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

(September 1, 2024)

Using complex-variable methods, Auger decay in core-ionized atoms and molecules can be described with CCSD or EOM-CCSD wave functions. ⁸⁵⁹ Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link ^, ⁸⁶⁰ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link From the difference of complex energies between ground-state and core-vacant state, the energy needed to form the core-vacancy and the its total decay width $\Gamma$ , proportional to its decay rate, can be obtained.

An estimation of how likely a certain electronic target state is obtained from Auger decay allows the simulation of Auger decay spectra as this probability is reflected in the heights of signals in such spectra. This requires the computation of 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 stem.

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.

(7.86)

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. ⁸⁵⁹ Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link From equation 7.86 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).

(7.87)

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. Thus, N_FROZEN_CORE has to 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 which are 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. ⁸⁵⁹ Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link ^, ⁸⁶⁰ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link

Example 7.7.73 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). ⁸⁶⁰ 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-EOMIP-CCSD calculations are accelerated by using the EOMIP-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 EOMIP-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. Thus, N_FROZEN_CORE has to 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. ⁸⁵⁹ Matz F., Jagau T.-C.
J. Chem. Phys.
(2022), 156, pp. 114117. Link ^, ⁸⁶⁰ Matz F., Jagau T.-C.
Mol. Phys.
(2023), 121, pp. e2105270. Link

Example 7.7.74 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

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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