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

7.14.1 Many-Body Methods for Core-Excited States

(April 13, 2024)

A highly robust and accurate strategy is to invoke many-body methods, such as EOM or ADC, together with the core-valence separation (CVS) scheme. ¹⁸⁹ Cederbaum L. S., Domcke W., Schirmer J.
Phys. Rev. A
(1980), 22, pp. 206. Link In this approach, the excitations involving core electrons are decoupled from the rest of the configurational space. This allows one to reduce computational costs and decouple the highly excited core states from the continuum. These methods are described in Sections 7.11.8 and 7.12.6; CVS can also be deployed within TDDFT as described in Section 7.14.2. Error introduced by the CVS approximation is negligible for K-edge (1s $\rightarrow$ virtual) excitations, ⁵⁰⁸ Herbst M. F., Fransson T.
J. Chem. Phys.
(2020), 153, pp. 054114. Link though its accuracy for other types of x-ray excitations is less certain.

An alternative highly accurate approach for finding core-excitation energies of closed-shell molecules is to use the restricted open-shell Kohn-Sham (ROKS) approach that is described in Section 7.9.3. ROKS is not systematically improvable like EOM or ADC methods, but is nonetheless quite accurate and modern density functionals are capable of predicting excitation energies to $<0.5$ eV error. ⁴⁵⁰ Hait D., Head-Gordon M.
J. Phys. Chem. Lett.
(2020), 11, pp. 775. Link The great strength of the ROKS approach is its computational efficiency: highly accurate results can be obtained for the same ${\cal{O}}({N^{3}})$ scaling as ground-state meta-GGAs, versus the ${\cal{O}}({N^{6}})$ scaling of EOM-CCSD or ${\cal{O}}({N^{5}})$ scaling of ADC(2). The basis set requirements of ROKS are also much more modest than wave function theories, with a mixed basis strategy being highly effective in practice. Details about using ROKS for core-level excitations can be found in Section 7.14.4.

Within EOM-CC formalism, one can also use an approximate EOM-EE/IP methods in which the target states are described by single excitations and double excitations are treated perturbatively; these methods are described in Section 7.11.16. While being moderately useful, these methods are less accurate than the CVS-EOM variants. ¹⁰⁷⁶ Sadybekov A., Krylov A. I.
J. Chem. Phys.
(2017), 147, pp. 014107. Link

In addition, one can use the $\Delta E$ approach, which amounts to a simple energy difference calculation in which core ionization is computed from energy differences computed for the neutral and core-ionized state. This procedure is illustrated by Example 7.14.1 below.

Example 7.154 Q-Chem input for calculating chemical shift for 1s-level of methane (CH ${}_{\text{4}}$ ). The first job is just an SCF calculation to obtain the orbitals and CCSD energy of the neutral. The second job solves the HF and CCSD equations for the core-ionized state.

$molecule
   0,1
   C      0.000000    0.000000    0.000000
   H      0.631339    0.631339    0.631339
   H     -0.631339   -0.631339    0.631339
   H     -0.631339    0.631339   -0.631339
   H      0.631339   -0.631339   -0.631339
$end

$rem
   EXCHANGE        = HF
   CORRELATION     = CCSD
   BASIS           = 6-31G*
   MAX_CIS_CYCLES  = 100
$end

@@@

$molecule
   +1,2
   C      0.000000    0.000000    0.000000
   H      0.631339    0.631339    0.631339
   H     -0.631339   -0.631339    0.631339
   H     -0.631339    0.631339   -0.631339
   H      0.631339   -0.631339   -0.631339
$end

$rem
   UNRESTRICTED    = TRUE
   EXCHANGE        = HF
   BASIS           = 6-31G*
   MAX_CIS_CYCLES  = 100
   SCF_GUESS       = read  Read MOs from previous job and use occupied as specified below
   CORRELATION     = CCSD
   MOM_START       = 1 Do not reorder orbitals in SCF procedure!
$end

$occupied
   1 2 3 4 5
   2 3 4 5
$end

In this job, we first compute the HF and CCSD energies of neutral CH ${}_{\text{4}}$ : $E_{\text{SCF}}=-40.1949062375~{}E_{h}$ and $E_{\text{CCSD}}=-40.35748087~{}E_{h}$ (HF orbital energy of the neutral gives the Koopmans IE, which is 11.210 hartree = 305.03 eV). In the second job, we do the same for core-ionized CH ${}_{\text{4}}$ . To obtain the desired SCF solution, MOM_START option and $occupied keyword are used. The resulting energies are $E_{\text{SCF}}=-29.4656758483~{}E_{h}$ ( $\langle\hat{S}^{2}\rangle$ = 0.7730) and $E_{\text{CCSD}}=-29.64793957~{}E_{h}$ . Thus, $\Delta E_{\text{CCSD}}=10.709~{}E_{h}=291.42$ eV.

This approach can be further extended to obtain multiple excited states involving core electrons by performing CIS, TDDFT, or EOM-EE calculations.

Note: This approach often leads to convergence problems in correlated calculations.

One can also use the following trick illustrated by Example 7.14.1.

Example 7.155 Q-Chem input for calculating chemical shift for 1s-level of methane using EOM-IP. Here we solve SCF as usual, then reorder the MOs such that the core orbital becomes the “HOMO”, then solve the CCSD and EOM-IP equations with all valence orbitals frozen and the core orbital being active.

$molecule
   0,1
   C      0.000000    0.000000    0.000000
   H      0.631339    0.631339    0.631339
   H     -0.631339   -0.631339    0.631339
   H     -0.631339    0.631339   -0.631339
   H      0.631339   -0.631339   -0.631339
$end

$rem
   EXCHANGE        = HF
   BASIS           = 6-31G*
   MAX_CIS_CYCLES  = 100
   CORRELATION     = CCSD
   CCMAN2          = false
   N_FROZEN_CORE   = 4 Freeze all valence orbitals
   IP_STATES       = [1,0,0,0] Find one EOM_IP state
$end

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

Here we use EOM-IP to compute core-ionized states. Since core states are very high in energy, we use “frozen core” trick to eliminate valence ionized states from the calculation. That is, we reorder MOs such that our core is the last occupied orbital and then freeze all the rest. The so computed EOM-IP energy is 245.57 eV. From the EOM-IP amplitude, we note that this state of a Koopmans character (dominated by single core ionization); thus, canonical HF MOs provide good representation of the correlated Dyson orbital. The same strategy can be used to compute core-excited states.

Note: The accuracy of this approach is rather poor and is similar to Koopmans’ approximation.

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7.14.1 Many-Body Methods for Core-Excited States