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

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

The core-valence separation (CVS) approximation155 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.

In Q-Chem, a slightly different version of CVS-EOM-EE-CCSD than the original theory by Coriani and Koch190 is implemented: the reference coupled-cluster amplitudes do not include core electrons944. To distinguish this method from the original190, below we refer to the Q-Chem implementation as frozen-core-ground-state/core-valence-separated EOM (FC-CVS-EOM) approach.944

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 strengths are obtained imposing that at least one index in the EOM excitation (and ionization) operators refer to a core occupied orbital.

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 7.8.5.2 and 7.8.5.2 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. Transition properties and Dyson orbitals can be computed either within CVS manifold or between CVS and valence manifolds (see Section 7.8.23 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.

7.8.5.1 Single-precision arithmetics in EOM-CC calculations

Similar to ground-state CCSD calculations described in Section 6.15, single precision can be used in EOM-CC and EOM-MP2 calculations 732. Currently, the following variants of EOM are supported: EE, SF,IP, EA; both in standard and RI/CD implementations, for energies and properties evaluation. If you wish to use single-precision version of EOM, please first read Section 6.15 for basic setup of single-precision coupled-cluster calculation. Here we describe only additional EOM-specific keywords.

Precision selection is controlled by the EOM_SINGLE_PREC keyword: 0 corresponds to double-precision calculation and 1 corresponds to single-precision calculation. EOM-specific convergence criteria are controlled by the same keywords as in the double precision, but the same rule as for CCSD applies: too tight thresholds may cause issues with convergence. The default Davidson threshold 10-5 works well for most cases 732.

The keyword CC_SP_DM controlls calculation of intermediates, density matrices, and S2 for EOM calculations in the same manner as for CCSD, which is described in Section 6.15.

Calculations of analytical gradients require solving amplitude-response equations, which can be done on single precision as well; this is activated by EOM_ARESP_SINGLE_PREC=1.

EOM_SINGLE_PREC
       Precision selection for EOM-CC/MP2 calculations. Available in CCMAN2 only.
TYPE:
       INTEGER
DEFAULT:
       0 double-precision calculation
OPTIONS:
       1 single-precision calculation 2 single-precision calculation is followed by double-precision clean-up iterations
RECOMMENDATION:
       Do not set too tight convergence criteria when use single precision

EOM_ARESP_SINGLE_PREC
       Precision selection for amplitude response EOM equations. Available in CCMAN2 only.
TYPE:
       INTEGER
DEFAULT:
       0 double-precision calculation
OPTIONS:
       1 single-precision calculation
RECOMMENDATION:
       NONE

Example 7.28  A job evaluating EOM-EA-CCSD energies for formaldehyde anion using single-precision execution combined with CD

$comment
Formaldehyde anion, single-precision calculation
$end

$molecule
0 1
C
H 1 1.127888
H 1 1.127888 2 100.546614
$end

$rem
basis = cc-pvdz
method = ccsd
cholesky_tol = 3
EA_STATES = [1,0,0,0]
cc_ref_prop = 1   Compute properties of the CCSD reference
!SP keywords
cc_single_prec = 1
cc_sp_t_conv = 4
cc_sp_e_conv = 6
cc_erase_dp_integrals = 0 ! set 1 to save disk space
cc_sp_dm = 1
!EOM-specific keyword
eom_single_prec = 1
$end

Example 7.29  Geometry optimization of a triplet excited state of uracil-water complex in single-precision setup

$molecule
0 1
N            .034130    -.986909    0.000000
N          -1.173397     .981920    0.000000
C          -1.218805    -.408164    0.000000
C           -.007302    1.702153    0.000000
C           1.196200    1.107045    0.000000
C           1.289085    -.345905    0.000000
O           2.310232    -.996874    0.000000
O          -2.257041   -1.026495    0.000000
H            .049329   -1.997961    0.000000
H          -2.070598    1.437050    0.000000
H           -.125651    2.776484    0.000000
H           2.111671    1.674079    0.000000
O           1.747914   -1.338382   -3.040233
H           2.180817   -1.817552   -2.333676
H           0.813180   -1.472188   -2.883392
$end

$rem
basis = cc-pvdz
job_type = opt
method = ccsd
cc_state_to_opt = [1,1]
mem_total = 30000
EE_TRIPLETS = [1]
cc_sp_t_conv = 4
cc_sp_e_conv = 6
cc_single_prec = 1
eom_single_prec = 1
CC_SP_DM = 1
CC_EOM_PROP = 1
EOM_ARESP_SINGLE_PREC = 1
$end

7.8.5.2 Examples

In example 7.8.5.2, 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 is activated by selecting CC_TRANS_PROP = 1 and the libwfa analysis is invoked by STATE_ANALYSIS = TRUE (see Section 11.2.6).

Example 7.8.5.2 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 7.8.5.2. 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 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 7.8.5.2 and 7.8.5.2 illustrate calculations of Dyson orbitals between core-excited and core-ionized states and between core-excited and valence-ionized states.

Example 7.30  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 7.31  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 7.32  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    =  true      ! required to activate a Dyson orbitals job
$end

Example 7.33  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    =  true        ! required to activate a Dyson orbitals job
$end

Example 7.34  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         ! required to activate a Dyson orbitals job
$end