Certain types of resonances can be described by using real-valued EOM-CC wave functions via Feshbach-Fano approach.318, 311 In this section we describe the application of Feshbach-Fano approach to core-excited and core-ionized states.1038, 1039 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.318, 311 The theory 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 as an antisymmetrized product of a stable channel state (described by an appropriate EOM-EE 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
and partial energy correction to the zero-order resonance position , defined as
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.1038, 1039 It is also possible to model 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. 1038.
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 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. (7.63) are evaluated analytically. Integration in Eq. (7.64) 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.
Examples 22.214.171.124 and 126.96.36.199 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 188.8.131.52 uses a plane-wave representation of the ejected electron. In example 184.108.40.206, 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