7.11 Coupled-Cluster Excited-State and Open-Shell Methods 7.11.27 Potential Energy Surface Crossing Minimization 7.11.29 Interpretation of EOM/CI Wave Functions and Orbital Numbering

7.11.28 Dyson Orbitals for Ionized or Attached States within the EOM-CCSD Formalism

(April 13, 2024)

Dyson orbitals can be used to compute total photodetachment/photoionization cross-sections, as well as angular distribution of photoelectrons. A Dyson orbital is the overlap between the N-electron molecular wave function and the $N-1$ / $N+1$ electron wave function of the corresponding cation/anion:

	$\displaystyle\phi^{d}(1)$	$\displaystyle=\frac{1}{N-1}\int\Psi^{N}(1,\ldots,n)\,\Psi^{N-1}(2,\ldots,n)\,d% 2\cdots dn$		(7.82)
	$\displaystyle\phi^{d}(1)$	$\displaystyle=\frac{1}{N+1}\int\Psi^{N}(2,\ldots,n+1),\Psi^{N+1}(1,\ldots,n+1)% \,d2\cdots d(n+1)$		(7.83)

For the Hartree-Fock wave functions and within Koopmans’ approximation, these are just the canonical HF orbitals. For correlated wave functions, Dyson orbitals are linear combinations of the reference molecular orbitals:

$\displaystyle\phi^{d}$	$\displaystyle=\sum_{p}\gamma_{p}\phi_{p}$	(7.84)
$\displaystyle\gamma_{p}$	$\displaystyle=\bigl{\langle}\Psi^{N}\bigl{\|}p^{+}\bigr{\|}\Psi^{N-1}\bigr{\rangle}$	(7.85)
$\displaystyle\gamma_{p}$	$\displaystyle=\bigl{\langle}\Psi^{N}\bigl{\|}p\bigr{\|}\Psi^{N+1}\bigr{\rangle}$	(7.86)

The calculation of Dyson orbitals is straightforward within the EOM-IP/EA-CCSD methods, where cation/anion and initial molecule states are defined with respect to the same MO basis. Since the left and right CC vectors are not the same, one can define correspondingly two Dyson orbitals (left and right):

\displaystyle\begin{aligned} \displaystyle\gamma_{p}^{R}&\displaystyle=\bigl{% \langle}\Phi_{0}e^{T_{1}+T_{2}}L^{EE}\bigl{|}p^{+}\bigr{|}R^{IP}e^{T_{1}+T_{2}% }\Phi_{0}\bigr{\rangle}\\ \displaystyle\gamma_{p}^{L}&\displaystyle=\bigl{\langle}\Phi_{0}e^{T_{1}+T_{2}% }L^{IP}\bigl{|}p\bigr{|}R^{EE}e^{T_{1}+T_{2}}\Phi_{0}\bigr{\rangle}\end{aligned}

(7.87)

The norm of these orbitals is proportional to the one-electron character of the transition.

Dyson orbitals also offer qualitative insight visualizing the difference between molecular and ionized/attached states. In ionization/photodetachment processes, these orbitals can be also interpreted as the wave function of the leaving electron. For additional details, see Refs. 897 Oana C. M., Krylov A. I.
J. Chem. Phys.
(2007), 127, pp. 234106. Link and 898 Oana C. M., Krylov A. I.
J. Chem. Phys.
(2009), 131, pp. 124114. Link . Dyson orbitals can be used for computing total and differential photoelectron cross-sections using a stand-alone ezDyson code. ⁴²⁵ Gozem S., Krylov A. I.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2022), 12, pp. e1546. Link Dyson orbitals can be computed both for valence states and core-level states; ¹²⁶² Vidal M. L., Krylov A. I., Coriani S.
Phys. Chem. Chem. Phys.
(2019), 22, pp. 2693. Link see Section 7.11.8 for calculations of Dyson orbitals within the FC-CVS-EOM framework.

The ezSpectra suite: An easy-to-use toolkit for spectroscopy modeling - Presented by Prof. Samer Gozem, Georgia State University

7.11.28.1 Dyson Orbitals Job Control

The calculation of Dyson orbitals is implemented for the ground (reference) and excited states ionization/electron attachment. To obtain the ground state Dyson orbitals one needs to run an EOM-IP/EA-CCSD calculation, request transition properties calculation by setting CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE. The Dyson orbitals decomposition in the MO basis is printed in the output, for all transitions between the reference and all IP/EA states. At the end of the file, also the coefficients of the Dyson orbitals in the AO basis are available.

Two implementations of Dyson orbitals are currently available: (i) the original implementation in CCMAN; and (ii) new implementation in CCMAN2. The CCMAN implementation is using a diffuse orbital trick (i.e., EOM_FAKE_IPEA will be automatically set to TRUE in these calculations). Note: this implementation has a bug affecting the values of norms of Dyson orbitals (the shapes are correct); thus, using this code is strongly discouraged. The CCMAN2 implementation has all types of initial states available: Dyson orbitals from ground CC, excited EOM-EE, and spin-flip EOM-SF states; it is fully compatible with all helper features for EOM calculations, like FNO, RI, Cholesky decomposition. The CCMAN2 implementation can use a user-specified EOM guess (using EOM_USER_GUESS keyword and $eom_user_guess section), which is recommended for highly excited states (such as core-ionized states). In addition, CCMAN2 can calculate Dyson orbitals involving meta-stable states (see Section 7.11.9) and core-level states (see Section 7.11.8).

For calculating Dyson orbitals between excited or spin-flip states from the reference configuration and IP/EA states, same CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE keywords have to be added to the combination of usual EOM-IP/EA-CCSD and EOM-EE-CCSD or EOM-SF-CCSD calculations. (However, note the separate keyword CC_DO_DYSON_EE = TRUE for CCMAN.) The IP_STATES keyword is used to specify the target ionized states. The attached states are specified by EA_STATES. The EA-SF states are specified by EOM_EA_BETA. The excited (or spin-flipped) states are specified by EE_STATES and SF_STATES. The Dyson orbital decomposition in MO and AO bases is printed for each EE/SF-IP/EA pair of states first for reference, then for all excited states in the order: CC-IP/EA1, CC-IP/EA2, $\ldots$ , EE/SF1 - IP/EA1, EE/SF1 - IP/EA2, $\ldots$ , EE/SF2 - IP/EA1, EE/SF2 - IP/EA2, $\ldots$ , and so on. CCMAN implementation keeps reference transitions separate, in accordance with separating keywords.

CC_DO_DYSON

CC_DO_DYSON
       CCMAN2: starts all types of Dyson orbitals calculations. Desired type is determined by requesting corresponding EOM-XX transitions CCMAN: whether the reference-state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations.
TYPE:
       LOGICAL
DEFAULT:
       FALSE (the option must be specified to run this calculation)
OPTIONS:
       TRUE/FALSE
RECOMMENDATION:
       none

CC_DO_DYSON_EE

CC_DO_DYSON_EE
       Whether excited-state or spin-flip state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations with CCMAN.
TYPE:
       LOGICAL
DEFAULT:
       FALSE (the option must be specified to run this calculation)
OPTIONS:
       TRUE/FALSE
RECOMMENDATION:
       none

Dyson orbitals are most easily visualized by setting IQMOL_FCHK = TRUE (equivalently, GUI = 2) and reading the resulting checkpoint file into IQmol. In addition to the canonical orbitals, the Dyson orbitals will appear under the Surfaces item in the Model View. For step-by-step instructions, see the ezDyson manual. Alternatively Dyson orbitals can be plotted using IANLTY = 200 and the $plots utility. Only the sizes of the box need to be specified, followed by a line of zeros:

$plots
   comment
   10   -2   2
   10   -2   2
   10   -2   2
   0     0   0    0
$plots

All Dyson orbitals on the Cartesian grid will be written in the resulting plot.mo file (only CCMAN). For RHF(UHF) reference, the columns order in plot.mo is: $\phi^{lr}_{1}\alpha\ (\phi^{lr}_{1}\beta)\ \phi^{rl}_{1}\alpha\ (\phi^{rl}_{1}% \beta)\ \phi^{lr}_{2}\alpha\ (\phi^{lr}_{2}\beta)\ \ldots$

In addition, setting the MAKE_CUBE_FILES keyword to TRUE will create cube files for Dyson orbitals which can be viewed with Visual Molecular Dynamics (VMD)^{W. Humphrey, A. Dalke, and K. Schulten (1996), 15} or other programs; see Section 10.5.5 for details. This option is available for CCMAN and CCMAN2. The Dyson orbitals will be written to files mo.1.cube, mo.2.cube, $\ldots$ in the order $\phi^{lr}_{1}\ \phi^{rl}_{1}\ \ \phi^{lr}_{2}\ \phi^{rl}_{2}\ldots$ . For meta-stable states, the real and imaginary parts of the Dyson orbitals are written to separate files in the order $\text{Re}(\phi^{lr}_{1})\ \text{Re}(\phi^{rl}_{1})\ \ \text{Re}(\phi^{lr}_{2})% \ \text{Re}(\phi^{rl}_{2})\ldots\text{Im}(\phi^{lr}_{1})\ \text{Im}(\phi^{rl}_% {1})\ \ \text{Im}(\phi^{lr}_{2})\ \text{Im}(\phi^{rl}_{2})\ldots$

Note: Visualization via the MolDen format is not available.

7.11.28.2 Examples

Example 7.119 Plotting grd-ex and ex-grd state Dyson orbitals for ionization of the oxygen molecule. The target states of the cation are ${}^{2}$ A ${}_{g}$ and ${}^{2}$ B ${}_{2u}$ . Works for CCMAN only.

$molecule
   0 3
   O    0.000  0.000  0.000
   O    1.222  0.000  0.000
$end

$rem
   BASIS          6-31G*
   METHOD         eom-ccsd
   IP_STATES      [1,0,0,0,0,0,1,0] Target EOM-IP states
   CC_TRANS_PROP  true  request transition OPDMs to be calculated
   CC_DO_DYSON    true  calculate Dyson orbitals
   IANLTY         200
$end

$plots
plots excited states densities and trans densities
   10   -2   2
   10   -2   2
   10   -2   2
   0     0   0    0
$plots

Example 7.120 Plotting ex-ex state Dyson orbitals between the 1st ${}^{2}A_{1}$ excited state of the HO radical and the the 1st A ${}_{1}$ and A ${}_{2}$ excited states of HO ${}^{-}$ . Works for CCMAN only.

$molecule
   -1 1
   H    0.000   0.000  0.000
   O    1.000   0.000  0.000
$end

$rem
   METHOD             eom-ccsd
   BASIS              6-31G*
   IP_STATES          [1,0,0,0]  states of HO radical
   EE_STATES          [1,1,0,0]  excited states of HO-
   CC_TRANS_PROP      2          calculate transition properties
   CC_DO_DYSON        true       calculate Dyson orbitals for ionization from ex. states
   IANLTY             200
$end

$plots
plot excited states densities and trans densities
   10   -2   2
   10   -2   2
   10   -2   2
   0     0   0    0
$plots

Example 7.121 Dyson orbitals for ionization of CO molecule; A ${}_{1}$ and B ${}_{1}$ ionized states requested.

$molecule
   0 1
   O
   C  O  1.131
$end

$rem
   CORRELATION           CCSD
   BASIS                 cc-pVDZ
   PURECART              111        5d, will be required for ezDyson
   IP_STATES             [1,0,1,0]  (A1,A2,B1,B2)
   CCMAN2                true
   CC_DO_DYSON           true
   CC_TRANS_PROP         true       necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS   true       will be required for ezDyson
$end

Example 7.122 Dyson orbitals for ionization of H ${}_{2}$ O; core (A ${}_{1}$ ) state requested — ionization from O(1s).

$molecule
   0 1
   O
   H1  O  0.955
   H2  O  0.955  H1  104.5
$end

$rem
   CORRELATION           CCSD
   BASIS                 cc-pVTZ
   PURECART              111        5d, will be required for ezDyson
   IP_STATES             [1,0,0,0]  (A1,A2,B1,B2)
   EOM_USER_GUESS        1          on, further defined in $eom_user_guess
   CCMAN2                true
   CC_DO_DYSON           true
   CC_TRANS_PROP         true       necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS   true       will be required for ezDyson
   N_FROZEN_CORE         false
$end

$eom_user_guess
   1
$end

Example 7.123 Dyson orbitals for ionization of NO molecule using EOM-EA and a closed-shell cation reference; A ${}_{1}$ and B ${}_{2}$ states requested.

$molecule
   +1 1
   N   0.00000  0.00000  0.00000
   O   0.00000  0.00000  1.02286
$end

$rem
   CORRELATION            CCSD
   BASIS                  aug-cc-pVTZ
   PURECART               111       5d, will be required for ezDyson
   EA_STATES              [1,0,0,1] (A1,A2,B1,B2)
   CCMAN2                 true
   CC_DO_DYSON            true
   CC_TRANS_PROP          true      necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS    true      will be required for ezDyson
$end

Example 7.7.124 Dyson orbitals for detachment from the meta-stable ${}^{2}\Pi_{g}$ state of N ${}_{2}^{-}$ .

$molecule
   0 1
   N   0.0   0.0    0.55
   N   0.0   0.0   -0.55
   GH  0.0   0.0    0.0
$end

$rem
   METHOD            EOM-CCSD
   EA_STATES         [0,0,2,0,0,0,0,0]
   CC_MEMORY         5000
   MEM_STATIC        1000
   BASIS             GEN
   COMPLEX_CCMAN     TRUE
   CC_TRANS_PROP     TRUE
   CC_DO_DYSON       TRUE
   MAKE_CUBE_FILES   TRUE
   IANLTY            200
$end

$complex_ccman
   CS_HF             1
   CAP_TYPE          1
   CAP_X             2760
   CAP_Y             2760
   CAP_Z             4880
   CAP_ETA           400
$end

$plots
plot Dyson orbitals
   50 -10.0 10.0
   50 -10.0 10.0
   50 -10.0 10.0
   0 0 0 0
$end

$basis
N    0
S    8    1.000000
   1.14200000E+04    5.23000000E-04
   1.71200000E+03    4.04500000E-03
   3.89300000E+02    2.07750000E-02
   1.10000000E+02    8.07270000E-02
   3.55700000E+01    2.33074000E-01
   1.25400000E+01    4.33501000E-01
   4.64400000E+00    3.47472000E-01
   5.11800000E-01   -8.50800000E-03
S    8    1.000000
   1.14200000E+04   -1.15000000E-04
   1.71200000E+03   -8.95000000E-04
   3.89300000E+02   -4.62400000E-03
   1.10000000E+02   -1.85280000E-02
   3.55700000E+01   -5.73390000E-02
   1.25400000E+01   -1.32076000E-01
   4.64400000E+00   -1.72510000E-01
   5.11800000E-01    5.99944000E-01
S    1    1.000000
   1.29300000E+00    1.00000000E+00
S    1    1.000000
   1.78700000E-01    1.00000000E+00
P    3    1.000000
   2.66300000E+01    1.46700000E-02
   5.94800000E+00    9.17640000E-02
   1.74200000E+00    2.98683000E-01
P    1    1.000000
   5.55000000E-01    1.00000000E+00
P    1    1.000000
   1.72500000E-01    1.00000000E+00
D    1    1.000000
   1.65400000E+00    1.00000000E+00
D    1    1.000000
   4.69000000E-01    1.00000000E+00
F    1    1.000000
   1.09300000E+00    1.00000000E+00
S    1    1.000000
   5.76000000E-02    1.00000000E+00
P    1    1.000000
   4.91000000E-02    1.00000000E+00
D    1    1.000000
   1.51000000E-01    1.00000000E+00
F    1    1.000000
   3.64000000E-01    1.00000000E+00
****
GH   0
S    1    1.000000
   2.88000000E-02    1.00000000E+00
S    1    1.000000
   1.44000000E-02    1.00000000E+00
S    1    1.000000
   0.72000000E-02    1.00000000E+00
S    1    1.000000
   0.36000000E-02    1.00000000E+00
S    1    1.000000
   0.18000000E-02    1.00000000E+00
S    1    1.000000
   0.09000000E-02    1.00000000E+00
P    1    1.000000
   2.45000000E-02    1.00000000E+00
P    1    1.000000
   1.22000000E-02    1.00000000E+00
P    1    1.000000
   0.61000000E-02    1.00000000E+00
P    1    1.000000
   0.305000000E-02    1.00000000E+00
P    1    1.000000
   0.152500000E-02    1.00000000E+00
P    1    1.000000
   0.076250000E-02    1.00000000E+00
D    1    1.000000
   0.755000000E-01    1.00000000E+00
D    1    1.000000
   0.377500000E-01    1.00000000E+00
D    1    1.000000
   0.188750000E-01    1.00000000E+00
D    1    1.000000
   0.094375000E-01    1.00000000E+00
D    1    1.000000
   0.047187500E-01    1.00000000E+00
D    1    1.000000
   0.023593750E-01    1.00000000E+00
****
$end

Example 7.125 Dyson orbitals for ionization of triplet O ${}_{2}$ and O ${}_{2}^{-}$ at slightly stretched (relative to the equilibrium O ${}_{2}$ geometry); B ${}_{3g}$ states are requested.

$comment
   EOM-IP-CCSD/6-311+G* and EOM-EA-CCSD/6-311+G* levels of theory,
   UHF reference.  Start from O2:
    1) detach electron - ionization of neutral (alpha IP).
    2) attach electron, use EOM-EA w.f. as initial state
       - ionization of anion (beta EA).
$end

$molecule
   0 3
   O   0.00000  0.00000  0.00000
   O   0.00000  0.00000  1.30000
$end

$rem
   CORRELATION           CCSD
   BASIS                 6-311(3+)G*
   PURECART              2222          6d, will be required for ezDyson
   EOM_IP_ALPHA          [0,0,0,1,0,0,0,0]  (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u)
   EOM_EA_BETA           [0,0,0,1,0,0,0,0]  (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u)
   CCMAN2                true
   CC_DO_DYSON           true
   CC_TRANS_PROP         true          necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS   true          will be required for ezDyson
$end

Example 7.126 Dyson orbitals for ionization of formaldehyde from the first excited state AND from the ground state.

$molecule
   0 1
   O     1.535338855      0.000000000     -0.438858006
   C     1.535331598     -0.000007025      0.767790994
   H     1.535342484      0.937663512      1.362651452
   H     1.535342484     -0.937656488      1.362672535
$end

$rem
   CORRELATION           CCSD
   BASIS                 6-31G*
   PURECART              2222    6d, will be required for ezDyson
   CCMAN2                true    new Dyson code
   EE_STATES             [1]
   EOM_IP_ALPHA          [1]
   EOM_IP_BETA           [1]
   CC_TRANS_PROP         true    necessary for Dyson orbitals job
   CC_DO_DYSON           true
   PRINT_GENERAL_BASIS   true    will be required for ezDyson
$end

Example 7.127 Dyson orbitals for core ionization of Li atom use Li ${}^{+}$ as a reference, get neutral atom via EOM-EA get 1st excitation for the cation via EOM-EE totally: core ionization AND 1st ionization of Li atom.

$molecule
   +1 1
   Li   0.00000  0.00000  0.00000
$end

$rem
   CORRELATION            CCSD
   BASIS                  6-311+G*
   PURECART               2222          6d, will be required for ezDyson
   CCMAN2                 true          new Dyson code
   EE_STATES              [1,0,0,0,0,0,0,0]
   EA_STATES              [1,0,0,0,0,0,0,0]
   EOM_NGUESS_SINGLES     5             to converge to the lowest EA state
   CC_TRANS_PROP          true          necessary for Dyson orbitals job
   CC_DO_DYSON            true
   PRINT_GENERAL_BASIS    true          will be required for ezDyson
$end

Example 7.128 Dyson orbitals for ionization of CH ${}_{2}$ from high-spin triplet reference and from the lowest SF state.

$molecule
   0 3
   C
   H  1 rCH
   H  1 rCH 2 aHCH

   rCH    = 1.1167
   aHCH   = 102.07
$end

$rem
   CORRELATION           CCSD
   BASIS                 6-31G*
   SCF_GUESS             core
   CCMAN2                true    new Dyson code
   CC_SYMMETRY           false
   SF_STATES             [1]
   EOM_IP_ALPHA          [2]     one should be careful to request
   EOM_EA_BETA           [2]     meaningful spin for EA/IP state(s)
   CC_TRANS_PROP         true    necessary for Dyson orbitals job
   CC_DO_DYSON           true
   IQMOL_FCHK            true    generate formatted checkpoint file for IQMol
$end

Example 7.129 Dyson orbitals for ionization of SO ${}^{-}$ using EOM-EA to describe the anion states and EOM-SF to describe the neutral; both sets of EOM states are generated using neutral triplet reference.

$comment
SO-, calculating Dyson orbitals using EOM-EA to describe
the anion states and EOM-SF to describe the neutral;
both sets of EOM states are generated using triplet reference.
$end

$molecule
0 3
S    0.0000000    0.0000000   -0.5241891
O    0.0000000    0.0000000    1.0676951
$end

$rem
JOBTYPE¯¯¯SP
METHOD¯          ¯EOM-CCSD
BASIS                  ¯6-31G*
PURECART¯¯111 needed for ezDyson
EA_BETA¯¯        [0,0,0,1]  anion state
SF_STATES               [2,2,0,0]  neutral states
CC_DO_DYSON¯¯true
CC_TRANS_PROP¯¯true
PRINT_GENERAL_BASIS¯true  needed for ezDyson
$end

$trans_prop
state_list
ea_beta   4 1      !state 1
sf_states 1 1    !state 2
sf_states 1 2    !state 3
sf_states 2 1
sf_states 2 2
end_list
state_pair_list
1 2   ! transition 1 <-> 2
1 3
1 4
1 5
end_pairs
calc dyson
$end

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7.11.28 Dyson Orbitals for Ionized or Attached States within the EOM-CCSD Formalism

7.11.28.1 Dyson Orbitals Job Control

7.11.28.2 Examples