7.10 Coupled-Cluster Excited-State and Open-Shell Methods 7.10.7 EOM-DEA-CCSD 7.10.9 EOM-CC Calculations of Metastable States

7.10.8 EOM-CC Calculations of Core-Level States

(July 14, 2022)

The core-valence separation (CVS) scheme ¹⁸⁵ Cederbaum L. S., Domcke W., Schirmer J.
Phys. Rev. A
(1980), 22, pp. 206. Link 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/SF/IP-CCSD for energies and transition properties (oscillator strengths, NTOs, Dyson orbitals, exciton descriptors, ECD moments). 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. ¹²²⁹ Vidal M. L., Krylov A. I., Coriani S.
Phys. Chem. Chem. Phys.
(2019), 22, pp. 2693. Link The calculation of non-linear spectra, such as RIXS, is also possible. L-edge spectra (XAS and XPS) can be described by using state-interaction approach in which spin-orbit coupling is evaluated using non-relativistic CVS-EOM-EE states ¹²³⁰ Vidal M. L. et al.
J. Phys. Chem. Lett.
(2020), 11, pp. 8314. Link . Auger spectra can be computed using CVS-EOM-EE combined with the explicit treatment of the continuum via Feshbach-Fano approach ¹¹⁰³ Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link (see Section 7.10.10).

In Q-Chem, a slightly different version of CVS-EOM-EE-CCSD than the original theory by Coriani and Koch ²³⁴ Coriani S., Koch H.
J. Chem. Phys.
(2015), 143, pp. 181103. Link is implemented: the reference coupled-cluster amplitudes do not include core electrons. ¹²²⁸ Vidal M. L. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 3117. Link To distinguish this method from the original, ²³⁴ Coriani S., Koch H.
J. Chem. Phys.
(2015), 143, pp. 181103. Link we refer in what follows to the Q-Chem implementation as frozen-core-ground-state/core-valence-separated EOM (FC-CVS-EOM) approach. ¹²²⁸ Vidal M. L. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 3117. Link

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. ⁸⁵⁵ Nanda K. D. et al.
Phys. Chem. Chem. Phys.
(2019), 22, pp. 2629. Link

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 7.10.8.2 and 7.10.8.2 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. Spin-flip version can be deployed by using CVS_SF_STATES (this may be needed for computing SOCs and L-edge spectra). 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 7.10.27 for definition of Dyson orbitals). CVS-EOM-CCSD is only available with CCMAN2.

Note: (1) 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. (2) The default setting (N_FROZEN_CORE = FC) does not work correctly in QM/MM calculations. One should specify the number of frozen core orbitals explicitly.

CVS_EOM_SHIFT

CVS_EOM_SHIFT
       Specifies energy shift in CVS-EOM calculations.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ corresponds to $n\times 10^{-3}$ hartree shift (i.e., 11000 = 11 hartree); solve for eigenstates around this value.
RECOMMENDATION:
       Improves the stability of the calculations.

CVS_EOM_PRECONV_SINGLES

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.

CVS_EE_SINGLETS

CVS_EE_SINGLETS
       Sets the number of singlet core-excited state roots to find. Valid only for closed-shell references.
TYPE:
       INTEGER/INTEGER ARRAY
DEFAULT:
       0 Do not look for any excited states.
OPTIONS:
       $[i,j,k\ldots]$ Find $i$ excited states in the first irrep, $j$ states in the second irrep etc.
RECOMMENDATION:
       None

CVS_EE_TRIPLETS

CVS_EE_TRIPLETS
       Sets the number of triplet core-excited state roots to find. Valid only for closed-shell references.
TYPE:
       INTEGER/INTEGER ARRAY
DEFAULT:
       0 Do not look for any excited states.
OPTIONS:
       $[i,j,k\ldots]$ Find $i$ excited states in the first irrep, $j$ states in the second irrep etc.
RECOMMENDATION:
       None

CVS_SF_STATES

CVS_SF_STATES
       Sets the number of core-level spin-flip target states roots to find.
TYPE:
       INTEGER/INTEGER ARRAY
DEFAULT:
       0 Do not look for any excited states.
OPTIONS:
       $[i,j,k\ldots]$ Find $i$ SF states in the first irrep, $j$ states in the second irrep etc.
RECOMMENDATION:
       None

CVS_IP_STATES

CVS_IP_STATES
       Sets the number of core-ionized states to find. By default, $\beta$ electron will be removed.
TYPE:
       INTEGER/INTEGER ARRAY
DEFAULT:
       0 Do not look for any IP states.
OPTIONS:
       [i,j,k…] Find $i$ ionized states in the first irrep, $j$ states in the second irrep etc.
RECOMMENDATION:
       None

CVS_IP_ALPHA

CVS_IP_ALPHA
       Sets the number of ionized target states derived by removing $\alpha$ electron ( $M_{S}=-\tfrac{1}{2}$ ).
TYPE:
       INTEGER/INTEGER ARRAY
DEFAULT:
       0 Do not look for any IP/ $\alpha$ states.
OPTIONS:
       $[i,j,k\ldots]$ Find $i$ ionized states in the first irrep, $j$ states in the second irrep etc.
RECOMMENDATION:
       None

CVS_IP_BETA

CVS_IP_BETA
       Sets the number of ionized target states derived by removing $\beta$ electron ( $M_{S}=\tfrac{1}{2}$ , default for CVS-IP).
TYPE:
       INTEGER/INTEGER ARRAY
DEFAULT:
       0 Do not look for any IP/ $\beta$ states.
OPTIONS:
       $[i,j,k\ldots]$ Find $i$ ionized states in the first irrep, $j$ states in the second irrep etc.
RECOMMENDATION:
       None

7.10.8.1 EOM-CC Calculations of RIXS

RIXS is a coherent two-photon process involving core-level states. ⁸⁵⁵ 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 7.10.20.6). 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. 853 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.

CC_EOM_RIXS

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 $\epsilon$ , 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 7.10.8.2 illustrates the setup of RIXS calculations.

Note: The solver of the response equations in RIXS is controlled by the same keywords as in 2PA: 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, increase CC_DIIS_SIZE (25 is recommended) and consider using a larger damping factor.

7.10.8.2 Examples

In example 7.10.8.2, 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 7.10.8.2 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 7.10.8.2. 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 7.10.8.2 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 ⁵⁸⁸ Kaufmann K., Baumeister W., Jungen M.
J. Phys. B: At. Mol. Opt. Phys.
(1989), 22, pp. 2223. Link .

Note: Whem 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 7.10.8.2 illustrates calculation of the XES spectrum of benzene. Examples 7.10.8.2 and 7.10.8.2 illustrate calculations of Dyson orbitals between core-excited and core-ionized states and between core-excited and valence-ionized states.

Example 7.10.8.2 illustrates the setup of RIXS calculations.

Calculations of L-edge spectra using state-interaction approach ¹²³⁰ 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 .

Example 7.42 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
   BASIS               = aug-cc-pVDZ
   CVS_EE_SINGLETS     = [3,0,2,1]
   N_FROZEN_CORE       = fc
   CC_TRANS_PROP       = true
   EOM_PRECONV_SINGLES = true
   STATE_ANALYSIS      = true !invoke libwa to compute NTOs and exciton descriptors
   MOLDEN_FORMAT       = true
   NTO_PAIRS           = 3
   POP_MULLIKEN        = true
$end

Example 7.43 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 7.7.44 Calculation of XAS spectrum of ammonia using additional Rydberg functions placed on the ghost atom.

$comment
NEXAFS AMMONIA / 6-311++G**+Rydberg functions
Optimized geometry: NH3_opt_coord.out
$end

$molecule
0 1
N      -0.0002568808    -0.2210075702    -0.0000000021
H       0.8838383819     0.5154343604     1.5287018716
H      -1.7658867303     0.5161842661     0.0000000000
H       0.8838383819     0.5154343604    -1.5287018716
Gh      0.0000000000     0.0000000000     0.0000000000
$end

$rem
    input_bohr = true
    jobtype = sp
    method = eom-ccsd
    cvs_ee_singlets    [5,5]
    basis = general    !User-defined basis to add extra Rydberg functions
    n_frozen_core = fc
    cc_trans_prop = true
    eom_preconv_singles = true
    eom_davidson_convergence = 4
    eom_shift = 14760
    FORCE_SYMMETRY_ON = true   Forces symmetry to be turned on
    MEM_TOTAL 30000   30GB of total memory
$end

$basis
N    0
6-311++G*
****
H    0
6-311++G*
****
Gh   0
S   1   1.00
        0.0112533427        1.0000000000
S   1   1.00
        0.0058583805        1.0000000000
S   1   1.00
        0.0033459739        1.0000000000
S   1   1.00
        0.0020484225        1.0000000000
S   1   1.00
        0.0013236424        1.0000000000
P   1   1.00
        0.0192542060        1.0000000000
P   1   1.00
        0.0099882106        1.0000000000
P   1   1.00
        0.0056893607        1.0000000000
P   1   1.00
        0.0034756797        1.0000000000
P   1   1.00
        0.0022420590        1.0000000000
D   1   1.00
        0.0274456919        1.0000000000
D   1   1.00
        0.0142043987        1.0000000000
D   1   1.00
        0.0080765930        1.0000000000
D   1   1.00
        0.0049271863        1.0000000000
D   1   1.00
        0.0031748110        1.0000000000
****
$end

Example 7.45 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 7.46 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 7.47 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 7.48 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
   BASIS           =  cc-pVDZ
   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        ! required to activate a Dyson orbitals job
$end

Example 7.49 Calculation of RIXS/REXS for benzene (10 excited states per 2PA active irrep)

$comment
RI-MP2/cc-pVTZ optimized geometry.
Pump XAS transition peak A at 285.97 eV, only one frequency point.
$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
   mem_total = 8500
$end

$rixs
   omega_1         2306503 500 1 0
   damped_epsilon  0.005
$end

Search Results

7.10.8 EOM-CC Calculations of Core-Level States

7.10.8.1 EOM-CC Calculations of RIXS

7.10.8.2 Examples