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# 7.13.2 Calculations of X-Ray Spectroscopy with TDDFT

(July 14, 2022)

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 setting TRNSS = TRUE in the $rem section, which triggers the use of TDDFT with a truncated excitation space as described in Section 7.3.2. The occupied core orbitals that the user desires to be active in such a calculation should be listed individually in the$alist input section, and the number of such orbitals must be specified using N_SOL in the $rem section. This invokes the CVS approximation, which for TDDFT amounts to freezing all of the occupied orbitals except for the ones that are listed in$alist, while using the full virtual space. Such calculations are not suited to describe the extended x-ray absorption fine structure (EXAFS) region, which corresponds to the scattering of the ionized electron by the neighboring atoms.

Standard exchange-correlation functionals (including hybrids) tend to severely underestimate core excitation energies, 106 Besley N. A., Peach M. J. G., Tozer D. J.
Phys. Chem. Chem. Phys.
(2009), 11, pp. 10350.
although chemical shifts (from one compound to the next) may still be valid. 348 Fransson T. et al.
J. Chem. Theory Comput.
(2021), 17, pp. 1618.
Q-Chem has short-range corrected (SRC) functionals available that are designed to predict K-edge core excitation energies accurately. 106 Besley N. A., Peach M. J. G., Tozer D. J.
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. Uses of SRC-TDDFT in conjunction with the CVS approximation is illustrated in Example 7.13.2. See Section 7.3.3 for basic TDDFT job control.

Example 7.149  Calculation of carbon K-edge [C(1s) $\rightarrow$ virtual excitations] using SRC-TDDFT within the CVS approximation.

$comment Carbon K-edge excitations for acetone. The only active occupied orbitals are three C(1s). The SRC1-R1 is parameterized for "first row" (C, N, O, ...)$end

$molecule 0 1 C -3.0219081 1.0061477 0.0000001 O -2.9337180 2.2246186 0.0000001 C -1.7817549 0.1662163 -0.0000003 C -4.3700966 0.3535647 0.0000004 H -0.8735407 0.8061311 -0.0000005 H -1.7663727 -0.4765415 -0.9049102 H -1.7663723 -0.4765416 0.9049094 H -5.1766925 1.1175964 0.0000006 H -4.4778785 -0.2802782 0.9049091 H -4.4778790 -0.2802781 -0.9049084$end

$rem exchange src1-r1 ! r1 = "first row" basis 6-31++G* cis_n_roots 25 cis_triplets false trnss true trtype 3 n_sol 3 ! no. of active orbs$end

$alist 2 3 4$end



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 Example 7.13.2 below. (Although scalar shift may be appropriate for K-edge excitations, but L-edge and M-edge excitations will be complicated by core-hole spin orbit coupling.)

REL_SHIFT

REL_SHIFT
Corrects the calculated TDDFT excitation energy for scalar relativistic effects.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
$Z$ Corresponding to the atomic number of the core-ionized element.
RECOMMENDATION:
None

Example 7.150  Calculation of core-excited states at the phosphorus $K$-edge, including a scalar relativistic shift.

$molecule 0 1 H 1.196206 0.000000 -0.469131 P 0.000000 0.000000 0.303157 H -0.598103 -1.035945 -0.469131 H -0.598103 1.035945 -0.469131$end

$rem EXCHANGE SRC2-R2 ! R2 = "second row" (Al, S, P, ...) BASIS 6-311(2+,2+)G** CIS_N_ROOTS 6 CIS_TRIPLETS false TRNSS true TRTYPE 3 N_SOL 1 REL_SHIFT 15$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. An implementation of TDDFT called “fTDDFTs” combines aggressive integral screening and a coarse DFT quadrature grid, which is especially efficient for the calculation of x-ray absorption spectra. 109 Besley N. A.
J. Chem. Theory Comput.
(2016), 12, pp. 5018.
, 110 Besley N. A.
Acc. Chem. Res.
(2020), 53, pp. 1306.
This approach may be fine-tuned using the $rem variables FAST_XAS, XAS_SCREEN_LEVEL and XAS_EDGE. 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. Job control variables for fTDDFTs are listed below and are illustrated in Example 7.13.2. Note: Currently fTDDFTs works only for restricted CIS/TDA calculations and is not parallelized. (Multiple threads can be used for the initial SCF calculation but the subsequent CIS/TDA calculation is performed in serial.) FAST_XAS FAST_XAS Controls whether fast TDDFT for core excitations is used. TYPE: LOGICAL DEFAULT: FALSE Normal TDDFT calculation. OPTIONS: TRUE Use fast TDDFT. RECOMMENDATION: None XAS_SCREEN_LEVEL XAS_SCREEN_LEVEL Sets the integral screening procedure for fast TDDFT. TYPE: INTEGER DEFAULT: No default OPTIONS: $1$ only evaluate integrals that include the inner core basis function on relevant atom(s). $2$ only evaluate integrals that include basis functions on relevant atom(s). RECOMMENDATION: 1 XAS_EDGE XAS_EDGE Specifies the nuclear charge of element being excited. TYPE: INTEGER DEFAULT: No default OPTIONS: $n$ Corresponding to the nuclear charge of element being excited. RECOMMENDATION: None TDDFT_NVIRT TDDFT_NVIRT Specifies the number of virtual orbitals included in the XAS TDDFT calculation. TYPE: INTEGER DEFAULT: No default OPTIONS: $n$ Corresponding to the lowest energy $n$ virtual orbitals. RECOMMENDATION: None Example 7.151 Fast, low-memory calculation of core-excited states at the oxygen K-edge of CO using fTDDFTs. $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. 1243 Wadey J. D., Besley N. A. J. Chem. Theory Comput. (2014), 10, pp. 4557. , 346 Fouda A. E. A., Besley N. A. 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 adapted basis sets to provide a good description of the core region. 345 Fouda A. E. A., Besley N. A. Theor. Chem. Acc. (2018), 137, pp. 6. , 447 Hanson-Heine M. W. D., George M. W., Besley N. A. Chem. Phys. Lett. (2018), 699, pp. 279. Example 7.152 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
$end$occupied