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
Phys. Rev. A
(1980), 22, pp. 206. 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. 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 -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.
J. Chem. Theory Comput.
(2016), 12, pp. 5018. , 104 Acc. Chem. Res.
(2020), 53, pp. 1306. 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.
J. Chem. Theory Comput.
(2014), 10, pp. 4557. , 323 J. Comput. Chem.
(2020), 41, pp. 1081. 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. , 409 Chem. Phys. Lett.
(2018), 699, pp. 279.
$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
and the oscillator strengths estimated from
where is a core orbital and 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.
Chem. Phys. Lett.
(2018), 696, pp. 119. 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.
Adv. Quantum Chem.
(1972), 6, pp. 1.
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