7.13 Core Ionization Energies and Core-Excited States 7.13.3 Methods Based on Kohn-Sham Eigenvalues 7.14 Visualization of Excited States

7.13.4 Calculations of Core Excitations with ROKS

(September 1, 2024)

The restricted open-shell Kohn-Sham (ROKS) approach is a highly accurate method for estimating core-excitation energies of closed-shell molecules, ⁴⁷² Hait D., Head-Gordon M.
J. Phys. Chem. Lett.
(2020), 11, pp. 775. Link as described in Section 7.8.3. Here, we briefly recapitulate the key aspects and refer the reader to Ref. 472 Hait D., Head-Gordon M.
J. Phys. Chem. Lett.
(2020), 11, pp. 775. Link for details ROKS with the SCAN functional is found to reproduce 40 experimental core excitation energies (from the 1s orbital, i.e., K-edge) of second-period elements (C, N, O, and F) to an RMS error of 0.2 eV and a maximum absolute error of 0.5 eV. The $\omega$ B97X-V functional provides similar (if a little worse) accuracy as well. Similar behavior is observed for the L ${}_{2,3}$ edges of third-period elements Si, P, S and Cl. Other widely used functionals like PBE fare somewhat worse, but still predict much lower error as compared to TDDFT using the same functionals. Recently, we extended the applicability for ROKS for core excited states of heavier elements by including scalar relativistic effects as described in Section 4.9.6. Accurate modeling of K-edge of elements up to $Z=24$ can be achieved with the SCAN functional. ²⁷⁵ Cunha L. A. et al.
J. Phys. Chem. Lett.
(2022), 13, pp. 3438. Link

That said, the ROKS approach is state-specific in that it can only predict a single state at a time and needs to be told which state to target (via the $reorder_mo section, as shown in Example 7.8.4). This makes it less black-box than TDDFT as the final orbital needs to be identified a priori, perhaps via a pilot TDDFT job if no other information is available. (For core-level excitations, the initial orbital is usually intuitively obvious.) ROKS can also be used for two-site doubly core-ionized states, or other systems with one broken electron pair in total.

The accuracy of ROKS stems from three factors: choice of density functional (SCAN or $\omega$ B97X-V), excited state orbital optimization (only available via SGM for core excitations, as described in Section 4.5.14) and a sufficiently flexible basis set. The last is key, as the split-core functions (as provided by basis sets like cc-pCV $n$ Z) are needed instead of standard basis sets like cc-pV $n$ Z that only have split valence functions. Indeed, a basis of triple- $\zeta$ quality like cc-pCVTZ is necessary to fully account for the core-hole relaxation and smaller basis sets lead to systematic overestimation of excitation energies. However, the highly local nature of the core-hole ensures that a large basis is only needed for the target atom of the ROKS calculation, and a smaller basis (of double- $\zeta$ quality, like cc-pVDZ) is adequate for all other atoms. An example of this mixed basis strategy is given below in Example 7.13.4. Details about using mixed basis sets in general can be found in Section 8.5.

The number of cycles needed for ROKS calculations can also be considerably reduced by decoupling the core hole relaxation from the rest of the orbital optimization. This entails a restricted open-shell $\Delta$ SCF calculation of the core-ionized state first, and use of those orbitals as guess for ROKS. Example 7.8.4 is a representative case for how such calculations should proceed.

The conjunction of high accuracy and low computational cost (due to the affordability of the SCAN meta-GGA and the mixed basis strategy) makes ROKS a very attractive approach for computing core spectra of large, closed-shell systems where more expensive wave function theories are unaffordable. Users are requested to cite Ref. 472 Hait D., Head-Gordon M.
J. Phys. Chem. Lett.
(2020), 11, pp. 775. Link when using ROKS for core excitations and Ref. 275 Cunha L. A. et al.
J. Phys. Chem. Lett.
(2022), 13, pp. 3438. Link when performing calculations that include scalar relativistic effects.

Example 7.187 RO- $\Delta$ SCF core-ionization at C for CO, using SGM. The mixed basis strategy is used as the core-hole is local to C.

$molecule
   0 1
   C 0.0000   0.0000   0.0000
   O 0.0000   0.0000   1.1282
$end

$rem
   METHOD     scan
   BASIS      gen
   BASIS2     aug-cc-pVDZ
   INTEGRAL_SYMMETRY false
$end

$basis
C
aug-cc-pCVTZ
****
O
aug-cc-pVDZ
****
$end

@@@@

$molecule
   1 2
   C 0.0000   0.0000   0.0000
   O 0.0000   0.0000   1.1282
$end

$rem
   METHOD          scan
   BASIS           gen
   UNRESTRICTED    false
   SCF_GUESS       read
   INTEGRAL_SYMMETRY false
   SCF_ALGORITHM   sgm
$end

$reorder_mo
   1 3 4 5 6 7 2
   1 3 4 5 6 7 2
$end

$basis
C
aug-cc-pCVTZ
****
O
aug-cc-pVDZ
****
$end

Example 7.7.188 Combined RO- $\Delta$ SCF core-ionization and $1s\rightarrow\text{LUMO}$ ROKS core-excited state for HCl including scalar relativistic effects. The uncontracted aug-pcX-2 basis has to be supplied by the user

$molecule
0 1
Cl  0.0000  0.0000  0.0000
H   0.0000  0.0000  1.2746
$end

$rem
method scan
basis  gen ! Decontracted aug-pcX-2 basis for X2C
rel_x2c 1  ! Do X2C relativistic one-electron Hamiltonian
rel_x2c_fd_displacement 100 ! Controls the finite difference step for W (default = 100 * 1e-6 = 1e-4)
integral_symmetry    false
point_group_symmetry false
$end

$basis
Cl     0
S   1   1.00
      0.216264D+06           1.00000000D+00
S   1   1.00
      0.324104D+05           1.00000000D+00
S   1   1.00
      0.737630D+04           1.00000000D+00
S   1   1.00
      0.208877D+04           1.00000000D+00
S   1   1.00
      0.681028D+03           1.00000000D+00
S   1   1.00
      0.245288D+03           1.00000000D+00
S   1   1.00
      0.949871D+02           1.00000000D+00
S   1   1.00
      0.385297D+02           1.00000000D+00
S   1   1.00
      0.158316D+02           1.00000000D+00
S   1   1.00
      0.610936D+01           1.00000000D+00
S   1   1.00
      0.244372D+01           1.00000000D+00
S   1   1.00
      0.603084D+00           1.00000000D+00
S   1   1.00
      0.209285D+00           1.00000000D+00
S   1   1.00
      0.679167D-01           1.00000000D+00
P   1   1.00
      0.121088D+04           1.00000000D+00
P   1   1.00
      0.287276D+03           1.00000000D+00
P   1   1.00
      0.924350D+02           1.00000000D+00
P   1   1.00
      0.343221D+02           1.00000000D+00
P   1   1.00
      0.139811D+02           1.00000000D+00
P   1   1.00
      0.590468D+01           1.00000000D+00
P   1   1.00
      0.250190D+01           1.00000000D+00
P   1   1.00
      0.928135D+00           1.00000000D+00
P   1   1.00
      0.348099D+00           1.00000000D+00
P   1   1.00
      0.117780D+00           1.00000000D+00
P   1   1.00
      0.373244D-01           1.00000000D+00
D   1   1.00
      0.241272D+01           1.00000000D+00
D   1   1.00
      0.544426D+00           1.00000000D+00
D   1   1.00
      0.111960D+00           1.00000000D+00
F   1   1.00
      0.749400D+00           1.00000000D+00
F   1   1.00
      0.149279D+00           1.00000000D+00
****
H     0
S   1   1.00
      0.754732D+02           1.00000000D+00
S   1   1.00
      0.113575D+02           1.00000000D+00
S   1   1.00
      0.260081D+01           1.00000000D+00
S   1   1.00
      0.735503D+00           1.00000000D+00
S   1   1.00
      0.231761D+00           1.00000000D+00
S   1   1.00
      0.741675D-01           1.00000000D+00
S   1   1.00
      0.223070D-01           1.00000000D+00
P   1   1.00
      0.160000D+01           1.00000000D+00
P   1   1.00
      0.450000D+00           1.00000000D+00
P   1   1.00
      0.669210D-01           1.00000000D+00
D   1   1.00
      0.125000D+01           1.00000000D+00
D   1   1.00
      0.892290D-01           1.00000000D+00
****
$end

@@@@@@@@@@@
$comment
RO core ionized state. Note that the $molecule section thus set up will read geometry from before, but not charge/spin, which is supplied
$end

$molecule
1 2
read
$end

$rem
method        scan
basis         gen
scf_algorithm sgm
scf_guess     read
roscf         true
rel_x2c 1 ! Do X2C relativistic one-electron Hamiltonian
rel_x2c_fd_displacement 100 ! Controls the finite difference step for W (default = 100 * 1e-6 = 1e-4)
integral_symmetry    false
point_group_symmetry false
$end

$reorder_mo ! push the 1s orbital to the frontier. For HCl we have 18 electrons, so 9 orbitals. Pushing 1 (1s) to the frontier (9) is done thus.
2 3 4 5 6 7 8 9 1
2 3 4 5 6 7 8 9 1
$end

$basis
Cl     0
S   1   1.00
      0.216264D+06           1.00000000D+00
S   1   1.00
      0.324104D+05           1.00000000D+00
S   1   1.00
      0.737630D+04           1.00000000D+00
S   1   1.00
      0.208877D+04           1.00000000D+00
S   1   1.00
      0.681028D+03           1.00000000D+00
S   1   1.00
      0.245288D+03           1.00000000D+00
S   1   1.00
      0.949871D+02           1.00000000D+00
S   1   1.00
      0.385297D+02           1.00000000D+00
S   1   1.00
      0.158316D+02           1.00000000D+00
S   1   1.00
      0.610936D+01           1.00000000D+00
S   1   1.00
      0.244372D+01           1.00000000D+00
S   1   1.00
      0.603084D+00           1.00000000D+00
S   1   1.00
      0.209285D+00           1.00000000D+00
S   1   1.00
      0.679167D-01           1.00000000D+00
P   1   1.00
      0.121088D+04           1.00000000D+00
P   1   1.00
      0.287276D+03           1.00000000D+00
P   1   1.00
      0.924350D+02           1.00000000D+00
P   1   1.00
      0.343221D+02           1.00000000D+00
P   1   1.00
      0.139811D+02           1.00000000D+00
P   1   1.00
      0.590468D+01           1.00000000D+00
P   1   1.00
      0.250190D+01           1.00000000D+00
P   1   1.00
      0.928135D+00           1.00000000D+00
P   1   1.00
      0.348099D+00           1.00000000D+00
P   1   1.00
      0.117780D+00           1.00000000D+00
P   1   1.00
      0.373244D-01           1.00000000D+00
D   1   1.00
      0.241272D+01           1.00000000D+00
D   1   1.00
      0.544426D+00           1.00000000D+00
D   1   1.00
      0.111960D+00           1.00000000D+00
F   1   1.00
      0.749400D+00           1.00000000D+00
F   1   1.00
      0.149279D+00           1.00000000D+00
****
H     0
S   1   1.00
      0.754732D+02           1.00000000D+00
S   1   1.00
      0.113575D+02           1.00000000D+00
S   1   1.00
      0.260081D+01           1.00000000D+00
S   1   1.00
      0.735503D+00           1.00000000D+00
S   1   1.00
      0.231761D+00           1.00000000D+00
S   1   1.00
      0.741675D-01           1.00000000D+00
S   1   1.00
      0.223070D-01           1.00000000D+00
P   1   1.00
      0.160000D+01           1.00000000D+00
P   1   1.00
      0.450000D+00           1.00000000D+00
P   1   1.00
      0.669210D-01           1.00000000D+00
D   1   1.00
      0.125000D+01           1.00000000D+00
D   1   1.00
      0.892290D-01           1.00000000D+00
****
$end

@@@@@@@
$comment
ROKS for core excitation
$end

$molecule
0 1
read
$end

$rem
method        scan
basis         gen
scf_algorithm sgm
scf_guess     read
os_roscf      true ! Turns ROKS on
scf_convergence 5
max_scf_cycles  5000
rel_x2c 1 ! Do X2C relativistic one-electron Hamiltonian
rel_x2c_fd_displacement 100 ! Controls the finite difference step for W (default = 100 * 1e-6 = 1e-4)
integral_symmetry    false
point_group_symmetry false
$end

$basis
Cl     0
S   1   1.00
      0.216264D+06           1.00000000D+00
S   1   1.00
      0.324104D+05           1.00000000D+00
S   1   1.00
      0.737630D+04           1.00000000D+00
S   1   1.00
      0.208877D+04           1.00000000D+00
S   1   1.00
      0.681028D+03           1.00000000D+00
S   1   1.00
      0.245288D+03           1.00000000D+00
S   1   1.00
      0.949871D+02           1.00000000D+00
S   1   1.00
      0.385297D+02           1.00000000D+00
S   1   1.00
      0.158316D+02           1.00000000D+00
S   1   1.00
      0.610936D+01           1.00000000D+00
S   1   1.00
      0.244372D+01           1.00000000D+00
S   1   1.00
      0.603084D+00           1.00000000D+00
S   1   1.00
      0.209285D+00           1.00000000D+00
S   1   1.00
      0.679167D-01           1.00000000D+00
P   1   1.00
      0.121088D+04           1.00000000D+00
P   1   1.00
      0.287276D+03           1.00000000D+00
P   1   1.00
      0.924350D+02           1.00000000D+00
P   1   1.00
      0.343221D+02           1.00000000D+00
P   1   1.00
      0.139811D+02           1.00000000D+00
P   1   1.00
      0.590468D+01           1.00000000D+00
P   1   1.00
      0.250190D+01           1.00000000D+00
P   1   1.00
      0.928135D+00           1.00000000D+00
P   1   1.00
      0.348099D+00           1.00000000D+00
P   1   1.00
      0.117780D+00           1.00000000D+00
P   1   1.00
      0.373244D-01           1.00000000D+00
D   1   1.00
      0.241272D+01           1.00000000D+00
D   1   1.00
      0.544426D+00           1.00000000D+00
D   1   1.00
      0.111960D+00           1.00000000D+00
F   1   1.00
      0.749400D+00           1.00000000D+00
F   1   1.00
      0.149279D+00           1.00000000D+00
****
H     0
S   1   1.00
      0.754732D+02           1.00000000D+00
S   1   1.00
      0.113575D+02           1.00000000D+00
S   1   1.00
      0.260081D+01           1.00000000D+00
S   1   1.00
      0.735503D+00           1.00000000D+00
S   1   1.00
      0.231761D+00           1.00000000D+00
S   1   1.00
      0.741675D-01           1.00000000D+00
S   1   1.00
      0.223070D-01           1.00000000D+00
P   1   1.00
      0.160000D+01           1.00000000D+00
P   1   1.00
      0.450000D+00           1.00000000D+00
P   1   1.00
      0.669210D-01           1.00000000D+00
D   1   1.00
      0.125000D+01           1.00000000D+00
D   1   1.00
      0.892290D-01           1.00000000D+00
****
$end

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7.13.4 Calculations of Core Excitations with ROKS