The Resonant and Normal Auger Spectra of Ozone
<p>Ozone geometry in the ground state.</p> "> Figure 2
<p><math display="inline"><semantics> <mrow> <mi>K</mi> <mo>−</mo> <mi>L</mi> <mi>L</mi> </mrow> </semantics></math> Auger spectrum of ozone, where <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> </semantics></math>. Green line: HF level of theory with 80 independent final states, obtained by diagonalization of the multichannel Hamiltonian (<a href="#FD24-symmetry-13-00516" class="html-disp-formula">24</a>). The black, red, and blue lines represent the spectral lineshape obtained with an active space of 27 molecular orbitals and a number of 20, 70, 140 independent final channels, respectively.</p> "> Figure 3
<p><math display="inline"><semantics> <mrow> <mi>K</mi> <mo>−</mo> <mi>L</mi> <mi>L</mi> </mrow> </semantics></math> Auger spectrum of ozone, where <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> <msub> <mi>b</mi> <mn>1</mn> </msub> </mrow> </semantics></math>. Green line: HF level of theory with 80 independent final states, obtained by diagonalization of the multichannel Hamiltonian (<a href="#FD24-symmetry-13-00516" class="html-disp-formula">24</a>). Black, red, and blue lines represent the spectral lineshape obtained with an active space of 27 molecular orbitals and a number of 20, 70, and 140 independent final channels, respectively.</p> "> Figure 4
<p><math display="inline"><semantics> <mrow> <mi>K</mi> <mo>−</mo> <mi>L</mi> <mi>L</mi> </mrow> </semantics></math> Auger spectrum of ozone, where <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> </semantics></math>. Green line: HF level of theory with 80 independent final states, obtained by diagonalization of the multichannel Hamiltonian (<a href="#FD24-symmetry-13-00516" class="html-disp-formula">24</a>). Black, red, and blue lines represent the spectral lineshape obtained with an active space of 27 molecular orbitals and a number of 20, 70, and 140 independent final channels, respectively.</p> "> Figure 5
<p>A comparison between the <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>−</mo> <mi>L</mi> <mi>L</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> </semantics></math>) Auger experimental spectrum of ozone (black line) [<a href="#B10-symmetry-13-00516" class="html-bibr">10</a>] and our first-principles simulation (green line) obtained with an active space of 27 molecular orbitals and 140 independent final channels. Our lineshapes were convolved via 0.8 eV Gaussian function to achieve the experimental broadening.</p> "> Figure 6
<p><math display="inline"><semantics> <mrow> <mi>O</mi> <mn>1</mn> <mi>s</mi> <mo>→</mo> <msup> <mi>σ</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> autoionization spectrum of ozone. Red, blue lines: CIS lineshapes with 140 final states obtained upon diagonalization of the multichannel Hamiltonian (<a href="#FD24-symmetry-13-00516" class="html-disp-formula">24</a>). The core-hole belongs to molecular symmetry orbitals, which are optimized using HF and extend all over the ozone molecule. However, upon CI the hole relocalizes in one of the two oxygen atoms (<math display="inline"><semantics> <msub> <mi>O</mi> <mi>T</mi> </msub> </semantics></math>) related by symmetry operations (see inset). Green line: the spectral lineshape for a core-hole relocalized after the CIS procedure in the central oxygen atom <math display="inline"><semantics> <msub> <mi>O</mi> <mi>C</mi> </msub> </semantics></math>.</p> "> Figure 7
<p><math display="inline"><semantics> <mrow> <mi>K</mi> <mo>−</mo> <mi>L</mi> <mi>L</mi> </mrow> </semantics></math> Auger spectrum of ozone obtained from CIS calculations after convolution with both Lorentzian functions having the theoretical widths and with 1 eV full width at half maximum (FWHM) Gaussian functions to reproduce the combined effect of the finite resolution of the spectrometer and of the nuclear vibrations. Black line: HF orbitals optimized starting from an atomic core-hole localized in one of the two <math display="inline"><semantics> <msub> <mi>O</mi> <mi>T</mi> </msub> </semantics></math> oxygen centers. Red line: HF orbitals are optimized with the initial core-hole “delocalized” in the molecular symmetry orbitals. Calculations were carried out with an active space of 27 molecular orbitals and 140 independent final channels, including the interchannel coupling (<a href="#FD24-symmetry-13-00516" class="html-disp-formula">24</a>).</p> ">
Abstract
:1. Introduction
2. Theoretical and Computational Methods
2.1. Autoionization and Auger Decay as Resonant Multichannel Processes
2.2. The Projected Potential Approach and the Many-Body Hamiltonian
3. Results and Discussion
3.1. The Ozone Molecular Geometry and Its Electronic Structure
3.2. The Auger Spectrum of Ozone
- ;
- ;
- .
3.3. The Autoionization Spectrum of Ozone
3.4. Core-Hole Orbital Symmetry and Localization
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Abbreviations
DFT | Density Functional Theory |
HF | Hartree–Fock |
CI | Configuration Interaction |
CIS | Configuration Interaction Single |
HGF | Hermite Gaussian Function |
FWHM | Full width at half maximum |
HGF | Hermite Gaussian Function |
PBE-GGA | Perdew–Burke–Ernzerhof generalized gradient approximation |
Appendix A. The Auger Spectrum of Ozone
Trans. State Occ. | Trans. En. | Trans. Prob. |
---|---|---|
2.00 | 2.00 | 1.97 | 1.59 | 1.42 | 1.64 | 0.09 | 1.71 | 495.17 | 0.1100 |
0.39 | 2.00 | 1.94 | 1.65 | 1.60 | 0.13 | 1.86 | ||
2.00 | 2.00 | 1.58 | 1.63 | 1.77 | 1.86 | 0.37 | 1.69 | 467.14 | 0.1818 |
0.15 | 2.00 | 1.58 | 1.67 | 1.84 | 0.25 | 1.61 | ||
2.00 | 2.00 | 2.00 | 1.95 | 1.63 | 1.53 | 0.19 | 1.38 | 502.3 | 0.0879 |
0.34 | 2.00 | 1.99 | 1.89 | 1.59 | 0.17 | 1.34 | ||
2.00 | 2.00 | 2.00 | 1.96 | 1.70 | 1.62 | 0.14 | 1.39 | 506.25 | 0.0767 |
0.54 | 2.00 | 1.99 | 1.90 | 1.68 | 0.05 | 1.01 | ||
2.00 | 2.00 | 1.99 | 1.94 | 1.80 | 1.53 | 0.34 | 1.21 | 500.02 | 0.0804 |
0.21 | 2.00 | 1.99 | 1.89 | 1.46 | 0.24 | 1.40 | ||
2.00 | 2.00 | 2.00 | 1.96 | 1.94 | 1.78 | 0.06 | 1.50 | 506.065 | 0.0697 |
0.55 | 2.00 | 1.98 | 1.31 | 1.91 | 0.03 | 0.98 | ||
2.00 | 2.00 | 1.99 | 1.93 | 1.74 | 1.54 | 0.14 | 1.44 | 506.03 | 0.0761 |
0.53 | 2.00 | 1.99 | 1.88 | 1.65 | 0.06 | 1.12 | ||
2.00 | 2.00 | 2.00 | 1.88 | 1.67 | 1.49 | 0.21 | 1.90 | 503.195 | 0.0596 |
0.18 | 2.00 | 1.98 | 1.33 | 1.70 | 0.12 | 1.55 | ||
2.00 | 2.00 | 1.69 | 1.67 | 1.45 | 1.83 | 0.19 | 1.79 | 476.33 | 0.0352 |
0.44 | 2.00 | 1.61 | 1.77 | 1.73 | 0.15 | 1.67 | ||
2.00 | 2.00 | 1.99 | 1.90 | 1.47 | 1.35 | 0.17 | 1.81 | 502.25 | 0.0693 |
0.26 | 2.00 | 1.97 | 1.73 | 1.50 | 0.27 | 1.58 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.36 | 1.22 | 0.03 | 1.95 | 508.36 | 0.0705 |
0.30 | 2.00 | 1.99 | 1.82 | 1.59 | 0.03 | 1.74 | ||
2.00 | 2.00 | 2.00 | 1.96 | 1.46 | 1.52 | 0.10 | 1.97 | 504.16 | 0.0518 |
0.54 | 2.00 | 1.99 | 1.67 | 1.21 | 0.11 | 1.47 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.13 | 0.02 | 1.54 | 512.03 | 0.0553 |
0.59 | 2.00 | 2.00 | 1.98 | 1.93 | 0.01 | 0.89 | ||
2.00 | 2.00 | 1.34 | 1.59 | 1.78 | 1.86 | 0.30 | 1.87 | 453.39 | 0.1166 |
0.29 | 2.00 | 1.41 | 1.76 | 1.72 | 0.27 | 1.81 | ||
2.00 | 2.00 | 1.99 | 1.73 | 1.67 | 1.59 | 0.09 | 1.77 | 503.62 | 0.0466 |
0.44 | 2.00 | 1.97 | 1.68 | 1.25 | 0.05 | 1.77 | ||
2.00 | 2.00 | 1.99 | 1.87 | 1.80 | 1.67 | 0.28 | 1.42 | 496.73 | 0.0512 |
0.28 | 2.00 | 1.99 | 1.63 | 1.50 | 0.26 | 1.31 | ||
2.00 | 2.00 | 2.00 | 1.95 | 1.95 | 1.49 | 0.49 | 1.83 | 502.45 | 0.0416 |
0.16 | 2.00 | 1.99 | 1.35 | 1.84 | 0.06 | 0.90 | ||
2.00 | 2.00 | 1.99 | 1.82 | 1.81 | 1.67 | 0.21 | 1.61 | 500.80 | 0.0432 |
0.23 | 2.00 | 1.99 | 1.60 | 1.59 | 0.27 | 1.21 | ||
Total Transition Probability () | A.U. |
Trans. State Occ. | Trans. En. | Trans. Prob. |
---|---|---|
2.00 | 2.00 | 2.00 | 1.99 | 1.98 | 1.92 | 0.04 | 1.93 | 510.751 | 0.2255 |
0.03 | 2.00 | 1.99 | 1.98 | 1.87 | 0.03 | 0.24 | ||
2.00 | 2.00 | 2.00 | 2.00 | 1.97 | 1.90 | 0.03 | 1.86 | 510.17 | 0.1713 |
0.08 | 2.00 | 1.99 | 1.98 | 1.10 | 0.02 | 1.07 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.46 | 1.58 | 0.05 | 1.89 | 502.71 | 0.1755 |
0.38 | 2.00 | 1.98 | 1.80 | 1.19 | 0.03 | 1.67 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.36 | 1.22 | 0.03 | 1.95 | 502.79 | 0.1453 |
0.30 | 2.00 | 1.99 | 1.82 | 1.59 | 0.03 | 1.74 | ||
2.00 | 2.00 | 2.00 | 1.97 | 1.49 | 1.94 | 0.03 | 1.78 | 504.44 | 0.1286 |
0.33 | 2.00 | 1.99 | 1.98 | 1.56 | 0.02 | 0.90 | ||
2.00 | 2.00 | 2.00 | 1.99 | 1.97 | 1.09 | 0.02 | 1.90 | 510.93 | 0.1439 |
0.07 | 2.00 | 2.00 | 1.98 | 1.94 | 0.02 | 1.03 | ||
2.00 | 2.00 | 2.00 | 1.97 | 1.84 | 1.29 | 0.02 | 1.95 | 508.79 | 0.1289 |
0.18 | 2.00 | 1.99 | 1.96 | 1.09 | 0.02 | 1.69 | ||
2.00 | 2.00 | 2.00 | 1.97 | 1.84 | 1.03 | 0.02 | 1.98 | 509.47 | 0.1359 |
0.19 | 2.00 | 2.00 | 1.97 | 1.16 | 0.01 | 1.82 | ||
2.00 | 2.00 | 2.00 | 1.99 | 1.98 | 1.81 | 0.04 | 1.10 | 504.35 | 0.1162 |
0.17 | 2.00 | 1.99 | 1.88 | 1.93 | 0.04 | 1.09 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.91 | 1.67 | 0.07 | 1.62 | 503.07 | 0.0798 |
0.33 | 2.00 | 1.99 | 1.52 | 1.82 | 0.03 | 1.07 | ||
2.00 | 2.00 | 2.00 | 1.92 | 1.87 | 1.69 | 0.02 | 1.92 | 503.99 | 0.0706 |
0.25 | 2.00 | 1.98 | 1.32 | 1.20 | 0.01 | 1.83 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.67 | 0.02 | 1.68 | 504.29 | 0.0608 |
0.29 | 2.00 | 1.98 | 1.52 | 1.86 | 0.01 | 1.06 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.13 | 0.02 | 1.54 | 506.46 | 0.0554 |
0.59 | 2.00 | 2.00 | 1.98 | 1.93 | 0.01 | 0.89 | ||
2.00 | 2.00 | 2.00 | 1.94 | 1.79 | 1.10 | 0.01 | 1.81 | 504.77 | 0.0777 |
0.47 | 2.00 | 1.99 | 1.59 | 1.65 | 0.01 | 1.64 | ||
2.00 | 2.00 | 1.91 | 1.62 | 1.55 | 1.65 | 0.29 | 1.83 | 472.09 | 0.0856 |
0.32 | 2.00 | 1.65 | 1.50 | 1.65 | 0.22 | 1.80 | ||
2.00 | 2.00 | 1.99 | 1.84 | 1.77 | 1.16 | 0.42 | 1.95 | 497.19 | 0.0339 |
0.28 | 2.00 | 1.96 | 1.59 | 1.23 | 0.04 | 1.76 | ||
2.00 | 2.00 | 1.73 | 1.70 | 1.70 | 1.57 | 0.34 | 1.91 | 467.51 | 0.0790 |
0.15 | 2.00 | 1.62 | 1.44 | 1.62 | 0.30 | 1.90 | ||
2.00 | 2.00 | 1.99 | 1.60 1.88 | 1.54 | 0.25 | 1.67 | 499.45 | 0.0032 |
0.18 | 2.00 | 1.99 | 1.83 | 1.87 | 0.05 | 1.15 | ||
2.00 | 2.00 | 1.68 | 1.70 | 1.68 | 1.68 | 0.19 | 1.68 | 468.17 | 0.0622 |
0.51 | 2.00 | 1.70 | 1.70 | 1.53 | 0.18 | 1.78 | ||
2.00 | 2.00 | 1.99 | 1.88 | 1.86 | 1.81 | 0.22 | 1.54 | 491.23 | 0.0032 |
0.37 | 2.00 | 1.73 | 1.62 | 1.74 | 0.14 | 1.09 | ||
Total Transition Probability () | A.U. |
Trans. State Occ. | Trans. En. | Trans. Prob. |
---|---|---|
2.00 | 2.00 | 2.00 | 1.99 | 1.98 | 1.92 | 0.04 | 1.93 | 510.07 | 0.2515 |
0.03 | 2.00 | 1.99 | 1.98 | 1.87 | 0.03 | 0.24 | ||
2.00 | 2.00 | 2.00 | 2.00 | 1.97 | 1.90 | 0.03 | 1.86 | 510.17 | 0.1693 |
| 0.08 | 2.00 | 1.99 | 1.98 | 1.10 | 0.02 | 1.07 | ||
2.00 | 2.00 2.00 | 1.98 | 1.46 | 1.58 | 0.05 | 1.89 | 502.71 | 0.1708 |
0.38 | 2.00 | 1.98 | 1.80 | 1.19 | 0.03 | 1.67 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.36 | 1.22 | 0.03 | 1.95 | 502.79 | 0.1478 |
0.30 | 2.00 | 1.99 | 1.82 | 1.59 | 0.03 | 1.74 | ||
2.00 | 2.00 | 2.00 | 1.97 | 1.49 | 1.94 | 0.03 | 1.78 | 504.44 | 0.1290 |
0.33 | 2.00 | 1.99 | 1.98 | 1.56 | 0.02 | 0.90 | ||
2.00 | 2.00 | 2.00 | 1.97 | 1.84 | 1.29 | 0.02 | 1.95 | 508.79 | 0.1401 |
0.18 | 2.00 | 1.99 | 1.96 | 1.09 | 0.02 | 1.69 | ||
2.00 | 2.00 2.00 | 1.99 | 1.97 | 1.09 | 0.02 | 1.90 | 510.92 | 0.1443 |
0.07 | 2.00 | 2.00 | 1.98 | 1.94 | 0.02 | 1.03 | ||
2.00 | 2.00 2.00 | 1.97 | 1.84 | 1.03 | 0.02 | 1.98 | 509.46 | 0.1225 |
0.19 | 2.00 | 2.00 | 1.97 | 1.16 | 0.01 | 1.82 | ||
2.00 | 2.00 | 2.00 | 1.99 | 1.98 | 1.81 | 0.04 | 1.10 | 504.35 | 0.1091 |
0.17 | 2.00 | 1.99 | 1.88 | 1.93 | 0.04 | 1.09 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.91 | 1.67 | 0.07 | 1.62 | 503.06 | 0.0797 |
0.33 | 2.00 | 1.99 | 1.52 | 1.82 | 0.03 | 1.07 | ||
2.00 | 2.00 | 2.00 | 1.92 | 1.87 | 1.69 | 0.02 | 1.92 | 503.99 | 0.0808 |
0.25 2.00 | 1.98 | 1.32 | 1.20 | 0.01 | 1.83 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.13 | 0.02 | 1.54 | 506.45 | 0.0552 |
0.59 | 2.00 | 2.00 | 1.98 | 1.93 | 0.01 | 0.89 | ||
2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.67 | 0.02 | 1.68 | 504.28 | 0.0610 |
0.29 | 2.00 | 1.98 | 1.52 | 1.86 | 0.01 | 1.06 | ||
2.00 | 2.00 | 2.00 | 1.94 | 1.79 | 1.10 | 0.01 | 1.81 | 504.77 | 0.0663 |
0.47 | 2.00 | 1.99 | 1.59 | 1.65 | 0.01 | 1.64 | ||
2.00 | 2.00 | 1.91 | 1.62 | 1.55 | 1.65 | 0.29 | 1.83 | 472.09 | 0.0878 |
0.32 | 2.00 | 1.65 | 1.50 | 1.65 | 0.22 | 1.80 | ||
2.00 | 2.00 | 1.99 | 1.84 | 1.77 | 1.16 | 0.42 | 1.95 | 497.19 | 0.030 |
0.28 | 2.00 | 1.96 | 1.59 | 1.23 | 0.04 | 1.76 | ||
2.00 | 2.00 | 1.99 | 1.60 | 1.88 | 1.54 | 0.25 | 1.67 | 499.45 | 0.0033 |
0.18 | 2.00 | 1.99 | 1.83 | 1.87 | 0.05 | 1.15 | ||
2.00 | 2.00 | 1.73 | 1.70 | 1.70 | 1.57 | 0.34 | 1.91 | 467.51 | 0.0799 |
0.15 | 2.00 | 1.62 | 1.44 | 1.62 | 0.30 | 1.90 | ||
2.00 | 2.00 | 1.68 | 1.70 | 1.68 | 1.68 | 0.19 | 1.68 | 468.17 | 0.0637 |
0.51 | 2.00 | 1.70 | 1.70 | 1.53 | 0.18 | 1.78 | ||
2.00 | 2.00 | 1.99 | 1.88 | 1.86 | 1.81 | 0.22 | 1.54 | 491.23 | 0.0032 |
0.37 | 2.00 | 1.73 | 1.62 | 1.74 | 0.14 | 1.09 | ||
Total Transition Probability () | A.U. |
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non rel. | −20.9179 | −20.7070 | −20.7070 | −1.7453 | −1.4281 | −1.0948 |
rel. | −20.9124 | −20.7015 | −20.7015 | −1.7460 | −1.4286 | −1.0957 |
non rel. | −0.8301 | −0.7983 | −0.7785 | −0.5640 | −0.5531 | −0.4870 |
rel. | −0.8296 | −0.7979 | −0.7778 | −0.5633 | −0.552 | −0.4865 |
non rel. | −224.356 | |||||
rel. | −224.408 |
Auger State | |||
---|---|---|---|
−203.705 (562.0) | −204.168 (549.4) | −204.268 (546.6) | |
−203.853 (557.9) | −204.373 (543.8) | −204.482 (540.8) | |
−203.854 (557.9) | −204.373 (543.8) | −204.482 (540.8) |
Number of Channels | |||
---|---|---|---|
Auger State | 20 | 70 | 140 |
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Taioli, S.; Simonucci, S. The Resonant and Normal Auger Spectra of Ozone. Symmetry 2021, 13, 516. https://doi.org/10.3390/sym13030516
Taioli S, Simonucci S. The Resonant and Normal Auger Spectra of Ozone. Symmetry. 2021; 13(3):516. https://doi.org/10.3390/sym13030516
Chicago/Turabian StyleTaioli, Simone, and Stefano Simonucci. 2021. "The Resonant and Normal Auger Spectra of Ozone" Symmetry 13, no. 3: 516. https://doi.org/10.3390/sym13030516