7.10.10 Auger Spectra and Lifetimes of Core-Level States

Certain types of resonances can be described by using real-valued EOM-CC wave functions via Feshbach-Fano approach. ³⁵⁸ Feshbach H.
Ann. Phys.
(1962), 19, pp. 287. Link ^{, 350} Fano U.
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. ¹¹⁷⁵ Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link ^{, 1176} Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084125. Link 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. ³⁵⁸ Feshbach H.
Ann. Phys.
(1962), 19, pp. 287. Link ^{, 350} Fano U.
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 7.10.8).

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 ${\varphi }_k$, ${\chi }_{\mu ,E_k}\sim 𝒜\{{\varphi }_k{\mathrm{\Psi }}_{\mu }\}$. 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

and partial energy correction ${\mathrm{\Delta }}_{\mu }$ to the zero-order resonance position $E_0$, defined as

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. ¹¹⁷⁵ Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link ^{, 1176} Skomorowski W., Krylov A. I.
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.  1175 Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link .

The implementation of Feshbach widths includes numerical integration over all possible directions of the emitted electron (k-vector). This integration over the sphere is carried out using Lebedev’s quadrature, with the default order of 5. For molecules with delocalized core-hole states (e.g., benzene), higher-order quadrature may be needed. The order of the quadrature is controlled by CC_FESHBACH_INT_ORDER.

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 the $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. (7.84) are evaluated analytically. Integration in Eq. (7.85) 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.

A molecular orbital description of the Auger process (or other two-electron decay processes such as intermolecular Coulomb and electron-transfer-mediated decay) can be obtained from the singular-value decompostion of the two-particle Dyson amplitudes (${\mathrm{\Gamma }}_{\mu }$) ⁵⁹⁴ Jayadev N. K., Skomorowski W., Krylov A. I.
J. Phys. Chem. Lett.
(2023), 14, pp. 8612. Link . The procedure yields several three-orbital sets, which represent the core-vacancy state and the valence decay states. These sets, called Natural Auger Orbitals, provide the most compact description of the two-electron decay process ⁵⁹⁴ Jayadev N. K., Skomorowski W., Krylov A. I.
J. Phys. Chem. Lett.
(2023), 14, pp. 8612. Link . The calculations of NAOs can be invoked within Feshbach calculations by adding CC_DO_NAO keyword. The NAOs are printed in the MolDen format in a subdirectory created in the working directory. The “energies” are the squares of singular values (not normalized), core-hole NAOs are assigned populations of 1, and valence decay NAOs are assigned populations of 0. The default threshold for the NAOs to be printed is 0.20. It can be controlled by the user using the WFA_ORB_THRESH keyword (see Section 10.2). This feature is illustated in examles 7.10.10.1, 7.10.10.1, and 7.10.10.1 below.

Note:  NAOs are computed using ${\mathrm{\Gamma }}_{r\mathbf{}\beta }^{p\mathbf{}\alpha \mathrm{,}q\mathbf{}\beta }$ block of the two-body Dyson amplitude. This means that in the non-resonant Auger decay calculations, the core hole should correspond to removing $\beta$-electron.