I.
INTRODUCTION
X-ray transient absorption spectroscopy is a powerful tool to study chemical dynamics in systems ranging from gas-phase molecules to solid-state materials.1,2 Subtle changes in electronic states are sensitively reflected in the shape of the core-to-valence absorption signals, which gives X-ray transient absorption spectroscopy unique capabilities to resolve charge-state, spin-state, and structural information. Recent developments in wavelength up-conversion through high-harmonic generation (HHG)3 have improved the time resolution of X-ray light sources from tens of femtoseconds down to hundreds of attoseconds.4–6 The past decade has witnessed great success of X-ray transient absorption spectroscopy in real-time tracking of ultrafast chemical dynamics. Examples include electronic coherence dynamics in rare-gas atoms,7,8 photodissociation or multi-mode vibrations of gas-phase molecules,9–14 and charge-carrier dynamics of solid state materials.15–18
Interpretation of X-ray transient absorption spectra requires comprehensive pictures of potential energy surfaces, in both valence and core-excited states. Experimental characterization of the core-excited landscapes is difficult due to the short autoionization lifetimes inherent to those highly excited states. Theoretical calculations are therefore needed to predict and explain the transitions, but there are several challenges in the computational treatment of core-excited states.19 First, core-excited states are embedded in an energy-level continuum lying above an ionization threshold, and a reduction of the many configuration state functions to a tractable number is necessary. Second, for core electrons, especially those in heavy elements, relativistic effects such as spin-orbit coupling are critical. Third, calculations have to be robust throughout the reaction coordinates, from the Franck-Condon region through transition states to the asymptotic limit. As ultrafast X-ray absorption spectroscopy is becoming a standard experimental technique, computational tools that can be widely applied for core-excited states are strongly desired.20–22
Here, we employ the recently developed method of spin-orbit general multiconfigurational quasidegenerate perturbation theory (SO-GMC-QDPT)23 and investigate the Br-3d core-excited electronic structures of hydrogen bromide (HBr). The M4,5–3d edge of bromine exhibits characteristic absorption peaks with photon energies in the range of 60–75 eV.24 These signals are readily accessible using HHG-based attosecond extreme ultraviolet (XUV) light sources, and a series of experiments on molecular dynamics have been reported based on the Br-3d-edge probing.9,25–28 The target molecule HBr serves as a benchmark for numerous spectroscopic studies owing to its simple structure and rich photochemical dynamics. The UV photolysis of HBr involves multiple electronic states that become spectroscopically bright due to intensity borrowing induced by spin-orbit coupling.29–31 In ionic HBr+, the ground X
II.
COMPUTATIONAL DETAILS
The electronic structures of HBr and HBr+ are computed using the SO-GMC-QDPT code23 implemented in the developer version of the GAMESS-US program package.38 The GMC-QDPT is a typical “perturb first, diagonalize second” method that includes both the non-dynamic and dynamic correlations.39–41 In the SO-GMC-QDPT scheme, the spin-free GMC-QDPT states are used as multi-electron basis states to calculate spin-orbit matrix elements. Diagonalization of the spin-orbit matrix results in energies and wave functions of the states that are perturbed by the spin-orbit interaction. In all computations, the ZFK-DK3 relativistic model core potential (MCP) and basis sets of triple-zeta quality42–45 are used. The MCPs are optimized to reproduce the integrals related to spin-orbit couplings, and they remove 12 core electrons from the Br atom. In the perturbation-treatment step, an energy-denominator shift of 0.01 Hartree is applied for intruder-state avoidance.46,47
A Hartree-Fock self-consistent field (SCF) computation is performed at the ground-state equilibrium internuclear distance of Re = 1.41 Å. The resultant molecular orbitals are used as initial orbitals for the subsequent state-averaged multi-configurational self-consistent field (SA-MCSCF) computations. Two active spaces are constructed based on the occupation-restricted multiple active space (ORMAS) scheme. The valence-active space is composed of the Br-4p and H-1s orbitals, containing 6 electrons in 4 orbitals (or 5 electrons in the ionic case). This is defined as a complete active space, i.e., all excitations are allowed in this space. The core-active space is composed of the Br-3d orbitals, and it is fully occupied containing 10 electrons in 5 orbitals. Single excitations from the core-active space to the valence-active space are allowed, giving the targeted core-to-valence excitations. Note that Rydberg states are not included in the active spaces. Since the Br-3d orbitals are highly contracted in space, they have little overlap with the diffuse orbitals, whose occupations are responsible for the Rydberg states. As a result, the transition dipole moments between the valence and core-excited states, those relevant for the XUV absorption spectrum, are unaffected by the exclusion of Rydberg states.
The valence electronic structures are computed using the valence-active space alone, and the core-excited electronic structures are computed using both the valence- and core-active spaces. The five Br-3d orbitals are frozen in the SA-MCSCF step to facilitate convergence.22 In order to obtain the correct spin-orbit energy splittings (3685 cm−1 for Br-4p and 8388 cm−1 for Br-3d orbitals48), effective-nuclear charges Zeff = 35.9 and 39.3 are used for Br in the valence- and the core-excited-state calculations, respectively. In the present calculations, the one-electron spin-orbit interaction is described by the first-order Douglas-Kroll approximation.49,50 The main body of the two-electron spin-orbit interaction is between the core and valence active electrons, which can be understood as screening of the one-electron spin-orbit interaction of the valence electrons and nuclei.51–54 Therefore, it is safe to incorporate this part of the two-electron spin-orbit interaction into the one-electron spin-orbit operator with an effective nuclear charge that reflects the screening. Furthermore, the potential energy curves of the neutral (ionic) core-excited states are shifted upward by 1.01 (1.05) eV with respect to those of the valence states so that the experimental 3d →4p excitation energies in the Br atom (cation) are reproduced.24 These constant energy shifts are needed because the basis sets are optimized only for the ground-state atomic energy, not to accurately reproduce the experimental core-to-valence transition energies.
III.
NEUTRAL ELECTRONIC STATES
In this section, we present the computed results for neutral HBr. Electronic structures of the valence and core-excited states are analyzed, after which the core-to-valence absorption strengths relevant to the UV photolysis are discussed.
A.
Valence states of HBr
Figures 1(a), 1(b) and 1(c), 1(d) show spin-orbit-free and spin-orbit-coupled potential energy curves of HBr, respectively. Molecular term symbols
FIG. 1.
Potential energy curves of HBr calculated (a) and (b) without and (c) and (d) with spin-orbit coupling. Main electronic configurations in the Franck-Condon region are denoted in brackets for the spin-orbit-free states. For the spin-orbit-coupled states, different colors are used to indicate their associated Ω quantum numbers (projection of the orbital angular momentum along the H-Br axis). In (c), there are 12 states (
The valence states of HBr [Figs. 1(b) and 1(d)] arising from the H(
Spectroscopic parameters (equilibrium internuclear distance Re, harmonic vibrational frequency ωe, and anharmonicity ωexe) of the bound
TABLE I.
Spectroscopic parameters determined for the bound electronic states in HBr and HBr+. Reference values are taken from previous experimental work. a:
State | Data source | ||||
---|---|---|---|---|---|
HBr | X | 1.40 | 2652.5 | 48.1 | This work |
1.41 | 2649.0 | 45.2 | Exp. a, c | ||
HBr+ | X 2Π3/2 | 1.45 | 2345.4 | 42.6 | This work |
1.45 | 2439.0 | 45.2 | Exp. a, c | ||
X 2Π1/2 | 1.45 | 2343.3 | 43.0 | This work | |
1.45 | 2431.3 | 44.0 | Exp. a, c | ||
A 2Σ1/2 | 1.68 | 1336.4 | 32.4 | This work | |
1.68 | 1322.8 | 40.3 | Exp. b, c | ||
HBr*+ | 2Δ5/2 | 1.44 | 2400.0 | 45.7 | This work |
2Π3/2 | 1.44 | 2403.4 | 46.8 | This work | |
2Σ1/2 | 1.44 | 2410.7 | 48.4 | This work | |
1.44 | 2401.2 | 46.5 | This work | ||
2Π1/2 | 1.44 | 2413.5 | 48.5 | This work |
B.
Core-excited states of HBr
The
The effect of spin-orbit coupling in the core-excited states is straightforward; it only splits the potential energy curves into two manifolds, each of which correlates with the Br(
C.
Core-to-valence absorption spectra of HBr
The core-to-valence absorption strengths from the five valence states (
FIG. 2.
Core-to-valence absorption strengths of HBr as a function of the internuclear distance calculated using the SO-GMC-QDPT results. The absorption strengths are computed from (a)
The Br-3d transition energies from the ground
Three of the lowest excited states,
A remarkable trend is observed in
IV.
IONIC ELECTRONIC STATES
We next consider the valence and core-excited states of HBr+. The singly charged ion exhibits both bound and predissociative states, and we will discuss the Br-3d edge probing of these electronic states.
A.
Valence states of HBr+
The singly charged Br+ ion with the
FIG. 3.
Potential energy curves of HBr+ calculated (a) and (b) without and (c) and (d) with spin-orbit coupling. Atomic term symbols are given for lower dissociation limits of the valence states. For the spin-orbit-coupled states, different colors are used to indicate their associated Ω quantum numbers.
Spectroscopic parameters calculated for the X 2Π3/2, X 2Π1/2, and A 2Σ1/2 states [Fig. 3(d)] are summarized in Table I. Overall, a good agreement with the experimental values33,35,56 is obtained, corroborating the accuracy of the present computational method.
B.
Core-excited states of HBr+
There are both bound and dissociative states in the core-excited configurations of HBr+ [Figs. 3(a) and 3(c)], in contrast to the neutral system where only dissociative states are formed [Figs. 1(a) and 1(c)]. The three lowest spin-orbit-free states,
Note that the three bound core-excited states are energetically separated [Fig. 3(a)], whereas in the neutral system, all six spin-orbit-free core-excited states are degenerate [Fig. 1(a)]. The origin of the energy splitting can be understood in terms of ligand-field splitting.59–64 In the ionic system, there is a field gradient along the bond axis that shifts the energies of the Br-3d orbitals. The ligand fields are mainly created by two contributions. One is the polarized density of the valence Br-4p electrons (valence term) distributed between the parallel (
Care must be taken in analyzing the energy splittings in the ionic states because both the spin-orbit coupling and the ligand-fields are making mixed contributions. In order to analyze the two interactions separately, we employ a model Hamiltonian59,65
(1)
which describes the spin-orbit coupling and the ligand-field splitting as additional effects to the Br-3d core-ionized states. In Eq. (1), the operatorsFigure 4 summarizes the fitted results of the model parameters from 0.8 Å to 2.4 Å. The splitting-free energy E3d [Fig. 4(a)] exhibits a bound-potential shape corresponding to the bonding
FIG. 4.
Fitted results of (a) the ionic state energy free of the splitting effects
C.
Core-to-valence absorption spectra of HBr+
The Br-3d core-to-valence absorption strengths in the ionic HBr+ are computed in the same way as in the neutral system using the SO-GMC-QDPT results. Figure 5(a) shows the core-to-valence absorption strengths calculated from the bound X 2Π3∕2 state. In the vicinity of the Franck-Condon region, the lower absorption signals corresponding to the
FIG. 5.
Core-to-valence absorption strengths of HBr+ as a function of the internuclear distance calculated using the SO-GMC-QDPT results. The absorption strengths are computed from (a) X 2Π3/2 and (b) A 2Σ1/2. The equilibrium internuclear distances as well as the location of the avoided crossing is indicated on the horizontal axis. The correspondence between the core-excited potentials in Fig. 3(c) is as follows: the
Figure 5(b) shows the core-to-valence absorption strengths calculated from the predissociative A 2Σ1∕2 state. One can clearly observe the drastic variation of the absorption signals around the avoided crossing (R ∼ 2.3 Å). In the bound-potential region (Re = 1.68 Å), the electronic state belongs to the
V.
COMPARISON WITH THE EXPERIMENTAL TRANSIENT-ABSORPTION SPECTRUM
In order to evaluate the accuracy of the calculated results, we make a comparison with an experimental XUV transient-absorption spectrum of HBr and HBr+ (Fig. 6). The Br-3d absorption spectra are simulated by numerically solving the time-dependent Schrödinger equation.66,67 The valence electronic states selected for the simulations are the neutral X
FIG. 6.
Comparison of the simulated and experimental absorption spectra. The experimental spectra of HBr and HBr+ (open circles, right axis) are recorded as differential optical density (ΔOD), which is the difference in absorbance when the ionizing laser pulse is on and off. The simulated absorption spectra (filled areas, left axis) are obtained by numerically solving the time-dependent Schrödinger equation for nuclear wave packets. The initial wave functions are taken to be the ground vibrational state of the (a) neutral X
In Fig. 6, simulated spectra are shown by filled areas, and the experimental spectrum is plotted by open circles. Note that the simulated signals from the neutral X
The Br-3d absorption signals originating from the neutral X
VI.
CONCLUSIONS
We have applied the SO-GMC-QDPT method to calculate the Br-3d core-to-valence absorption signals in HBr and HBr+. In neutral HBr, five valence states involved in the UV photolysis (
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Abstract
Ultrafast X-ray/XUV transient absorption spectroscopy is a powerful tool for real-time probing of chemical dynamics. Interpretation of the transient absorption spectra requires knowledge of core-excited potentials, which necessitates assistance from high-level electronic-structure computations. In this study, we investigate Br-3d core-excited electronic structures of hydrogen bromide (HBr) using spin-orbit general multiconfigurational quasidegenerate perturbation theory (SO-GMC-QDPT). Potential energy curves and transition dipole moments are calculated from the Franck-Condon region to the asymptotic limit and used to construct core-to-valence absorption strengths for five electronic states of HBr (
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer