7.13 Core Ionization Energies and Core-Excited States

In experiments using high-energy radiation (such as X-ray spectroscopy, EXAFS, NEXAFS, XAS, XES, RIXS, REXS, etc) 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 Feshbach resonances⁸³⁸).

A highly robust and accurate strategy is to invoke many-body methods, such as EOM or ADC, together with the core-valence separation (CVS) scheme¹⁴². 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 7.10.6 and 7.11.4; CVS can also be deployed within TDDFT by using TRNSS (see Sections 7.3.2 and 7.13.1).

An alternative highly accurate approach for finding core-excitation energies of closed-shell molecules is to use the Restricted Open-Shell Kohn-Sham approach described in Section 7.7. ROKS is not systematically improvable like EOM or ADC methods, but is nonetheless quite accurate, with modern density functionals being capable of predicting excitation energies to eV error³³³. The great strength of the ROKS approach is its computational efficiency—highly accurate results can be obtained for the same $O(N^3)$ scaling as ground-state meta-GGAs, vs the $O(N^6)$ scaling of EOM-CCSD or $O(N^5)$ 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-excitations is supplied at 7.13.3.

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 7.10.12. While being moderately useful, these methods are less accurate than the CVS-EOM variants⁸³⁸.

In addition, one can use the $\mathrm{\Delta }E$ 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 7.13 below.

In this job, we first compute the HF and CCSD energies of neutral CH${}_4$: $E_{\mathrm{SCF}}=-40.1949062375$ and $E_{\mathrm{CCSD}}=-40.35748087$ (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 $E_{\mathrm{SCF}}=-29.4656758483$ ($⟨S^2⟩$ = 0.7730) and $E_{\mathrm{CCSD}}=-29.64793957$. Thus, $\mathrm{\Delta }{E}_{\mathrm{CCSD}}=(40.357481-29.647940)=10.709$ 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.

One can also use the following trick illustrated by example 7.13.

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.

Finally, one can use Koopmans’ theorem to compute the transitions involving core orbitals. While the direct application of Koopmans theorem yields rather large errors for core-ionized states and, consequently, the transitions involving these orbitals (such as in XES or XAS), there are certain tricks that can deliver considerably improved results. Within TDDFT, one can obtain reasonable estimates of the transitions between core and valence orbitals (as in XES) by simply using SRC functionals; this is illustrated by Example 7.13.2 below and discussed in Ref. 343 (the evaluation of energy loss spectra as in RIXS is also possible by using this feature together with MOM). The keywords NCORE_XES and NVAL_XES specify which transitions to compute.

Note:  This feature is only available with GEN_SCFMAN = FALSE .

Another approach of partial account of strong orbital relaxation is called transition potential (TP-)DFT.^{917, 970} This approach uses Kohn-Sham orbital eigenvalue differences to approximate core-level excitation energies, based on a Kohn-Sham calculation with partial occupations of the orbitals involved in the transitions. This can be justified based on a Taylor expansion in terms of the orbital occupations, as originally suggested by Slater.⁸⁸⁷

Note:  This is an experimental feature, only energies are currently implemented.