Iteration Scheme for Solving the System of Coupled Integro-Differential Equations for Excited and Ionized States of Molecular Systems
Abstract
:1. Introduction
2. Materials and Methods
2.1. Basic Equation of Iteration Scheme
2.2. The Inclusion of Higher Spherical Harmonics in SC Expansion of MO
2.3. The Expansion of the Wave Functions of the Electron Shells of an Atom in a Series of Spherical Harmonics
3. Solution Procedure
3.1. Scheme for Numerical Solution for the System of Equations for MO
3.2. Starting Wave Function MO and Stopping Criteria
3.3. Summary of the Iterative Scheme
4. Results and Discussions
- A relatively small number of molecule shells allows the exploration of the possibility of the developed iterative scheme and the procedure for taking into account the higher harmonics in the SC expansion of MO when they are implemented on the ECM with limited CPU time;
- The spectrum of free states and the structure of the wave functions have been thoroughly investigated [34].
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- Dutoi, A.D.; Leone, S.R. Simulation of X-ray transient absorption for following vibrations in coherently ionized F2 molecules. Chem. Phys. 2017, 482, 249–264. [Google Scholar] [CrossRef]
- Granados-Castro, C.M.; Ancarani, L.U.; Gasaneo, G.; Mitnik, D.M. A Sturmian Approach to Photoionization of Molecules. In Advances in Quantum Chemistry, 1st ed.; Hoggan, P.E., Ozdogan, T., Eds.; Elsevier: Amsterdam, The Netherlands, 2016; Volume 73, pp. 3–57. ISBN 978-0-12-803060-8. [Google Scholar]
- Yavna, V.; Hopersky, A.; Nadolinsky, A.; Yavna, S. Orientation effects in elastic scattering of polarized X-rays by linear molecules. J. Synchrotron Radiat. 2001, 8, 240–242. [Google Scholar] [CrossRef] [PubMed]
- Gel’mukhanov, F.; Gren, H. Resonant inelastic x-ray scattering with symmetry-selective excitation. Phys. Rev. A 1994, 49, 4378–4389. [Google Scholar] [CrossRef] [PubMed]
- Roothaan, C.C.J. Self–Consistent Field Theory for Open Shell of Electronic Systems. Rev. Mod. Phys. 1960, 32, 179–185. [Google Scholar] [CrossRef]
- Bishop, D.M. Single-Center Molecular Wave Function. In Advances in Quantum Chemistry, 1st ed.; Löwdin, P.-O., Ed.; Elsevier: Amsterdam, The Netherlands, 1967; Volume 3, pp. 25–59. ISBN 978-0-12-034803-9. [Google Scholar]
- Pople, J.A. Molecular orbital methods in organic chemistry. Acc. Chem. Res. 1970, 3, 217–223. [Google Scholar] [CrossRef]
- Wilson, S. Electron Correlation in Molecules, 2nd ed.; Dover Publications: Mineola, NY, USA, 2007; pp. 1–304. ISBN 978-0-48-645879-3. [Google Scholar]
- Bunge, C.F.; Barrientos, J.A.; Bunge, A.V.; Cogordan, J.A. Hartree-Fock and Roothaan-Hartree-Fock energies for the ground states of He through Xe. Phys. Rev. A 1992, 46, 3691–3696. [Google Scholar] [CrossRef] [PubMed]
- Huzinaga, S. Gaussian–Type Functions for the Polyatomic Systems. J. Chem. Phys. 1965, 42, 1293–1302. [Google Scholar] [CrossRef]
- Roothaan, C.C.J. New Development in Molecular Orbital Theory. Rev. Mod. Phys. 1951, 23, 69–89. [Google Scholar] [CrossRef]
- Schmidt, M.W.; Baldridge, K.K.; Boatz, J.A.; Elbert, S.T.; Gordon, M.S.; Jensen, J.H.; Koseki, S.; Matsunaga, N.; Nguyen, K.A.; Su, S.; et al. General atomic and molecular electronic structure system. J. Comput. Chem. 1993, 14, 1347–1363. [Google Scholar] [CrossRef]
- Lowe, J.P. Quantum Chemistry, 1st ed.; Academic Press: Cambridge, MA, USA, 1978; pp. 1–599. ISBN 978-0-12-457550-9. [Google Scholar]
- Faisal, F.H.M. Electron-Molecular Interactions: I-Single Center Wave-Functions and Potentials. J. Phys. B 1970, 3, 636–640. [Google Scholar] [CrossRef]
- Yavna, V.A.; Nadolinsky, A.M.; Hopersky, A.N. Theoretical study of processes of multiple excitation/ionization in 2s-photoabsorption of the CO molecule. J. Electron Spectrosc. Relat. Phenom. 1998, 94, 49–57. [Google Scholar] [CrossRef]
- Yavna, V.A.; Nadolinsky, A.M.; Demekhin, V.F. Theoretical study of inner shell photoabsorption spectra of simple molecules. J. Electron Spectrosc. Relat. Phenom. 1994, 68, 267–275. [Google Scholar] [CrossRef]
- Demekhin, P.V.; Omel’yanenko, D.V.; Lagutin, B.M.; Sukhorukov, V.L.; Werner, L.; Ehresmann, A.; Schartner, K.-H.; Schmoranzer, H. Investigation of photoionization and photodissociation of an oxygen molecule by the method of coupled differential equations. Opt. Spectrosc. 2007, 103, 318–329. [Google Scholar] [CrossRef]
- Lagutin, B.M.; Migal, Y.F. Single-center method of calculation for clusters and molecules other than hydrides. Theor. Exp. Chem. 1987, 25, 10–17. [Google Scholar] [CrossRef]
- Demekhin, P.V.; Ehresmann, A.; Sukhorukov, V.L. Single center method: A computational tool for ionization and electronic excitation studies of molecules. J. Chem. Phys. 2011, 134, 024113. [Google Scholar] [CrossRef] [PubMed]
- Demekhin, P.V.; Lagutin, B.M.; Petrov, I.D. Theoretical study of angular-resolved two-photon ionization of H2. Phys. Rev. A 2012, 85, 023416. [Google Scholar] [CrossRef]
- Inhester, L.; Burmeister, C.F.; Groenhof, G.; Grubmüller, H. Auger spectrum of a water molecule after single and double core ionization. J. Chem. Phys. 2012, 136, 144304. [Google Scholar] [CrossRef] [PubMed]
- Galitskiy, S.A.; Artemyev, A.N.; Jänkälä, K.; Lagutin, B.M.; Demekhin, P.V. Hartree-Fock calculation of the differential photoionization cross sections of small Li clusters. J. Chem. Phys. 2015, 142, 034306. [Google Scholar] [CrossRef] [PubMed]
- Stern, M.S. Comparison of numerical solutions of the partial-wave Schrödinger differential and integral equations. J. Comput. Phys. 1977, 25, 56–70. [Google Scholar] [CrossRef]
- Gavnholt, J.; Olsen, T.; Engelund, M.; Schiøtz, J. Δ self-consistent field method to obtain potential energy surfaces of excited molecules on surfaces. Phys. Rev. B 2008, 78, 075441. [Google Scholar] [CrossRef]
- Yavna, V.A.; Nadolinsky, A.M.; Demekhin, V.P. Theoretical study of photoabsorption cross sections in the regions of CK- and NK-ionization thresholds of CO and N2 molecules. Opt. Spectrosc. 1990, 69, 1278–1284. [Google Scholar]
- Yavna, V.A.; Nadolinsky, A.M.; Demekhin, V.P. Theoretical study of XANES 2σ-shells CO. Opt. Spectrosc. 1992, 73, 1081–1085. [Google Scholar]
- Cacelli, I. The Hartree–Fock Method. In Reference Module in Chemistry, Molecular Sciences and Chemical Engineering, 1st ed.; Reedijk, J., Ed.; Elsevier: Amsterdam, The Netherlands, 2015; pp. 1–14. ISBN 978-0-124-09547-2. [Google Scholar]
- Sobel’Man, I.I. Introduction to the Theory of Atomic Spectra, 1st ed.; Pergamon Press: Oxford, UK, 1972; pp. 1–626. ISBN 978-1-483-15972-0. [Google Scholar]
- Varshalovich, D.A.; Moskalev, A.N.; Khersonskii, V.K. Quantum Theory of Angular Momentum, 1st ed.; World Scientific: Singapore, 1988; pp. 1–526. ISBN 978-9-971-50107-5. [Google Scholar]
- Hairer, E.; Nørsett, S.P.; Wanner, G. Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 1993; pp. 1–528. ISBN 978-3-540-56670-0. [Google Scholar]
- Numerov, B. Note on the numerical integration of d2x/dt2 = f(x, t). Astron. Nachr. 1927, 230, 359–364. [Google Scholar] [CrossRef]
- Ahuja, P. Tridiagonal matrix algorithm (TDMA). In Introduction to Numerical Methods in Chemical Engineering, 1st ed.; Ahuja, P., Ed.; PHI Learning: Delhi, India, 2010; pp. 1–304. ISBN 978-8-120-34018-3. [Google Scholar]
- The Program Code for Calculation the Wave Function of a Molecular Systems. Available online: http://www.rgups.ru/site/assets/files/39146/software_code.zip (accessed on 20 November 2017).
- Yavna, V.A.; Popov, V.A.; Yavna, S.A. Correlation and vibronic effects in K-photoabsorption of HF and HCl. Opt. Spectrosc. 1993, 75, 39–46. [Google Scholar]
- Kane, P.P. Inelastic scattering of X-rays and gamma rays by inner shell electrons. Phys. Rep. 1992, 218, 67–139. [Google Scholar] [CrossRef]
Type | 1σ | 2σ | 3σ | 1π | Total Energy |
---|---|---|---|---|---|
a | −26.2923 | −1.5990 | −0.7674 | −0.6464 | −100.0577 |
b | −26.2905 | −1.5995 | −0.7656 | −0.6492 | −100.0594 |
c | −26.2915 | −1.5995 | −0.7656 | −0.6492 | −100.0616 |
Total Energy (a) | Total Energy (b) | |
---|---|---|
2 | −101.0902 | −101.0951 |
3 | −102.8639 | −102.9189 |
4 | −105.1593 | −105.4553 |
5 | −107.7773 | −108.5403 |
Ionized Shell | 1σ | 2σ | 3σ | 1π | Total Energy |
---|---|---|---|---|---|
−30.1430 | −2.3366 | −1.5016 | −1.4186 | −74.6050 | |
−26.9639 | −2.3036 | −1.3277 | −1.2266 | −98.5775 | |
−26.9128 | −2.1432 | −1.4587 | −1.2196 | −99.3889 |
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Kasprzhitskii, A.; Lazorenko, G.; Yavna, V. Iteration Scheme for Solving the System of Coupled Integro-Differential Equations for Excited and Ionized States of Molecular Systems. Algorithms 2018, 11, 1. https://doi.org/10.3390/a11010001
Kasprzhitskii A, Lazorenko G, Yavna V. Iteration Scheme for Solving the System of Coupled Integro-Differential Equations for Excited and Ionized States of Molecular Systems. Algorithms. 2018; 11(1):1. https://doi.org/10.3390/a11010001
Chicago/Turabian StyleKasprzhitskii, Anton, Georgy Lazorenko, and Victor Yavna. 2018. "Iteration Scheme for Solving the System of Coupled Integro-Differential Equations for Excited and Ionized States of Molecular Systems" Algorithms 11, no. 1: 1. https://doi.org/10.3390/a11010001