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. ³⁷⁰ Feshbach H.
Ann. Phys.
(1962), 19, pp. 287. Link ^{, 362} 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. ¹²⁰⁸ Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link ^{, 1209} 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. The energy of the Auger electrons are computed as differences between the intial core-hole state and final decay states. For example, for regular Auger decay:

where $E_n^{\text{CoreIP}}$ and $E_{\mu }^{\text{DIP}}$ are energies of the initial (core-hole) and final (doubly ionized valence) states, respectively. The intensities of each decay channel are given by partial decay widths, ${\mathrm{\Gamma }}_{\mu n}$. In the crudest approximation, one can assign each decay channel unit intensity—so-computed Auger spectra reflect density of doubly ionized states of the system. For more rigorous treatments, the decay widths can be computed by either the Feshbach–Fano or complex-variable approaches ¹²⁰⁸ Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link for electronic resonances (the latter approach is described in sections 7.10.10 and 7.10.11).

The Feshbach–Fano theory ³⁷⁰ Feshbach H.
Ann. Phys.
(1962), 19, pp. 287. Link ^{, 362} Fano U.
Phys. Rev.
(1961), 124, pp. 1866. Link 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 by an antisymmetrized product of a stable channel state ${\mathrm{\Psi }}_{\mu }$ (described by an appropriate EOM-CC 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. ¹²⁰⁸ Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link ^{, 1209} 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 an additional input section $coulomb_wave, which provides the Coulomb wave parameters for the expansion of the Coulomb wave in terms of product of a plane wave and Gaussian-type functions (PW-CGTOs), as detailed in Ref.  1208 Skomorowski W., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 084124. Link . The parameters are listed below.

$coulomb_wave
   lmax   =   4      ! Maximum angular momentum in the pseudo partial wave expansion
   ncoeff =   6      ! Number of contraction coefficient for each \emph{l}
   Rmin   =   0.05   ! PW-CGTO grid in atomic unit
   Rmax   =   10
   Rmid   =   2
   Charge =   6.2    ! Effective charge of the Coulomb wave
   cwave_center = 1  ! Atom index for the position of Coulomb wave. Starts from 1.
   print_cw = false  ! For printing the partial wave and the PW-CGTO basis
$end

The Coulomb wave is approximated by the linear combination of PW-CGTOs. Typically, lmax = 4-6 and ncoeff = 6-8 provide a good approximation; increasing these values significantly increases computational cost. The grid is set up by the Rmin, Rmax, and Rmid parameters. If Rmid is specified, the grid spacing is set to 0.05 au from Rmin to Rmid and 0.1 au from Rmid to Rmax. If Rmid is not provided, a uniform grid spacing of 0.1 au is used. The grid should be large enough to accurately represent the continuum function in the space where bound-domain orbitals involved in the decay have considerable amplitudes. The cwave_center sets the position of the Coulomb wave. By default, it is at the origin (cwave_center = 0). One can place it at a specific atom or any other location, such as the midpoint of a bond, by defining a ghost atom (Gh) in the input and setting its index as the cwave_center. The print_cw option enables printing the Coulomb wave, its fitted representation (for each l), and the corresponding Coulomb basis. The user can check the Fit square error: in the qchem output file and adjust these parameters accordingly by varying ncoeff.

It is also possible to provide the direct expansion of the Coulomb wave in terms of PW-CGTOs from external optimization (for the given effective charge and kinetic energy) in the input section $coulomb_wave. This is enabled by setting CC_FESHBACH_CW = 3.

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. Examples 7.10.10.1 and 7.10.10.1 below illustrate calculations of auger decay rates using plane-wave and Coulomb wave treatments.

For non-resonant Auger decay, the initial state can be conveniently computed by CVS-EOM-IP-CCSD and its stable decay channels can be computed by EOM-DIP-CCSD. A 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 and the corresponding decay channels can be computed by EOM-IP-CCSD. 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. (7.94) are evaluated analytically. Integration in Eq. (7.95) 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 (${\mathrm{\Gamma }}_{\mu }$) ⁶¹² 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 ⁶¹² 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 the 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.