by selecting CC_TRANS_PROP = 1 and the libwfa analysis is invoked by STATE_ANALYSIS = TRUE (see Section 10.2.6).

Example 7.9.6.3 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 7.9.6.3. 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 7.9.6.2 illustrates calculation of the XES spectrum of benzene. Examples 7.9.6.3 and 7.9.6.2 illustrate calculations of Dyson orbitals between core-excited and core-ionized states and between core-excited and valence-ionized states.

Example 7.9.6.2 illustrates the setup of RIXS calculations.

$molecule 0 1 O 0.0000 0.0000 0.1173 H 0.0000 0.7572 -0.4692 H 0.0000 -0.7572 -0.4692 $end $rem method = eom-ccsd cvs_ee_singlets = [3,0,2,1] basis = aug-cc-pVDZ n_frozen_core = fc CC_TRANS_PROP = true eom_preconv_singles = true state_analysis = true !invoke libwa to compute NTOs and exciton descriptors ! libwa controls below molden_format = true nto_pairs = 3 pop_mulliken = true $end

$comment CO, carbon edge $end $molecule 0 1 O 0.0000 0.0000 0.913973 C 0.0000 0.0000 -1.218243 $end $rem METHOD = eom-ccsd BASIS = aug-cc-pVDZ INPUT_BOHR = true CVS_EE_SINGLETS = [2,0,2,2] N_FROZEN_CORE = fc EOM_PRECONV_SINGLES = true CC_TRANS_PROP = true $end @@@ $comment CO, oxygen edge $end $molecule read $end $rem METHOD = eom-ccsd BASIS = aug-cc-pVDZ CVS_EE_SINGLETS = [2,0,2,2] N_FROZEN_CORE = 1 EOM_PRECONV_SINGLES = true CC_TRANS_PROP = true $end

$comment RI-MP2/cc-pVTZ optimized geometry. XES calculation. $end $molecule 0 1 H 2.4750347531 0.0000000000 0.0000000000 C 1.3935929418 0.0000000000 0.0000000000 C 0.6967964709 1.2068868901 0.0000000000 H 1.2375173766 2.1434429715 0.0000000000 C -0.6967964709 1.2068868901 0.0000000000 H -1.2375173766 2.1434429715 0.0000000000 C -1.3935929418 0.0000000000 0.0000000000 H -2.4750347531 0.0000000000 0.0000000000 C -0.6967964709 -1.2068868901 0.0000000000 H -1.2375173766 -2.1434429715 0.0000000000 C 0.6967964709 -1.2068868901 0.0000000000 H 1.2375173766 -2.1434429715 0.0000000000 $end $rem BASIS = 6-31G* METHOD = eom-ccsd IP_STATES = [3,2,1,1,0,1,2,2] !All valence Koompans-like ionized states except for 3 lowest ones CVS_IP_STATES = [2,1,0,0,0,0,1,2] !All core-ionized states CC_TRANS_PROP = 2 !Compute transitions between all pairs of EOM states CC_MEMORY = 8000 !8 GB $end

$comment CVS-IP/CVS-EE Dyson orbitals, formaldehyde $end $molecule 0 1 C H 1 1.096135 H 1 1.096135 2 116.191164 O 1 1.207459 2 121.904418 3 -180.000000 0 $end $rem METHOD = eom-ccsd BASIS = cc-pVDZ ! Please do not use BASIS2 SCF_CONVERGENCE = 8 CVS_IP_STATES = [1,0,0,0] CVS_EE_STATES = [1,0,1,0] CC_DO_DYSON = true CC_TRANS_PROP = 2 !Compute all EOM-to-EOM transitions $end

$comment IP/CVS-EE Dyson orbitals, formaldehyde $end $molecule 0 1 C H 1 1.096135 H 1 1.096135 2 116.191164 O 1 1.207459 2 121.904418 3 -180.000000 0 $end $rem METHOD = eom-ccsd BASIS = cc-pVDZ SCF_CONVERGENCE = 8 IP_STATES = [1,0,0,0] ! Valence a1 hole CVS_EE_STATES = [1,0,0,0] CC_DO_DYSON = true CC_TRANS_PROP = 2 !Compute all EOM-to-EOM transitions $end

$comment CVS-IP/CVS-EE Dyson orbitals, formaldehyde $end $rem BASIS = cc-pVDZ JOB_TYPE = SP SCF_CONVERGENCE = 8 METHOD = eom-ccsd IP_states = [1,0,0,0] Valence a1 hole CVS_EE_states = [1,0,0,0] cc_do_dyson = true cc_trans_prop = true i ! required to activate a Dyson orbitals job $end

$comment RI-MP2/cc-pVTZ optimized geometry. 6-31(+,+)G** Pump XAS transition peak A at 285.97 eV, only one frequency point. This is closed-shell calculation. $end $molecule 0 1 H 2.4750347531 0.0000000000 0.0000000000 C 1.3935929418 0.0000000000 0.0000000000 C 0.6967964709 1.2068868901 0.0000000000 H 1.2375173766 2.1434429715 0.0000000000 C -0.6967964709 1.2068868901 0.0000000000 H -1.2375173766 2.1434429715 0.0000000000 C -1.3935929418 0.0000000000 0.0000000000 H -2.4750347531 0.0000000000 0.0000000000 C -0.6967964709 -1.2068868901 0.0000000000 H -1.2375173766 -2.1434429715 0.0000000000 C 0.6967964709 -1.2068868901 0.0000000000 H 1.2375173766 -2.1434429715 0.0000000000 $end $rem basis = 6-31(+,+)G** method = eom-ccsd cvs_ee_states = [0,0,0,0,0,0,0,0] just to invoke CVS ee_states = [10,10,10,10,0,0,0,0] 10 states in each 2PA active irrep cc_ref_prop = 1 ! Calculate REXS in addition to RIXS cc_eom_rixs = 1 ! Activate RIXS calculation using fc-CVS-EOM-EE-CCSD cc_diis_size = 25 ! Use for better convergence of response equations cc_memory = 8000 !8 GB $end $rixs omega_1 2306503 500 1 0 damped_epsilon 0.005 $end

Examples 7.9.6.3 and 7.9.6.2 illustrate calculations of Dyson orbitals between core-excited and core-ionized states and between core-excited and valence-ionized states.