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(May 16, 2021)

In experiments using high-energy radiation (such as X-ray spectroscopy, EXAFS,
NEXAFS, XAS, XES, RIXS, REXS, etc) core electrons can be ionized or excited to
low-lying virtual orbitals. There are several ways to compute ionization or
excitation energies of core electrons in Q-Chem. Standard approaches for
excited and ionized states need to be modified to tackle core-level states,
because these states have very high energies and are embedded in the ionization
continuum (*i.e.*, they are Feshbach resonances
^{
955
}
J. Chem. Phys.

(2017),
147,
pp. 014107.
Link
).

A highly robust and accurate strategy is to invoke many-body methods, such as
EOM or ADC, together with the core-valence separation (CVS)
scheme
^{
174
}
Phys. Rev. A

(1980),
22,
pp. 206.
Link
. In this approach, the excitations involving core
electrons are decoupled from the rest of the configurational space. This allows
one to reduce computational costs and decouple the highly excited core states
from the continuum. These methods are described in Sections 7.10.7
and 7.11.5; CVS can also be deployed within TDDFT by using
TRNSS (see Sections 7.3.2 and
7.13.1).

An alternative highly accurate approach for finding core-excitation energies of
closed-shell molecules is to use the Restricted Open-Shell Kohn-Sham approach
described in Section 7.8.1. ROKS is not systematically improvable
like EOM or ADC methods, but is nonetheless quite accurate, with modern density
functionals being capable of predicting excitation energies to $$ eV
error
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397
}
J. Phys. Chem. Lett.

(2020),
11,
pp. 775.
Link
. The great strength of the ROKS approach is its
computational efficiency—highly accurate results can be obtained for the same
$O({N}^{3})$ scaling as ground-state meta-GGAs, vs the $O({N}^{6})$ scaling of EOM-CCSD
or $O({N}^{5})$ scaling of ADC(2). The basis set requirements of ROKS are also much
more modest than wave function theories, with a mixed basis strategy being
highly effective in practice. Details about using ROKS for core-excitations is
supplied at 7.13.3.

Within EOM-CC formalism, one can also use an approximate EOM-EE/IP methods in
which the target states are described by single excitations and double
excitations are treated perturbatively; these methods are described in
Section 7.10.14. While being moderately useful, these methods are
less accurate than the CVS-EOM variants
^{
955
}
J. Chem. Phys.

(2017),
147,
pp. 014107.
Link
.

In addition, one can use the $\mathrm{\Delta}E$ approach, which amounts to a simple energy difference calculation in which core ionization is computed from energy differences computed for the neutral and core-ionized state. It is illustrated by example 7.13 below.

$molecule 0,1 C 0.000000 0.000000 0.000000 H 0.631339 0.631339 0.631339 H -0.631339 -0.631339 0.631339 H -0.631339 0.631339 -0.631339 H 0.631339 -0.631339 -0.631339 $end $rem EXCHANGE = HF CORRELATION = CCSD BASIS = 6-31G* MAX_CIS_CYCLES = 100 $end @@@ $molecule +1,2 C 0.000000 0.000000 0.000000 H 0.631339 0.631339 0.631339 H -0.631339 -0.631339 0.631339 H -0.631339 0.631339 -0.631339 H 0.631339 -0.631339 -0.631339 $end $rem UNRESTRICTED = TRUE EXCHANGE = HF BASIS = 6-31G* MAX_CIS_CYCLES = 100 SCF_GUESS = read Read MOs from previous job and use occupied as specified below CORRELATION = CCSD MOM_START = 1 Do not reorder orbitals in SCF procedure! $end $occupied 1 2 3 4 5 2 3 4 5 $end

In this job, we first compute the HF and CCSD energies of neutral CH${}_{4}$:
${E}_{\mathrm{SCF}}=-40.1949062375$ and ${E}_{\mathrm{CCSD}}=-40.35748087$ (HF orbital energy
of the neutral gives the Koopmans IE, which is 11.210 hartree = 305.03 eV). In the
second job, we do the same for core-ionized CH${}_{4}$. To obtain the desired SCF
solution, MOM_START option and *$occupied* keyword are used. The
resulting energies are ${E}_{\mathrm{SCF}}=-29.4656758483$ ($\u27e8{S}^{2}\u27e9$ =
0.7730) and ${E}_{\mathrm{CCSD}}=-29.64793957$. Thus, $\mathrm{\Delta}{E}_{\mathrm{CCSD}}=(40.357481-29.647940)=10.709$ hartree = 291.42 eV.

This approach can be further extended to obtain multiple excited states involving core electrons by performing CIS, TDDFT, or EOM-EE calculations.

Note: This approach often leads to convergence problems in correlated calculations.

One can also use the following trick illustrated by example 7.13.

$molecule 0,1 C 0.000000 0.000000 0.000000 H 0.631339 0.631339 0.631339 H -0.631339 -0.631339 0.631339 H -0.631339 0.631339 -0.631339 H 0.631339 -0.631339 -0.631339 $end $rem EXCHANGE = HF BASIS = 6-31G* MAX_CIS_CYCLES = 100 CORRELATION = CCSD CCMAN2 = false N_FROZEN_CORE = 4 Freeze all valence orbitals IP_STATES = [1,0,0,0] Find one EOM_IP state $end $reorder_mo 5 2 3 4 1 5 2 3 4 1 $end

Here we use EOM-IP to compute core-ionized states. Since core states are very high in energy, we use “frozen core” trick to eliminate valence ionized states from the calculation. That is, we reorder MOs such that our core is the last occupied orbital and then freeze all the rest. The so computed EOM-IP energy is 245.57 eV. From the EOM-IP amplitude, we note that this state of a Koopmans character (dominated by single core ionization); thus, canonical HF MOs provide good representation of the correlated Dyson orbital. The same strategy can be used to compute core-excited states.

Note: The accuracy of this approach is rather poor and is similar to Koopmans’ approximation.