7.10 Coupled-Cluster Excited-State and Open-Shell Methods 7.10.10 Auger Spectra and Lifetimes of Core-Level States 7.10.12 Charge Stabilization for EOM-DIP and Other Methods

7.10.11 Partial Auger Decay Widths from Complex-Variable Calculations

(May 21, 2025)

Auger decay in core-ionized atoms and molecules can be described by Feshbach–Fano approach (see Section 7.10.10) as well as by complex-variable extentions of the CCSD or EOM-CCSD methods. ⁸⁸² Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link ^, ⁸⁸³ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link In this latter approach, energies and widths $\Gamma$ (which are proportional to the decay rates) of core-hole states are computed as the difference of complex energies of ground and core-vacant states.

Both in Feshbach–Fano and complex-variable calculations of the AES, the Auger intensities are proportional to the rate of adecay into a particular channel. Hence, one need to compute partial decay widths $\Gamma_{ij}$ , which add up to the total width $\Gamma$ . Therein, $i$ and $j$ refer to the valence orbitals from which the electrons involved in the decay process are removed. Below we describe two different ways of computing partial decay width using complex-variable CC/EOM-CC approaches.

7.10.11.1 Decomposition of the Coupled-Cluster energy

The energy of a complex-variable coupled-cluster singles and doubles wave function can be written as

E-\mathrm{i}\frac{\Gamma}{2}=E_{\mathrm{CCSD}}=E_{\mathrm{HF}}+\sum_{ijab}% \left(\frac{1}{4}t_{ij}^{ab}+\frac{1}{2}t_{i}^{a}t_{j}^{b}\right)\langle ij||% ab\rangle.

(7.96)

If the reference state has a core-hole, Auger decay-like transitions leading to doubly ionized states are double excitations from two valence orbitals $i$ and $j$ to the core-hole $a$ and a virtual orbital $b$ , which represents emission into the continuum when using a complex-variable method. ⁸⁸² Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link From equation 7.96 it is evident that we can obtain the contribution from one of these valence orbital combinations, i.e., the partial width, from the amplitude and two-electron integral tensors as

\frac{\Gamma_{ij}}{2}=-\mathrm{Im}\left(\sum_{b}\left(t_{ij}^{ab}+2t_{i}^{a}t_% {j}^{b}\right)\langle ij||ab\rangle\right).

(7.97)

Since these quantities are determined in every CCSD calculation, the computational cost for such a partial width calculation is negligible. This procedure is implemented in the ccman2 module of Q-Chem and can be invoked by setting the CC_PW variable to 1. Open decay channels are determined by comparing the orbital energies. The output contains a list of all combinations of two valence orbitals and their partial widths.

If the core-vacancy is produced through core-ionization in the closed-shell ground state of a molecule or atom, the combination of $i_{\alpha}$ and $j_{\beta}$ describes the same target state as $i_{\beta}$ and $j_{\alpha}$ . In the current implementation, these two channels are automatically combined to a single decay width: in the output, one of the orbitals characterizing the decay channel is always an alpha orbital and the inverted spin case is implicitly contained.

Note: Core electrons must not be frozen in such calculations—N_FROZEN_CORE should be set to 0. The core hole must be in a $\beta$ orbital.

CC_PW

CC_PW
       Activates calculation of partial Auger decay widths via decomposition of the imaginary part of the coupled-cluster energy of a complex-variable CCSD calculation on a core-ionized state. Currently, this is implemented for states resulting from ionization of a $\beta$ core electron of a closed-shell system.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 do not invoke energy decomposition into partial Auger decay widths 1 invoke energy decomposition into partial Auger decay widths
RECOMMENDATION:
       Use to compute partial widths for a complex-variable calculation on a core-vacant state. An appropriate complex-scaled basis set has to be chosen in order to capture Auger decay and the optimal scaling angle needs to be determined. ⁸⁸² Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link ^, ⁸⁸³ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link

Example 7.7.77 Calculation of Auger partial decay widths of the core-ionized neon atom.

$molecule
0 1
Ne 0 0 0
$end

$rem
BASIS cc-pCVDZ
COMPLEX_THETA = 200
method = hf
n_frozen_core = 0
complex_basis gen
complex_exponents 1
complex_scf 1
complex_scf_guess 1
$end

$complex_ccman
cs_alpha 1000
CS_THETA 0
$end

$zbasis
Ne     0
S   1   1.00
      4.3306000              1.0000000
S   1   1.00
      1.4028562              1.0000000
P   1   1.00
     17.4312839              1.0000000
P   1   1.00
      5.6513946              1.0000000
D   1   1.00
     23.7130337              1.0000000
D   1   1.00
      4.1919117              1.0000000
S    8   1.00
  17880.0000000              0.0007380
   2683.0000000              0.0056770
    611.5000000              0.0288830
    173.5000000              0.1085400
     56.6400000              0.2909070
     20.4200000              0.4483240
      7.8100000              0.2580260
      1.6530000              0.0150630
S    8   1.00
  17880.0000000             -0.0001720
   2683.0000000             -0.0013570
    611.5000000             -0.0067370
    173.5000000             -0.0276630
     56.6400000             -0.0762080
     20.4200000             -0.1752270
      7.8100000             -0.1070380
      1.6530000              0.5670500
S    1   1.00
     12.8540000              1.0000000
S    1   1.00
      0.4869000              1.0000000
P    3   1.00
     28.3900000              0.0460870
      6.2700000              0.2401810
      1.6950000              0.5087440
P    1   1.00
     40.1840000              1.0000000
P    1   1.00
      0.4317000              1.0000000
D    1   1.00
      2.2020000              1.0000000
****
$end

@@@

$molecule
+1 2
Ne 0 0 0
$end

$rem
BASIS cc-pCVDZ
COMPLEX_THETA = 200
SCF_GUESS = READ
method = ccsd
n_frozen_core = 0
MOM_START = 1
cs_ccsd 1
complex_ccman 1
complex_basis gen
complex_exponents 1
complex_scf 2
complex_scf_guess 1
cc_pw 1
$end

$complex_ccman
cs_alpha 1000
CS_THETA 0
$end

$occupied
 1 2 3 4 5
 2 3 4 5
$end

$zbasis
Ne     0
S   1   1.00
      4.3306000              1.0000000
S   1   1.00
      1.4028562              1.0000000
P   1   1.00
     17.4312839              1.0000000
P   1   1.00
      5.6513946              1.0000000
D   1   1.00
     23.7130337              1.0000000
D   1   1.00
      4.1919117              1.0000000
S    8   1.00
  17880.0000000              0.0007380
   2683.0000000              0.0056770
    611.5000000              0.0288830
    173.5000000              0.1085400
     56.6400000              0.2909070
     20.4200000              0.4483240
      7.8100000              0.2580260
      1.6530000              0.0150630
S    8   1.00
  17880.0000000             -0.0001720
   2683.0000000             -0.0013570
    611.5000000             -0.0067370
    173.5000000             -0.0276630
     56.6400000             -0.0762080
     20.4200000             -0.1752270
      7.8100000             -0.1070380
      1.6530000              0.5670500
S    1   1.00
     12.8540000              1.0000000
S    1   1.00
      0.4869000              1.0000000
P    3   1.00
     28.3900000              0.0460870
      6.2700000              0.2401810
      1.6950000              0.5087440
P    1   1.00
     40.1840000              1.0000000
P    1   1.00
      0.4317000              1.0000000
D    1   1.00
      2.2020000              1.0000000
****
$end

7.10.11.2 Auger Channel Projectors

An alternative recipe to obtain partial decay widths using complex-variable methods is to restrict the excitation manifold so that excitations describing decay via a certain channel are no longer included. The projectors which accomplish this have been dubbed Auger Channel Projectors (ACP). ⁸⁸³ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link A calculation with a decay channel projected out yields a different energy and decay width than one with the channel present, and the difference in the decay width represents the partial decay width of that channel.

ACP-EOM-IP-CCSD calculations are accelerated by using the EOM-IP-CCSD solution with the full excitation manifold as a guess. For this purpose, the Maximum Overlap Method is used to ensure convergence to the same roots as in the initial EOM-IP-CCSD calculation. This is invoked in the ccman2 module of Q-Chem by setting the EOM_ACP variable to 1. The output contains a list of all open decay channels and their partial widths for each core-ionized state.

Note: Core electrons must not be frozen in such calculations—N_FROZEN_CORE should be set to 0. The core-valence separation must not be invoked.

EOM_ACP

EOM_ACP
       Activates calculation of partial Auger decay widths by recomputation of the EOM-CCSD state with an Auger Channel Projector applied. Currently, this is implemented for EOMIP-CCSD calculations with a closed-shell reference.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 do not run ACP-EOM-CCSD calculations 1 determine partial Auger decay widths by running ACP-EOM-CCSD calculations
RECOMMENDATION:
       Use to compute partial widths for a complex-variable calculation which produces a core-vacant state. An appropriate complex-scaled basis set has to be chosen in order to capture Auger decay and the optimal scaling angle needs to be determined. ⁸⁸² Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link ^, ⁸⁸³ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link

Example 7.7.78 Calculation of Auger partial decay widths of the core-ionized neon atom.

$molecule
0 1
Ne 0 0 0
$end

$rem
MEM_TOTAL = 172421
BASIS cc-pCVDZ
COMPLEX_THETA = 200
method = ccsd
n_frozen_core = 0
complex_ccman 1
complex_basis gen
complex_exponents 1
complex_scf 1
complex_scf_guess 1
ip_states = [1,0,0,0,0,0,0,0]
eom_shift = 32000
eom_acp 1
$end

$complex_ccman
cs_alpha 1000
CS_THETA 0
$end

$zbasis
Ne     0
S   1   1.00
      4.3306000              1.0000000
S   1   1.00
      1.4028562              1.0000000
P   1   1.00
     17.4312839              1.0000000
P   1   1.00
      5.6513946              1.0000000
D   1   1.00
     23.7130337              1.0000000
D   1   1.00
      4.1919117              1.0000000
S    8   1.00
  17880.0000000              0.0007380
   2683.0000000              0.0056770
    611.5000000              0.0288830
    173.5000000              0.1085400
     56.6400000              0.2909070
     20.4200000              0.4483240
      7.8100000              0.2580260
      1.6530000              0.0150630
S    8   1.00
  17880.0000000             -0.0001720
   2683.0000000             -0.0013570
    611.5000000             -0.0067370
    173.5000000             -0.0276630
     56.6400000             -0.0762080
     20.4200000             -0.1752270
      7.8100000             -0.1070380
      1.6530000              0.5670500
S    1   1.00
     12.8540000              1.0000000
S    1   1.00
      0.4869000              1.0000000
P    3   1.00
     28.3900000              0.0460870
      6.2700000              0.2401810
      1.6950000              0.5087440
P    1   1.00
     40.1840000              1.0000000
P    1   1.00
      0.4317000              1.0000000
D    1   1.00
      2.2020000              1.0000000
****
$end

Example 7.7.79 Calculation of Auger partial decay widths using an open-shell reference and frozen core.

$molecule
0 3
O         0.0000000000    0.0000000000   -0.5800000000
O         0.0000000000    0.0000000000    0.5800000000
$end

$rem
BASIS gen
COMPLEX_THETA = 110
method = ccsd
scf_convergence = 9
cc_convergence = 9
eom_davidson_convergence = 9
eom_davidson_max_iter = 200
ip_states = [0,0,0,0,0,1,0,0]
scf_max_cycles = 400
n_frozen_core = 1
cs_ccsd 1
complex_ccman 1
complex_spin_state 3
complex_basis gen
complex_spin_state 3
complex_scf_guess 1
complex_exponents 1
complex_scf 2
complex_metscf 1
thresh = 14
eom_acp 1
eom_user_guess 1
$end

$eom_user_guess
2
$end

$complex_ccman
cs_alpha 1000
CS_THETA 0
$end

$zbasis
O 0
S   1   1.00
8.00 1.00
P   1   1.00
8.00 1.00
S   6   1.00
      0.5484671660D+04       0.1831074430D-02
      0.8252349460D+03       0.1395017220D-01
      0.1880469580D+03       0.6844507810D-01
      0.5296450000D+02       0.2327143360D+00
      0.1689757040D+02       0.4701928980D+00
      0.5799635340D+01       0.3585208530D+00
SP   3   1.00
      0.1553961625D+02      -0.1107775495D+00       0.7087426823D-01
      0.3599933586D+01      -0.1480262627D+00       0.3397528391D+00
      0.1013761750D+01       0.1130767015D+01       0.7271585773D+00
SP   1   1.00
      0.2700058226D+00       0.1000000000D+01       0.1000000000D+01
****
$end

$basis
O     0
S   6   1.00
      0.5484671660D+04       0.1831074430D-02
      0.8252349460D+03       0.1395017220D-01
      0.1880469580D+03       0.6844507810D-01
      0.5296450000D+02       0.2327143360D+00
      0.1689757040D+02       0.4701928980D+00
      0.5799635340D+01       0.3585208530D+00
SP   3   1.00
      0.1553961625D+02      -0.1107775495D+00       0.7087426823D-01
      0.3599933586D+01      -0.1480262627D+00       0.3397528391D+00
      0.1013761750D+01       0.1130767015D+01       0.7271585773D+00
SP   1   1.00
      0.2700058226D+00       0.1000000000D+01       0.1000000000D+01
****
$end

Search Results

7.10.11 Partial Auger Decay Widths from Complex-Variable Calculations

7.10.11.1 Decomposition of the Coupled-Cluster energy

7.10.11.2 Auger Channel Projectors