Certain types of resonances can be described by using real-valued EOM-CC
wave functions via Feshbach–Fano approach.
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pp. 287.
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,
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(1961),
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pp. 1866.
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In this section we describe
the application of Feshbach–Fano approach to core-excited and
core-ionized states.
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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. 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:
(7.93) |
where and 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, .
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
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for electronic resonances (the latter approach is described in sections
7.10.10 and 7.10.11).
The Feshbach–Fano theory
370
Ann. Phys.
(1962),
19,
pp. 287.
Link
,
362
Phys. Rev.
(1961),
124,
pp. 1866.
Link
invokes
two projection operators, and , 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 is a core-hole ionized or core-hole excited state, which can be described by CVS-EOM-CC. The final (continuum-like) state is represented by an antisymmetrized product of a stable channel state (described by an appropriate EOM-CC model) and a continuum orbital , . Note that is a state with one electron less than . Two essential parameters defining AES are the rate of the decay into a channel , given as
(7.94) |
and partial energy correction to the zero-order resonance position , defined as
(7.95) |
In the expressions above is the electronic Hamiltonian,
is the energy of the channel state ,
is the energy of the ejected electron (),
superscripts denote left and right
EOM-CCSD wavefunctions, and stands for the Cauchy principle value.
Calculations of are activated with the CC_DO_FESHBACH keyword.
By default, the continuum orbital is approximated with a plane
wave.
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It is also possible to model 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.
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. 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 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 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 . The upper limits of those intervals
are set to default values related to . 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 ()
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. 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
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. 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 block of the two-body Dyson amplitude. This means that in the non-resonant Auger decay calculations, the core hole should correspond to removing -electron.
CC_DO_FESHBACH
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
CC_FESHBACH_CW
Activates Coulomb wave description of the ejected electron.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Use plane wave
1
Use Coulomb wave generated internally
3
Use Coulomb basis provided by the user in the $coulomb_wave section
RECOMMENDATION:
Additional details need to be specified in $coulomb_wave section.
CC_FESHBACH_INT_ORDER
CC_FESHBACH_INT_ORDER
Controls k-vector integration grid in calculations of resonance widths using Feshbach-Fano approach.
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
corresponds to the Lebedev quadrature order
RECOMMENDATION:
Use default, unless tighter convergence is desired (16 gives fully converged widths).
CC_FESHBACH_DELTA_INTB
CC_FESHBACH_DELTA_INTB
Specifies integration limits in calculation of energy shift in Feshbach-Fano calculations.
TYPE:
INTEGER
DEFAULT:
Preset
OPTIONS:
corresponds to energy limit in eV
RECOMMENDATION:
Use default.
CC_FESHBACH_DELTA_INTC
CC_FESHBACH_DELTA_INTC
Specifies integration limits in calculation of energy shift in Feshbach-Fano calculations.
TYPE:
INTEGER
DEFAULT:
Preset
OPTIONS:
corresponds to energy limit in eV
RECOMMENDATION:
Use default.
CC_DO_NAO
CC_DO_NAO
Activates calculation of NAOs within Feshbach–Fano calculation of the decay widths.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
do not compute NAOs
1
compute NAOs
RECOMMENDATION:
Initial and final states should be correctly specified.
Examples 7.10.10.1 and 7.10.10.1 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 7.10.10.1 uses a plane-wave representation of the ejected electron. In example 7.10.10.1, the autoionizing electron is described by the Coulomb wave, represented by a pseudo-partial wave expansion over PW-CGTO functions. Examples 7.10.10.1, 7.10.10.1, and 7.10.10.1 illustarte calculations of NAOs for regular and resonant Auger decay.
Example 7.72 Calculation of Auger decay rates of core-ionized water molecule to selected singlet and triplet final states. Continuum orbital is a plane wave.
$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
Example 7.73 Calculation of Auger decay rates of core-ionized water molecule to selected singlet and triplet final states. Continuum orbital is approximated by a Coulomb wave.
$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 lmax = 4 ncoeff = 6 Rmin = 0.05 Rmax = 10 Rmid = 2 Charge = 4.9 cwave_center = 1 !CW is centered on oxygen (atom #1) print_cw = false $end
Example 7.74 Calculation of Natural Auger Orbitals for regular (non-resonant) Auger decay in water for both singlet and triplet decay channels.
$comment NAO calculation - regular Auger decay in water $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 jobtype sp method eom-ccsd basis 6-31g cvs_eom_ip_beta [1,0,0,0] !Initial core-hole state dip_singlets [1,1,0,0] !Final singlet decay channels dip_triplets [0,1,0,0] !Final triplet decay channels cc_do_dyson 1 !Needed for Feshbach-type calculations cc_do_feshbach 1 !Needed for Feshbach-type calculations cc_do_nao 1 !Needed for natural Auger orbitals wfa_orb_thresh 1 !Setting the threshold for singular value dceomposition mem_total 4000 $end
Example 7.75 Calculation of Natural Auger Orbitals for regular (non-resonant) Auger decay in benzene for both singlet and triplet decay channels, and for several core-hole states.
$comment NAO calculation - regular Auger decay in benzene $end $molecule 0 1 H 2.4750347531 0.0000000000 0.0000000000 C 1.3935929418 0.0000000000 0.0000000000 C 0.6967964709 1.2068868901 0.0000000000 H 1.2375173766 2.1434429715 0.0000000000 C -0.6967964709 1.2068868901 0.0000000000 H -1.2375173766 2.1434429715 0.0000000000 C -1.3935929418 0.0000000000 0.0000000000 H -2.4750347531 0.0000000000 0.0000000000 C -0.6967964709 -1.2068868901 0.0000000000 H -1.2375173766 -2.1434429715 0.0000000000 C 0.6967964709 -1.2068868901 0.0000000000 H 1.2375173766 -2.1434429715 0.0000000000 $end $rem jobtype sp method ccsd basis 6-31g cvs_eom_ip_beta [1,0,0,0,0,0,1,1] !Initial core-hole state dip_singlets [2,0,0,1,0,0,0,0] !Final singlet decay channel dip_triplets [1,0,0,1,0,0,0,0] !Final triplet decay channel cc_do_dyson 1 !Needed for Feshbach-type calculation cc_do_feshbach 1 !Needed for Feshbach-type calculation cc_do_nao 1 !Needed for natural Auger orbital calculation mem_total 8000 $end
Example 7.76 Calculation of Natural Auger Orbitals for resonant Auger decay in water.
$comment NAO calculation - resonant Auger decay in water $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 jobtype sp method eom-ccsd basis 6-31g cvs_ee_states [1,0,0,0] !Initial core-hole state ip_states [1,0,1,0] !Final decay channels cc_do_dyson 1 !Needed for Feshbach-type calculations cc_do_feshbach 1 !Needed for Feshbach-type calculations cc_do_nao 1 !Needed for natural Auger orbitals mem_total 4000 $end