In this article, we have focused on several fundamental issues of angular momentum (ring current) of π electrons in a low-symmetry aromatic ring molecule. One of the issues is on characteristics of the angular momentum generated by ultrashort polarized laser pulse(s). The generated angular momentum, called coherent angular momentum, is time-dependent, i.e., its
Figure 16.(a) Expectation values of π-electron angular momentum for the free rotations after the [AO]
localization. (b) Results of the optimal control for maintaining the localization. The control time is tf= 30 fs. (c) Absolute values of the optimal electric field. The inset figure is the field in the restricted time domain. Reprinted from [91] with permission of the American Chemical Society.
Appl. Sci. 2018, 8, 2347 31 of 37
The simulations in this section were performed using a simplified model omitting both the effects of vibrational motions and those of heat bath modes. In real aromatic ring molecules these effects influence on π-electron rotations. Mineo et al. recently found that intramolecular vibrational effects make significant contributions to dephasing of π-electron rotations under finite temperature conditions [95] as well as bath mode effects in condensed phases beyond the Markov approximation [96]. It has been also shown in Section 2.3that nonadiabatic transitions strongly influence π-electron rotations [97]. Application of optimal control theory to such open systems is one of the important subsequent issues.
4. Summary and Outlook
In this article, we have focused on several fundamental issues of angular momentum (ring current) of π electrons in a low-symmetry aromatic ring molecule. One of the issues is on characteristics of the angular momentum generated by ultrashort polarized laser pulse(s). The generated angular momentum, called coherent angular momentum, is time-dependent, i.e., its direction changes in time, the duration of which is inversely proportional to the energy difference between quasi-degenerate, two electronic eigenstates. The phase in the oscillatory behavior of angular momentum, which is directly related to the relative quantum phase of the superposed quasi-degenerate states, can be controlled by the ellipticity and orientation of the incident laser. This is contrasted with the angular momentum generated in a highly symmetric aromatic molecule, which is time-independent, i.e., unidirectional. That is, stationary angular momentum is generated by an excitation of a degenerate excited state by a circularly polarized laser.
Another issue is the nonadiabatic coupling between laser-driven π-electron ring currents and molecular vibration. The coherent angular momentum is gradually attenuated by the nonadiabatic coupling. On the other hand, the amplitude of induced molecular vibration is prominently dependent on the direction of linear polarization vectors but insensitive to the helicity of circular polarization.
The characteristic feature in vibrational amplitudes is ascribed to the WP interference effects in nonadiabatic transition. This offers a new possibility of attosecond/several-femtosecond polarized laser pulses as a promising tool to manipulate molecular vibration (and chemical reactions) through the WP interference in nonadiabatic transition.
Recently, there has been new progress on the issue of coherent angular momentum. Yamaki et al.
have performed optimal control simulations of coherent π-electron rotations in (P)-2,20-biphenol by taking into account two types of control targets: one is generation of the maximum angular momentum of the x component, which corresponds to CC or AA rotation of π electrons shown in Figure12, and the other is the maintaining of the generated unidirectional angular momentum during a setting time duration [98]. The optimal control pulse for each target was designed. The analysis of the simulation results shows that the effective maintaining of unidirectional angular momentum can be realized by applying the 2π pulse to one of the two excited states forming a coherent electronic state. The 2π pulse prevents the reverse rotation of π electrons by dumping the excited state population to the ground state and subsequently by pumping the population back to the excited state. In addition, Yamaki et al. have found an effectively stationary angular momentum in a low-symmetry molecule [99]. This is realized by applying two linearly polarized lasers with the same frequency and with an appropriate relative phase to quasi-degenerate two excited states. This is due to creation of a dressed state with equal populations for the two excited eigenstates. Mineo et al. have proposed another laser-control scenario of generation of unidirectional angular momentum in a low-symmetry aromatic ring molecule [100].
This is intuitively explained in terms of dynamic Stark shifts of two relevant excited states, which are induced by using two linearly polarized stationary lasers: Each laser is set to selectively interact with one of the two excited states. The lower and higher excited states are shifted up and down with the same rate, respectively, and the two excited states become degenerate at their midpoint.
The control scenario with the ground state as the initial condition was applied to toluene. The derived time-dependent angular momentum consists of a train of unidirectional angular momentum. Further
Appl. Sci. 2018, 8, 2347 32 of 37
theoretical investigations on coherent π-electron rotation dynamics of aromatic ring molecules having no degenerate excited states are required by taking into account vibrational modes.
So far, we took simple low-symmetry molecules of a single aromatic ring or two aromatic rings for demonstration of fundamental features of coherent π-electron dynamics. From the viewpoint of practical application, organic materials having a large number of aromatic rings such as polycyclic aromatic hydrocarbons (PAHs) are much more interesting. Attention has already been paid to PAHs as organic molecular electronic materials, particularly for field-effect transistor devices [101]. It is important to explore the possibility of such an optical response in PAHs. Mineo and Fujimura have theoretically demonstrated the control of π electrons in PAHs by lasers to create ring currents and current-induced magnetic fields [102,103]. This is expected to serve as a fundamental research for next-generation organic optical switching devices.
The theoretical approach proposed in this article is useful to investigate not only the polarization dependence of intramolecular electron dynamics caused by single-photon absorption but also that of multiphoton electronic excitation. Hertel et al. have experimentally discovered that the multiphoton ionization probabilities of a xenon atom and C60fullerene irradiated by an intense femtosecond near-IR laser greatly changes with the ellipticity of the laser [104,105]. The application of the present approach to the analysis of multiphoton excitation processes induced by polarized light has been reported elsewhere [106].
Author Contributions:Methodology, software, validation, formal analysis, data curation, and visualization, M.K.;
Conceptualization, supervision, and project administration, Y.F.; Writing—original draft preparation, M.K. and Y.F.; Investigation, resources, writing—review and editing, and funding acquisition, M.K., H.K., and Y.F.
Funding:This work was supported in part by JSPS KAKENHI Grant Numbers JP26810002 and JP16H04091.
Conflicts of Interest:The authors declare no conflict of interest.
References
1. Polanyi, J.C.; Zewail, A.H. Direct Observation of the Transition State. Acc. Chem. Res 1995, 28, 119–132.
[CrossRef]
2. Zewail, A.H. Femtochemistry: Ultrafast Dynamics of the Chemical Bond; World Scientific: Singapore, 1994;
Volume 1.
3. Zewail, A.H. Femtochemistry: Ultrafast Dynamics of the Chemical Bond; World Scientific: Singapore, 1994;
Volume 2.
4. Zewail, A.H. Femtochemistry: Atomic-Scale Dynamics of the Chemical Bond. J. Phys. Chem. A 2000, 104, 5660–5694. [CrossRef]
5. Zewail, A.H. Femtochemistry: Atomic-Scale Dynamics of the Chemical Bond Using Ultrafast Lasers.
Angew. Chem. Int. Ed. 2000, 39, 2587–2631. [CrossRef]
6. Agostini, P. Observation of a Train of Attosecond Pulses from High Harmonic Generation. Science 2001, 292, 1689–1692.
7. Hentschel, M.; Kienberger, R.; Spielmann, C.; Reider, G.A.; Milosevic, N.; Brabec, T.; Corkum, P.;
Heinzmann, U.; Drescher, M.; Krausz, F. Attosecond metrology. Nature 2001, 414, 509–513. [CrossRef]
[PubMed]
8. Tzallas, P.; Charalambidis, D.; Papadogiannis, N.A.; Witte, K.; Tsakiris, G.D. Direct observation of attosecond light bunching. Nature 2003, 426, 267–271. [CrossRef] [PubMed]
9. Mairesse, Y.; de Bohan, A.; Frasinski, L.J.; Merdji, H.; Dinu, L.C.; Monchicourt, P.; Breger, P.; Kovaˇcev, M.;
Taïeb, R.; Carré, B.; et al. Attosecond Synchronization of High-Harmonic Soft X-rays. Science 2003, 302, 1540–1543. [CrossRef] [PubMed]
10. Kienberger, R.; Goulielmakis, E.; Uiberacker, M.; Baltuska, A.; Yakovlev, V.; Bammer, F.; Scrinzi, A.;
Westerwalbesloh, T.; Kleineberg, U.; Heinzmann, U.; et al. Atomic transient recorder. Nature 2004, 427, 817–821. [CrossRef] [PubMed]
11. Sekikawa, T.; Kosuge, A.; Kanai, T.; Watanabe, S. Nonlinear optics in the extreme ultraviolet. Nature 2004, 432, 605–608. [CrossRef] [PubMed]
Appl. Sci. 2018, 8, 2347 33 of 37
12. Nabekawa, Y.; Shimizu, T.; Okino, T.; Furusawa, K.; Hasegawa, H.; Yamanouchi, K.; Midorikawa, K.
Interferometric Autocorrelation of an Attosecond Pulse Train in the Single-Cycle Regime. Phys. Rev. Lett.
2006, 97, 153904. [CrossRef] [PubMed]
13. Sansone, G.; Benedetti, E.; Calegari, F.; Vozzi, C.; Avaldi, L.; Flammini, R.; Poletto, L.; Villoresi, P.; Altucci, C.;
Velotta, R.; et al. Isolated Single-Cycle Attosecond Pulses. Science 2006, 314, 443–446. [CrossRef] [PubMed]
14. Goulielmakis, E.; Schultze, M.; Hofstetter, M.; Yakovlev, N.S.; Gagnon, J.; Uiberacker, M.; Aquila, A.L.;
Gullikson, E.M.; Attwood, D.T.; Kienberger, R.; et al. Single-Cycle Nonlinear Optics. Science 2008, 320, 1614–1617. [CrossRef] [PubMed]
15. Krausz, F.; Ivanov, M. Attosecond physics. Rev. Mod. Phys. 2009, 81, 163–234. [CrossRef]
16. Lépine, F.; Sansone, G.; Vrakking, M.J.J. Molecular applications of attosecond laser pulses. Chem. Phys. Lett.
2013, 578, 1–14. [CrossRef]
17. McPherson, A.; Gibson, G.; Jara, H.; Johann, U.; Luk, T.S.; McIntyre, I.A.; Boyer, K.; Rhodes, C.K. Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases. J. Opt. Soc. Am. B 1987, 4, 595–601.
[CrossRef]
18. Ferray, M.; L’Huillier, A.; Li, X.F.; Lompré, L.A.; Mainfray, G.; Manus, C. Multiple-harmonic conversion of 1064 nm radiation in rare gases. J. Phys. B 1988, 21, L31–L35. [CrossRef]
19. Sarukura, N.; Hata, K.; Adachi, T.; Nodomi, R.; Watanabe, M.; Watanabe, S. Coherent soft-x-ray generation by the harmonics of an ultrahigh-power KrF laser. Phys. Rev. A 1991, 43, 1669–1672. [CrossRef] [PubMed]
20. Chang, Z.; Rundquist, A.; Wang, H.; Murnane, M.M.; Kapteyn, H.C. Generation of Coherent Soft X Rays at 2.7 nm Using High Harmonics. Phys. Rev. Lett. 1997, 79, 2967–2970. [CrossRef]
21. Spielmann, C.; Burnett, N.H.; Sartania, S.; Koppitsch, R.; Schnürer, M.; Kan, C.; Lenzner, M.; Wobrauschek, P.;
Krausz, F. Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses. Science 1997, 278, 661–664. [CrossRef]
22. Seres, J.; Seres, E.; Verhoef, A.J.; Tempea, G.; Streli, C.; Wobrauschek, P.; Yakovlev, V.; Scrinzi, A.; Spielmann, C.;
Krausz, F. Source of coherent kiloelectronvolt X-rays. Nature 2005, 433, 596. [CrossRef] [PubMed]
23. Krause, P.; Klamroth, T.; Saalfrank, P. Time-dependent configuration-interaction calculations of laser-pulse-driven many-electron dynamics: Controlled dipole switching in lithium cyanide. J. Chem. Phys.
2005, 123, 074105. [CrossRef] [PubMed]
24. Krause, P.; Klamroth, T.; Saalfrank, P. Molecular response properties from explicitly time-dependent configuration interaction methods. J. Chem. Phys. 2007, 127, 034107. [CrossRef] [PubMed]
25. Remacle, F.; Kienberger, R.; Krausz, F.; Levine, R.D. On the feasibility of an ultrafast purely electronic reorganization in lithium hydride. Chem. Phys. 2007, 338, 342–347. [CrossRef]
26. Remacle, F.; Nest, M.; Levine, R.D. Laser Steered Ultrafast Quantum Dynamics of Electrons in LiH.
Phys. Rev. Lett. 2007, 99, 183902. [CrossRef] [PubMed]
27. Nest, M.; Remacle, F.; Levine, R.D. Pump and probe ultrafast electron dynamics in LiH: A computational study. New J. Phys. 2008, 10, 025019. [CrossRef]
28. Barth, I.; Manz, J.; Serrano-Andrés, L. Quantum simulations of toroidal electric ring currents and magnetic fields in linear molecules induced by circularly polarized laser pulses. Chem. Phys. 2008, 347, 263–271.
[CrossRef]
29. Barth, I.; Serrano-Andrés, L.; Seideman, T. Nonadiabatic orientation, toroidal current, and induced magnetic field in BeO molecules. J. Chem. Phys. 2008, 129, 164303. [CrossRef] [PubMed]
30. Barth, I.; Serrano-Andrés, L.; Seideman, T. Erratum: “Nonadiabatic orientation, toroidal current, and induced magnetic field in BeO molecules” [J. Chem. Phys. 129, 164303 (2008)]. J. Chem. Phys. 2009, 130, 109901.
[CrossRef]
31. Yonehara, T.; Takatsuka, K. Characterization of electron-deficient chemical bonding of diborane with attosecond electron wavepacket dynamics and laser response. Chem. Phys. 2009, 366, 115–128. [CrossRef]
32. Takatsuka, K.; Yonehara, T. Exploring dynamical electron theory beyond the Born–Oppenheimer framework:
From chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field. Phys. Chem. Chem. Phys. 2011, 13, 4987–5016. [CrossRef] [PubMed]
33. Mignolet, B.; Gijsbertsen, A.; Vrakking, M.J.J.; Levine, R.D.; Remacle, F. Stereocontrol of attosecond time-scale electron dynamics in ABCU using ultrafast laser pulses: A computational study. Phys. Chem. Chem. Phys.
2011, 13, 8331–8344. [CrossRef] [PubMed]
Appl. Sci. 2018, 8, 2347 34 of 37
34. Ulusoy, I.S.; Nest, M. Correlated Electron Dynamics: How Aromaticity Can Be Controlled. J. Am. Chem. Soc.
2011, 133, 20230–20236. [CrossRef] [PubMed]
35. Hermann, G.; Liu, C.; Manz, J.; Paulus, B.; Pérez-Torres, J.F.; Pohl, V.; Tremblay, J.C. Multidirectional Angular Electronic Flux during Adiabatic Attosecond Charge Migration in Excited Benzene. J. Phys. Chem. A 2016, 120, 5360–5369. [CrossRef] [PubMed]
36. Jia, D.; Manz, J.; Paulus, B.; Pohl, V.; Tremblay, J.C.; Yang, Y. Quantum control of electronic fluxes during adiabatic attosecond charge migration in degenerate superposition states of benzene. Chem. Phys. 2017, 482, 146–159. [CrossRef]
37. Hermann, G.; Liu, C.; Manz, J.; Paulus, B.; Pohl, V.; Tremblay, J.C. Attosecond angular flux of partial charges on the carbon atoms of benzene in non-aromatic excited state. Chem. Phys. Lett. 2017, 683, 553–558.
[CrossRef]
38. Cederbaum, L.S.; Zobeley, J. Ultrafast charge migration by electron correlation. Chem. Phys. Lett. 1999, 307, 205–210. [CrossRef]
39. Breidbach, J.; Cederbaum, L.S. Migration of holes: Formalism, mechanisms, and illustrative applications.
J. Chem. Phys. 2003, 118, 3983–3996. [CrossRef]
40. Bandrauk, A.D.; Chelkowski, S.; Nguyen, H.S. Attosecond localization of electrons in molecules. Int. J.
Quantum Chem. 2004, 100, 834–844. [CrossRef]
41. Hennig, H.; Breidbach, J.; Cederbaum, L.S. Electron Correlation as the Driving Force for Charge Transfer:
Charge Migration Following Ionization in N-Methyl Acetamide. J. Phys. Chem. A 2005, 109, 409–414.
[CrossRef] [PubMed]
42. Breidbach, J.; Cederbaum, L.S. Universal Attosecond Response to the Removal of an Electron. Phys. Rev. Lett.
2005, 94, 033901. [CrossRef] [PubMed]
43. Kuleff, A.I.; Breidbach, J.; Cederbaum, L.S. Multielectron wave-packet propagation: General theory and application. J. Chem. Phys. 2005, 123, 044111. [CrossRef] [PubMed]
44. Remacle, F.; Levine, R.D. An electronic time scale in chemistry. Proc. Natl. Acad. Sci. USA 2006, 103, 6793–6798. [CrossRef] [PubMed]
45. Kuleff, A.I.; Cederbaum, L.S. Charge migration in different conformers of glycine: The role of nuclear geometry. Chem. Phys. 2007, 338, 320–328. [CrossRef]
46. Lünnemann, S.; Kuleff, A.I.; Cederbaum, L.S. Charge migration following ionization in systems with chromophore-donor and amine-acceptor sites. J. Chem. Phys. 2008, 129, 104305. [CrossRef] [PubMed]
47. Kuleff, A.I.; Cederbaum, L.S. Radiation Generated by the Ultrafast Migration of a Positive Charge Following the Ionization of a Molecular System. Phys. Rev. Lett. 2011, 106, 053001. [CrossRef] [PubMed]
48. Calegari, F.; Ayuso, D.; Trabattoni, A.; Belshaw, L.; De Camillis, S.; Anumula, S.; Frassetto, F.; Poletto, L.;
Palacios, A.; Decleva, P.; et al. Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses.
Science 2014, 346, 336–339. [CrossRef] [PubMed]
49. Ueda, K. To catch and smash charge on the hop. Science 2015, 350, 740–741. [CrossRef] [PubMed]
50. Kraus, P.M.; Mignolet, B.; Baykusheva, D.; Rupenyan, A.; Horný, L.; Penka, E.F.; Grassi, G.; Tolstikhin, O.I.;
Schneider, J.; Jensen, F.; et al. Measurement and laser control of attosecond charge migration in ionized iodoacetylene. Science 2015, 350, 790–795. [CrossRef] [PubMed]
51. Nobusada, K.; Yabana, K. Photoinduced electric currents in ring-shaped molecules by circularly polarized laser pulses. Phys. Rev. A 2007, 75, 032518. [CrossRef]
52. Barth, I.; Manz, J. Periodic Electron Circulation Induced by Circularly Polarized Laser Pulses: Quantum Model Simulations for Mg Porphyrin. Angew. Chem. Int. Ed. 2006, 45, 2962–2965. [CrossRef] [PubMed]
53. Barth, I.; Manz, J.; Shigeta, Y.; Yagi, K. Unidirectional Electronic Ring Current Driven by a Few Cycle Circularly Polarized Laser Pulse: Quantum Model Simulations for Mg–Porphyrin. J. Am. Chem. Soc. 2006, 128, 7043–7049. [CrossRef] [PubMed]
54. Barth, I.; Manz, J. Quantum Switching of Magnetic Fields by Circularly Polarized Re-Optimized π Laser Pulses: From One-Electron Atomic Ions to Molecules. In Progress in Ultrafast Intense Laser Science;
Yamanouchi, K., Gerber, G., Bandrauk, A.D., Eds.; Springer: Berlin, Germany, 2010; Volume 6, pp. 21–44.
55. Kanno, M.; Kono, H.; Fujimura, Y. Control of π-Electron Rotations in Chiral Aromatic Molecules Using Intense Laser Pulses. In Progress in Ultrafast Intense Laser Science; Yamanouchi, K., Charalambidis, D., Normand, D., Eds.; Springer: Berlin, Germany, 2011; Volume 7, pp. 53–78.
Appl. Sci. 2018, 8, 2347 35 of 37
56. Kanno, M.; Ono, Y.; Kono, H.; Fujimura, Y. Laser-Polarization Effects on Coherent Vibronic Excitation of Molecules with Quasi-Degenerate Electronic States. J. Phys. Chem. A 2012, 116, 11260–11272. [CrossRef]
[PubMed]
57. Kanno, M.; Kono, H.; Lin, S.H.; Fujimura, Y. Laser-Induced Electronic and Nuclear Coherent Motions in Chiral Aromatic Molecules. In Quantum Systems in Chemistry and Physics: Progress in Methods and Applications;
Nishikawa, K., Maruani, J., Brändas, E.J., Delgado-Barrio, G., Piecuch, P., Eds.; Progress in Theoretical Chemistry and Physics; Springer: Amsterdam, The Netherlands, 2012; Volume 26, pp. 121–148.
58. Kanno, M.; Ono, Y.; Kono, H.; Fujimura, Y. Nonadiabatically Coupled π-Electron Rotation and Molecular Vibration in Aromatic Molecules Excited by Polarized UV/Vis Laser Pulses. Chin. J. Phys. 2014, 52, 617–651.
59. Tannor, D.J. Introduction to Quantum Mechanics: A Time-Dependent Perspective; University Science Books:
Mill Valley, CA, USA, 2007; pp. 479–482.
60. Kanno, M.; Kono, H.; Fujimura, Y. Control of π-electron rotation in chiral aromatic molecules by nonhelical laser pulses. Angew. Chem. Int. Ed. 2006, 45, 7995–7998. [CrossRef] [PubMed]
61. Kanno, M.; Hoki, K.; Kono, H.; Fujimura, Y. Quantum optimal control of electron ring currents in chiral aromatic molecules. J. Chem. Phys. 2007, 127, 204314. [CrossRef] [PubMed]
62. Kanno, M.; Kono, H.; Fujimura, Y.; Lin, S.H. Nonadiabatic Response Model of Laser-Induced Ultrafast π-Electron Rotations in Chiral Aromatic Molecules. Phys. Rev. Lett. 2010, 104, 108302. [CrossRef] [PubMed]
63. Selle, R.; Nuernberger, P.; Langhojer, F.; Dimler, F.; Fechner, S.; Gerber, G.; Brixner, T. Generation of polarization-shaped ultraviolet femtosecond pulses. Opt. Lett. 2008, 33, 803–805. [CrossRef] [PubMed]
64. Nuernberger, P.; Selle, R.; Langhojer, F.; Dimler, F.; Fechner, S.; Gerber, G.; Brixner, T. Polarization-shaped femtosecond laser pulses in the ultraviolet. J. Opt. A 2009, 11, 085202. [CrossRef]
65. Seidel, M.T.; Zhang, Z.; Yan, S.; Tan, H.-S. Ultraviolet polarization pulse shaping using sum-frequency generation. J. Opt. Soc. Am. B 2011, 28, 1146–1151. [CrossRef]
66. Seidel, M.T.; Zhang, Z.; Yan, S.; Wells, K.L.; Tan, H.-S. Characterization of polarization shaped ultraviolet femtosecond laser pulses. J. Opt. Soc. Am. B 2011, 28, 2718–2725. [CrossRef]
67. Yuan, K.-J.; Bandrauk, A.D. Circularly polarized attosecond pulses from molecular high-order harmonic generation by ultrashort intense bichromatic circularly and linearly polarized laser pulses. J. Phys. B 2012, 45, 074001. [CrossRef]
68. Seideman, T. Revival Structure of Aligned Rotational Wave Packets. Phys. Rev. Lett. 1999, 83, 4971–4974.
[CrossRef]
69. Stapelfeldt, H.; Seideman, T. Aligning molecules with strong laser pulses. Rev. Mod. Phys. 2003, 75, 543–557.
[CrossRef]
70. Levine, I.N. Quantum Chemistry, 6th ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2009; pp. 471–635.
71. Werner, H.-J.; Knowles, P.J.; Lindh, R.; Manby, F.R.; Schütz, M.; Celani, P.; Korona, T.; Rauhut, G.; Amos, R.D.;
Bernhardsson, A.; et al. MOLPRO, version 2006.1; Cardiff, UK, 2006.
72. Hampel, C.; Peterson, K.; Werner, H.-J. A comparison of the efficiency and accuracy of the quadratic configuration interaction (QCISD), coupled cluster (CCSD), and Brueckner coupled cluster (BCCD) methods.
Chem. Phys. Lett. 1992, 190, 1–12. [CrossRef]
73. Werner, H.-J.; Knowles, P.J. A second order multiconfiguration SCF procedure with optimum convergence.
J. Chem. Phys. 1985, 82, 5053–5063. [CrossRef]
74. Knowles, P.J.; Werner, H.-J. An efficient second-order MCSCF method for long configuration expansions.
Chem. Phys. Lett. 1985, 115, 259–267. [CrossRef]
75. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 9th ed.; Abramowitz, M.;
Stegun, I.A., Eds.; Dover: New York, NY, USA, 1970; p. 896.
76. Werner, H.-J. Third-order multireference perturbation theory The CASPT3 method. Mol. Phys. 1996, 89, 645–661. [CrossRef]
77. Celani, P.; Werner, H.-J. Multireference perturbation theory for large restricted and selected active space reference wave functions. J. Chem. Phys. 2000, 112, 5546–5557. [CrossRef]
78. Baer, M. Beyond Born-Oppenheimer; Wiley: Hoboken, NJ, USA, 2006; pp. 26–57.
79. Sarkar, B.; Adhikari, S. A rigorous approach to the formulation of extended Born-Oppenheimer equation for a three-state system. Int. J. Quantum Chem. 2009, 109, 650–667. [CrossRef]
80. Simah, D.; Hartke, B.; Werner, H.-J. Photodissociation dynamics of H2S on new coupled ab initio potential energy surfaces. J. Chem. Phys. 1999, 111, 4523–4534. [CrossRef]
Appl. Sci. 2018, 8, 2347 36 of 37
81. Ohtsuki, Y.; Nakagami, K.; Fujimura, Y. Quantum Control of Molecular Dynamics. In Advances in Multi-Photon Processes and Spectroscopy; Lin, S.H., Villaeys, A.A., Fujimura, Y., Eds.; World Scientific:
Singapore, 2001; Volume 13, pp. 1–127.
82. Gross, P.; Neuhauser, D.; Rabitz, H. Optimal control of curve-crossing systems. J. Chem. Phys. 1992, 96, 2834–2845. [CrossRef]
83. Born, M.; Oppenheimer, J.R. Zur Quantentheorie der Molekeln. Ann. Phys. 1927, 84, 457–484. [CrossRef]
84. Tannor, D.J. Introduction to Quantum Mechanics: A Time-Dependent Perspective; University Science Books:
Mill Valley, CA, USA, 2007; pp. 81–86.
85. Fujimura, Y.; Sakai, H. Electronic and Nuclear Dynamics in Molecular Systems; World Scientific: Singapore, 2011;
pp. 117–132.
86. Mineo, H.; Yamaki, M.; Teranishi, Y.; Hayashi, M.; Lin, S.H.; Fujimura, Y. Quantum Switching of π-Electron Rotations in a Nonplanar Chiral Molecule by Using Linearly Polarized UV Laser Pulses. J. Am. Chem. Soc.
2012, 134, 14279–14282. [CrossRef] [PubMed]
87. Mineo, H.; Lin, S.H.; Fujimura, Y. Coherent π-electron dynamics of (P)-2,20-biphenol induced by ultrashort linearly polarized UV pulses: Angular momentum and ring current. J. Chem. Phys. 2013, 138, 074304.
[CrossRef] [PubMed]
88. Fujimura, Y.; Kono, H.; Nakajima, T.; Lin, S.H. A theoretical study of resonance Raman scattering from molecules. III. Resonance Raman scattering and resonance fluorescence. J. Chem. Phys. 1981, 75, 99–106.
[CrossRef]
89. Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.;
Mennucci, B.; Petersson, G.A.; et al. Gaussian 09, Revision E.01; Gaussian, Inc.: Wallingford, CT, USA, 2009.
90. Baskin, J.S.; Felker, P.M.; Zewail, A.H. Doppler-free time-resolved polarization spectroscopy of large molecules: Measurement of excited state rotational constants. J. Chem. Phys. 1986, 84, 4708–4710. [CrossRef]
91. Yamaki, M.; Mineo, H.; Teranishi, Y.; Hayashi, M.; Fujimura, Y.; Nakamura, H.; Lin, S.H. Quantum Localization of Coherent π-Electron Angular Momentum in (P)-2,20-Biphenol. J. Phys. Chem. Lett. 2014, 5, 2044–2049. [CrossRef] [PubMed]
92. Zhu, W.; Botina, J.; Rabitz, H. Rapidly convergent iteration methods for quantum optimal control of population. J. Chem. Phys. 1998, 108, 1953–1963. [CrossRef]
93. Ohtsuki, Y.; Zhu, W.; Rabitz, H. Monotonically convergent algorithm for quantum optimal control with dissipation. J. Chem. Phys. 1999, 110, 9825–9832. [CrossRef]
94. Umeda, H.; Fujimura, Y. Quantum control of chemical reaction dynamics in a classical way. J. Chem. Phys.
2000, 113, 3510–3518. [CrossRef]
95. Mineo, H.; Lin, S.H.; Fujimura, Y. Vibrational effects on UV/Vis laser-driven π-electron ring currents in aromatic ring molecules. Chem. Phys. 2014, 442, 103–110. [CrossRef]
96. Mineo, H.; Lin, S.H.; Fujimura, Y.; Xu, J.; Xu, R.X.; Yan, Y.J. Non-Markovian response of ultrafast coherent electronic ring currents in chiral aromatic molecules in a condensed phase. J. Chem. Phys. 2013, 139, 214306.
[CrossRef] [PubMed]
97. Mineo, H.; Kanno, M.; Kono, H.; Chao, S.D.; Lin, S.H.; Fujimura, Y. Ultrafast coherent dynamics of nonadiabatically coupled quasi-degenerate excited states in molecules: Population and vibrational coherence transfers. Chem. Phys. 2012, 392, 136–142. [CrossRef]
98. Yamaki, M.; Mineo, H.; Teranishi, Y.; Lin, S.H.; Fujimura, Y. Quantum Control of Coherent π-Electron Dynamics in Chiral Aromatic Molecules. J. Chin. Chem. Soc. 2016, 63, 87–92. [CrossRef]
99. Yamaki, M.; Teranishi, Y.; Nakamura, H.; Lin, S.H.; Fujimura, Y. The generation of stationary π-electron rotations in chiral aromatic ring molecules possessing non-degenerate excited states. Phys. Chem. Chem. Phys.
2016, 18, 1570–1577. [CrossRef] [PubMed]
100. Mineo, H.; Yamaki, M.; Kim, G.-S.; Teranishi, Y.; Lin, S.H.; Fujimura, Y. Induction of unidirectional π-electron rotations in low-symmetry aromatic ring molecules using two linearly polarized stationary lasers. Phys. Chem.
Chem. Phys. 2016, 18, 26786–26795. [CrossRef] [PubMed]
101. Anthony, J.E. The Larger Acenes: Versatile Organic Semiconductors. Angew. Chem. Int. Ed. 2008, 47, 452–483.
[CrossRef] [PubMed]
102. Mineo, H.; Fujimura, Y. Quantum Design of π-Electron Ring Currents in Polycyclic Aromatic Hydrocarbons:
Parallel and Antiparallel Ring Currents in Naphthalene. J. Phys. Chem. Lett. 2017, 8, 2019–2025. [CrossRef]
[PubMed]
Appl. Sci. 2018, 8, 2347 37 of 37
103. Mineo, H.; Fujimura, Y. Quantum control of coherent π-electron ring currents in polycyclic aromatic hydrocarbons. J. Chem. Phys. 2017, 147, 224301. [CrossRef] [PubMed]
104. Hertel, I.V.; Shchatsinin, I.; Laarmann, T.; Zhavoronkov, N.; Ritze, H.-H.; Schulz, C.P. Fragmentation and Ionization Dynamics of C60 in Elliptically Polarized Femtosecond Laser Fields. Phys. Rev. Lett. 2009, 102, 023003. [CrossRef] [PubMed]
105. Shchatsinin, I.; Ritze, H.-H.; Schulz, C.P.; Hertel, I.V. Multiphoton excitation and ionization by elliptically polarized, intense short laser pulses: Recognizing multielectron dynamics and doorway states in C60vs. Xe.
Phys. Rev. A 2009, 79, 053414. [CrossRef]
106. Kanno, M.; Inada, N.; Kono, H. Single-active-electron analysis of laser-polarization effects on atomic/
molecular multiphoton excitation. J. Chem. Phys. 2017, 147, 154310. [CrossRef] [PubMed]
© 2018 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/).