Core Hole Effect to Valence Excitations: Tracking and Visualizing the Same Excitation in XPS Shake-Up Satellites and UV Absorption Spectra

Introducing a core hole significantly alters the electronic structure of a molecule, and various X-ray spectroscopy techniques are available for probing the valence electronic structure in the presence of a core hole. In this study, we visually demonstrate the influence of a core hole on valence excitations by computing the ultraviolet absorption spectra and the shake-up satellites in X-ray photoelectron spectra for pyrrole, furan, and thiophene, as complemented by the natural transition orbital (NTO) analysis over transitions with and without a core hole. Employing equivalent core hole time-dependent density functional theory (ECH-TDDFT) and TDDFT methods, we achieved balanced accuracy in both spectra for reliable comparative analysis. We tracked the same involved valence transition in both spectra, offering a vivid illustration of the core hole effect via the change in corresponding particle NTOs introduced by a 1s core hole on a Cα, Cβ, or O atom. Our analysis deepens the understanding of the core hole effect on valence transitions, a phenomenon ubiquitously observed in general X-ray spectroscopic analyses.

The core ionization energies calculated by delta SCF and Slater's transition state theory

The core ionization energies of the second-period and third-period elements are studied by ΔSCF and Slater's transition state (STS) theory by using Hartree-Fock (HF) and Kohn-Sham (KS) approximations. Electron correlation increases the estimated core ionization energies, while the self-interaction error (SIE) decreases them, especially for the third-period elements and is a more significant factor. As a result, while HF lacks electron correlation, it is free of SIE and reasonably predicts the core ionization energies. The core ionization energies calculated by HF STS are very close to those calculated by HF ΔSCF, showing that STS reasonably describes the relaxation of the core hole. The core ionization energies calculated by KS are particularly sensitive to the SIE of the functional used, with functionals having less SIE yielding more accurate ΔSCF core ionization energies. Consequently, BH&HLYP gives better results than B3LYP and LC-BOP since BH&HLYP is the hybrid functional with high proportion of the exact HF exchange. Although the core ionization energies are underestimated by ΔSCF due to SIE, STS gives larger core ionization energies than ΔSCF due to a concave behavior of the error curves of STS, which is also related to SIE. The mean absolute deviations of STS relative to ΔSCF, and relative to the experiment, are almost constant regardless of the nuclei among the element in the second period, and likewise among those in the third period. The systematic nature suggests that shifting the STS core ionization energies may be useful. We propose the shifted STS (1) for reproducing ΔSCF values, and the shifted STS (2) to reproduce the observed ones for KS calculations. Both schemes work quite well. The calculated results of KS ΔSCF and STS vary depending on the functional. However, the variation of each species' shifted STS (2) is very small, and all shifted STS (2) values are close to the observed ones. As the shifted STS require only one SCF calculation, they are simple and practical for predicting the core ionization energies.

An improved Slater's transition state approximation

We have extended Slater's transition state concept for the approximation of the difference in total energies of the initial and final states by three orbital energies of initial, final, and half-way Slater's transition states of the system. Numerical validation was performed with the ionization energies for H₂O, CO, and pyrrole by calculation using Hartree-Fock (HF) and Kohn-Sham (KS) theories with the B3LYP and LCgau-core-BOP functionals. The present extended method reproduces full ΔSCF very accurately for all occupied orbitals obtained with HF and for valence orbitals obtained with KS. KS core orbitals have some errors due to the self-interaction errors, but the present method significantly improves the core electron binding energies. In its current form, the newly derived theory may not yet be practically useful, but it is simple and conceptually useful for gaining improved understanding of SCF-type orbital theories.

Koopmans'-Type Theorem in Kohn-Sham Theory with Optimally Tuned Long-Range-Corrected (LC) Functionals

In the present study, we have investigated the applicability of long-range-corrected (LC) functionals to a Kohn-Sham (KS) Koopmans'-type theorem. Specifically, we have examined the performance of optimally tuned LCgau-core functionals (in combination with BOP and PW86-PW91 exchange-correlation functionals) by calculating the ionization potential (IP) within the context of Koopmans' prediction. In the LC scheme, the electron repulsion operator, 1/r₁₂, is divided into short-range and long-range components using a standard error function, with a range separation parameter μ determining the weight of the two ranges. For each system that we have examined (H₂O, CO, benzene, N₂, HF, H₂CO, C₂H₄, and five-membered ring compounds cyclo-C₄H₄X, with X = CH₂, NH, O, and S, and pyridine), the value of μ is optimized to minimize the deviation of the negative HOMO energy from the experimental IP. Our results demonstrate the utility of optimally tuned LC functionals in predicting the IP of outer valence levels. The accuracy is comparable to that of highly accurate ab initio theory. However, our Koopmans' method is less accurate for the inner valence and core levels. Overall, our results support the notion that orbitals in KS-DFT, when obtained with the LC functional, provide an accurate one-electron energy spectrum. This method represents a one-electron orbital theory that is attractive in its simple formulation and effective in its practical application.

Computational Study of the Electron Spectra of Vapor-Phase Indole and Four Azaindoles

After geometry optimization, the electron spectra of indole and four azaindoles are calculated by density functional theory. Available experimental photoemission and excitation data for indole and 7-azaindole are used to compare with the theoretical values. The results for the other azaindoles are presented as predictions to help the interpretation of experimental spectra when they become available.

Computational study of the structures and photoelectron spectra of 12 azabenzenes

The molecular structures of 12 azabenzenes have been optimized with the Gaussian09 package at the level of coupled cluster singles and doubles with the basis set cc-pVTZ. The optimized geometry of each is used in the ADF13 program for the calculation of the vertical ionization energies of all the electrons. For both outer-shell and inner-shell valence electrons, the 2009 method of ΔPBE0(SAOP) is used, whereas the 1999 method of ΔPW86PW91 + Crel is employed for the core electrons. For degenerate orbitals, the alternative method chosen is to use localized orbitals, keeping the integer number of electrons, while giving up proper symmetry, rather than to keep symmetry with fractional electrons. The success of the computed results of valence ionization potentials of pyridine and the diazabenzenes gives confidence for the predicted values for the higher azabenzenes. The calculated results for core-electron binding energies provide incentive to experimentalists to measure them with X-ray photoelectron spectrometers and (or) synchrotron facilities.

Electronic structure and VUV photoabsorption measurements of thiophene

The absolute photoabsorption cross sections for thiophene in the 5.0-10.7 eV range were measured using synchrotron radiation. New theoretical calculations performed at the time-dependent density functional theory level were used to qualitatively interpret the recorded photoabsorption spectrum. The calculations facilitated a re-analysis of the observed vibronic and Rydberg structures in the photoabsorption spectrum. Here a number of features have been re-assigned, while a number of other features have been assigned for the first time. This represents the most comprehensive and self-consistent assignment of the thiophene high-resolution photoabsorption spectrum to date.

Density functional theory study of allopurinol

Allopurinol vapour is studied with density functional theory. Using the best method from past experience for each property, we predict the equilibrium geometry, vibrational spectrum, dipole moment, average dipole polarizability, UV absorption spectrum, vertical ionization energies of valence electrons, and core-electron binding energies.