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(May 16, 2021)

X-ray absorption spectroscopy can be calculated using TDDFT
by restricting the excitation space to include excitations from a
set of core-orbitals. This is achieved
by using TRNSS and *$alist* which invoke the
core-valence separation
^{
174
}
Phys. Rev. A

(1980),
22,
pp. 206.
Link
within TDDFT for calculations of core-excited states. Note that these
calculations are not suited to describe the extended X-ray absorption fine structure
(EXAFS) region which corresponds to the scattering of the ionised electron by the
neighboring atoms. GGA
and hybrid exchange-correlation functionals tend to underestimate
core-excitation energies and Q-Chem has short-range corrected (SRC) functionals
available that are designed to predict K-edge core-excitation energies accurately.
^{
100
}
Phys. Chem. Chem. Phys.

(2009),
11,
pp. 10350.
Link
These functionals are a modification of the more familiar long-range corrected
functionals (discussed in Section 5.6). However, in SRC-DFT the
short-range component of the Coulomb operator is predominantly Hartree-Fock
exchange, while the mid to long-range component is primarily treated with
standard DFT exchange.
Relativistic effects become increasingly significant for calculation
of X-ray absorption spectra at the K-edge of heavier elements. The REL_SHIFT keyword
introduces a correction to the calculated excitation energies to account for these
effects. This is illustrated in the example below. Calculations for
$L$-shell excitations will also complicated by core-hole spin orbit coupling.

$molecule 0 1 H 1.196206 0.000000 -0.469131 P 0.000000 0.000000 0.303157 H -0.598103 -1.035945 -0.469131 H -0.598103 1.035945 -0.469131 $end $rem EXCHANGE SRC2-R2 BASIS 6-311(2+,2+)G** CIS_N_ROOTS 6 CIS_TRIPLETS false TRNSS true TRTYPE 3 N_SOL 1 REL_SHIFT 15 $end $alist 1 $end

Despite the relatively low computational cost of TDDFT, it can become challenging to calculate
X-ray absorption spectra for large systems. The high density of core-excited states makes
simulating spectra more computationally expensive than comparable calculations of the
UV/vis spectra. This is particularly the case when excitations from many core-orbitals are
required, which is often the situation when studying the carbon K-edge of organic molecules.
There are two aspects to the computational cost, firstly the CPU time required and secondly
the memory required. Q-Chem has available an implementation of TDDFT that is particularly
efficient for the calculation of X-ray absorption spectra.
^{
103
}
J. Chem. Theory Comput.

(2016),
12,
pp. 5018.
Link
^{,}
^{
104
}
Acc. Chem. Res.

(2020),
53,
pp. 1306.
Link
This
approach greatly increases the speed
of the calculations through integral screening controlled by the XAS_SCREEN_LEVEL
and XAS_EDGE keywords, while also reducing the memory required. The memory required
for these calculations can be reduced further through the TDDFT_NVIRT keyword that
reduces the number of virtual orbitals included in the TDDFT calculation.

$molecule 0 1 C 0.000000 0.000000 -0.648906 O 0.000000 0.000000 0.486357 $end $rem EXCHANGE SRC1-R1 BASIS 6-311G* CIS_N_ROOTS 6 CIS_TRIPLETS false TRNSS true TRTYPE 3 N_SOL 1 FAST_XAS true XAS_EDGE 6 XAS_SCREEN_LEVEL 1 $end $alist 1 $end

It is also possible to compute X-ray emission spectroscopy using TDDFT. This is achieved
by using a reference determinant with a core-hole.
^{
1142
}
J. Chem. Theory Comput.

(2014),
10,
pp. 4557.
Link
^{,}
^{
323
}
J. Comput. Chem.

(2020),
41,
pp. 1081.
Link
The calculated
excitation energies can be quite sensitive to the choice of basis set, and for the K-edge
of heavier elements it can be necessary to use large or specially adpated basis sets
to provide a good description of the core region.
^{
322
}
Theor. Chem. Acc.

(2018),
137,
pp. 6.
Link
^{,}
^{
409
}
Chem. Phys. Lett.

(2018),
699,
pp. 279.
Link

$molecule 0 1 O 0.0000 0.0000 0.1168 H 0.0000 0.7629 -0.4672 H 0.0000 -0.7629 -0.4672 $end $rem method cam-b3lyp basis cc-pvdz $end @@@ $molecule +1 2 O 0.0000 0.0000 0.1168 H 0.0000 0.7629 -0.4672 H 0.0000 -0.7629 -0.4672 $end $rem method cam-b3lyp basis cc-pvdz scf_guess read mom_start 1 cis_n_roots 5 cis_triplets false $end $occupied 1:5 2:5 $end

Alternatively, an XES spectrum can be determined directly from the DFT wherein the transitions energies are determinded from the energy difference between the orbital energies of the neutral ground state molecule

$$\mathrm{\Delta}E={\u03f5}_{v}-{\u03f5}_{c}$$ | (7.117) |

and the oscillator strengths estimated from

$$f\propto {|\u27e8{\varphi}_{c}|\widehat{\mu}|{\varphi}_{v}\u27e9|}^{2}$$ | (7.118) |

where ${\varphi}_{c}$ is a core orbital and ${\varphi}_{v}$ is a valence orbital.
The critical benefit from this approach is that only a calculation for the
ground state is required, however as a consequence no account of the orbital
relaxation for the core-ionised state is included. It has been shown that using
this approach in conjunction with SRC functionals can lead to
reasonable estimates of the transition energies and this is discussed in Ref. 407,
and this approach can be applied to study large systems.
^{
408
}
Chem. Phys. Lett.

(2018),
696,
pp. 119.
Link
This approach to calculating XES is illustrated by Example 7.13.1 and extension of this approach
to resonant X-ray emission spectroscopy is possible by using this feature together with MOM.
The keywords NCORE_XES and NVAL_XES specify which transitions to compute.

Note: This feature is only available with GEN_SCFMAN = FALSE .

$molecule 0 1 C 0.0000000000 0.0000000000 0.5121520001 O 0.0000000000 0.0000000000 -0.6942567610 H 0.9377642813 0.0000000000 1.1074358558 H -0.9377642813 0.0000000000 1.1074358558 $end $rem METHOD src1r1 BASIS 6-311G** NCORE_XES 2 NVAL_XES 4 GEN_SCFMAN false $end

Another approach of partial account of strong orbital relaxation is called
transition potential (TP-)DFT.
This approach uses Kohn-Sham orbital eigenvalue differences to approximate core-level excitation
energies, based on a Kohn-Sham calculation with
partial occupations of the orbitals involved in the transitions. This can be justified based on a
Taylor expansion in terms of the orbital occupations, as originally suggested by Slater.
^{
1011
}
Adv. Quantum Chem.

(1972),
6,
pp. 1.
Link

Note: This is an experimental feature, only energies are currently implemented.

$molecule 0 1 O 0.0000000000 0.0000000000 -0.1239093563 H 0.0000000000 1.4299372840 0.9832657567 H 0.0000000000 -1.4299372840 0.9832657567 $end $rem METHOD b3lyp BASIS aug-cc-pCVQZ INPUT_BOHR true $end @@@ $molecule read $end $rem METHOD b3lyp BASIS aug-cc-pCVQZ INPUT_BOHR true UNRESTRICTED true TPDFT_ATOM 1 TPDFT_FRAC 50 TPDFT_LUMO 0 $end

$molecule 0 1 O 0.0000000000 0.0000000000 -0.1239093563 H 0.0000000000 1.4299372840 0.9832657567 H 0.0000000000 -1.4299372840 0.9832657567 $end $rem METHOD b3lyp BASIS aug-cc-pCVQZ INPUT_BOHR true $end @@@ $molecule read $end $rem METHOD b3lyp BASIS aug-cc-pCVQZ INPUT_BOHR true UNRESTRICTED true TPDFT_ATOM 1 TPDFT_FRAC 50 $end