7.13 Core Ionization Energies and Core-Excited States

7.13.1 Calculations of X-ray spectroscopy with (TD)DFT

(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 Cederbaum L. S., Domcke W., Schirmer J.
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 Besley N. A., Peach M. J. G., Tozer D. J.
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.

Example 7.133  This example shows a calculation of the core-excited states at the phosphorus K-edge with a short-range corrected functional.

$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

View output

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 Besley N. A.
J. Chem. Theory Comput.
(2016), 12, pp. 5018.
Link
, 104 Besley N. A.
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.

Example 7.134  This example shows a calculation of the core-excited states at the oxygen K-edge of CO with a short-range corrected functional using fast/low memory TDDFT.

$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

View output

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 Wadey J. D., Besley N. A.
J. Chem. Theory Comput.
(2014), 10, pp. 4557.
Link
, 323 Fouda A.E.A., Besley N. A.
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 Fouda A.E.A., Besley N. A.
Theor. Chem. Acc.
(2018), 137, pp. 6.
Link
, 409 Hanson-Heine M. W. D., George M. W., Besley N. A.
Chem. Phys. Lett.
(2018), 699, pp. 279.
Link

Example 7.135  This example shows a calculation of the XES spectrum of water using TDDFT+MOM.

$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

View output

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

ΔE=ϵv-ϵc (7.117)

and the oscillator strengths estimated from

f|ϕc|μ^|ϕv|2 (7.118)

where ϕc is a core orbital and ϕv 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 Hanson-Heine M. W. D., George M. W., Besley N. A.
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 .

Example 7.136  The calculation of the XES spectrum of water using Koopmans’ theorem within KS-DFT with a short-range corrected functional.

$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

View output

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 Slater J. C.
Adv. Quantum Chem.
(1972), 6, pp. 1.
Link

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

Example 7.137  This example shows a calculation using TP-DFT.

$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

View output

Example 7.138  This example shows a calculation using TP-DFT.

$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

View output