7.14 Core Ionization Energies and Core-Excited States 7.14.2 Calculations of X-Ray Spectroscopy with TDDFT 7.14.4 Calculations of Core Excitations with ROKS

7.14.3 Methods Based on Kohn-Sham Eigenvalues

(April 13, 2024)

7.14.3.1 Koopmans’ Approach

A simpler alternative to TDDFT for x-ray emission is to use Kohn-Sham eigenvalue differences,

\Delta E=\epsilon_{v}-\epsilon_{c}\;,

(7.127)

where $\epsilon_{v}$ and $\epsilon_{c}$ are valence and core energy levels, respectively. Oscillator strengths are obtained from the corresponding transition dipole matrix elements,

f_{c\to v}\propto\|\langle\phi_{c}|\hat{\bm{\mu}}|\phi_{v}\rangle\|^{2}\;.

(7.128)

This Koopmans’ theorem-type approach is somewhat crude, as there is no account for orbital relaxation in the core-excited state, but it has the benefit that only a ground-state calculation is required and therefore this approach is applicable to large systems, ⁴⁶¹ Hanson-Heine M. W. D., George M. W., Besley N. A.
Chem. Phys. Lett.
(2018), 696, pp. 119. Link and in conjunction with SRC functionals even this simple procedure can afford reasonable estimates of the transition energies. ⁴⁶⁰ Hanson-Heine M. W. D., George M. W., Besley N. A.
J. Chem. Phys.
(2017), 146, pp. 094106. Link The method is controlled by $rem variables NCORE_XES and NVAL_XES that specify the number of core ( $c$ ) and valence ( $v$ ) levels to consider, and an example is given in Example 7.14.3.1. Extension of this approach to resonant x-ray emission spectroscopy (involving an excited electronic state) is possible by modeling that state as a non-aufbau solution of the SCF equations, e.g., using algorithms such as (Section 4.5.13), SGM (Section 4.5.14), or STEP (4.5.15).

NCORE_XES

NCORE_XES
       Specifies how many core levels to use in a Koopmans-type XES calculation.
TYPE:
       INTEGER
DEFAULT:
       NONE
OPTIONS:
       $n$ Compute transition dipoles corresponding to the first (lowest energy) $n$ core orbitals, $\phi_{c}$ .
RECOMMENDATION:
       None

NVAL_XES

NVAL_XES
       Specifies how many valence virtual levels to use in a Koopmans-type XES calculation.
TYPE:
       INTEGER
DEFAULT:
       NONE
OPTIONS:
       $n$ Compute transition dipoles corresponding to the highest $n$ occupied orbitals, $\phi_{v}$ .
RECOMMENDATION:
       Setting $n=1$ will include the HOMO in the occupied space, $n=2$ will include HOMO and $\text{HOMO}-1$ , etc.

Example 7.160 The calculation of the XES spectrum of water using the Koopmans approach 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
$end

7.14.3.2 Transition-Potential DFT

The transition potential (TP-)DFT method ¹¹⁷⁸ Stener M., Lisini A., Decleva P.
Chem. Phys.
(1995), 191, pp. 141. Link is an alternative approach that accounts for some orbital relaxation yet retains a framework based on Kohn-Sham eigenvalues, requiring only a ground-state calculation. This approach is based on Slater’s transition concept, ¹¹⁴⁰ Slater J. C.
Adv. Quantum Chem.
(1972), 6, pp. 1. Link ^, ⁵⁶² Jana S., Herbert J. M.
J. Chem. Phys.
(2023), 158, pp. 094111. Link in which an SCF calculation with a fractional electron (originally $n_{i}=1/2$ ) is removed from occupied orbital $\phi_{i}$ , then the ionization energy for that MO is approximated as

\text{IE}_{i}\approx-\epsilon_{i}(n_{i}=1/2)\;.

(7.129)

This can be justified based on a Taylor expansion in terms of the orbital occupations. ¹¹⁴⁰ Slater J. C.
Adv. Quantum Chem.
(1972), 6, pp. 1. Link ^, ⁵⁶² Jana S., Herbert J. M.
J. Chem. Phys.
(2023), 158, pp. 094111. Link Excitation energies are approximated as eigenvalue differences $\epsilon_{a}-\epsilon_{i}$ obtained from a fractional-electron SCF calculation in which $n_{i}=1/2$ electron is promoted from $\phi_{i}$ into the LUMO:

\Delta E_{i\to a}=\epsilon_{a}(n_{i}=1/2,n_{\text{LUMO}}=1/2)-\epsilon_{i}(n_{% i}=1/2,n_{\text{LUMO}}=1/2)\;,

(7.130)

with oscillator strengths $f_{i\to a}\propto|\langle\phi_{i}|\hat{\bm{\mu}}|\phi_{a}\rangle|^{2}$ , as in Eq. (7.128).

TP-DFT calculations in Q-Chem are setup to remove 1/2 electron from the lowest core orbital of a given atom that is specified using TPDFT_ATOM. (For more flexible and general fractional-electron SCF schemes, see Section 7.14.3.3.) Optionally, one may use TPDFT_LUMO to occupy the LUMO, corresponding to an excitation energy calculation [Eq. (7.130)], or omit this variable to compute the core-level electron binding energy [Eq. (7.129)].

TPDFT_ATOM

TPDFT_ATOM
       Activate TP-DFT by specifying the atom from which to remove an electron.
TYPE:
       INTEGER
DEFAULT:
       NONE
OPTIONS:
       $n$ Remove an electron from the lowest-energy orbital on the atom whose index is $n$ .
RECOMMENDATION:
       Be sure to set UNRESTRICTED = TRUE for TP-DFT calculations.

TPDFT_FRAC

TPDFT_FRAC
       Specify the fractional value of $n_{i}$ to be removed.
TYPE:
       INTEGER
DEFAULT:
       NONE
OPTIONS:
       $n$ Remove $n/100$ of an electron from the orbital specified using TPDFT_ATOM.
RECOMMENDATION:
       None

TPDFT_LUMO

TPDFT_LUMO
       Specify the fractional value of $n_{\text{LUMO}}$ to be added.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ Add $n/100$ of an electron to the LUMO.
RECOMMENDATION:
       Leave TPDFT_LUMO = 0 for core-level binding energy calculations [Eq. (7.129)] or use TPDFT_LUMO = 50 to implement Eq. (7.130).

Example 7.161 TP-DFT excitation energy calculation, removing $n_{i}=1/2$ from O(1s) and placing it in the LUMO.

$molecule
   0 1
   O
   H 1 0.95
   H 2 0.95 2 104.5
$end

$rem
   METHOD       b3lyp
   BASIS        aug-cc-pCVQZ
   UNRESTRICTED true
   TPDFT_ATOM   1
   TPDFT_FRAC   50
   TPDFT_LUMO   50 ! set to 0 for IE calculation
$end

7.14.3.3 Slater-Type Fractional-Electron Methods

A more general set of fractional-electron methods for both core-level ionization (i.e., XPS) and core-level excitation (XAS and also XES) has been explored by Jana and Herbert. ⁵⁶² Jana S., Herbert J. M.
J. Chem. Phys.
(2023), 158, pp. 094111. Link These methods generalize Slater’s transition concept, ¹¹⁴⁰ Slater J. C.
Adv. Quantum Chem.
(1972), 6, pp. 1. Link ^, ⁵⁶² Jana S., Herbert J. M.
J. Chem. Phys.
(2023), 158, pp. 094111. Link and allow an arbitrary fraction of an electron to be removed from a core MO and (optionally) placed into a virtual MO, as specified by the user. The use of these generalized Slater-type methods is controlled by the $rem variables that are described below and illustrated in examples that follow. For XAS, oscillator strengths are computed according to Eq. (7.128), and the (occupied, virtual) orbital pairs ( $\phi_{i},\phi_{a}$ ) for which the transition dipole moment is computed are specified using NCORE_XAS and NVAL_XAS, as described in Section 7.14.3.1. To converge the fractional-electron state, it may be necessary to use an algorithm such as MOM (Section 4.5.13), SGM (Section 4.5.14), or STEP (4.5.15). Consult Refs. 562 Jana S., Herbert J. M.
J. Chem. Phys.
(2023), 158, pp. 094111. Link and for best practices regarding which generalized Slater-type method to use.

FRACTIONAL_ELECTRON

FRACTIONAL_ELECTRON
       Specify the fraction of an electron to be removed from the occupied space.
TYPE:
       INTEGER
DEFAULT:
       NONE
OPTIONS:
       $-n$ Remove $n/1000$ of an electron.
RECOMMENDATION:
       The original Slater method ( $n_{i}=1/2$ ) corresponds to FRACTIONAL_ELECTRON = $-$ 500 but there can be other choices.

FRAC_VIR_ELEC

FRAC_VIR_ELEC
       Specify the fraction of an electron to place into the occupied space.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ Add $n/1000$ of an electron.
RECOMMENDATION:
       A value $>0$ should be used for excitation (XAS or XES), whereas the default is appropriate for ionization (XPS).

FRAC_ELEC_ORB

FRAC_ELEC_ORB
       Specify the occupied orbital from which the fractional electron should be removed.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ Remove from $\phi_{n}$ .
RECOMMENDATION:
       None

FRAC_VIR_ELEC_ORB

FRAC_VIR_ELEC_ORB
       Specify the virtual orbital to which the fractional electron should be added.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ Add to $\phi_{n}$ .
RECOMMENDATION:
       Use this only if FRAC_VIR_ELEC $>0$ .

Example 7.162 Original Slater transition method (STM) for oxygen K-shell XPS, removing $n_{i}=1/2$ electron from O(1s) core orbital. A normal SCF calculation is used to obtain the initial set of orbitals, followed by a fractional-electron calculation that is converged using the IMOM algorithm.

$molecule
0 1
C  -0.00000000  0.00000000  0.07378202
O   0.00000000  0.00000000  1.20921798
$end

$rem
      UNRESTRICTED  TRUE
      BASIS         def2-QZVP
      METHOD        B3LYP
point_group_symmetry False
integral_symmetry FALSE
$end

@@@

$molecule
read
$end

$rem
      SCF_GUESS     READ
      UNRESTRICTED  TRUE
      BASIS         def2-QZVP
      METHOD        B3LYP
point_group_symmetry False
integral_symmetry FALSE
      FRAC_ELEC_ORB        1  ! lowest-energy orbital is O(1s)
      FRAC_VIR_ELEC_ORB    1
      FRACTIONAL_ELECTRON -500
      FRAC_VIR_ELEC        0
      MOM_START            1
      MOM_METHOD           IMOM
$end

Example 7.163 XAS at carbon K-edge, using Slater’s transition method with $n_{i}=1/2$ electrons removed from C(1s) and $n_{a}=1/2$ electron placed in the LUMO.

$molecule
 0 1
C  -0.00000000  0.00000000  0.07378202
O   0.00000000  0.00000000  1.20921798
$end

$rem
      UNRESTRICTED  TRUE
      BASIS         def2-QZVP
      METHOD        B3LYP
point_group_symmetry False
integral_symmetry FALSE
$end

@@@

$molecule
read
$end

$rem
      SCF_GUESS     READ
      UNRESTRICTED  TRUE
      BASIS         def2-QZVP
      METHOD        B3LYP
point_group_symmetry False
integral_symmetry FALSE
      FRAC_ELEC_ORB        2  ! C(1s)
      FRAC_VIR_ELEC_ORB    1
      FRACTIONAL_ELECTRON -500
      FRAC_VIR_ELEC        500
      MOM_START            1
      MOM_METHOD           IMOM
      NCORE_XAS            2
      NVAL_XAS             2
$end

Search Results

7.14.3 Methods Based on Kohn-Sham Eigenvalues

7.14.3.1 Koopmans’ Approach

7.14.3.2 Transition-Potential DFT

7.14.3.3 Slater-Type Fractional-Electron Methods