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(May 16, 2021)

Certain types of resonances can be described by using real-valued EOM-CC
wave functions via Feshbach-Fano approach.
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Ann. Phys.

(1962),
19,
pp. 287.
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^{,}
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Phys. Rev.

(1961),
124,
pp. 1866.
Link
In this section we describe
the application of Feshbach-Fano approach to core-excited and
core-ionized states.
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J. Chem. Phys.

(2021),
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pp. 084124.
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J. Chem. Phys.

(2021),
154,
pp. 084125.
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Core-hole states, which are Feshbach resonances, are subject to autoionization—commonly known
as Auger decay. Auger Electron Spectroscopy (AES) measures kinetic energy and intensity of
ejected electrons. Theoretical description of AES can be formulated using
Feshbach-Fano approach for electronic resonances.
^{
310
}
Ann. Phys.

(1962),
19,
pp. 287.
Link
^{,}
^{
304
}
Phys. Rev.

(1961),
124,
pp. 1866.
Link
The theory invokes
two projection operators, $\widehat{Q}$ and $\widehat{P}$, which
decompose the total wavefunction into bound-like and continuum-like
components. In the case of core-level states this separation
is enabled by invoking the CVS scheme and frozen-core approximation in the calculations of initial and final
states in the Auger process (more details about CVS can be found in Section 7.10.7).

The initial (bound-like) state ${\mathrm{\Psi}}_{0}$ is a core-hole ionized or core-hole excited state, which can be described by CVS-EOM-CC. The final (continuum-like) state ${\chi}_{\mu ,{E}_{k}}$ is represented as an antisymmetrized product of a stable channel state ${\mathrm{\Psi}}_{\mu}$ (described by an appropriate EOM-EE model) and a continuum orbital ${\varphi}_{k}$, ${\chi}_{\mu ,{E}_{k}}\sim \mathcal{A}\{{\varphi}_{k}{\mathrm{\Psi}}_{\mu}\}$. Note that ${\mathrm{\Psi}}_{\mu}$ is a state with one electron less than ${\mathrm{\Psi}}_{0}$. Two essential parameters defining AES are the rate of the decay into a channel $\mu $, given as

$${\mathrm{\Gamma}}_{\mu}=2\pi \u27e8{\mathrm{\Psi}}_{0}^{L}|\widehat{H}-{E}_{0}|{\chi}_{\mu ,{E}_{k}}^{R}\u27e9\u27e8{\chi}_{\mu ,{E}_{k}}^{L}|\widehat{H}-{E}_{0}|{\mathrm{\Psi}}_{0}^{R}\u27e9,$$ | (7.62) |

and partial energy correction ${\mathrm{\Delta}}_{\mu}$ to the zero-order resonance position ${E}_{0}$, defined as

$${\mathrm{\Delta}}_{\mu}=P.V.{\int}_{0}^{\mathrm{\infty}}\frac{\u27e8{\mathrm{\Psi}}_{0}^{L}|\widehat{H}-{E}_{0}|{\chi}_{\mu ,E}^{R}\u27e9\u27e8{\chi}_{\mu ,E}^{L}|\widehat{H}-{E}_{0}|{\mathrm{\Psi}}_{0}^{R}\u27e9}{{E}_{0}-{E}_{\mu}-E}\mathit{d}E.$$ | (7.63) |

In the expressions above $\widehat{H}$ is the electronic Hamiltonian,
${E}_{\mu}$ is the energy of the channel state ${\mathrm{\Psi}}_{\mu}$,
${E}_{k}$ is the energy of the ejected electron (${E}_{k}={E}_{0}-{E}_{\mu}$),
$L/R$ superscripts denote left and right
EOM-CCSD wavefunctions, and $P.V.$ stands for the Cauchy principle value.
Calculations of ${\mathrm{\Gamma}}_{\mu}$ are activated with the CC_DO_FESHBACH keyword.
By default, the continuum orbital ${\varphi}_{k}$ is approximated with a plane
wave.
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1006
}
J. Chem. Phys.

(2021),
154,
pp. 084124.
Link
^{,}
^{
1007
}
J. Chem. Phys.

(2021),
154,
pp. 084125.
Link
It is also possible to model ${\varphi}_{k}$ with a Coulomb wave by setting CC_FESHBACH_CW = 1.
This option requires to include in the input an additional input section *$coulomb_wave*,
which provides an expansion of the Coulomb wave (for the given
effective charge and kinetic energy) in terms of products of a plane wave and Gaussian-type functions, as detailed
in Ref. 1006.

For non-resonant Auger decay, the initial state can be conveniently computed by CVS-EOM-IP-CCSD, whereas its stable decay channels can be obtained from EOM-DIP-CCSD calculations. Section of the input invoking Auger decay rates calculation for an atom can be given as:

$rem jobtype = sp method = eom-ccsd basis = 6-31G* cvs_eom_ip_beta [1,0,0,0,0,0,0,0] !This is the initial core-hole state dip_triplets [0,0,0,0,0,1,1,1] !These are the final triplet decay channels dip_singlets [3,1,1,1,0,1,1,1] !These are the final singlet decay channels cc_do_dyson 1 !Needed for Feshbach-type calculations cc_do_feshbach 1 $end

In resonant Auger decay, the initial state can be computed by
CVS-EOM-EE-CCSD, whereas the corresponding decay channels can be obtained from EOM-IP-CCSD calculations.
By default, Feshbach calculations are performed for all possible state pairs that
include an energetically allowed decay channel. This is not practical
if, for example, the core-hole state of interest is not the lowest state in the
given symmetry, or when the Coulomb wave is used to model the continuum orbital.
In such a case, the user can specify pairs of states for Feshbach calculations
using *$trans_prop* section with $dyson$ as the requested property:

$trans_prop state_list cvs_ip_beta 1 1 !state 1: CVS_IP with irrep = 1 and istate = 1 dip_singlets 1 3 !state 2: DIP_SINGLET state with irrep = 1 and istate = 3 dip_triplets 6 1 !state 3: DIP_TRIPLET state with irrep = 6 and istate = 1 end_list state_pair_list 1 2 ! transition 1 <-> 2 1 3 ! transition 1 <-> 3 end_pairs calc dyson $end

Calculations of energy correction ${\mathrm{\Delta}}_{\mu}$ are invoked by setting CC_DO_FESHBACH = 2, and are currently available only within the plane-wave approximation.

The integrals in Eq. (7.62) are evaluated analytically. Integration in Eq. (7.63) is done numerically, and is split into two or three intervals to bypass the singularity at $E={E}_{0}-{E}_{\mu}$. The upper limits of those intervals are set to default values related to ${E}_{0}$. They can also be customized (except for the first interval) by setting CC_FESHBACH_DELTA_INTB = XX and/or CC_FESHBACH_DELTA_INTC = YY where XX and/or YY are desired upper integration limits in units of eV.

CC_DO_FESHBACH

Activates calculation of resonance widths using Feshbach-Fano approach.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
do not invoke Feshbach-Fano calculation
1
invoke Feshbach-Fano calculation of the resonance width
2
invoke Feshbach-Fano calculation of the resonance width and resonance shift

RECOMMENDATION:

Initial and final states should be correctly specified.

CC_FESHBACH_CW

Activates Coulomb wave description of the ejected electron.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
Use plane wave
1
Use Coulomb wave

RECOMMENDATION:

Additional details need to be specified in *$coulomb_wave* section.

CC_FESHBACH_DELTA_INTB

Specifies integration limits in calculation of energy shift in Feshbach-Fano calculations.

TYPE:

INTEGER

DEFAULT:

Preset

OPTIONS:

$n$
corresponds to energy limit in eV

RECOMMENDATION:

Use default.

CC_FESHBACH_DELTA_INTC

Specifies integration limits in calculation of energy shift in Feshbach-Fano calculations.

TYPE:

INTEGER

DEFAULT:

Preset

OPTIONS:

$n$
corresponds to energy limit in eV

RECOMMENDATION:

Use default.

Examples 7.10.9.1 and 7.10.9.1 illustrate calculation of resonant Auger decay of core-ionized water molecule. The initial state is described by CVS-EOM-IP-CCSD and the decay channels are described by EOM-DIP-CCSD. Example 7.10.9.1 uses a plane-wave representation of the ejected electron. In example 7.10.9.1, the autoionizing electron is described by the Coulomb wave, represented by a pseudo-partial wave expansion over PW-CGTO functions.

$molecule 0 1 O 0.0000 0.000 0.0000 H -0.7528 0.000 -0.5917 H 0.7528 0.000 -0.5917 $end $rem METHOD ccsd BASIS 6-311+G(3df) CVS_EOM_IP_BETA [1,0,0,0] DIP_SINGLETS [4,1,2,2] DIP_TRIPLETS [1,1,2,2] CC_DO_DYSON 1 CC_DO_FESHBACH 1 $end

$molecule 0 1 O 0.0000 0.000 0.0000 H -0.7528 0.000 -0.5917 H 0.7528 0.000 -0.5917 $end $rem METHOD ccsd BASIS 6-311+G(3df) CVS_EOM_IP_BETA [1,0,0,0] DIP_TRIPLETS [1,1,2,2] CC_DO_DYSON 1 CC_DO_FESHBACH 1 CC_FESHBACH_CW 1 $end $trans_prop state_list cvs_ip_beta 1 1 dip_triplets 3 2 end_list state_pair_list 1 2 ! transition 1 <-> 2 end_pairs calc dyson $end $coulomb_wave !This PW-CGTO expansion of CW is optimized for Z = 4.9 and Ek = 475.7 eV !CW is centered on oxygen (atom #1), has Lmax = 2, and n = 4 GTOs for each L 1 2 4 !List of GTO exponents for each consecutive pseudo-partial wave from L = 0 to Lmax 33.92543607 0.85503320 0.03878479 0.00464513 10.09805405 0.75935967 0.06727680 0.00646507 6.96653113 0.94413668 0.11599464 0.01425085 !List of corresponding GTO contraction coefficients - real and imaginary parts 1.15237075 -1.28233348 0.96764647 -0.30588374 0.94868507 0.99338435 -1.18258037 -0.06876149 -0.62304129 0.90336892 -0.14457938 0.18631218 -0.07528422 0.01001695 -0.00950295 -0.02658981 0.22796804 -0.19298801 0.01268528 -0.03579628 0.00369451 -0.00318780 0.00068338 0.00016431 $end