7.9 Coupled-Cluster Excited-State and Open-Shell Methods

7.9.6 EOM-CC Calculations of Core-Level States: Core-Valence Separation within EOM-CCSD

The core-valence separation (CVS) schemeCederbaum: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 KochCoriani: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 n10-3 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.

7.9.6.1 EOM-CC Calculations of RIXS

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 σ-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 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 7.9.6.2 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.

7.9.6.2 Examples

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 1s 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.

Example 7.32  FC-CVS-EOM-EE-CCSD calculation of the first six dipole-allowed core excitation energies and their intensities at the oxygen edge of water. Wave-function analysis is also performed.

$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

Example 7.33  FC-CVS-EOM-EE-CCSD calculations of the first two dipole-allowed core excitation energies per irreducible representation and their intensities at the carbon and oxygen edges of carbon monoxide.

$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

Example 7.34  Calculation of XES spectrum of benzene

$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

Example 7.35  Calculation of Dyson orbitals between FC-CVS-EOM-EE-CCSD and FC-CVS-EOM-IP-CCSD manifolds.

$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

Example 7.36  Calculation of Dyson orbitals between FC-CVS-EOM-EE-CCSD and EOM-IP-CCSD manifolds.

$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

Example 7.37  Calculation of Dyson orbitals between FC-CVS-EOM-EE-CCSD and FC-CVS-EOM-IP-CCSD manifolds.

$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

Example 7.38  Calculation of RIXS/REXS for benzene (10 excited states per 2PA active irrep)

$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

7.9.6.3 Examples

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 1s 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.

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 7.39  FC-CVS-EOM-CCSD calculation of the first six dipole-allowed core excitation energies and their intensities at the oxygen edge of water. Wave-function analysis is also performed.

$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

Example 7.40  FC-CVS-EOM-EE-CCSD calculations of the first two dipole allowed core excitation energies per irreducible representation and their intensities at the carbon and oxygen edges of carbon monoxide.

$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

Example 7.41  Calculation of Dyson orbitals between FC-CVS-EOM-EE-CCSD and FC-CVS-EOM-IP-CCSD manifolds.

$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