7.10 Core Ionization Energies and Core-Excited States

In experiments using high-energy radiation (such as X-ray spectroscopy, EXAFS, NEXAFS, XAS) core electrons can be ionized or excited to low-lying virtual orbitals. There are several ways to compute ionization or excitation energies of core electrons in Q-Chem. Standard approaches for excited and ionized states need to be modified to tackle core-level states, because these states have very high energies and are embedded in the ionization continuum (i.e., they are Feschbach resonances[Sadybekov and Krylov(2017)]).

The most robust and accurate strategy is to invoke many-body methods, such as EOM or ADC, together with the core-valence separation (CVS) approximation[Cederbaum et al.(1980)Cederbaum, Domcke, and Schirmer]. 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 7.7.5 and 7.8.4.

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 7.7.10. While being moderately useful, these methods are less accurate than the CVS-EOM variants[Sadybekov and Krylov(2017)].

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. It is illustrated by example 7.10 below.

Example 7.194 Q-Chem input for calculating chemical shift for 1 $s$ -level of methane (CH $_4$ ). 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 $_4$ : $E_{\rm SCF}=-40.1949062375$ and $E_{\rm CCSD}=-40.35748087$ (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 $_4$ . To obtain the desired SCF solution, MOM_START option and $occupied keyword are used. The resulting energies are $E_{\rm SCF}=-29.4656758483$ ( $\langle {S}^2\rangle$ = 0.7730) and $E_{\rm CCSD}=-29.64793957$ . Thus, $\Delta E_{\rm CCSD}=(40.357481-29.647940)=10.709$ hartree = 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.

Finally, one can also use the following trick illustrated by example 7.10.

Example 7.195 Q-Chem input for calculating chemical shift for 1 $s$ -level of methane (CH $_4$ ) 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 the Koopmans approximation.

7.10.1 Calculations of States Involving Core Excitation/Ionization with (TD)DFT

TDDFT is not suited to describe the extended X-ray absorption fine structure (EXAFS) region, wherein the core electron is ejected and scattered by the neighboring atoms. Core-excitation energies computed with TDDFT with standard hybrid functionals are many electron volts too low compared with experiment. Exchange-correlation functionals specifically designed to treat core excitations are available in Q-Chem. These short-range corrected (SRC) 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. These functionals can be invoked by using the SRC_DFT rem. In addition, a number of parameters (OMEGA, OMEGA2, HF_LR, HF_SR) that control the shape of the short and long-range Hartree-Fock components need to be specified. Full details of these functionals and appropriate values for the parameters can be found in Refs. Besley:2009a,Besley:2010. An example of how to use these functionals is given below. For the $K$ -shell of heavy elements (2nd row of the periodic table) relativistic effects become increasing important and a further correction for these effects is required. Also calculations for $L$ -shell excitations are complicated by core-hole spin orbit coupling.